Subgroups — C₆₇×#34609 #30713860

Order 17152.

2782 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#532×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×#41×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2635	C₂	1	4	yes	maximal, complemented
4	C₆₇×#2632	C₂	1	2	yes	maximal, complemented
5	C₆₇×#2631	C₂	1	2	yes	maximal, complemented
6	C₆₇×#2628	C₂	1	2	yes	maximal, complemented
7	C₆₇×#2624	C₂	1	2	yes	maximal, complemented
8	C₆₇×#2619	C₂	1	2	yes	maximal, complemented
9	C₆₇×#2615	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	C₆₇×#2388	C₂	1	4	yes	maximal, complemented
13	C₄×#121×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	C₂²×#41×C₆₇	C₂²	1	2	yes	—
17	#532×C₆₇	—	2	8	no	—
18	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×#124×C₆₇	C₂²	1	2	yes	—
21	C₂×#121×C₆₇	C₂²	1	4	yes	—
22	C₂×#121×C₆₇	C₂²	1	2	yes	—
23	C₂×#121×C₆₇	C₂²	1	2	yes	—
24	C₄×#41×C₆₇	—	2	8	no	—
25	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
27	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
28	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
29	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
30	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
31	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
32	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
33	C₂×C₄×D₄×C₆₇	C₂²	1	8	yes	—
34	C₂×#117×C₆₇	C₂²	1	2	yes	—
35	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
36	C₂²×C₄²×C₆₇	C₂²	1	2	yes	—
37	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
38	#400×C₆₇	C₂²	1	4	yes	—
39	#399×C₆₇	C₂²	1	4	yes	—
40	#399×C₆₇	C₂²	1	4	yes	—
41	#397×C₆₇	C₂²	1	4	yes	—
42	#395×C₆₇	C₂²	1	4	yes	—
43	#394×C₆₇	C₂²	1	4	yes	—
44	#394×C₆₇	C₂²	1	4	yes	—
45	#394×C₆₇	C₂²	1	8	yes	—
46	#393×C₆₇	C₂²	1	4	yes	—
47	#393×C₆₇	C₂²	1	4	yes	—
48	#392×C₆₇	C₂²	1	4	yes	—
49	#392×C₆₇	C₂²	1	8	yes	—
50	#392×C₆₇	C₂²	1	4	yes	—
51	#392×C₆₇	C₂²	1	4	yes	—
52	#390×C₆₇	C₂²	1	4	yes	—
53	#389×C₆₇	C₂²	1	4	yes	—
54	#388×C₆₇	C₂²	1	8	yes	—
55	#386×C₆₇	C₂²	1	8	yes	—
56	#383×C₆₇	C₂²	1	2	yes	—
57	#382×C₆₇	C₂²	1	2	yes	—
58	#381×C₆₇	C₂²	1	2	yes	—
59	#380×C₆₇	C₂²	1	2	yes	—
60	#380×C₆₇	C₂²	1	4	yes	—
61	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
62	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
63	C₄×C₂²⋊C₄×C₆₇	C₂²	1	8	yes	—
64	#376×C₆₇	C₂²	1	2	yes	—
65	C₂×#41×C₆₇	—	2	4	no	—
66	C₂×#41×C₆₇	—	2	8	no	—
67	C₂×#41×C₆₇	—	2	8	no	—
68	C₂×#41×C₆₇	—	4	8	no	—
69	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
70	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
71	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
72	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
73	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
74	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
75	C₂²×D₄×C₆₇	C₂³	1	8	yes	—
76	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
77	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
78	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
79	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
80	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
81	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
82	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
