Subgroups — C₆₇×#34936 #30714187

Order 17152.

4138 conjugacy classes of subgroups.

The subgroup lattice has 554 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×#525×C₆₇	C₂	1	4	yes	maximal, complemented
3	C₂²×C₄×D₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2542	C₂	1	2	yes	maximal, complemented
5	C₆₇×#2541	C₂	1	2	yes	maximal, complemented
6	C₆₇×#2540	C₂	1	4	yes	maximal, complemented
7	C₆₇×#2539	C₂	1	4	yes	maximal, complemented
8	C₆₇×#2501	C₂	1	2	yes	maximal, complemented
9	C₆₇×#2433	C₂	1	2	yes	maximal, complemented
10	C₆₇×#2397	C₂	1	4	yes	maximal, complemented
11	C₄×#127×C₆₇	C₂	1	4	yes	maximal, complemented
12	C₄×#117×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×#41×C₆₇	C₂²	1	2	yes	—
14	C₂³×D₄×C₆₇	C₂²	1	1	yes	characteristic
15	C₂⁴×C₄×C₆₇	C₂²	1	2	yes	—
16	#532×C₆₇	—	2	8	no	—
17	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
18	C₂×#127×C₆₇	C₂²	1	4	yes	—
19	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic, complemented
20	C₂×#124×C₆₇	C₂²	1	2	yes	complemented
21	#525×C₆₇	—	2	32	no	—
22	C₂×#123×C₆₇	C₂²	1	8	yes	—
23	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	8	yes	complemented
24	C₂×#121×C₆₇	C₂²	1	2	yes	—
25	C₂×#121×C₆₇	C₂²	1	2	yes	—
26	C₂×#121×C₆₇	C₂²	1	8	yes	complemented
27	C₂×#120×C₆₇	C₂²	1	8	yes	—
28	C₂×C₄×D₄×C₆₇	—	2	8	no	—
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	2	yes	—
32	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
33	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	complemented
34	C₂×C₄×D₄×C₆₇	C₂²	1	16	yes	complemented
35	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	complemented
36	C₂×#117×C₆₇	C₂²	1	2	yes	—
37	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
38	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
39	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
40	#399×C₆₇	C₂²	1	8	yes	complemented
41	#399×C₆₇	C₂²	1	8	yes	complemented
42	#392×C₆₇	C₂²	1	8	yes	—
43	#392×C₆₇	C₂²	1	8	yes	complemented
44	#390×C₆₇	C₂²	1	4	yes	—
45	#390×C₆₇	C₂²	1	4	yes	complemented
46	#390×C₆₇	C₂²	1	16	yes	complemented
47	#384×C₆₇	C₂²	1	4	yes	complemented
48	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	complemented
49	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	complemented
50	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
51	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
52	#376×C₆₇	C₂²	1	2	yes	complemented
53	C₄³×C₆₇	C₂²	1	2	yes	complemented
54	C₂⁵×C₆₇	C₂³	1	2	yes	—
55	C₂×#41×C₆₇	—	2	16	no	—
56	C₂×#41×C₆₇	—	2	8	no	—
57	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
58	C₂²×D₄×C₆₇	—	2	8	no	—
59	C₂²×D₄×C₆₇	—	2	4	no	—
60	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
61	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
62	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
