Subgroups — C₆₇×#34963 #30714214

Order 17152.

3270 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#41×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₄×D₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2598	C₂	1	4	yes	maximal, complemented
4	C₆₇×#2580	C₂	1	2	yes	maximal, complemented
5	C₆₇×#2535	C₂	1	2	yes	maximal, complemented
6	C₆₇×#2487	C₂	1	2	yes	maximal, complemented
7	C₆₇×#2482	C₂	1	2	yes	maximal, complemented
8	C₆₇×#2408	C₂	1	2	yes	maximal, complemented
9	C₆₇×#2404	C₂	1	2	yes	maximal, complemented
10	C₆₇×#2398	C₂	1	2	yes	maximal, complemented
11	C₆₇×#2397	C₂	1	2	yes	maximal, complemented
12	C₄×#121×C₆₇	C₂	1	4	yes	maximal, complemented
13	C₄×#120×C₆₇	C₂	1	4	yes	maximal, complemented
14	C₆₇×#2283	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
16	C₂³×D₄×C₆₇	C₂²	1	1	yes	characteristic
17	C₂⁴×C₄×C₆₇	C₂²	1	2	yes	—
18	C₂×#121×C₆₇	C₂²	1	4	yes	—
19	C₂×#120×C₆₇	C₂²	1	4	yes	—
20	C₄×#41×C₆₇	—	2	8	no	—
21	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
22	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
23	C₂×C₄×D₄×C₆₇	—	2	8	no	—
24	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
25	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
26	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
27	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
28	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
29	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
30	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic, complemented
31	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic, complemented
32	C₂×C₄×D₄×C₆₇	C₂²	1	8	yes	complemented
33	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
34	C₂×#117×C₆₇	C₂²	1	2	yes	complemented
35	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
36	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
37	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
38	C₂²×C₄²×C₆₇	C₂²	1	2	yes	complemented
39	#400×C₆₇	C₂²	1	4	yes	complemented
40	#399×C₆₇	C₂²	1	4	yes	complemented
41	#399×C₆₇	C₂²	1	4	yes	complemented
42	#397×C₆₇	C₂²	1	4	yes	complemented
43	#394×C₆₇	C₂²	1	4	yes	—
44	#393×C₆₇	C₂²	1	4	yes	complemented
45	#392×C₆₇	C₂²	1	4	yes	—
46	#392×C₆₇	C₂²	1	4	yes	complemented
47	#390×C₆₇	C₂²	1	4	yes	—
48	#389×C₆₇	C₂²	1	4	yes	complemented
49	#388×C₆₇	C₂²	1	8	yes	complemented
50	#386×C₆₇	C₂²	1	4	yes	—
51	#386×C₆₇	C₂²	1	8	yes	complemented
52	#386×C₆₇	C₂²	1	8	yes	complemented
53	#385×C₆₇	C₂²	1	4	yes	—
54	#385×C₆₇	C₂²	1	8	yes	complemented
55	#383×C₆₇	C₂²	1	2	yes	complemented
56	#382×C₆₇	C₂²	1	2	yes	complemented
57	#381×C₆₇	C₂²	1	2	yes	—
58	#380×C₆₇	C₂²	1	2	yes	—
59	#380×C₆₇	C₂²	1	4	yes	complemented
60	#379×C₆₇	C₂²	1	4	yes	—
61	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	complemented
62	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
63	C₄×C₂²⋊C₄×C₆₇	C₂²	1	8	yes	complemented
64	#376×C₆₇	C₂²	1	2	yes	complemented
65	C₂⁵×C₆₇	C₂³	1	2	yes	—
66	C₂×#41×C₆₇	—	2	8	no	—
67	C₂×#41×C₆₇	—	2	4	no	—
68	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
69	C₂²×D₄×C₆₇	—	2	8	no	—
70	C₂²×D₄×C₆₇	—	2	4	no	—
71	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
72	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