83	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
84	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
85	C₄⋊Q₈×C₆₇	—	2	4	no	—
86	#127×C₆₇	—	2	4	no	—
87	#124×C₆₇	—	2	8	no	—
88	#121×C₆₇	—	2	8	no	—
89	#121×C₆₇	—	2	8	no	—
90	#121×C₆₇	—	2	16	no	—
91	C₄×Q₈×C₆₇	—	2	4	no	—
92	C₄×Q₈×C₆₇	—	2	4	no	—
93	C₄×D₄×C₆₇	—	2	8	no	—
94	C₄×D₄×C₆₇	—	2	8	no	—
95	C₄×D₄×C₆₇	—	2	4	no	—
96	C₄×D₄×C₆₇	—	2	4	no	—
97	C₄×D₄×C₆₇	—	2	8	no	—
98	C₄×D₄×C₆₇	—	2	8	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	C₂³	1	2	yes	—
103	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
104	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
105	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
106	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
107	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
108	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
109	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
110	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
111	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
112	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
113	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
114	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
115	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
116	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
117	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
118	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
119	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
120	C₂×C₄²×C₆₇	—	2	8	no	—
121	C₂×C₄²×C₆₇	D₄	1	8	yes	complemented
122	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
123	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
124	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
125	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
126	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
127	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
128	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
129	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
130	#95×C₆₇	C₂³	1	4	yes	—
131	#95×C₆₇	C₂³	1	4	yes	—
132	#95×C₆₇	C₂³	1	8	yes	—
133	#95×C₆₇	C₂³	1	4	yes	—
134	#95×C₆₇	C₂³	1	8	yes	—
135	#95×C₆₇	C₂³	1	4	yes	—
136	C₂⁴×C₆₇	C₂⁴	1	4	yes	—
137	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
138	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
139	#41×C₆₇	—	2	32	no	—
140	#41×C₆₇	—	4	32	no	—
141	C₂×Q₈×C₆₇	—	2	2	no	—
142	C₂×Q₈×C₆₇	—	2	4	no	—
143	C₂×Q₈×C₆₇	—	2	4	no	—
144	C₂×Q₈×C₆₇	—	4	4	no	—
145	C₂×D₄×C₆₇	—	2	4	no	—
146	C₂×D₄×C₆₇	—	2	4	no	—
147	C₂×D₄×C₆₇	—	2	4	no	—
148	C₂×D₄×C₆₇	—	2	8	no	—
149	C₂×D₄×C₆₇	—	2	16	no	—
150	C₂×D₄×C₆₇	—	2	2	no	—
151	C₂×D₄×C₆₇	—	2	4	no	—
152	C₂×D₄×C₆₇	—	2	4	no	—
153	C₂×D₄×C₆₇	—	2	4	no	—
154	C₂×D₄×C₆₇	—	2	4	no	—
155	C₂×D₄×C₆₇	—	2	8	no	—
156	C₂×D₄×C₆₇	—	4	8	no	—
157	C₂×D₄×C₆₇	—	4	4	no	—
158	C₂×D₄×C₆₇	—	4	8	no	—
159	C₂×D₄×C₆₇	—	4	4	no	—
160	C₂×D₄×C₆₇	—	4	8	no	—
161	C₂²×C₄×C₆₇	—	2	4	no	—
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	8	no	—
168	C₂²×C₄×C₆₇	—	2	8	no	—
169	C₂²×C₄×C₆₇	—	2	16	no	—
170	C₂²×C₄×C₆₇	—	4	2	no	—
171	C₂²×C₄×C₆₇	—	4	4	no	—