63	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
64	C₂²×D₄×C₆₇	C₂³	1	16	yes	—
65	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
66	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
67	C₂³×C₄×C₆₇	—	2	8	no	—
68	C₂³×C₄×C₆₇	—	2	16	no	—
69	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
70	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
71	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
72	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
73	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
74	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
75	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
76	C₄⋊Q₈×C₆₇	—	2	4	no	—
77	#127×C₆₇	—	2	16	no	—
78	#127×C₆₇	—	2	4	no	—
79	#124×C₆₇	—	2	8	no	—
80	#123×C₆₇	—	2	32	no	—
81	C₂²⋊Q₈×C₆₇	—	2	32	no	—
82	#121×C₆₇	—	2	8	no	—
83	#121×C₆₇	—	2	8	no	—
84	#121×C₆₇	—	2	32	no	—
85	#120×C₆₇	—	2	32	no	—
86	C₄×D₄×C₆₇	—	2	8	no	—
87	C₄×D₄×C₆₇	—	2	8	no	—
88	C₄×D₄×C₆₇	—	2	8	no	—
89	C₄×D₄×C₆₇	—	2	8	no	—
90	C₄×D₄×C₆₇	—	2	8	no	—
91	C₄×D₄×C₆₇	—	2	8	no	—
92	C₄×D₄×C₆₇	—	2	64	no	—
93	#117×C₆₇	D₄	1	16	yes	complemented
94	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
95	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
96	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
97	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
98	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
99	C₂×C₄⋊C₄×C₆₇	C₂³	1	16	yes	—
100	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
101	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
102	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
103	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
104	C₂×C₂²⋊C₄×C₆₇	C₂³	1	16	yes	—
105	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
106	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
107	C₂×C₂²⋊C₄×C₆₇	C₂³	1	16	yes	—
108	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
109	C₂×C₄²×C₆₇	D₄	1	8	yes	complemented
110	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
111	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
112	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
113	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
114	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
115	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
116	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
117	#95×C₆₇	C₂³	1	4	yes	—
118	#95×C₆₇	C₂³	1	4	yes	—
119	C₂⁴×C₆₇	—	2	4	no	—
120	C₂⁴×C₆₇	—	2	8	no	—
121	C₂⁴×C₆₇	—	2	8	no	—
122	C₂⁴×C₆₇	—	4	4	no	—
123	C₂⁴×C₆₇	C₂⁴	1	4	yes	—
124	C₂⁴×C₆₇	C₂⁴	1	4	yes	—
125	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
126	#41×C₆₇	—	4	32	no	—
127	C₂×Q₈×C₆₇	—	2	8	no	—
128	C₂×Q₈×C₆₇	—	2	4	no	—
129	C₂×D₄×C₆₇	—	2	32	no	—
130	C₂×D₄×C₆₇	—	2	8	no	—
131	C₂×D₄×C₆₇	—	2	16	no	—
132	C₂×D₄×C₆₇	—	2	8	no	—
