Subgroups — C₆₇×#25940 #30705191

Order 17152.

1098 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#554×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#489×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#463×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#2592	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#2026	C₂	1	2	yes	maximal, complemented
6	C₆₇×#2025	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1988	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1985	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#1879	C₂	1	2	yes	maximal
10	C₆₇×#1877	C₂	1	2	yes	maximal, complemented
11	C₆₇×#1762	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×SD₁₆×C₆₇	C₂²	1	1	yes	characteristic
13	C₂²×D₈×C₆₇	C₂²	1	1	yes	characteristic
14	#554×C₆₇	—	2	4	no	—
15	#554×C₆₇	—	2	4	no	—
16	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×#125×C₆₇	C₂²	1	2	yes	—
18	C₂×#124×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×C₂²⋊Q₈×C₆₇	—	2	1	no	—
20	C₂×C₂²⋊Q₈×C₆₇	—	2	2	no	—
21	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
23	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
24	C₂²×C₄⋊C₄×C₆₇	—	2	1	no	—
25	#489×C₆₇	—	2	4	no	—
26	#463×C₆₇	—	2	4	no	—
27	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
28	C₂×#103×C₆₇	C₂²	1	1	yes	characteristic
29	C₂×#103×C₆₇	C₂²	1	2	yes	—
30	C₂×#102×C₆₇	C₂²	1	1	yes	characteristic
31	C₂×#102×C₆₇	C₂²	1	2	yes	—
32	C₂×#102×C₆₇	C₂²	1	2	yes	—
33	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
34	#397×C₆₇	—	2	1	no	—
35	#393×C₆₇	—	2	1	no	—
36	#391×C₆₇	C₂²	1	2	yes	—
37	#390×C₆₇	C₂²	1	2	yes	—
38	#388×C₆₇	—	2	2	no	—
39	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
40	#340×C₆₇	C₂²	1	2	yes	—
41	#340×C₆₇	C₂²	1	2	yes	—
42	#340×C₆₇	C₂²	1	2	yes	—
43	#336×C₆₇	C₂²	1	2	yes	—
44	#336×C₆₇	C₂²	1	2	yes	—
45	C₂²×Q₈×C₆₇	—	2	1	no	—
46	C₂²×Q₈×C₆₇	D₄	1	1	yes	characteristic
47	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
48	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
49	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
50	C₂³×C₄×C₆₇	—	2	1	no	—
51	C₂³×C₄×C₆₇	—	2	2	no	—
52	C₂×SD₁₆×C₆₇	—	2	2	no	—
53	C₂×SD₁₆×C₆₇	—	2	4	no	—
54	C₂×D₈×C₆₇	—	2	2	no	—
55	C₂×D₈×C₆₇	—	2	4	no	—
56	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
57	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
58	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
59	C₄⋊Q₈×C₆₇	—	2	2	no	—
60	C₄⋊Q₈×C₆₇	D₄	1	4	yes	—
61	#125×C₆₇	—	2	4	no	—
62	#125×C₆₇	—	2	4	no	—
63	#124×C₆₇	—	2	2	no	—
64	#124×C₆₇	—	2	2	no	—
65	C₂²⋊Q₈×C₆₇	—	2	8	no	—
66	C₂²⋊Q₈×C₆₇	—	2	16	no	—
67	C₄×Q₈×C₆₇	—	2	2	no	—
68	C₄×Q₈×C₆₇	—	2	2	no	—
69	C₄×D₄×C₆₇	—	2	4	no	—
70	C₄×D₄×C₆₇	—	2	4	no	—
71	C₄×D₄×C₆₇	—	2	2	no	—
72	C₄×D₄×C₆₇	D₄	1	4	yes	—
73	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