172	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
173	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
174	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
175	C₂²×C₄×C₆₇	#41	1	8	yes	complemented
176	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
177	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
178	C₂²×C₄×C₆₇	C₂⁴	1	8	yes	—
179	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
180	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
181	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
182	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
183	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
184	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
185	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
186	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
187	C₄⋊C₄×C₆₇	—	2	4	no	—
188	C₄⋊C₄×C₆₇	—	2	4	no	—
189	C₄⋊C₄×C₆₇	—	2	2	no	—
190	C₄⋊C₄×C₆₇	—	2	2	no	—
191	C₄⋊C₄×C₆₇	—	2	4	no	—
192	C₄⋊C₄×C₆₇	—	2	4	no	—
193	C₄⋊C₄×C₆₇	—	2	8	no	—
194	C₄⋊C₄×C₆₇	—	4	8	no	—
195	C₂²⋊C₄×C₆₇	—	2	4	no	—
196	C₂²⋊C₄×C₆₇	—	2	4	no	—
197	C₂²⋊C₄×C₆₇	—	2	4	no	—
198	C₂²⋊C₄×C₆₇	—	2	8	no	—
199	C₂²⋊C₄×C₆₇	—	2	8	no	—
200	C₂²⋊C₄×C₆₇	—	2	16	no	—
201	C₂²⋊C₄×C₆₇	—	4	8	no	—
202	C₂²⋊C₄×C₆₇	—	4	4	no	—
203	C₂²⋊C₄×C₆₇	—	4	8	no	—
204	C₂²⋊C₄×C₆₇	—	4	8	no	—
205	C₄²×C₆₇	—	2	4	no	—
206	C₄²×C₆₇	—	2	4	no	—
207	C₄²×C₆₇	—	2	4	no	—
208	C₄²×C₆₇	—	2	8	no	—
209	C₄²×C₆₇	C₂×D₄	1	4	yes	—
210	C₄²×C₆₇	C₂×D₄	1	4	yes	—
211	C₄²×C₆₇	C₂×D₄	1	4	yes	—
212	C₄²×C₆₇	C₂×D₄	1	4	yes	—
213	C₂³×C₆₇	—	2	2	no	—
214	C₂³×C₆₇	—	2	2	no	—
215	C₂³×C₆₇	—	2	2	no	—
216	C₂³×C₆₇	—	2	4	no	—
217	C₂³×C₆₇	—	2	2	no	—
218	C₂³×C₆₇	—	2	2	no	—
219	C₂³×C₆₇	—	2	4	no	—
220	C₂³×C₆₇	—	2	4	no	—
221	C₂³×C₆₇	—	2	8	no	—
222	C₂³×C₆₇	—	2	8	no	—
223	C₂³×C₆₇	—	2	4	no	—
224	C₂³×C₆₇	—	2	4	no	—
225	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
226	C₂³×C₆₇	C₂×#41	1	4	yes	—
227	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
228	Q₈×C₆₇	—	2	8	no	—
229	Q₈×C₆₇	—	4	8	no	—
230	D₄×C₆₇	—	2	8	no	—
231	D₄×C₆₇	—	2	16	no	—
232	D₄×C₆₇	—	4	8	no	—
233	D₄×C₆₇	—	4	8	no	—
234	D₄×C₆₇	—	4	32	no	—
235	D₄×C₆₇	—	4	16	no	—
236	D₄×C₆₇	—	4	8	no	—
237	C₂×C₄×C₆₇	—	2	2	no	—
238	C₂×C₄×C₆₇	—	2	2	no	—
239	C₂×C₄×C₆₇	—	2	2	no	—
240	C₂×C₄×C₆₇	—	2	2	no	—
241	C₂×C₄×C₆₇	—	2	2	no	—
242	C₂×C₄×C₆₇	—	2	2	no	—
243	C₂×C₄×C₆₇	—	2	8	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	4	no	—
249	C₂×C₄×C₆₇	—	2	8	no	—
250	C₂×C₄×C₆₇	—	2	8	no	—
251	C₂×C₄×C₆₇	—	2	4	no	—
252	C₂×C₄×C₆₇	—	2	8	no	—
253	C₂×C₄×C₆₇	—	2	4	no	—
254	C₂×C₄×C₆₇	—	2	8	no	—
255	C₂×C₄×C₆₇	—	4	4	no	—
256	C₂×C₄×C₆₇	—	4	8	no	—
257	C₂×C₄×C₆₇	—	4	8	no	—
258	C₂×C₄×C₆₇	—	4	16	no	—
259	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
260	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
261	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
262	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
263	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
264	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
265	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
266	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
267	C₂×C₄×C₆₇	H(5,𝔽₂)	1	4	yes	—
268	C₂²×C₆₇	—	2	2	no	—
269	C₂²×C₆₇	—	2	4	no	—