133	C₂×D₄×C₆₇	—	2	32	no	—
134	C₂×D₄×C₆₇	—	2	4	no	—
135	C₂×D₄×C₆₇	—	2	4	no	—
136	C₂×D₄×C₆₇	—	2	32	no	—
137	C₂×D₄×C₆₇	—	2	4	no	—
138	C₂×D₄×C₆₇	—	2	4	no	—
139	C₂×D₄×C₆₇	—	2	8	no	—
140	C₂×D₄×C₆₇	—	2	4	no	—
141	C₂×D₄×C₆₇	—	2	4	no	—
142	C₂×D₄×C₆₇	—	2	8	no	—
143	C₂×D₄×C₆₇	—	4	16	no	—
144	C₂×D₄×C₆₇	—	4	4	no	—
145	C₂×D₄×C₆₇	#41	1	8	yes	complemented
146	C₂²×C₄×C₆₇	—	2	4	no	—
147	C₂²×C₄×C₆₇	—	2	4	no	—
148	C₂²×C₄×C₆₇	—	2	8	no	—
149	C₂²×C₄×C₆₇	—	2	8	no	—
150	C₂²×C₄×C₆₇	—	2	8	no	—
151	C₂²×C₄×C₆₇	—	2	16	no	—
152	C₂²×C₄×C₆₇	—	2	8	no	—
153	C₂²×C₄×C₆₇	—	2	16	no	—
154	C₂²×C₄×C₆₇	—	2	16	no	—
155	C₂²×C₄×C₆₇	—	4	2	no	—
156	C₂²×C₄×C₆₇	—	4	32	no	—
157	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
158	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
159	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
160	C₂²×C₄×C₆₇	C₂×D₄	1	8	yes	—
161	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
162	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
163	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
164	C₂²×C₄×C₆₇	C₂⁴	1	8	yes	—
165	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
166	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
167	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
168	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
169	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
170	C₄⋊C₄×C₆₇	—	2	4	no	—
171	C₄⋊C₄×C₆₇	—	2	4	no	—
172	C₄⋊C₄×C₆₇	—	2	4	no	—
173	C₄⋊C₄×C₆₇	—	2	4	no	—
174	C₄⋊C₄×C₆₇	—	2	32	no	—
175	C₄⋊C₄×C₆₇	C₂×D₄	1	16	yes	complemented
176	C₂²⋊C₄×C₆₇	—	2	8	no	—
177	C₂²⋊C₄×C₆₇	—	2	8	no	—
178	C₂²⋊C₄×C₆₇	—	2	32	no	—
179	C₂²⋊C₄×C₆₇	—	2	32	no	—
180	C₂²⋊C₄×C₆₇	—	2	8	no	—
181	C₂²⋊C₄×C₆₇	—	2	8	no	—
182	C₂²⋊C₄×C₆₇	C₂×D₄	1	16	yes	—
183	C₄²×C₆₇	—	2	8	no	—
184	C₄²×C₆₇	—	4	4	no	—
185	C₄²×C₆₇	C₂×D₄	1	4	yes	—
186	C₄²×C₆₇	C₂×D₄	1	4	yes	—
187	C₄²×C₆₇	C₂×D₄	1	4	yes	—
188	C₄²×C₆₇	C₂×D₄	1	4	yes	—
189	C₄²×C₆₇	C₂×D₄	1	16	yes	complemented
190	C₂³×C₆₇	—	2	2	no	—
191	C₂³×C₆₇	—	2	2	no	—
192	C₂³×C₆₇	—	2	16	no	—
193	C₂³×C₆₇	—	2	4	no	—
194	C₂³×C₆₇	—	2	8	no	—
195	C₂³×C₆₇	—	2	4	no	—
196	C₂³×C₆₇	—	2	8	no	—
197	C₂³×C₆₇	—	2	4	no	—
198	C₂³×C₆₇	—	2	8	no	—
199	C₂³×C₆₇	—	2	8	no	—
200	C₂³×C₆₇	—	2	8	no	—
201	C₂³×C₆₇	—	4	32	no	—
202	C₂³×C₆₇	—	4	8	no	—
203	C₂³×C₆₇	—	4	8	no	—
204	C₂³×C₆₇	C₂²×D₄	1	4	yes	—
205	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
206	C₂³×C₆₇	C₂×#41	1	8	yes	—
207	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
208	Q₈×C₆₇	—	4	8	no	—
209	D₄×C₆₇	—	2	16	no	—
210	D₄×C₆₇	—	2	16	no	—
211	D₄×C₆₇	—	4	64	no	—
212	D₄×C₆₇	—	4	8	no	—
213	D₄×C₆₇	—	4	16	no	—
214	D₄×C₆₇	—	4	8	no	—
215	D₄×C₆₇	—	4	8	no	—
216	C₂×C₄×C₆₇	—	2	8	no	—
217	C₂×C₄×C₆₇	—	2	2	no	—
218	C₂×C₄×C₆₇	—	2	2	no	—
219	C₂×C₄×C₆₇	—	2	8	no	—
220	C₂×C₄×C₆₇	—	2	2	no	—
221	C₂×C₄×C₆₇	—	2	8	no	—
222	C₂×C₄×C₆₇	—	2	4	no	—
223	C₂×C₄×C₆₇	—	2	8	no	—