74	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
75	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
76	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
77	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
78	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
79	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
80	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
81	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
82	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
83	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
84	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
85	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
86	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
87	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
88	C₂×C₄²×C₆₇	—	2	1	no	—
89	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
90	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
91	C₄⋊C₈×C₆₇	—	2	2	no	—
92	#103×C₆₇	—	2	2	no	—
93	#103×C₆₇	—	2	4	no	—
94	#102×C₆₇	—	2	4	no	—
95	#102×C₆₇	—	2	2	no	—
96	#102×C₆₇	—	2	4	no	—
97	#97×C₆₇	—	2	2	no	—
98	#95×C₆₇	—	2	1	no	—
99	#95×C₆₇	—	2	2	no	—
100	#95×C₆₇	—	2	2	no	—
101	C₂⁴×C₆₇	—	2	1	no	—
102	C₂⁴×C₆₇	—	2	1	no	—
103	C₂×Q₈×C₆₇	—	2	2	no	—
104	C₂×Q₈×C₆₇	—	2	4	no	—
105	C₂×Q₈×C₆₇	—	2	1	no	—
106	C₂×Q₈×C₆₇	—	2	1	no	—
107	C₂×Q₈×C₆₇	—	2	2	no	—
108	C₂×Q₈×C₆₇	—	2	2	no	—
109	C₂×Q₈×C₆₇	—	4	2	no	—
110	C₂×Q₈×C₆₇	—	8	1	no	—
111	C₂×Q₈×C₆₇	—	8	1	no	—
112	C₂×D₄×C₆₇	—	2	1	no	—
113	C₂×D₄×C₆₇	—	2	1	no	—
114	C₂×D₄×C₆₇	—	2	1	no	—
115	C₂×D₄×C₆₇	—	2	2	no	—
116	C₂×D₄×C₆₇	—	2	2	no	—
117	C₂×D₄×C₆₇	—	2	2	no	—
118	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
119	C₂×D₄×C₆₇	#41	1	4	yes	—
120	C₂²×C₄×C₆₇	—	2	1	no	—
121	C₂²×C₄×C₆₇	—	2	2	no	—
122	C₂²×C₄×C₆₇	—	2	8	no	—
123	C₂²×C₄×C₆₇	—	2	4	no	—
124	C₂²×C₄×C₆₇	—	2	2	no	—
125	C₂²×C₄×C₆₇	—	2	1	no	—
126	C₂²×C₄×C₆₇	—	4	4	no	—
127	C₂²×C₄×C₆₇	—	8	2	no	—
128	C₂²×C₄×C₆₇	—	8	2	no	—
129	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
130	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
131	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
132	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
133	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
134	SD₁₆×C₆₇	—	4	4	no	—
135	D₈×C₆₇	—	4	4	no	—
136	C₂×C₈×C₆₇	—	2	2	no	—
137	C₂×C₈×C₆₇	—	2	1	no	—
138	C₂×C₈×C₆₇	—	2	1	no	—
139	C₂×C₈×C₆₇	—	2	2	no	—
140	C₂×C₈×C₆₇	—	2	2	no	—
141	C₂×C₈×C₆₇	—	2	2	no	—
142	C₂×C₈×C₆₇	—	2	2	no	—
143	C₄⋊C₄×C₆₇	—	2	8	no	—
144	C₄⋊C₄×C₆₇	—	2	8	no	—
145	C₄⋊C₄×C₆₇	—	2	2	no	—
146	C₄⋊C₄×C₆₇	—	2	4	no	—
147	C₄⋊C₄×C₆₇	—	2	4	no	—
148	C₄⋊C₄×C₆₇	—	2	1	no	—
149	C₄⋊C₄×C₆₇	—	2	2	no	—
150	C₄⋊C₄×C₆₇	—	2	2	no	—
151	C₄⋊C₄×C₆₇	—	2	2	no	—
152	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