270	C₂²×C₆₇	—	2	2	no	—
271	C₂²×C₆₇	—	2	2	no	—
272	C₂²×C₆₇	—	2	8	no	—
273	C₂²×C₆₇	—	4	2	no	—
274	C₂²×C₆₇	—	4	2	no	—
275	C₂²×C₆₇	—	4	4	no	—
276	C₂²×C₆₇	—	4	4	no	—
277	C₂²×C₆₇	—	4	4	no	—
278	C₂²×C₆₇	—	4	4	no	—
279	C₂²×C₆₇	—	4	4	no	—
280	C₂²×C₆₇	—	4	8	no	—
281	C₂²×C₆₇	—	4	8	no	—
282	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
283	C₂²×C₆₇	C₂²×#41	1	2	yes	—
284	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	2	yes	—
285	C₂²×C₆₇	#585	1	2	yes	—
286	C₄×C₆₇	—	2	2	no	—
287	C₄×C₆₇	—	2	4	no	—
288	C₄×C₆₇	—	2	2	no	—
289	C₄×C₆₇	—	2	2	no	—
290	C₄×C₆₇	—	2	2	no	—
291	C₄×C₆₇	—	4	2	no	—
292	C₄×C₆₇	—	4	4	no	—
293	C₄×C₆₇	—	4	2	no	—
294	C₄×C₆₇	—	4	8	no	—
295	C₄×C₆₇	#525	1	4	yes	—
296	C₄×C₆₇	#535	1	4	yes	—
297	#34609	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
298	C₂×C₆₇	—	4	2	no	—
299	C₂×C₆₇	—	4	4	no	—
300	C₂×C₆₇	—	8	4	no	—
301	C₂×C₆₇	C₂×#525	1	1	yes	characteristic
302	C₂×C₆₇	C₂×#535	1	1	yes	characteristic
303	C₂×C₆₇	#3449	1	1	yes	characteristic
304	C₂×C₆₇	#3460	1	4	yes	—
305	C₂×#532	C₂×C₆₇	1	1	yes	characteristic, complemented
306	C₂×C₄×#41	C₂×C₆₇	1	1	yes	characteristic, complemented
307	#2635	C₂×C₆₇	1	4	yes	complemented
308	#2632	C₂×C₆₇	1	2	yes	complemented
309	#2631	C₂×C₆₇	1	2	yes	complemented
310	#2628	C₂×C₆₇	1	2	yes	complemented
311	#2624	C₂×C₆₇	1	2	yes	complemented
312	#2619	C₂×C₆₇	1	2	yes	complemented
313	#2615	C₂×C₆₇	1	2	yes	complemented
314	#2398	C₂×C₆₇	1	2	yes	complemented
315	#2397	C₂×C₆₇	1	2	yes	complemented
316	#2388	C₂×C₆₇	1	4	yes	complemented
317	C₄×#121	C₂×C₆₇	1	4	yes	complemented
318	#2283	C₂×C₆₇	1	1	yes	characteristic, complemented
319	C₆₇	#34609	1	1	yes	characteristic, direct product factor, Sylow
320	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
321	C₂²×#41	C₂²×C₆₇	1	2	yes	—
322	#532	—	2	8	no	—
323	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
324	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
325	C₂×#124	C₂²×C₆₇	1	2	yes	—
326	C₂×#121	C₂²×C₆₇	1	4	yes	—
327	C₂×#121	C₂²×C₆₇	1	2	yes	—
328	C₂×#121	C₂²×C₆₇	1	2	yes	—
329	C₄×#41	—	2	8	no	—
330	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
331	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
332	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
333	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
334	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
335	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
336	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
337	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
338	C₂×C₄×D₄	C₂²×C₆₇	1	8	yes	—
339	C₂×#117	C₂²×C₆₇	1	2	yes	—
340	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
341	C₂²×C₄²	C₂²×C₆₇	1	2	yes	—
342	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
343	#400	C₂²×C₆₇	1	4	yes	—
344	#399	C₂²×C₆₇	1	4	yes	—
345	#399	C₂²×C₆₇	1	4	yes	—
346	#397	C₂²×C₆₇	1	4	yes	—
347	#395	C₂²×C₆₇	1	4	yes	—
348	#394	C₂²×C₆₇	1	4	yes	—
349	#394	C₂²×C₆₇	1	4	yes	—
350	#394	C₂²×C₆₇	1	8	yes	—
351	#393	C₂²×C₆₇	1	4	yes	—
352	#393	C₂²×C₆₇	1	4	yes	—
353	#392	C₂²×C₆₇	1	4	yes	—
354	#392	C₂²×C₆₇	1	8	yes	—
355	#392	C₂²×C₆₇	1	4	yes	—
356	#392	C₂²×C₆₇	1	4	yes	—
357	#390	C₂²×C₆₇	1	4	yes	—
358	#389	C₂²×C₆₇	1	4	yes	—