224	C₂×C₄×C₆₇	—	2	4	no	—
225	C₂×C₄×C₆₇	—	2	8	no	—
226	C₂×C₄×C₆₇	—	4	4	no	—
227	C₂×C₄×C₆₇	—	4	8	no	—
228	C₂×C₄×C₆₇	—	4	16	no	—
229	C₂×C₄×C₆₇	—	4	16	no	—
230	C₂×C₄×C₆₇	—	4	16	no	—
231	C₂×C₄×C₆₇	#127	1	16	yes	complemented
232	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
233	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
234	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
235	C₂×C₄×C₆₇	C₂²×D₄	1	16	yes	—
236	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
237	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
238	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
239	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
240	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
241	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
242	C₂²×C₆₇	—	2	2	no	—
243	C₂²×C₆₇	—	2	2	no	—
244	C₂²×C₆₇	—	2	8	no	—
245	C₂²×C₆₇	—	2	8	no	—
246	C₂²×C₆₇	—	2	16	no	—
247	C₂²×C₆₇	—	4	32	no	—
248	C₂²×C₆₇	—	4	4	no	—
249	C₂²×C₆₇	—	4	4	no	—
250	C₂²×C₆₇	—	4	8	no	—
251	C₂²×C₆₇	—	4	4	no	—
252	C₂²×C₆₇	—	4	4	no	—
253	C₂²×C₆₇	—	4	4	no	—
254	C₂²×C₆₇	—	4	16	no	—
255	C₂²×C₆₇	C₂×#127	1	4	yes	—
256	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
257	C₂²×C₆₇	C₂³×D₄	1	2	yes	—
258	C₂²×C₆₇	C₂²×#41	1	2	yes	—
259	C₂²×C₆₇	#585	1	2	yes	—
260	C₄×C₆₇	—	2	2	no	—
261	C₄×C₆₇	—	2	2	no	—
262	C₄×C₆₇	—	2	2	no	—
263	C₄×C₆₇	—	2	16	no	—
264	C₄×C₆₇	—	4	4	no	—
265	C₄×C₆₇	—	4	4	no	—
266	C₄×C₆₇	#525	1	4	yes	complemented
267	C₄×C₆₇	C₂×#127	1	4	yes	—
268	C₄×C₆₇	C₂×#127	1	4	yes	—
269	#34936	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
270	C₂×C₆₇	—	2	4	no	—
271	C₂×C₆₇	—	4	8	no	—
272	C₂×C₆₇	—	8	4	no	—
273	C₂×C₆₇	C₂×#525	1	1	yes	characteristic
274	C₂×C₆₇	C₂²×#127	1	1	yes	characteristic
275	C₂×C₆₇	#3449	1	1	yes	characteristic
276	C₂×C₆₇	D₄×#41	1	4	yes	—
277	C₂×#532	C₂×C₆₇	1	1	yes	characteristic, complemented
278	C₂×#525	C₂×C₆₇	1	4	yes	complemented
279	C₂²×C₄×D₄	C₂×C₆₇	1	1	yes	characteristic, complemented
280	#2542	C₂×C₆₇	1	2	yes	complemented
281	#2541	C₂×C₆₇	1	2	yes	complemented
282	#2540	C₂×C₆₇	1	4	yes	complemented
283	#2539	C₂×C₆₇	1	4	yes	complemented
284	#2501	C₂×C₆₇	1	2	yes	complemented
285	#2433	C₂×C₆₇	1	2	yes	complemented
286	#2397	C₂×C₆₇	1	4	yes	complemented
287	C₄×#127	C₂×C₆₇	1	4	yes	complemented
288	C₄×#117	C₂×C₆₇	1	1	yes	characteristic, complemented
289	C₆₇	#34936	1	1	yes	characteristic, direct product factor, Sylow
290	C₂²×#41	C₂²×C₆₇	1	2	yes	—
291	C₂³×D₄	C₂²×C₆₇	1	1	yes	characteristic
292	C₂⁴×C₄	C₂²×C₆₇	1	2	yes	—
293	#532	—	2	8	no	—
294	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
295	C₂×#127	C₂²×C₆₇	1	4	yes	—
296	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic, complemented
297	C₂×#124	C₂²×C₆₇	1	2	yes	complemented
298	#525	—	2	32	no	—
299	C₂×#123	C₂²×C₆₇	1	8	yes	—
300	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	8	yes	complemented