73	C₂²×D₄×C₆₇	C₂³	1	8	yes	—
74	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
75	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
76	C₂³×C₄×C₆₇	—	2	8	no	—
77	C₂³×C₄×C₆₇	—	4	8	no	—
78	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
79	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
80	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
81	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
82	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
83	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
84	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
85	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
86	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
87	#121×C₆₇	—	2	16	no	—
88	#120×C₆₇	—	2	16	no	—
89	C₄×Q₈×C₆₇	—	2	4	no	—
90	C₄×Q₈×C₆₇	—	2	4	no	—
91	C₄×D₄×C₆₇	—	2	8	no	—
92	C₄×D₄×C₆₇	—	2	8	no	—
93	C₄×D₄×C₆₇	—	2	8	no	—
94	C₄×D₄×C₆₇	—	2	8	no	—
95	C₄×D₄×C₆₇	—	2	8	no	—
96	C₄×D₄×C₆₇	—	2	8	no	—
97	C₄×D₄×C₆₇	—	2	4	no	—
98	C₄×D₄×C₆₇	—	2	4	no	—
99	C₄×D₄×C₆₇	—	4	16	no	—
100	#117×C₆₇	—	2	4	no	—
101	#117×C₆₇	—	2	8	no	—
102	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
103	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
104	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
105	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
106	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
107	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
108	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
109	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
110	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
111	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
112	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
113	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
114	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
115	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
116	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
117	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
118	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
119	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
120	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
121	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
122	C₂×C₄²×C₆₇	—	2	8	no	—
123	C₂×C₄²×C₆₇	D₄	1	8	yes	complemented
124	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
125	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
126	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
127	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
128	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
129	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
130	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
131	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
132	#95×C₆₇	C₂³	1	4	yes	—
133	#95×C₆₇	C₂³	1	4	yes	—
134	#95×C₆₇	C₂³	1	4	yes	—
135	#95×C₆₇	C₂³	1	8	yes	—
136	#95×C₆₇	C₂³	1	8	yes	—
137	#95×C₆₇	C₂³	1	4	yes	—
138	C₂⁴×C₆₇	—	2	4	no	—
139	C₂⁴×C₆₇	—	2	8	no	—
140	C₂⁴×C₆₇	—	2	8	no	—
141	C₂⁴×C₆₇	—	4	4	no	—
142	C₂⁴×C₆₇	C₂⁴	1	4	yes	—
143	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
144	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