359	#388	C₂²×C₆₇	1	8	yes	—
360	#386	C₂²×C₆₇	1	8	yes	—
361	#383	C₂²×C₆₇	1	2	yes	—
362	#382	C₂²×C₆₇	1	2	yes	—
363	#381	C₂²×C₆₇	1	2	yes	—
364	#380	C₂²×C₆₇	1	2	yes	—
365	#380	C₂²×C₆₇	1	4	yes	—
366	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
367	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
368	C₄×C₂²⋊C₄	C₂²×C₆₇	1	8	yes	—
369	#376	C₂²×C₆₇	1	2	yes	—
370	C₂×#41	—	2	4	no	—
371	C₂×#41	—	2	8	no	—
372	C₂×#41	—	2	8	no	—
373	C₂×#41	—	4	8	no	—
374	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
375	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
376	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
377	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
378	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
379	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
380	C₂²×D₄	C₂³×C₆₇	1	8	yes	—
381	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
382	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
383	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
384	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
385	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
386	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
387	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
388	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
389	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
390	C₄⋊Q₈	—	2	4	no	—
391	#127	—	2	4	no	—
392	#124	—	2	8	no	—
393	#121	—	2	8	no	—
394	#121	—	2	8	no	—
395	#121	—	2	16	no	—
396	C₄×Q₈	—	2	4	no	—
397	C₄×Q₈	—	2	4	no	—
398	C₄×D₄	—	2	8	no	—
399	C₄×D₄	—	2	8	no	—
400	C₄×D₄	—	2	4	no	—
401	C₄×D₄	—	2	4	no	—
402	C₄×D₄	—	2	8	no	—
403	C₄×D₄	—	2	8	no	—
404	C₄×D₄	—	4	16	no	—
405	#117	—	2	4	no	—
406	#117	—	2	8	no	—
407	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
408	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
409	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
410	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
411	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
412	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
413	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
414	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
415	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
416	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
417	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
418	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
419	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
420	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
421	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
422	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
423	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
424	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
425	C₂×C₄²	—	2	8	no	—
426	C₂×C₄²	D₄×C₆₇	1	8	yes	complemented
427	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
428	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
429	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
430	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
431	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