301	C₂×#121	C₂²×C₆₇	1	2	yes	—
302	C₂×#121	C₂²×C₆₇	1	2	yes	—
303	C₂×#121	C₂²×C₆₇	1	8	yes	complemented
304	C₂×#120	C₂²×C₆₇	1	8	yes	—
305	C₂×C₄×D₄	—	2	8	no	—
306	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
307	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
308	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
309	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
310	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	complemented
311	C₂×C₄×D₄	C₂²×C₆₇	1	16	yes	complemented
312	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	complemented
313	C₂×#117	C₂²×C₆₇	1	2	yes	—
314	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
315	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
316	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
317	#399	C₂²×C₆₇	1	8	yes	complemented
318	#399	C₂²×C₆₇	1	8	yes	complemented
319	#392	C₂²×C₆₇	1	8	yes	—
320	#392	C₂²×C₆₇	1	8	yes	complemented
321	#390	C₂²×C₆₇	1	4	yes	—
322	#390	C₂²×C₆₇	1	4	yes	complemented
323	#390	C₂²×C₆₇	1	16	yes	complemented
324	#384	C₂²×C₆₇	1	4	yes	complemented
325	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	complemented
326	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	complemented
327	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
328	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
329	#376	C₂²×C₆₇	1	2	yes	complemented
330	C₄³	C₂²×C₆₇	1	2	yes	complemented
331	C₂⁵	C₂³×C₆₇	1	2	yes	—
332	C₂×#41	—	2	16	no	—
333	C₂×#41	—	2	8	no	—
334	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
335	C₂²×D₄	—	2	8	no	—
336	C₂²×D₄	—	2	4	no	—
337	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
338	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
339	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
340	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
341	C₂²×D₄	C₂³×C₆₇	1	16	yes	—
342	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
343	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
344	C₂³×C₄	—	2	8	no	—
345	C₂³×C₄	—	2	16	no	—
346	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
347	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
348	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
349	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
350	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
351	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
352	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
353	C₄⋊Q₈	—	2	4	no	—
354	#127	—	2	16	no	—
355	#127	—	2	4	no	—
356	#124	—	2	8	no	—
357	#123	—	2	32	no	—
358	C₂²⋊Q₈	—	2	32	no	—
359	#121	—	2	8	no	—
360	#121	—	2	8	no	—
361	#121	—	2	32	no	—
362	#120	—	2	32	no	—
363	C₄×D₄	—	2	8	no	—
364	C₄×D₄	—	2	8	no	—
365	C₄×D₄	—	2	8	no	—
366	C₄×D₄	—	2	8	no	—
367	C₄×D₄	—	2	8	no	—
368	C₄×D₄	—	2	8	no	—