145	#41×C₆₇	—	2	32	no	—
146	C₂×Q₈×C₆₇	—	2	2	no	—
147	C₂×Q₈×C₆₇	—	2	4	no	—
148	C₂×D₄×C₆₇	—	2	32	no	—
149	C₂×D₄×C₆₇	—	2	4	no	—
150	C₂×D₄×C₆₇	—	2	8	no	—
151	C₂×D₄×C₆₇	—	2	4	no	—
152	C₂×D₄×C₆₇	—	2	4	no	—
153	C₂×D₄×C₆₇	—	2	16	no	—
154	C₂×D₄×C₆₇	—	2	16	no	—
155	C₂×D₄×C₆₇	—	2	4	no	—
156	C₂×D₄×C₆₇	—	2	4	no	—
157	C₂×D₄×C₆₇	—	2	2	no	—
158	C₂×D₄×C₆₇	—	2	4	no	—
159	C₂×D₄×C₆₇	—	4	8	no	—
160	C₂×D₄×C₆₇	#41	1	8	yes	—
161	C₂×D₄×C₆₇	#41	1	8	yes	complemented
162	C₂²×C₄×C₆₇	—	2	4	no	—
163	C₂²×C₄×C₆₇	—	2	4	no	—
164	C₂²×C₄×C₆₇	—	2	4	no	—
165	C₂²×C₄×C₆₇	—	2	4	no	—
166	C₂²×C₄×C₆₇	—	2	4	no	—
167	C₂²×C₄×C₆₇	—	2	4	no	—
168	C₂²×C₄×C₆₇	—	2	4	no	—
169	C₂²×C₄×C₆₇	—	2	4	no	—
170	C₂²×C₄×C₆₇	—	2	4	no	—
171	C₂²×C₄×C₆₇	—	2	8	no	—
172	C₂²×C₄×C₆₇	—	2	8	no	—
173	C₂²×C₄×C₆₇	—	2	8	no	—
174	C₂²×C₄×C₆₇	—	2	8	no	—
175	C₂²×C₄×C₆₇	—	2	16	no	—
176	C₂²×C₄×C₆₇	—	4	8	no	—
177	C₂²×C₄×C₆₇	—	4	4	no	—
178	C₂²×C₄×C₆₇	—	4	32	no	—
179	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
180	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
181	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
182	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
183	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
184	C₂²×C₄×C₆₇	C₂⁴	1	8	yes	—
185	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
186	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
187	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
188	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
189	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
190	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
191	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
192	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
193	C₄⋊C₄×C₆₇	—	2	4	no	—
194	C₄⋊C₄×C₆₇	—	2	4	no	—
195	C₄⋊C₄×C₆₇	—	2	4	no	—
196	C₄⋊C₄×C₆₇	—	2	4	no	—
197	C₄⋊C₄×C₆₇	—	2	2	no	—
198	C₄⋊C₄×C₆₇	—	2	2	no	—
199	C₄⋊C₄×C₆₇	—	2	8	no	—
200	C₄⋊C₄×C₆₇	—	4	8	no	—
201	C₂²⋊C₄×C₆₇	—	2	4	no	—
202	C₂²⋊C₄×C₆₇	—	2	8	no	—
203	C₂²⋊C₄×C₆₇	—	2	16	no	—
204	C₂²⋊C₄×C₆₇	—	2	8	no	—
205	C₂²⋊C₄×C₆₇	—	2	8	no	—
206	C₂²⋊C₄×C₆₇	—	2	4	no	—
207	C₂²⋊C₄×C₆₇	—	2	4	no	—
208	C₂²⋊C₄×C₆₇	—	2	16	no	—
209	C₂²⋊C₄×C₆₇	—	4	8	no	—
210	C₂²⋊C₄×C₆₇	—	4	8	no	—
211	C₄²×C₆₇	—	2	4	no	—
212	C₄²×C₆₇	—	2	4	no	—
213	C₄²×C₆₇	—	2	4	no	—
214	C₄²×C₆₇	—	2	8	no	—
215	C₄²×C₆₇	C₂×D₄	1	4	yes	—
216	C₄²×C₆₇	C₂×D₄	1	4	yes	—
217	C₄²×C₆₇	C₂×D₄	1	4	yes	—
218	C₄²×C₆₇	C₂×D₄	1	4	yes	—
219	C₂³×C₆₇	—	2	2	no	—
220	C₂³×C₆₇	—	2	2	no	—
221	C₂³×C₆₇	—	2	16	no	—
222	C₂³×C₆₇	—	2	2	no	—
223	C₂³×C₆₇	—	2	4	no	—
224	C₂³×C₆₇	—	2	8	no	—
225	C₂³×C₆₇	—	2	2	no	—
226	C₂³×C₆₇	—	2	4	no	—
227	C₂³×C₆₇	—	2	4	no	—
228	C₂³×C₆₇	—	2	4	no	—
229	C₂³×C₆₇	—	2	8	no	—
230	C₂³×C₆₇	—	4	32	no	—
231	C₂³×C₆₇	—	4	8	no	—
232	C₂³×C₆₇	—	4	8	no	—
233	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
234	C₂³×C₆₇	C₂×#41	1	8	yes	—
235	C₂³×C₆₇	C₂×#41	1	4	yes	—
236	C₂³×C₆₇	C₂×#41	1	4	yes	—
237	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
238	Q₈×C₆₇	—	2	8	no	—
239	D₄×C₆₇	—	2	16	no	—
240	D₄×C₆₇	—	2	8	no	—
241	D₄×C₆₇	—	2	16	no	—
242	D₄×C₆₇	—	2	16	no	—
243	D₄×C₆₇	—	4	32	no	—
244	C₂×C₄×C₆₇	—	2	16	no	—
245	C₂×C₄×C₆₇	—	2	8	no	—
246	C₂×C₄×C₆₇	—	2	2	no	—
247	C₂×C₄×C₆₇	—	2	2	no	—
248	C₂×C₄×C₆₇	—	2	2	no	—
249	C₂×C₄×C₆₇	—	2	2	no	—
