Subgroups — C₆₇×#21414 #30700665

Order 17152.

1198 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#540×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#405×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2314	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#1992	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1991	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#1961	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1957	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1895	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1893	C₂	1	2	yes	maximal, complemented
10	C₆₇×#1762	C₂	1	2	yes	maximal, complemented
11	C₄×#102×C₆₇	C₂	1	2	yes	maximal, complemented
12	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
13	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
14	#540×C₆₇	—	2	4	no	—
15	#540×C₆₇	C₄	1	8	yes	complemented
16	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×#125×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×#123×C₆₇	—	2	2	no	—
19	C₂×#121×C₆₇	—	2	2	no	—
20	C₂×#121×C₆₇	C₂²	1	2	yes	—
21	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
22	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×#102×C₆₇	C₂²	1	2	yes	—
24	C₂×#102×C₆₇	C₂²	1	2	yes	—
25	C₂×#102×C₆₇	C₂²	1	2	yes	—
26	C₂×#102×C₆₇	C₂²	1	2	yes	—
27	#405×C₆₇	—	2	4	no	—
28	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
29	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
31	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
32	#388×C₆₇	—	2	2	no	—
33	#387×C₆₇	—	2	2	no	—
34	#384×C₆₇	C₂²	1	1	yes	characteristic
35	#382×C₆₇	C₂²	1	1	yes	characteristic
36	#380×C₆₇	C₂²	1	2	yes	—
37	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
38	#340×C₆₇	C₂²	1	2	yes	—
39	#340×C₆₇	C₂²	1	2	yes	—
40	#340×C₆₇	C₂²	1	2	yes	—
41	#340×C₆₇	C₂²	1	2	yes	—
42	C₂²×D₄×C₆₇	—	2	2	no	—
43	C₂²×D₄×C₆₇	—	2	2	no	—
44	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
45	C₂³×C₄×C₆₇	—	2	2	no	—
46	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
47	C₂×M₁₆×C₆₇	—	2	2	no	—
48	C₂×M₁₆×C₆₇	—	2	2	no	—
49	C₂×M₁₆×C₆₇	—	2	2	no	—
50	C₂²×C₈×C₆₇	—	2	2	no	—
51	C₂²×C₈×C₆₇	—	2	2	no	—
52	C₂²×C₈×C₆₇	—	2	2	no	—
53	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
54	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
55	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
56	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
57	#127×C₆₇	—	2	2	no	—
58	#127×C₆₇	C₂×C₄	1	4	yes	—
59	#125×C₆₇	—	2	2	no	—
60	#125×C₆₇	C₂×C₄	1	4	yes	—
61	#123×C₆₇	—	2	16	no	—
62	#121×C₆₇	—	2	4	no	—
63	#121×C₆₇	—	2	16	no	—
64	#121×C₆₇	C₂×C₄	1	8	yes	—
65	C₄×D₄×C₆₇	—	2	4	no	—
66	C₄×D₄×C₆₇	C₂×C₄	1	8	yes	—
67	#117×C₆₇	C₂×C₄	1	4	yes	—
68	#117×C₆₇	D₄	1	4	yes	—
69	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
70	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