153	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
154	C₂²⋊C₄×C₆₇	—	2	8	no	—
155	C₂²⋊C₄×C₆₇	—	2	4	no	—
156	C₂²⋊C₄×C₆₇	—	4	4	no	—
157	C₂²⋊C₄×C₆₇	—	4	2	no	—
158	C₂²⋊C₄×C₆₇	—	4	2	no	—
159	C₂²⋊C₄×C₆₇	—	8	2	no	—
160	C₄²×C₆₇	—	2	2	no	—
161	C₄²×C₆₇	—	2	2	no	—
162	C₄²×C₆₇	—	8	1	no	—
163	C₄²×C₆₇	C₂×D₄	1	2	yes	—
164	C₄²×C₆₇	C₂×D₄	1	2	yes	—
165	C₂³×C₆₇	—	2	2	no	—
166	C₂³×C₆₇	—	2	4	no	—
167	C₂³×C₆₇	—	4	2	no	—
168	C₂³×C₆₇	—	4	1	no	—
169	C₂³×C₆₇	—	4	1	no	—
170	C₂³×C₆₇	—	4	1	no	—
171	C₂³×C₆₇	—	4	2	no	—
172	C₂³×C₆₇	—	4	2	no	—
173	C₂³×C₆₇	—	4	1	no	—
174	C₂³×C₆₇	—	4	1	no	—
175	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
176	Q₈×C₆₇	—	4	4	no	—
177	Q₈×C₆₇	—	4	1	no	—
178	Q₈×C₆₇	—	4	1	no	—
179	Q₈×C₆₇	—	4	2	no	—
180	Q₈×C₆₇	—	8	4	no	—
181	D₄×C₆₇	—	2	2	no	—
182	D₄×C₆₇	—	2	4	no	—
183	D₄×C₆₇	—	4	1	no	—
184	D₄×C₆₇	—	4	1	no	—
185	D₄×C₆₇	—	4	2	no	—
186	D₄×C₆₇	#124	1	4	yes	—
187	C₂×C₄×C₆₇	—	2	2	no	—
188	C₂×C₄×C₆₇	—	2	1	no	—
189	C₂×C₄×C₆₇	—	2	1	no	—
190	C₂×C₄×C₆₇	—	2	2	no	—
191	C₂×C₄×C₆₇	—	2	2	no	—
192	C₂×C₄×C₆₇	—	2	2	no	—
193	C₂×C₄×C₆₇	—	2	4	no	—
194	C₂×C₄×C₆₇	—	2	2	no	—
195	C₂×C₄×C₆₇	—	2	2	no	—
196	C₂×C₄×C₆₇	—	2	2	no	—
197	C₂×C₄×C₆₇	—	2	2	no	—
198	C₂×C₄×C₆₇	—	4	8	no	—
199	C₂×C₄×C₆₇	—	4	2	no	—
200	C₂×C₄×C₆₇	—	4	2	no	—
201	C₂×C₄×C₆₇	—	4	2	no	—
202	C₂×C₄×C₆₇	—	4	1	no	—
203	C₂×C₄×C₆₇	—	4	2	no	—
204	C₂×C₄×C₆₇	—	4	2	no	—
205	C₂×C₄×C₆₇	—	4	2	no	—
206	C₂×C₄×C₆₇	—	4	2	no	—
207	C₂×C₄×C₆₇	—	8	8	no	—
208	C₂×C₄×C₆₇	#120	1	2	yes	—
209	C₂×C₄×C₆₇	#120	1	2	yes	—
210	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
211	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
212	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
213	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
214	C₈×C₆₇	—	4	1	no	—
215	C₈×C₆₇	—	4	1	no	—
216	C₈×C₆₇	—	4	2	no	—
217	C₂²×C₆₇	—	2	4	no	—
218	C₂²×C₆₇	—	4	4	no	—
219	C₂²×C₆₇	—	4	4	no	—
220	C₂²×C₆₇	—	4	2	no	—
221	C₂²×C₆₇	—	4	2	no	—
222	C₂²×C₆₇	—	8	2	no	—
223	C₂²×C₆₇	—	8	2	no	—
224	C₂²×C₆₇	—	8	1	no	—
225	C₂²×C₆₇	—	8	1	no	—
226	C₂²×C₆₇	—	8	1	no	—
227	C₂²×C₆₇	C₂×#120	1	1	yes	characteristic
228	C₂²×C₆₇	#548	1	2	yes	—
229	C₂²×C₆₇	#575	1	2	yes	—
230	C₂²×C₆₇	#576	1	1	yes	characteristic
231	C₂²×C₆₇	#577	1	1	yes	characteristic
232	C₄×C₆₇	—	2	2	no	—
233	C₄×C₆₇	—	2	2	no	—
234	C₄×C₆₇	—	4	2	no	—
235	C₄×C₆₇	—	4	2	no	—
236	C₄×C₆₇	—	4	2	no	—
237	C₄×C₆₇	—	4	4	no	—
238	C₄×C₆₇	—	8	2	no	—
239	C₄×C₆₇	—	8	1	no	—
240	C₄×C₆₇	C₂×#124	1	1	yes	characteristic
241	C₄×C₆₇	#536	1	1	yes	characteristic
242	C₄×C₆₇	#548	1	2	yes	—
243	#25940	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
244	C₂×C₆₇	—	4	4	no	—
245	C₂×C₆₇	—	16	1	no	—
246	C₂×C₆₇	#2436	1	1	yes	characteristic
247	C₂×C₆₇	#3005	1	1	yes	characteristic
248	C₂×C₆₇	#3012	1	1	yes	characteristic
249	C₂×C₆₇	#3370	1	2	yes	—
250	C₂×C₆₇	#3376	1	2	yes	—