250	C₂×C₄×C₆₇	—	2	2	no	—
251	C₂×C₄×C₆₇	—	2	2	no	—
252	C₂×C₄×C₆₇	—	2	8	no	—
253	C₂×C₄×C₆₇	—	2	8	no	—
254	C₂×C₄×C₆₇	—	2	2	no	—
255	C₂×C₄×C₆₇	—	2	2	no	—
256	C₂×C₄×C₆₇	—	2	4	no	—
257	C₂×C₄×C₆₇	—	2	4	no	—
258	C₂×C₄×C₆₇	—	2	8	no	—
259	C₂×C₄×C₆₇	—	2	8	no	—
260	C₂×C₄×C₆₇	—	2	8	no	—
261	C₂×C₄×C₆₇	—	2	4	no	—
262	C₂×C₄×C₆₇	—	2	8	no	—
263	C₂×C₄×C₆₇	—	4	16	no	—
264	C₂×C₄×C₆₇	—	4	8	no	—
265	C₂×C₄×C₆₇	—	4	8	no	—
266	C₂×C₄×C₆₇	—	4	8	no	—
267	C₂×C₄×C₆₇	—	4	16	no	—
268	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
269	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
270	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
271	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
272	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
273	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
274	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
275	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
276	C₂×C₄×C₆₇	H(5,𝔽₂)	1	4	yes	—
277	C₂²×C₆₇	—	2	2	no	—
278	C₂²×C₆₇	—	2	2	no	—
279	C₂²×C₆₇	—	2	8	no	—
280	C₂²×C₆₇	—	2	4	no	—
281	C₂²×C₆₇	—	2	8	no	—
282	C₂²×C₆₇	—	2	8	no	—
283	C₂²×C₆₇	—	2	16	no	—
284	C₂²×C₆₇	—	4	32	no	—
285	C₂²×C₆₇	—	4	2	no	—
286	C₂²×C₆₇	—	4	4	no	—
287	C₂²×C₆₇	—	4	4	no	—
288	C₂²×C₆₇	—	4	2	no	—
289	C₂²×C₆₇	—	4	4	no	—
290	C₂²×C₆₇	—	4	8	no	—
291	C₂²×C₆₇	#525	1	4	yes	—
292	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
293	C₂²×C₆₇	C₂²×#41	1	2	yes	—
294	C₂²×C₆₇	C₂²×#41	1	2	yes	—
295	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	2	yes	—
296	C₄×C₆₇	—	2	2	no	—
297	C₄×C₆₇	—	2	2	no	—
298	C₄×C₆₇	—	2	2	no	—
299	C₄×C₆₇	—	2	4	no	—
300	C₄×C₆₇	—	2	2	no	—
301	C₄×C₆₇	—	4	2	no	—
302	C₄×C₆₇	—	4	2	no	—
303	C₄×C₆₇	—	4	4	no	—
304	C₄×C₆₇	—	4	8	no	—
305	C₄×C₆₇	#525	1	4	yes	—
306	C₄×C₆₇	#535	1	4	yes	—
307	#34963	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
308	C₂×C₆₇	—	2	4	no	—
309	C₂×C₆₇	—	4	4	no	—
310	C₂×C₆₇	—	4	8	no	—
311	C₂×C₆₇	C₂×#525	1	1	yes	characteristic
312	C₂×C₆₇	C₂×#525	1	1	yes	characteristic
313	C₂×C₆₇	C₂×#535	1	1	yes	characteristic
314	C₂×C₆₇	#3481	1	4	yes	—
315	C₂×C₄×#41	C₂×C₆₇	1	1	yes	characteristic, complemented
316	C₂²×C₄×D₄	C₂×C₆₇	1	1	yes	characteristic, complemented
317	#2598	C₂×C₆₇	1	4	yes	complemented
318	#2580	C₂×C₆₇	1	2	yes	complemented
319	#2535	C₂×C₆₇	1	2	yes	complemented
320	#2487	C₂×C₆₇	1	2	yes	complemented
321	#2482	C₂×C₆₇	1	2	yes	complemented
322	#2408	C₂×C₆₇	1	2	yes	complemented
323	#2404	C₂×C₆₇	1	2	yes	complemented
324	#2398	C₂×C₆₇	1	2	yes	complemented
325	#2397	C₂×C₆₇	1	2	yes	complemented
326	C₄×#121	C₂×C₆₇	1	4	yes	complemented
327	C₄×#120	C₂×C₆₇	1	4	yes	complemented
328	#2283	C₂×C₆₇	1	1	yes	characteristic, complemented
329	C₆₇	#34963	1	1	yes	characteristic, direct product factor, Sylow
330	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
331	C₂³×D₄	C₂²×C₆₇	1	1	yes	characteristic
332	C₂⁴×C₄	C₂²×C₆₇	1	2	yes	—
333	C₂×#121	C₂²×C₆₇	1	4	yes	—
334	C₂×#120	C₂²×C₆₇	1	4	yes	—
335	C₄×#41	—	2	8	no	—
336	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
337	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
338	C₂×C₄×D₄	—	2	8	no	—
339	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
340	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