369	C₄×D₄	—	2	64	no	—
370	#117	D₄×C₆₇	1	16	yes	complemented
371	C₂×C₄⋊C₄	—	2	4	no	—
372	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
373	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
374	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
375	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
376	C₂×C₄⋊C₄	C₂³×C₆₇	1	16	yes	—
377	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
378	C₂×C₂²⋊C₄	—	2	8	no	—
379	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
380	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
381	C₂×C₂²⋊C₄	C₂³×C₆₇	1	16	yes	—
382	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
383	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
384	C₂×C₂²⋊C₄	C₂³×C₆₇	1	16	yes	—
385	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
386	C₂×C₄²	D₄×C₆₇	1	8	yes	complemented
387	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
388	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
389	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
390	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
391	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
392	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
393	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
394	#95	C₂³×C₆₇	1	4	yes	—
395	#95	C₂³×C₆₇	1	4	yes	—
396	C₂⁴	—	2	4	no	—
397	C₂⁴	—	2	8	no	—
398	C₂⁴	—	2	8	no	—
399	C₂⁴	—	4	4	no	—
400	C₂⁴	C₂⁴×C₆₇	1	4	yes	—
401	C₂⁴	C₂⁴×C₆₇	1	4	yes	—
402	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
403	#41	—	4	32	no	—
404	C₂×Q₈	—	2	8	no	—
405	C₂×Q₈	—	2	4	no	—
406	C₂×D₄	—	2	32	no	—
407	C₂×D₄	—	2	8	no	—
408	C₂×D₄	—	2	16	no	—
409	C₂×D₄	—	2	8	no	—
410	C₂×D₄	—	2	32	no	—
411	C₂×D₄	—	2	4	no	—
412	C₂×D₄	—	2	4	no	—
413	C₂×D₄	—	2	32	no	—
414	C₂×D₄	—	2	4	no	—
415	C₂×D₄	—	2	4	no	—
416	C₂×D₄	—	2	8	no	—
417	C₂×D₄	—	2	4	no	—
418	C₂×D₄	—	2	4	no	—
419	C₂×D₄	—	2	8	no	—
420	C₂×D₄	—	4	16	no	—
421	C₂×D₄	—	4	4	no	—
422	C₂×D₄	#41×C₆₇	1	8	yes	complemented
423	C₂²×C₄	—	2	4	no	—
424	C₂²×C₄	—	2	4	no	—
425	C₂²×C₄	—	2	8	no	—
426	C₂²×C₄	—	2	8	no	—
427	C₂²×C₄	—	2	8	no	—
428	C₂²×C₄	—	2	16	no	—
429	C₂²×C₄	—	2	8	no	—
430	C₂²×C₄	—	2	16	no	—
431	C₂²×C₄	—	2	16	no	—
432	C₂²×C₄	—	4	2	no	—
433	C₂²×C₄	—	4	32	no	—
434	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
435	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
436	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
437	C₂²×C₄	C₂×D₄×C₆₇	1	8	yes	—
438	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
439	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
440	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
441	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
442	C₂²×C₄	C₂⁴×C₆₇	1	8	yes	—
443	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
444	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