71	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
72	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
73	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
74	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
75	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
76	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
77	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
78	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
79	C₂×C₄²×C₆₇	—	2	2	no	—
80	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
81	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
82	#102×C₆₇	—	2	4	no	—
83	#102×C₆₇	—	2	4	no	—
84	#102×C₆₇	—	2	4	no	—
85	#102×C₆₇	—	2	4	no	—
86	#97×C₆₇	—	2	2	no	—
87	#97×C₆₇	—	2	2	no	—
88	C₄×C₈×C₆₇	—	2	2	no	—
89	C₄×C₈×C₆₇	—	2	2	no	—
90	#95×C₆₇	—	2	2	no	—
91	#95×C₆₇	—	2	2	no	—
92	#95×C₆₇	—	2	2	no	—
93	C₂⁴×C₆₇	—	2	2	no	—
94	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
95	C₂×D₄×C₆₇	—	2	8	no	—
96	C₂×D₄×C₆₇	—	2	8	no	—
97	C₂×D₄×C₆₇	—	2	2	no	—
98	C₂×D₄×C₆₇	—	2	2	no	—
99	C₂×D₄×C₆₇	—	2	2	no	—
100	C₂×D₄×C₆₇	—	2	2	no	—
101	C₂×D₄×C₆₇	—	2	2	no	—
102	C₂×D₄×C₆₇	—	8	2	no	—
103	C₂×D₄×C₆₇	—	8	2	no	—
104	C₂×D₄×C₆₇	—	8	2	no	—
105	C₂×D₄×C₆₇	—	8	2	no	—
106	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
107	C₂²×C₄×C₆₇	—	2	1	no	—
108	C₂²×C₄×C₆₇	—	2	1	no	—
109	C₂²×C₄×C₆₇	—	2	1	no	—
110	C₂²×C₄×C₆₇	—	2	1	no	—
111	C₂²×C₄×C₆₇	—	2	2	no	—
112	C₂²×C₄×C₆₇	—	2	8	no	—
113	C₂²×C₄×C₆₇	—	2	2	no	—
114	C₂²×C₄×C₆₇	—	2	2	no	—
115	C₂²×C₄×C₆₇	—	8	2	no	—
116	C₂²×C₄×C₆₇	—	8	2	no	—
117	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
118	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
119	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
120	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
121	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
122	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
123	M₁₆×C₆₇	—	2	8	no	—
124	C₂×C₈×C₆₇	—	2	8	no	—
125	C₂×C₈×C₆₇	—	2	1	no	—
126	C₂×C₈×C₆₇	—	2	1	no	—
127	C₂×C₈×C₆₇	—	2	1	no	—
128	C₂×C₈×C₆₇	—	2	1	no	—
129	C₂×C₈×C₆₇	—	2	1	no	—
130	C₂×C₈×C₆₇	—	2	1	no	—
131	C₂×C₈×C₆₇	—	2	1	no	—
132	C₂×C₈×C₆₇	—	2	1	no	—
133	C₂×C₈×C₆₇	#41	1	2	yes	—
134	C₂×C₈×C₆₇	#41	1	2	yes	—
135	C₂×C₈×C₆₇	#41	1	2	yes	—
136	C₂×C₈×C₆₇	#41	1	2	yes	—
137	C₄⋊C₄×C₆₇	—	2	8	no	—
138	C₄⋊C₄×C₆₇	—	2	2	no	—
139	C₄⋊C₄×C₆₇	—	2	2	no	—
140	C₄⋊C₄×C₆₇	—	2	2	no	—
141	C₄⋊C₄×C₆₇	—	2	2	no	—
142	C₄⋊C₄×C₆₇	—	2	2	no	—
143	C₄⋊C₄×C₆₇	—	8	2	no	—
144	C₄⋊C₄×C₆₇	C₂²⋊C₄	1	8	yes	—
145	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
146	C₂²⋊C₄×C₆₇	—	2	8	no	—
147	C₂²⋊C₄×C₆₇	—	2	8	no	—
148	C₂²⋊C₄×C₆₇	—	8	2	no	—
149	C₂²⋊C₄×C₆₇	C₂²⋊C₄	1	8	yes	—
150	C₄²×C₆₇	—	8	2	no	—
151	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
152	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
153	C₄²×C₆₇	C₂×D₄	1	2	yes	—
154	C₄²×C₆₇	C₂×D₄	1	2	yes	—
155	C₂³×C₆₇	—	2	1	no	—
156	C₂³×C₆₇	—	2	1	no	—