341	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
342	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
343	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
344	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
345	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic, complemented
346	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic, complemented
347	C₂×C₄×D₄	C₂²×C₆₇	1	8	yes	complemented
348	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
349	C₂×#117	C₂²×C₆₇	1	2	yes	complemented
350	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
351	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
352	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
353	C₂²×C₄²	C₂²×C₆₇	1	2	yes	complemented
354	#400	C₂²×C₆₇	1	4	yes	complemented
355	#399	C₂²×C₆₇	1	4	yes	complemented
356	#399	C₂²×C₆₇	1	4	yes	complemented
357	#397	C₂²×C₆₇	1	4	yes	complemented
358	#394	C₂²×C₆₇	1	4	yes	—
359	#393	C₂²×C₆₇	1	4	yes	complemented
360	#392	C₂²×C₆₇	1	4	yes	—
361	#392	C₂²×C₆₇	1	4	yes	complemented
362	#390	C₂²×C₆₇	1	4	yes	—
363	#389	C₂²×C₆₇	1	4	yes	complemented
364	#388	C₂²×C₆₇	1	8	yes	complemented
365	#386	C₂²×C₆₇	1	4	yes	—
366	#386	C₂²×C₆₇	1	8	yes	complemented
367	#386	C₂²×C₆₇	1	8	yes	complemented
368	#385	C₂²×C₆₇	1	4	yes	—
369	#385	C₂²×C₆₇	1	8	yes	complemented
370	#383	C₂²×C₆₇	1	2	yes	complemented
371	#382	C₂²×C₆₇	1	2	yes	complemented
372	#381	C₂²×C₆₇	1	2	yes	—
373	#380	C₂²×C₆₇	1	2	yes	—
374	#380	C₂²×C₆₇	1	4	yes	complemented
375	#379	C₂²×C₆₇	1	4	yes	—
376	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	complemented
377	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
378	C₄×C₂²⋊C₄	C₂²×C₆₇	1	8	yes	complemented
379	#376	C₂²×C₆₇	1	2	yes	complemented
380	C₂⁵	C₂³×C₆₇	1	2	yes	—
381	C₂×#41	—	2	8	no	—
382	C₂×#41	—	2	4	no	—
383	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
384	C₂²×D₄	—	2	8	no	—
385	C₂²×D₄	—	2	4	no	—
386	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
387	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
388	C₂²×D₄	C₂³×C₆₇	1	8	yes	—
389	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
390	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
391	C₂³×C₄	—	2	8	no	—
392	C₂³×C₄	—	4	8	no	—
393	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
394	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
395	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
396	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
397	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
398	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
399	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
400	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
401	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
402	#121	—	2	16	no	—
403	#120	—	2	16	no	—
404	C₄×Q₈	—	2	4	no	—
405	C₄×Q₈	—	2	4	no	—
406	C₄×D₄	—	2	8	no	—
407	C₄×D₄	—	2	8	no	—
408	C₄×D₄	—	2	8	no	—
409	C₄×D₄	—	2	8	no	—
410	C₄×D₄	—	2	8	no	—
411	C₄×D₄	—	2	8	no	—
412	C₄×D₄	—	2	4	no	—
413	C₄×D₄	—	2	4	no	—
414	C₄×D₄	—	4	16	no	—
415	#117	—	2	4	no	—
416	#117	—	2	8	no	—
417	C₂×C₄⋊C₄	—	2	4	no	—
418	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
419	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