432	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
433	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
434	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
435	#95	C₂³×C₆₇	1	4	yes	—
436	#95	C₂³×C₆₇	1	4	yes	—
437	#95	C₂³×C₆₇	1	8	yes	—
438	#95	C₂³×C₆₇	1	4	yes	—
439	#95	C₂³×C₆₇	1	8	yes	—
440	#95	C₂³×C₆₇	1	4	yes	—
441	C₂⁴	C₂⁴×C₆₇	1	4	yes	—
442	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
443	C₂⁴	C₂⁴×C₆₇	1	2	yes	—
444	#41	—	2	32	no	—
445	#41	—	4	32	no	—
446	C₂×Q₈	—	2	2	no	—
447	C₂×Q₈	—	2	4	no	—
448	C₂×Q₈	—	2	4	no	—
449	C₂×Q₈	—	4	4	no	—
450	C₂×D₄	—	2	4	no	—
451	C₂×D₄	—	2	4	no	—
452	C₂×D₄	—	2	4	no	—
453	C₂×D₄	—	2	8	no	—
454	C₂×D₄	—	2	16	no	—
455	C₂×D₄	—	2	2	no	—
456	C₂×D₄	—	2	4	no	—
457	C₂×D₄	—	2	4	no	—
458	C₂×D₄	—	2	4	no	—
459	C₂×D₄	—	2	4	no	—
460	C₂×D₄	—	2	8	no	—
461	C₂×D₄	—	4	8	no	—
462	C₂×D₄	—	4	4	no	—
463	C₂×D₄	—	4	8	no	—
464	C₂×D₄	—	4	4	no	—
465	C₂×D₄	—	4	8	no	—
466	C₂²×C₄	—	2	4	no	—
467	C₂²×C₄	—	2	4	no	—
468	C₂²×C₄	—	2	4	no	—
469	C₂²×C₄	—	2	4	no	—
470	C₂²×C₄	—	2	4	no	—
471	C₂²×C₄	—	2	4	no	—
472	C₂²×C₄	—	2	8	no	—
473	C₂²×C₄	—	2	8	no	—
474	C₂²×C₄	—	2	16	no	—
475	C₂²×C₄	—	4	2	no	—
476	C₂²×C₄	—	4	4	no	—
477	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
478	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
479	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
480	C₂²×C₄	#41×C₆₇	1	8	yes	complemented
481	C₂²×C₄	C₂⁴×C₆₇	1	8	yes	—
482	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
483	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
484	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
485	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
486	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
487	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
488	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
489	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
490	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
491	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
492	C₄⋊C₄	—	2	4	no	—
493	C₄⋊C₄	—	2	4	no	—
494	C₄⋊C₄	—	2	2	no	—
495	C₄⋊C₄	—	2	2	no	—
496	C₄⋊C₄	—	2	4	no	—
497	C₄⋊C₄	—	2	4	no	—
498	C₄⋊C₄	—	2	8	no	—
499	C₄⋊C₄	—	4	8	no	—
500	C₂²⋊C₄	—	2	4	no	—
501	C₂²⋊C₄	—	2	4	no	—
502	C₂²⋊C₄	—	2	4	no	—
503	C₂²⋊C₄	—	2	8	no	—
504	C₂²⋊C₄	—	2	8	no	—
505	C₂²⋊C₄	—	2	16	no	—
506	C₂²⋊C₄	—	4	8	no	—
507	C₂²⋊C₄	—	4	8	no	—
508	C₂²⋊C₄	—	4	8	no	—
509	C₂²⋊C₄	—	4	4	no	—
510	C₄²	—	2	4	no	—
511	C₄²	—	2	4	no	—
512	C₄²	—	2	4	no	—
513	C₄²	—	2	8	no	—
514	C₄²	C₂×D₄×C₆₇	1	4	yes	—
515	C₄²	C₂×D₄×C₆₇	1	4	yes	—
516	C₄²	C₂×D₄×C₆₇	1	4	yes	—
517	C₄²	C₂×D₄×C₆₇	1	4	yes	—
518	C₂³	—	2	2	no	—
519	C₂³	—	2	2	no	—
520	C₂³	—	2	2	no	—
521	C₂³	—	2	4	no	—
522	C₂³	—	2	2	no	—
523	C₂³	—	2	2	no	—
524	C₂³	—	2	4	no	—
525	C₂³	—	2	4	no	—
526	C₂³	—	2	8	no	—
527	C₂³	—	2	8	no	—
528	C₂³	—	2	4	no	—
529	C₂³	—	2	4	no	—
530	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
531	C₂³	C₂×#41×C₆₇	1	4	yes	—
532	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
533	Q₈	—	2	8	no	—
534	Q₈	—	4	8	no	—
535	D₄	—	2	8	no	—
536	D₄	—	2	16	no	—
537	D₄	—	4	8	no	—
538	D₄	—	4	8	no	—
539	D₄	—	4	16	no	—
540	D₄	—	4	32	no	—
541	D₄	—	4	8	no	—
542	C₂×C₄	—	2	2	no	—
543	C₂×C₄	—	2	2	no	—
544	C₂×C₄	—	2	2	no	—