445	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
446	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
447	C₄⋊C₄	—	2	4	no	—
448	C₄⋊C₄	—	2	4	no	—
449	C₄⋊C₄	—	2	4	no	—
450	C₄⋊C₄	—	2	4	no	—
451	C₄⋊C₄	—	2	32	no	—
452	C₄⋊C₄	C₂×D₄×C₆₇	1	16	yes	complemented
453	C₂²⋊C₄	—	2	8	no	—
454	C₂²⋊C₄	—	2	8	no	—
455	C₂²⋊C₄	—	2	32	no	—
456	C₂²⋊C₄	—	2	32	no	—
457	C₂²⋊C₄	—	2	8	no	—
458	C₂²⋊C₄	—	2	8	no	—
459	C₂²⋊C₄	C₂×D₄×C₆₇	1	16	yes	—
460	C₄²	—	2	8	no	—
461	C₄²	—	4	4	no	—
462	C₄²	C₂×D₄×C₆₇	1	4	yes	—
463	C₄²	C₂×D₄×C₆₇	1	4	yes	—
464	C₄²	C₂×D₄×C₆₇	1	4	yes	—
465	C₄²	C₂×D₄×C₆₇	1	4	yes	—
466	C₄²	C₂×D₄×C₆₇	1	16	yes	complemented
467	C₂³	—	2	2	no	—
468	C₂³	—	2	2	no	—
469	C₂³	—	2	16	no	—
470	C₂³	—	2	4	no	—
471	C₂³	—	2	8	no	—
472	C₂³	—	2	4	no	—
473	C₂³	—	2	8	no	—
474	C₂³	—	2	4	no	—
475	C₂³	—	2	8	no	—
476	C₂³	—	2	8	no	—
477	C₂³	—	2	8	no	—
478	C₂³	—	4	32	no	—
479	C₂³	—	4	8	no	—
480	C₂³	—	4	8	no	—
481	C₂³	C₂²×D₄×C₆₇	1	4	yes	—
482	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
483	C₂³	C₂×#41×C₆₇	1	8	yes	—
484	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
485	Q₈	—	4	8	no	—
486	D₄	—	2	16	no	—
487	D₄	—	2	16	no	—
488	D₄	—	4	64	no	—
489	D₄	—	4	8	no	—
490	D₄	—	4	16	no	—
491	D₄	—	4	8	no	—
492	D₄	—	4	8	no	—
493	C₂×C₄	—	2	8	no	—
494	C₂×C₄	—	2	2	no	—
495	C₂×C₄	—	2	2	no	—
496	C₂×C₄	—	2	8	no	—
497	C₂×C₄	—	2	2	no	—
498	C₂×C₄	—	2	8	no	—
499	C₂×C₄	—	2	4	no	—
500	C₂×C₄	—	2	8	no	—
501	C₂×C₄	—	2	4	no	—
502	C₂×C₄	—	2	8	no	—
503	C₂×C₄	—	4	4	no	—
504	C₂×C₄	—	4	8	no	—
505	C₂×C₄	—	4	16	no	—
506	C₂×C₄	—	4	16	no	—
507	C₂×C₄	—	4	16	no	—
508	C₂×C₄	#127×C₆₇	1	16	yes	complemented
509	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
510	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
511	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
512	C₂×C₄	C₂²×D₄×C₆₇	1	16	yes	—
513	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
514	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
515	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
516	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
517	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
518	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
519	C₂²	—	2	2	no	—
520	C₂²	—	2	2	no	—
521	C₂²	—	2	8	no	—
522	C₂²	—	2	8	no	—
523	C₂²	—	2	16	no	—
524	C₂²	—	4	32	no	—
525	C₂²	—	4	4	no	—
526	C₂²	—	4	4	no	—
527	C₂²	—	4	8	no	—
528	C₂²	—	4	4	no	—
529	C₂²	—	4	4	no	—
530	C₂²	—	4	4	no	—
531	C₂²	—	4	16	no	—
532	C₂²	C₂×#127×C₆₇	1	4	yes	—
533	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
534	C₂²	C₂³×D₄×C₆₇	1	2	yes	—
535	C₂²	C₂²×#41×C₆₇	1	2	yes	—
536	C₂²	#585×C₆₇	1	2	yes	—
537	C₄	—	2	2	no	—
538	C₄	—	2	2	no	—
539	C₄	—	2	2	no	—
540	C₄	—	2	16	no	—
541	C₄	—	4	4	no	—
542	C₄	—	4	4	no	—
543	C₄	#525×C₆₇	1	4	yes	complemented
544	C₄	C₂×#127×C₆₇	1	4	yes	—
545	C₄	C₂×#127×C₆₇	1	4	yes	—
546	C₂	—	2	4	no	—