251	C₂×#554	C₂×C₆₇	1	1	yes	characteristic, complemented
252	C₂×#489	C₂×C₆₇	1	1	yes	characteristic, complemented
253	C₂×#463	C₂×C₆₇	1	1	yes	characteristic
254	#2592	C₂×C₆₇	1	1	yes	characteristic, complemented
255	#2026	C₂×C₆₇	1	2	yes	complemented
256	#2025	C₂×C₆₇	1	2	yes	complemented
257	#1988	C₂×C₆₇	1	1	yes	characteristic, complemented
258	#1985	C₂×C₆₇	1	1	yes	characteristic, complemented
259	#1879	C₂×C₆₇	1	2	yes	—
260	#1877	C₂×C₆₇	1	2	yes	complemented
261	#1762	C₂×C₆₇	1	1	yes	characteristic, complemented
262	C₆₇	#25940	1	1	yes	characteristic, direct product factor, Sylow
263	C₂²×SD₁₆	C₂²×C₆₇	1	1	yes	characteristic
264	C₂²×D₈	C₂²×C₆₇	1	1	yes	characteristic
265	#554	—	2	4	no	—
266	#554	—	2	4	no	—
267	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
268	C₂×#125	C₂²×C₆₇	1	2	yes	—
269	C₂×#124	C₂²×C₆₇	1	1	yes	characteristic
270	C₂×C₂²⋊Q₈	—	2	1	no	—
271	C₂×C₂²⋊Q₈	—	2	2	no	—
272	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
273	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
274	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
275	C₂²×C₄⋊C₄	—	2	1	no	—
276	#489	—	2	4	no	—
277	#463	—	2	4	no	—
278	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
279	C₂×#103	C₂²×C₆₇	1	2	yes	—
280	C₂×#103	C₂²×C₆₇	1	1	yes	characteristic
281	C₂×#102	C₂²×C₆₇	1	2	yes	—
282	C₂×#102	C₂²×C₆₇	1	2	yes	—
283	C₂×#102	C₂²×C₆₇	1	1	yes	characteristic
284	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
285	#397	—	2	1	no	—
286	#393	—	2	1	no	—
287	#391	C₂²×C₆₇	1	2	yes	—
288	#390	C₂²×C₆₇	1	2	yes	—
289	#388	—	2	2	no	—
290	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
291	#340	C₂²×C₆₇	1	2	yes	—
292	#340	C₂²×C₆₇	1	2	yes	—
293	#340	C₂²×C₆₇	1	2	yes	—
294	#336	C₂²×C₆₇	1	2	yes	—
295	#336	C₂²×C₆₇	1	2	yes	—
296	C₂²×Q₈	—	2	1	no	—
297	C₂²×Q₈	D₄×C₆₇	1	1	yes	characteristic
298	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
299	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
300	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
301	C₂³×C₄	—	2	1	no	—
302	C₂³×C₄	—	2	2	no	—
303	C₂×SD₁₆	—	2	2	no	—
304	C₂×SD₁₆	—	2	4	no	—
305	C₂×D₈	—	2	2	no	—
306	C₂×D₈	—	2	4	no	—
307	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
308	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
309	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
310	C₄⋊Q₈	—	2	2	no	—
311	C₄⋊Q₈	D₄×C₆₇	1	4	yes	—
312	#125	—	2	4	no	—
313	#125	—	2	4	no	—
314	#124	—	2	2	no	—
315	#124	—	2	2	no	—
316	C₂²⋊Q₈	—	2	8	no	—
317	C₂²⋊Q₈	—	2	16	no	—
318	C₄×Q₈	—	2	2	no	—
319	C₄×Q₈	—	2	2	no	—
320	C₄×D₄	—	2	4	no	—
321	C₄×D₄	—	2	4	no	—
322	C₄×D₄	—	2	2	no	—
323	C₄×D₄	D₄×C₆₇	1	4	yes	—
324	C₂×C₄⋊C₄	—	2	8	no	—
325	C₂×C₄⋊C₄	—	2	2	no	—
326	C₂×C₄⋊C₄	—	2	1	no	—
327	C₂×C₄⋊C₄	—	2	2	no	—
328	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