420	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
421	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
422	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
423	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
424	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
425	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
426	C₂×C₂²⋊C₄	—	2	8	no	—
427	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
428	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
429	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
430	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
431	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
432	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
433	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
434	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
435	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
436	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
437	C₂×C₄²	—	2	8	no	—
438	C₂×C₄²	D₄×C₆₇	1	8	yes	complemented
439	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
440	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
441	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
442	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
443	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
444	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
445	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
446	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
447	#95	C₂³×C₆₇	1	4	yes	—
448	#95	C₂³×C₆₇	1	4	yes	—
449	#95	C₂³×C₆₇	1	4	yes	—
450	#95	C₂³×C₆₇	1	8	yes	—
451	#95	C₂³×C₆₇	1	8	yes	—
452	#95	C₂³×C₆₇	1	4	yes	—
453	C₂⁴	—	2	4	no	—
454	C₂⁴	—	2	8	no	—
455	C₂⁴	—	2	8	no	—
456	C₂⁴	—	4	4	no	—
457	C₂⁴	C₂⁴×C₆₇	1	4	yes	—
458	C₂⁴	C₂⁴×C₆₇	1	2	yes	—
459	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
460	#41	—	2	32	no	—
461	C₂×Q₈	—	2	2	no	—
462	C₂×Q₈	—	2	4	no	—
463	C₂×D₄	—	2	32	no	—
464	C₂×D₄	—	2	4	no	—
465	C₂×D₄	—	2	8	no	—
466	C₂×D₄	—	2	4	no	—
467	C₂×D₄	—	2	4	no	—
468	C₂×D₄	—	2	16	no	—
469	C₂×D₄	—	2	16	no	—
470	C₂×D₄	—	2	4	no	—
471	C₂×D₄	—	2	4	no	—
472	C₂×D₄	—	2	2	no	—
473	C₂×D₄	—	2	4	no	—
474	C₂×D₄	—	4	8	no	—
475	C₂×D₄	#41×C₆₇	1	8	yes	—
476	C₂×D₄	#41×C₆₇	1	8	yes	complemented
477	C₂²×C₄	—	2	4	no	—
478	C₂²×C₄	—	2	4	no	—
479	C₂²×C₄	—	2	4	no	—
480	C₂²×C₄	—	2	4	no	—
481	C₂²×C₄	—	2	4	no	—
482	C₂²×C₄	—	2	4	no	—
483	C₂²×C₄	—	2	4	no	—
484	C₂²×C₄	—	2	4	no	—
485	C₂²×C₄	—	2	4	no	—
486	C₂²×C₄	—	2	8	no	—
487	C₂²×C₄	—	2	8	no	—
488	C₂²×C₄	—	2	8	no	—
489	C₂²×C₄	—	2	8	no	—
490	C₂²×C₄	—	2	16	no	—
491	C₂²×C₄	—	4	8	no	—
492	C₂²×C₄	—	4	4	no	—
493	C₂²×C₄	—	4	32	no	—
494	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
495	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
496	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
497	C₂²×C₄	C₂⁴×C₆₇	1	8	yes	—
498	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
499	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
500	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
501	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
502	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
503	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