547	C₂	—	4	8	no	—
548	C₂	—	8	4	no	—
549	C₂	C₂×#525×C₆₇	1	1	yes	characteristic
550	C₂	C₂²×#127×C₆₇	1	1	yes	characteristic
551	C₂	C₆₇×#3449	1	1	yes	characteristic
552	C₂	D₄×#41×C₆₇	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×#525×C₆₇	C₂	semidirect
3. C₂²×C₄×D₄×C₆₇	C₂	semidirect
4. C₆₇×#2542	C₂	semidirect
5. C₆₇×#2541	C₂	semidirect
6. C₆₇×#2540	C₂	semidirect
7. C₆₇×#2539	C₂	semidirect
8. C₆₇×#2501	C₂	semidirect
9. C₆₇×#2433	C₂	semidirect
10. C₆₇×#2397	C₂	semidirect
11. C₄×#127×C₆₇	C₂	semidirect
12. C₄×#117×C₆₇	C₂	semidirect
17. C₂×C₄⋊Q₈×C₆₇	C₂²	semidirect
19. C₂×#127×C₆₇	C₂²	semidirect
20. C₂×#124×C₆₇	C₂²	semidirect
23. C₂×C₂²⋊Q₈×C₆₇	C₂²	semidirect
26. C₂×#121×C₆₇	C₂²	semidirect
33. C₂×C₄×D₄×C₆₇	C₂²	semidirect
34. C₂×C₄×D₄×C₆₇	C₂²	semidirect
35. C₂×C₄×D₄×C₆₇	C₂²	semidirect
40. #399×C₆₇	C₂²	semidirect
41. #399×C₆₇	C₂²	semidirect
43. #392×C₆₇	C₂²	semidirect
45. #390×C₆₇	C₂²	semidirect
46. #390×C₆₇	C₂²	semidirect
47. #384×C₆₇	C₂²	semidirect
48. C₄×C₄⋊C₄×C₆₇	C₂²	semidirect
49. C₄×C₄⋊C₄×C₆₇	C₂²	semidirect
52. #376×C₆₇	C₂²	semidirect
53. C₄³×C₆₇	C₂²	semidirect
93. #117×C₆₇	D₄	semidirect
109. C₂×C₄²×C₆₇	D₄	semidirect
145. C₂×D₄×C₆₇	#41	semidirect
175. C₄⋊C₄×C₆₇	C₂×D₄	semidirect
189. C₄²×C₆₇	C₂×D₄	semidirect
231. C₂×C₄×C₆₇	#127	semidirect
266. C₄×C₆₇	#525	semidirect
269. #34936	C₆₇	direct (N × H)
277. C₂×#532	C₂×C₆₇	semidirect
278. C₂×#525	C₂×C₆₇	semidirect
279. C₂²×C₄×D₄	C₂×C₆₇	semidirect
280. #2542	C₂×C₆₇	semidirect
281. #2541	C₂×C₆₇	semidirect
282. #2540	C₂×C₆₇	semidirect
283. #2539	C₂×C₆₇	semidirect
284. #2501	C₂×C₆₇	semidirect
285. #2433	C₂×C₆₇	semidirect
286. #2397	C₂×C₆₇	semidirect
287. C₄×#127	C₂×C₆₇	semidirect
288. C₄×#117	C₂×C₆₇	semidirect
289. C₆₇	#34936	direct (N × H)
294. C₂×C₄⋊Q₈	C₂²×C₆₇	semidirect
296. C₂×#127	C₂²×C₆₇	semidirect
297. C₂×#124	C₂²×C₆₇	semidirect
300. C₂×C₂²⋊Q₈	C₂²×C₆₇	semidirect
303. C₂×#121	C₂²×C₆₇	semidirect
310. C₂×C₄×D₄	C₂²×C₆₇	semidirect
311. C₂×C₄×D₄	C₂²×C₆₇	semidirect
312. C₂×C₄×D₄	C₂²×C₆₇	semidirect
317. #399	C₂²×C₆₇	semidirect
318. #399	C₂²×C₆₇	semidirect
320. #392	C₂²×C₆₇	semidirect
322. #390	C₂²×C₆₇	semidirect
323. #390	C₂²×C₆₇	semidirect
324. #384	C₂²×C₆₇	semidirect
325. C₄×C₄⋊C₄	C₂²×C₆₇	semidirect
326. C₄×C₄⋊C₄	C₂²×C₆₇	semidirect
329. #376	C₂²×C₆₇	semidirect
330. C₄³	C₂²×C₆₇	semidirect
370. #117	D₄×C₆₇	semidirect
386. C₂×C₄²	D₄×C₆₇	semidirect
422. C₂×D₄	#41×C₆₇	semidirect
452. C₄⋊C₄	C₂×D₄×C₆₇	semidirect
466. C₄²	C₂×D₄×C₆₇	semidirect
508. C₂×C₄	#127×C₆₇	semidirect
543. C₄	#525×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×#525×C₆₇	2
C₂²×C₄×D₄×C₆₇	2
C₆₇×#2542	2
C₆₇×#2541	2
C₆₇×#2540	2
C₆₇×#2539	2
C₆₇×#2501	2
C₆₇×#2433	2
C₆₇×#2397	2
C₄×#127×C₆₇	2
C₄×#117×C₆₇	2
#34936	67

Subgroups — C67×#34936 #30714187

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#34936 #30714187