545	C₂×C₄	—	2	2	no	—
546	C₂×C₄	—	2	2	no	—
547	C₂×C₄	—	2	2	no	—
548	C₂×C₄	—	2	8	no	—
549	C₂×C₄	—	2	16	no	—
550	C₂×C₄	—	2	8	no	—
551	C₂×C₄	—	2	2	no	—
552	C₂×C₄	—	2	2	no	—
553	C₂×C₄	—	2	4	no	—
554	C₂×C₄	—	2	8	no	—
555	C₂×C₄	—	2	8	no	—
556	C₂×C₄	—	2	4	no	—
557	C₂×C₄	—	2	8	no	—
558	C₂×C₄	—	2	4	no	—
559	C₂×C₄	—	2	8	no	—
560	C₂×C₄	—	4	4	no	—
561	C₂×C₄	—	4	8	no	—
562	C₂×C₄	—	4	16	no	—
563	C₂×C₄	—	4	8	no	—
564	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
565	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
566	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
567	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
568	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
569	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
570	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
571	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
572	C₂×C₄	H(5,𝔽₂)×C₆₇	1	4	yes	—
573	C₂²	—	2	2	no	—
574	C₂²	—	2	4	no	—
575	C₂²	—	2	2	no	—
576	C₂²	—	2	2	no	—
577	C₂²	—	2	8	no	—
578	C₂²	—	4	2	no	—
579	C₂²	—	4	2	no	—
580	C₂²	—	4	4	no	—
581	C₂²	—	4	4	no	—
582	C₂²	—	4	4	no	—
583	C₂²	—	4	4	no	—
584	C₂²	—	4	4	no	—
585	C₂²	—	4	8	no	—
586	C₂²	—	4	8	no	—
587	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
588	C₂²	C₂²×#41×C₆₇	1	2	yes	—
589	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	2	yes	—
590	C₂²	#585×C₆₇	1	2	yes	—
591	C₄	—	2	2	no	—
592	C₄	—	2	4	no	—
593	C₄	—	2	2	no	—
594	C₄	—	2	2	no	—
595	C₄	—	2	2	no	—
596	C₄	—	4	4	no	—
597	C₄	—	4	2	no	—
598	C₄	—	4	2	no	—
599	C₄	—	4	8	no	—
600	C₄	#525×C₆₇	1	4	yes	—
601	C₄	#535×C₆₇	1	4	yes	—
602	C₂	—	4	2	no	—
603	C₂	—	4	4	no	—
604	C₂	—	8	4	no	—
605	C₂	C₂×#525×C₆₇	1	1	yes	characteristic
606	C₂	C₂×#535×C₆₇	1	1	yes	characteristic
607	C₂	C₆₇×#3449	1	1	yes	characteristic
608	C₂	C₆₇×#3460	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×#532×C₆₇	C₂	semidirect
2. C₂×C₄×#41×C₆₇	C₂	semidirect
3. C₆₇×#2635	C₂	semidirect
4. C₆₇×#2632	C₂	semidirect
5. C₆₇×#2631	C₂	semidirect
6. C₆₇×#2628	C₂	semidirect
7. C₆₇×#2624	C₂	semidirect
8. C₆₇×#2619	C₂	semidirect
9. C₆₇×#2615	C₂	semidirect
10. C₆₇×#2398	C₂	semidirect
11. C₆₇×#2397	C₂	semidirect
12. C₆₇×#2388	C₂	semidirect
13. C₄×#121×C₆₇	C₂	semidirect
14. C₆₇×#2283	C₂	semidirect
121. C₂×C₄²×C₆₇	D₄	semidirect
175. C₂²×C₄×C₆₇	#41	semidirect
297. #34609	C₆₇	direct (N × H)
305. C₂×#532	C₂×C₆₇	semidirect
306. C₂×C₄×#41	C₂×C₆₇	semidirect
307. #2635	C₂×C₆₇	semidirect
308. #2632	C₂×C₆₇	semidirect
309. #2631	C₂×C₆₇	semidirect
310. #2628	C₂×C₆₇	semidirect
311. #2624	C₂×C₆₇	semidirect
312. #2619	C₂×C₆₇	semidirect
313. #2615	C₂×C₆₇	semidirect
314. #2398	C₂×C₆₇	semidirect
315. #2397	C₂×C₆₇	semidirect
316. #2388	C₂×C₆₇	semidirect
317. C₄×#121	C₂×C₆₇	semidirect
318. #2283	C₂×C₆₇	semidirect
319. C₆₇	#34609	direct (N × H)
426. C₂×C₄²	D₄×C₆₇	semidirect
480. C₂²×C₄	#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×#532×C₆₇	2
C₂×C₄×#41×C₆₇	2
C₆₇×#2635	2
C₆₇×#2632	2
C₆₇×#2631	2
C₆₇×#2628	2
C₆₇×#2624	2
C₆₇×#2619	2
C₆₇×#2615	2
C₆₇×#2398	2
C₆₇×#2397	2
C₆₇×#2388	2
C₄×#121×C₆₇	2
C₆₇×#2283	2
#34609	67

Subgroups — C67×#34609 #30713860

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#34609 #30713860