329	C₂×C₄⋊C₄	D₄×C₆₇	1	2	yes	—
330	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
331	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
332	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
333	C₂×C₂²⋊C₄	—	2	2	no	—
334	C₂×C₂²⋊C₄	—	2	1	no	—
335	C₂×C₂²⋊C₄	—	2	1	no	—
336	C₂×C₂²⋊C₄	—	2	1	no	—
337	C₂×C₂²⋊C₄	—	2	2	no	—
338	C₂×C₂²⋊C₄	—	2	2	no	—
339	C₂×C₄²	—	2	1	no	—
340	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
341	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
342	C₄⋊C₈	—	2	2	no	—
343	#103	—	2	2	no	—
344	#103	—	2	4	no	—
345	#102	—	2	4	no	—
346	#102	—	2	2	no	—
347	#102	—	2	4	no	—
348	#97	—	2	2	no	—
349	#95	—	2	1	no	—
350	#95	—	2	2	no	—
351	#95	—	2	2	no	—
352	C₂⁴	—	2	1	no	—
353	C₂⁴	—	2	1	no	—
354	C₂×Q₈	—	2	2	no	—
355	C₂×Q₈	—	2	4	no	—
356	C₂×Q₈	—	2	1	no	—
357	C₂×Q₈	—	2	1	no	—
358	C₂×Q₈	—	2	2	no	—
359	C₂×Q₈	—	2	2	no	—
360	C₂×Q₈	—	4	2	no	—
361	C₂×Q₈	—	8	1	no	—
362	C₂×Q₈	—	8	1	no	—
363	C₂×D₄	—	2	1	no	—
364	C₂×D₄	—	2	1	no	—
365	C₂×D₄	—	2	1	no	—
366	C₂×D₄	—	2	2	no	—
367	C₂×D₄	—	2	2	no	—
368	C₂×D₄	—	2	2	no	—
369	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
370	C₂×D₄	#41×C₆₇	1	4	yes	—
371	C₂²×C₄	—	2	1	no	—
372	C₂²×C₄	—	2	2	no	—
373	C₂²×C₄	—	2	8	no	—
374	C₂²×C₄	—	2	4	no	—
375	C₂²×C₄	—	2	1	no	—
376	C₂²×C₄	—	2	2	no	—
377	C₂²×C₄	—	4	4	no	—
378	C₂²×C₄	—	8	2	no	—
379	C₂²×C₄	—	8	2	no	—
380	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
381	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
382	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
383	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
384	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
385	SD₁₆	—	4	4	no	—
386	D₈	—	4	4	no	—
387	C₂×C₈	—	2	2	no	—
388	C₂×C₈	—	2	1	no	—
389	C₂×C₈	—	2	1	no	—
390	C₂×C₈	—	2	2	no	—
391	C₂×C₈	—	2	2	no	—
392	C₂×C₈	—	2	2	no	—
393	C₂×C₈	—	2	2	no	—
394	C₄⋊C₄	—	2	8	no	—
395	C₄⋊C₄	—	2	8	no	—
396	C₄⋊C₄	—	2	2	no	—
397	C₄⋊C₄	—	2	4	no	—
398	C₄⋊C₄	—	2	4	no	—
399	C₄⋊C₄	—	2	1	no	—
400	C₄⋊C₄	—	2	2	no	—
401	C₄⋊C₄	—	2	2	no	—
402	C₄⋊C₄	—	2	2	no	—
403	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
404	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
405	C₂²⋊C₄	—	2	8	no	—
406	C₂²⋊C₄	—	2	4	no	—
407	C₂²⋊C₄	—	4	4	no	—
408	C₂²⋊C₄	—	4	2	no	—
409	C₂²⋊C₄	—	4	2	no	—
410	C₂²⋊C₄	—	8	2	no	—
411	C₄²	—	2	2	no	—
412	C₄²	—	2	2	no	—
413	C₄²	—	8	1	no	—
414	C₄²	C₂×D₄×C₆₇	1	2	yes	—
415	C₄²	C₂×D₄×C₆₇	1	2	yes	—
416	C₂³	—	2	2	no	—
417	C₂³	—	2	4	no	—
418	C₂³	—	4	2	no	—
419	C₂³	—	4	1	no	—
420	C₂³	—	4	1	no	—
421	C₂³	—	4	1	no	—
422	C₂³	—	4	2	no	—
423	C₂³	—	4	2	no	—
424	C₂³	—	4	1	no	—
425	C₂³	—	4	1	no	—
426	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
427	Q₈	—	4	4	no	—
428	Q₈	—	4	1	no	—
429	Q₈	—	4	1	no	—
430	Q₈	—	4	2	no	—
431	Q₈	—	8	4	no	—
432	D₄	—	2	2	no	—
433	D₄	—	2	4	no	—