157	C₂³×C₆₇	—	2	2	no	—
158	C₂³×C₆₇	—	2	2	no	—
159	C₂³×C₆₇	—	2	4	no	—
160	C₂³×C₆₇	—	4	2	no	—
161	C₂³×C₆₇	—	4	2	no	—
162	C₂³×C₆₇	—	4	2	no	—
163	C₂³×C₆₇	—	4	2	no	—
164	C₂³×C₆₇	—	4	2	no	—
165	C₂³×C₆₇	—	4	2	no	—
166	C₂³×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
167	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
168	D₄×C₆₇	—	2	4	no	—
169	D₄×C₆₇	—	2	4	no	—
170	D₄×C₆₇	—	4	2	no	—
171	D₄×C₆₇	—	4	2	no	—
172	D₄×C₆₇	—	8	8	no	—
173	D₄×C₆₇	—	8	8	no	—
174	C₂×C₄×C₆₇	—	2	1	no	—
175	C₂×C₄×C₆₇	—	2	1	no	—
176	C₂×C₄×C₆₇	—	2	2	no	—
177	C₂×C₄×C₆₇	—	2	2	no	—
178	C₂×C₄×C₆₇	—	2	2	no	—
179	C₂×C₄×C₆₇	—	2	2	no	—
180	C₂×C₄×C₆₇	—	2	2	no	—
181	C₂×C₄×C₆₇	—	2	2	no	—
182	C₂×C₄×C₆₇	—	2	2	no	—
183	C₂×C₄×C₆₇	—	2	2	no	—
184	C₂×C₄×C₆₇	—	2	4	no	—
185	C₂×C₄×C₆₇	—	4	2	no	—
186	C₂×C₄×C₆₇	—	4	2	no	—
187	C₂×C₄×C₆₇	—	4	2	no	—
188	C₂×C₄×C₆₇	—	4	2	no	—
189	C₂×C₄×C₆₇	—	4	2	no	—
190	C₂×C₄×C₆₇	—	4	2	no	—
191	C₂×C₄×C₆₇	—	4	2	no	—
192	C₂×C₄×C₆₇	—	4	2	no	—
193	C₂×C₄×C₆₇	—	8	8	no	—
194	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
195	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
196	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
197	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
198	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
199	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
200	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
201	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
202	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
203	C₈×C₆₇	—	2	2	no	—
204	C₈×C₆₇	—	2	2	no	—
205	C₈×C₆₇	—	2	2	no	—
206	C₈×C₆₇	—	2	2	no	—
207	C₂²×C₆₇	—	2	4	no	—
208	C₂²×C₆₇	—	2	2	no	—
209	C₂²×C₆₇	—	4	1	no	—
210	C₂²×C₆₇	—	4	1	no	—
211	C₂²×C₆₇	—	4	2	no	—
212	C₂²×C₆₇	—	4	4	no	—
213	C₂²×C₆₇	—	4	4	no	—
214	C₂²×C₆₇	—	4	4	no	—
215	C₂²×C₆₇	—	8	2	no	—
216	C₂²×C₆₇	—	8	2	no	—
217	C₂²×C₆₇	—	8	2	no	—
218	C₂²×C₆₇	—	8	2	no	—
219	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
220	C₂²×C₆₇	#532	1	1	yes	characteristic
221	C₂²×C₆₇	#532	1	1	yes	characteristic
222	C₂²×C₆₇	#576	1	2	yes	—
223	C₂²×C₆₇	#577	1	2	yes	—
224	C₄×C₆₇	—	2	4	no	—
225	C₄×C₆₇	—	2	4	no	—
226	C₄×C₆₇	—	2	2	no	—
227	C₄×C₆₇	—	4	4	no	—
228	C₄×C₆₇	—	8	2	no	—
229	C₄×C₆₇	—	8	2	no	—
230	C₄×C₆₇	C₄×#41	1	1	yes	characteristic
231	C₄×C₆₇	C₄×#41	1	1	yes	characteristic
232	C₄×C₆₇	#525	1	1	yes	characteristic
233	C₄×C₆₇	#525	1	1	yes	characteristic
234	#21414	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
235	C₂×C₆₇	—	4	2	no	—
236	C₂×C₆₇	—	8	4	no	—
237	C₂×C₆₇	#2297	1	1	yes	characteristic
238	C₂×C₆₇	#2897	1	2	yes	—
239	C₂×C₆₇	#3174	1	1	yes	characteristic
240	C₂×C₆₇	#3176	1	1	yes	characteristic
241	C₂×C₆₇	#3179	1	2	yes	—
242	C₂×#540	C₂×C₆₇	1	1	yes	characteristic
243	C₂×#405	C₂×C₆₇	1	1	yes	characteristic, complemented
244	#2314	C₂×C₆₇	1	1	yes	characteristic, complemented
245	#1992	C₂×C₆₇	1	1	yes	characteristic, complemented