504	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
505	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
506	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
507	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
508	C₄⋊C₄	—	2	4	no	—
509	C₄⋊C₄	—	2	4	no	—
510	C₄⋊C₄	—	2	4	no	—
511	C₄⋊C₄	—	2	4	no	—
512	C₄⋊C₄	—	2	2	no	—
513	C₄⋊C₄	—	2	2	no	—
514	C₄⋊C₄	—	2	8	no	—
515	C₄⋊C₄	—	4	8	no	—
516	C₂²⋊C₄	—	2	4	no	—
517	C₂²⋊C₄	—	2	8	no	—
518	C₂²⋊C₄	—	2	16	no	—
519	C₂²⋊C₄	—	2	8	no	—
520	C₂²⋊C₄	—	2	8	no	—
521	C₂²⋊C₄	—	2	4	no	—
522	C₂²⋊C₄	—	2	4	no	—
523	C₂²⋊C₄	—	2	16	no	—
524	C₂²⋊C₄	—	4	8	no	—
525	C₂²⋊C₄	—	4	8	no	—
526	C₄²	—	2	4	no	—
527	C₄²	—	2	4	no	—
528	C₄²	—	2	4	no	—
529	C₄²	—	2	8	no	—
530	C₄²	C₂×D₄×C₆₇	1	4	yes	—
531	C₄²	C₂×D₄×C₆₇	1	4	yes	—
532	C₄²	C₂×D₄×C₆₇	1	4	yes	—
533	C₄²	C₂×D₄×C₆₇	1	4	yes	—
534	C₂³	—	2	2	no	—
535	C₂³	—	2	2	no	—
536	C₂³	—	2	16	no	—
537	C₂³	—	2	2	no	—
538	C₂³	—	2	4	no	—
539	C₂³	—	2	8	no	—
540	C₂³	—	2	2	no	—
541	C₂³	—	2	4	no	—
542	C₂³	—	2	4	no	—
543	C₂³	—	2	4	no	—
544	C₂³	—	2	8	no	—
545	C₂³	—	4	32	no	—
546	C₂³	—	4	8	no	—
547	C₂³	—	4	8	no	—
548	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
549	C₂³	C₂×#41×C₆₇	1	8	yes	—
550	C₂³	C₂×#41×C₆₇	1	4	yes	—
551	C₂³	C₂×#41×C₆₇	1	4	yes	—
552	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
553	Q₈	—	2	8	no	—
554	D₄	—	2	16	no	—
555	D₄	—	2	8	no	—
556	D₄	—	2	16	no	—
557	D₄	—	2	16	no	—
558	D₄	—	4	32	no	—
559	C₂×C₄	—	2	16	no	—
560	C₂×C₄	—	2	8	no	—
561	C₂×C₄	—	2	2	no	—
562	C₂×C₄	—	2	2	no	—
563	C₂×C₄	—	2	2	no	—
564	C₂×C₄	—	2	2	no	—
565	C₂×C₄	—	2	2	no	—
566	C₂×C₄	—	2	2	no	—
567	C₂×C₄	—	2	8	no	—
568	C₂×C₄	—	2	8	no	—
569	C₂×C₄	—	2	2	no	—
570	C₂×C₄	—	2	2	no	—
571	C₂×C₄	—	2	4	no	—
572	C₂×C₄	—	2	4	no	—
573	C₂×C₄	—	2	8	no	—
574	C₂×C₄	—	2	8	no	—
575	C₂×C₄	—	2	8	no	—
576	C₂×C₄	—	2	4	no	—
577	C₂×C₄	—	2	8	no	—
578	C₂×C₄	—	4	16	no	—
579	C₂×C₄	—	4	8	no	—
580	C₂×C₄	—	4	8	no	—
581	C₂×C₄	—	4	16	no	—
582	C₂×C₄	—	4	8	no	—
583	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
584	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
585	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
586	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
587	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
588	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
589	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
590	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
591	C₂×C₄	H(5,𝔽₂)×C₆₇	1	4	yes	—
592	C₂²	—	2	2	no	—
593	C₂²	—	2	2	no	—
594	C₂²	—	2	8	no	—
595	C₂²	—	2	4	no	—
596	C₂²	—	2	8	no	—
597	C₂²	—	2	8	no	—
598	C₂²	—	2	16	no	—
599	C₂²	—	4	32	no	—
600	C₂²	—	4	2	no	—
601	C₂²	—	4	4	no	—
602	C₂²	—	4	4	no	—
603	C₂²	—	4	2	no	—
604	C₂²	—	4	8	no	—
605	C₂²	—	4	4	no	—
606	C₂²	#525×C₆₇	1	4	yes	—
607	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
608	C₂²	C₂²×#41×C₆₇	1	2	yes	—
609	C₂²	C₂²×#41×C₆₇	1	2	yes	—
610	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	2	yes	—
611	C₄	—	2	2	no	—
612	C₄	—	2	2	no	—
613	C₄	—	2	2	no	—
614	C₄	—	2	4	no	—
615	C₄	—	2	2	no	—
616	C₄	—	4	2	no	—
617	C₄	—	4	4	no	—
618	C₄	—	4	2	no	—
619	C₄	—	4	8	no	—
620	C₄	#525×C₆₇	1	4	yes	—
621	C₄	#535×C₆₇	1	4	yes	—
622	C₂	—	2	4	no	—
623	C₂	—	4	8	no	—
624	C₂	—	4	4	no	—