434	D₄	—	4	1	no	—
435	D₄	—	4	1	no	—
436	D₄	—	4	2	no	—
437	D₄	#124×C₆₇	1	4	yes	—
438	C₂×C₄	—	2	2	no	—
439	C₂×C₄	—	2	1	no	—
440	C₂×C₄	—	2	1	no	—
441	C₂×C₄	—	2	2	no	—
442	C₂×C₄	—	2	2	no	—
443	C₂×C₄	—	2	2	no	—
444	C₂×C₄	—	2	4	no	—
445	C₂×C₄	—	2	2	no	—
446	C₂×C₄	—	2	2	no	—
447	C₂×C₄	—	2	2	no	—
448	C₂×C₄	—	2	2	no	—
449	C₂×C₄	—	4	8	no	—
450	C₂×C₄	—	4	2	no	—
451	C₂×C₄	—	4	2	no	—
452	C₂×C₄	—	4	2	no	—
453	C₂×C₄	—	4	1	no	—
454	C₂×C₄	—	4	2	no	—
455	C₂×C₄	—	4	2	no	—
456	C₂×C₄	—	4	2	no	—
457	C₂×C₄	—	4	2	no	—
458	C₂×C₄	—	8	8	no	—
459	C₂×C₄	#120×C₆₇	1	2	yes	—
460	C₂×C₄	#120×C₆₇	1	2	yes	—
461	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
462	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
463	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
464	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
465	C₈	—	4	1	no	—
466	C₈	—	4	1	no	—
467	C₈	—	4	2	no	—
468	C₂²	—	2	4	no	—
469	C₂²	—	4	2	no	—
470	C₂²	—	4	4	no	—
471	C₂²	—	4	4	no	—
472	C₂²	—	4	2	no	—
473	C₂²	—	8	1	no	—
474	C₂²	—	8	2	no	—
475	C₂²	—	8	2	no	—
476	C₂²	—	8	1	no	—
477	C₂²	—	8	1	no	—
478	C₂²	C₂×#120×C₆₇	1	1	yes	characteristic
479	C₂²	#548×C₆₇	1	2	yes	—
480	C₂²	#575×C₆₇	1	2	yes	—
481	C₂²	#576×C₆₇	1	1	yes	characteristic
482	C₂²	#577×C₆₇	1	1	yes	characteristic
483	C₄	—	2	2	no	—
484	C₄	—	2	2	no	—
485	C₄	—	4	2	no	—
486	C₄	—	4	4	no	—
487	C₄	—	4	2	no	—
488	C₄	—	4	2	no	—
489	C₄	—	8	2	no	—
490	C₄	—	8	1	no	—
491	C₄	C₂×#124×C₆₇	1	1	yes	characteristic
492	C₄	#536×C₆₇	1	1	yes	characteristic
493	C₄	#548×C₆₇	1	2	yes	—
494	C₂	—	4	4	no	—
495	C₂	—	16	1	no	—
496	C₂	C₆₇×#2436	1	1	yes	characteristic
497	C₂	C₆₇×#3005	1	1	yes	characteristic
498	C₂	C₆₇×#3012	1	1	yes	characteristic
499	C₂	C₆₇×#3370	1	2	yes	—
500	C₂	C₆₇×#3376	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂²×D₈×C₆₇	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂×C₄×Q₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#554×C₆₇	C₂	semidirect
2. C₂×#489×C₆₇	C₂	semidirect
4. C₆₇×#2592	C₂	semidirect
5. C₆₇×#2026	C₂	semidirect
6. C₆₇×#2025	C₂	semidirect
7. C₆₇×#1988	C₂	semidirect
8. C₆₇×#1985	C₂	semidirect
10. C₆₇×#1877	C₂	semidirect
11. C₆₇×#1762	C₂	semidirect
243. #25940	C₆₇	direct (N × H)
251. C₂×#554	C₂×C₆₇	semidirect
252. C₂×#489	C₂×C₆₇	semidirect
254. #2592	C₂×C₆₇	semidirect
255. #2026	C₂×C₆₇	semidirect
256. #2025	C₂×C₆₇	semidirect
257. #1988	C₂×C₆₇	semidirect
258. #1985	C₂×C₆₇	semidirect
260. #1877	C₂×C₆₇	semidirect
261. #1762	C₂×C₆₇	semidirect
262. C₆₇	#25940	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#554×C₆₇	2
C₂×#489×C₆₇	2
C₂×#463×C₆₇	2
C₆₇×#2592	2
C₆₇×#2026	2
C₆₇×#2025	2
C₆₇×#1988	2
C₆₇×#1985	2
C₆₇×#1879	2
C₆₇×#1877	2
C₆₇×#1762	2
#25940	67

Subgroups — C67×#25940 #30705191

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#25940 #30705191