246	#1991	C₂×C₆₇	1	1	yes	characteristic, complemented
247	#1961	C₂×C₆₇	1	1	yes	characteristic, complemented
248	#1957	C₂×C₆₇	1	1	yes	characteristic, complemented
249	#1895	C₂×C₆₇	1	2	yes	complemented
250	#1893	C₂×C₆₇	1	2	yes	complemented
251	#1762	C₂×C₆₇	1	2	yes	complemented
252	C₄×#102	C₂×C₆₇	1	2	yes	complemented
253	C₆₇	#21414	1	1	yes	characteristic, direct product factor, Sylow
254	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
255	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
256	#540	—	2	4	no	—
257	#540	C₄×C₆₇	1	8	yes	complemented
258	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
259	C₂×#125	C₂²×C₆₇	1	1	yes	characteristic
260	C₂×#123	—	2	2	no	—
261	C₂×#121	—	2	2	no	—
262	C₂×#121	C₂²×C₆₇	1	2	yes	—
263	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
264	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
265	C₂×#102	C₂²×C₆₇	1	2	yes	—
266	C₂×#102	C₂²×C₆₇	1	2	yes	—
267	C₂×#102	C₂²×C₆₇	1	2	yes	—
268	C₂×#102	C₂²×C₆₇	1	2	yes	—
269	#405	—	2	4	no	—
270	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
271	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
272	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
273	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
274	#388	—	2	2	no	—
275	#387	—	2	2	no	—
276	#384	C₂²×C₆₇	1	1	yes	characteristic
277	#382	C₂²×C₆₇	1	1	yes	characteristic
278	#380	C₂²×C₆₇	1	2	yes	—
279	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
280	#340	C₂²×C₆₇	1	2	yes	—
281	#340	C₂²×C₆₇	1	2	yes	—
282	#340	C₂²×C₆₇	1	2	yes	—
283	#340	C₂²×C₆₇	1	2	yes	—
284	C₂²×D₄	—	2	2	no	—
285	C₂²×D₄	—	2	2	no	—
286	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
287	C₂³×C₄	—	2	2	no	—
288	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
289	C₂×M₁₆	—	2	2	no	—
290	C₂×M₁₆	—	2	2	no	—
291	C₂×M₁₆	—	2	2	no	—
292	C₂²×C₈	—	2	2	no	—
293	C₂²×C₈	—	2	2	no	—
294	C₂²×C₈	—	2	2	no	—
295	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
296	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
297	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
298	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
299	#127	—	2	2	no	—
300	#127	C₂×C₄×C₆₇	1	4	yes	—
301	#125	—	2	2	no	—
302	#125	C₂×C₄×C₆₇	1	4	yes	—
303	#123	—	2	16	no	—
304	#121	—	2	4	no	—
305	#121	—	2	16	no	—
306	#121	C₂×C₄×C₆₇	1	8	yes	—
307	C₄×D₄	—	2	4	no	—
308	C₄×D₄	C₂×C₄×C₆₇	1	8	yes	—
309	#117	C₂×C₄×C₆₇	1	4	yes	—
310	#117	D₄×C₆₇	1	4	yes	—
311	C₂×C₄⋊C₄	—	2	2	no	—
312	C₂×C₄⋊C₄	—	2	2	no	—
313	C₂×C₄⋊C₄	D₄×C₆₇	1	2	yes	—
314	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
315	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
316	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
317	C₂×C₂²⋊C₄	—	2	2	no	—
318	C₂×C₂²⋊C₄	—	2	2	no	—
319	C₂×C₂²⋊C₄	—	2	2	no	—
320	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
321	C₂×C₄²	—	2	2	no	—