625	C₂	C₂×#525×C₆₇	1	1	yes	characteristic
626	C₂	C₂×#525×C₆₇	1	1	yes	characteristic
627	C₂	C₂×#535×C₆₇	1	1	yes	characteristic
628	C₂	C₆₇×#3481	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₆₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₆₇
socle	C₂³×C₆₇	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₆₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂×C₄×#41×C₆₇	C₂	semidirect
2. C₂²×C₄×D₄×C₆₇	C₂	semidirect
3. C₆₇×#2598	C₂	semidirect
4. C₆₇×#2580	C₂	semidirect
5. C₆₇×#2535	C₂	semidirect
6. C₆₇×#2487	C₂	semidirect
7. C₆₇×#2482	C₂	semidirect
8. C₆₇×#2408	C₂	semidirect
9. C₆₇×#2404	C₂	semidirect
10. C₆₇×#2398	C₂	semidirect
11. C₆₇×#2397	C₂	semidirect
12. C₄×#121×C₆₇	C₂	semidirect
13. C₄×#120×C₆₇	C₂	semidirect
14. C₆₇×#2283	C₂	semidirect
21. C₂×C₄×Q₈×C₆₇	C₂²	semidirect
22. C₂×C₄×Q₈×C₆₇	C₂²	semidirect
30. C₂×C₄×D₄×C₆₇	C₂²	semidirect
31. C₂×C₄×D₄×C₆₇	C₂²	semidirect
32. C₂×C₄×D₄×C₆₇	C₂²	semidirect
34. C₂×#117×C₆₇	C₂²	semidirect
38. C₂²×C₄²×C₆₇	C₂²	semidirect
39. #400×C₆₇	C₂²	semidirect
40. #399×C₆₇	C₂²	semidirect
41. #399×C₆₇	C₂²	semidirect
42. #397×C₆₇	C₂²	semidirect
44. #393×C₆₇	C₂²	semidirect
46. #392×C₆₇	C₂²	semidirect
48. #389×C₆₇	C₂²	semidirect
49. #388×C₆₇	C₂²	semidirect
51. #386×C₆₇	C₂²	semidirect
52. #386×C₆₇	C₂²	semidirect
54. #385×C₆₇	C₂²	semidirect
55. #383×C₆₇	C₂²	semidirect
56. #382×C₆₇	C₂²	semidirect
59. #380×C₆₇	C₂²	semidirect
61. C₄×C₄⋊C₄×C₆₇	C₂²	semidirect
63. C₄×C₂²⋊C₄×C₆₇	C₂²	semidirect
64. #376×C₆₇	C₂²	semidirect
123. C₂×C₄²×C₆₇	D₄	semidirect
161. C₂×D₄×C₆₇	#41	semidirect
307. #34963	C₆₇	direct (N × H)
315. C₂×C₄×#41	C₂×C₆₇	semidirect
316. C₂²×C₄×D₄	C₂×C₆₇	semidirect
317. #2598	C₂×C₆₇	semidirect
318. #2580	C₂×C₆₇	semidirect
319. #2535	C₂×C₆₇	semidirect
320. #2487	C₂×C₆₇	semidirect
321. #2482	C₂×C₆₇	semidirect
322. #2408	C₂×C₆₇	semidirect
323. #2404	C₂×C₆₇	semidirect
324. #2398	C₂×C₆₇	semidirect
325. #2397	C₂×C₆₇	semidirect
326. C₄×#121	C₂×C₆₇	semidirect
327. C₄×#120	C₂×C₆₇	semidirect
328. #2283	C₂×C₆₇	semidirect
329. C₆₇	#34963	direct (N × H)
336. C₂×C₄×Q₈	C₂²×C₆₇	semidirect
337. C₂×C₄×Q₈	C₂²×C₆₇	semidirect
345. C₂×C₄×D₄	C₂²×C₆₇	semidirect
346. C₂×C₄×D₄	C₂²×C₆₇	semidirect
347. C₂×C₄×D₄	C₂²×C₆₇	semidirect
349. C₂×#117	C₂²×C₆₇	semidirect
353. C₂²×C₄²	C₂²×C₆₇	semidirect
354. #400	C₂²×C₆₇	semidirect
355. #399	C₂²×C₆₇	semidirect
356. #399	C₂²×C₆₇	semidirect
357. #397	C₂²×C₆₇	semidirect
359. #393	C₂²×C₆₇	semidirect
361. #392	C₂²×C₆₇	semidirect
363. #389	C₂²×C₆₇	semidirect
364. #388	C₂²×C₆₇	semidirect
366. #386	C₂²×C₆₇	semidirect
367. #386	C₂²×C₆₇	semidirect
369. #385	C₂²×C₆₇	semidirect
370. #383	C₂²×C₆₇	semidirect
371. #382	C₂²×C₆₇	semidirect
374. #380	C₂²×C₆₇	semidirect
376. C₄×C₄⋊C₄	C₂²×C₆₇	semidirect
378. C₄×C₂²⋊C₄	C₂²×C₆₇	semidirect
379. #376	C₂²×C₆₇	semidirect
438. C₂×C₄²	D₄×C₆₇	semidirect
476. C₂×D₄	#41×C₆₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₄×#41×C₆₇	2
C₂²×C₄×D₄×C₆₇	2
C₆₇×#2598	2
C₆₇×#2580	2
C₆₇×#2535	2
C₆₇×#2487	2
C₆₇×#2482	2
C₆₇×#2408	2
C₆₇×#2404	2
C₆₇×#2398	2
C₆₇×#2397	2
C₄×#121×C₆₇	2
C₄×#120×C₆₇	2
C₆₇×#2283	2
#34963	67

Subgroups — C67×#34963 #30714214

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#34963 #30714214