322	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
323	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
324	#102	—	2	4	no	—
325	#102	—	2	4	no	—
326	#102	—	2	4	no	—
327	#102	—	2	4	no	—
328	#97	—	2	2	no	—
329	#97	—	2	2	no	—
330	C₄×C₈	—	2	2	no	—
331	C₄×C₈	—	2	2	no	—
332	#95	—	2	2	no	—
333	#95	—	2	2	no	—
334	#95	—	2	2	no	—
335	C₂⁴	—	2	2	no	—
336	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
337	C₂×D₄	—	2	8	no	—
338	C₂×D₄	—	2	8	no	—
339	C₂×D₄	—	2	2	no	—
340	C₂×D₄	—	2	2	no	—
341	C₂×D₄	—	2	2	no	—
342	C₂×D₄	—	2	2	no	—
343	C₂×D₄	—	2	2	no	—
344	C₂×D₄	—	8	2	no	—
345	C₂×D₄	—	8	2	no	—
346	C₂×D₄	—	8	2	no	—
347	C₂×D₄	—	8	2	no	—
348	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
349	C₂²×C₄	—	2	1	no	—
350	C₂²×C₄	—	2	1	no	—
351	C₂²×C₄	—	2	1	no	—
352	C₂²×C₄	—	2	1	no	—
353	C₂²×C₄	—	2	2	no	—
354	C₂²×C₄	—	2	8	no	—
355	C₂²×C₄	—	2	2	no	—
356	C₂²×C₄	—	2	2	no	—
357	C₂²×C₄	—	8	2	no	—
358	C₂²×C₄	—	8	2	no	—
359	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
360	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
361	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
362	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
363	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
364	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
365	M₁₆	—	2	8	no	—
366	C₂×C₈	—	2	8	no	—
367	C₂×C₈	—	2	1	no	—
368	C₂×C₈	—	2	1	no	—
369	C₂×C₈	—	2	1	no	—
370	C₂×C₈	—	2	1	no	—
371	C₂×C₈	—	2	1	no	—
372	C₂×C₈	—	2	1	no	—
373	C₂×C₈	—	2	1	no	—
374	C₂×C₈	—	2	1	no	—
375	C₂×C₈	#41×C₆₇	1	2	yes	—
376	C₂×C₈	#41×C₆₇	1	2	yes	—
377	C₂×C₈	#41×C₆₇	1	2	yes	—
378	C₂×C₈	#41×C₆₇	1	2	yes	—
379	C₄⋊C₄	—	2	8	no	—
380	C₄⋊C₄	—	2	2	no	—
381	C₄⋊C₄	—	2	2	no	—
382	C₄⋊C₄	—	2	2	no	—
383	C₄⋊C₄	—	2	2	no	—
384	C₄⋊C₄	—	2	2	no	—
385	C₄⋊C₄	—	8	2	no	—
386	C₄⋊C₄	C₂²⋊C₄×C₆₇	1	8	yes	—
387	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
388	C₂²⋊C₄	—	2	8	no	—
389	C₂²⋊C₄	—	2	8	no	—
390	C₂²⋊C₄	—	8	2	no	—
391	C₂²⋊C₄	C₂²⋊C₄×C₆₇	1	8	yes	—
392	C₄²	—	8	2	no	—
393	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
394	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
395	C₄²	C₂×D₄×C₆₇	1	2	yes	—
396	C₄²	C₂×D₄×C₆₇	1	2	yes	—
397	C₂³	—	2	1	no	—
398	C₂³	—	2	1	no	—
399	C₂³	—	2	2	no	—
400	C₂³	—	2	2	no	—
401	C₂³	—	2	4	no	—
402	C₂³	—	4	2	no	—
403	C₂³	—	4	2	no	—
404	C₂³	—	4	2	no	—
405	C₂³	—	4	2	no	—
406	C₂³	—	4	2	no	—
407	C₂³	—	4	2	no	—
408	C₂³	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
409	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
410	D₄	—	2	4	no	—
411	D₄	—	2	4	no	—
412	D₄	—	4	2	no	—
413	D₄	—	4	2	no	—
414	D₄	—	8	8	no	—
415	D₄	—	8	8	no	—
416	C₂×C₄	—	2	1	no	—
417	C₂×C₄	—	2	1	no	—
418	C₂×C₄	—	2	2	no	—
419	C₂×C₄	—	2	2	no	—
420	C₂×C₄	—	2	2	no	—
421	C₂×C₄	—	2	2	no	—
422	C₂×C₄	—	2	2	no	—
423	C₂×C₄	—	2	2	no	—
424	C₂×C₄	—	2	2	no	—
425	C₂×C₄	—	2	2	no	—
426	C₂×C₄	—	2	4	no	—
427	C₂×C₄	—	4	2	no	—
428	C₂×C₄	—	4	2	no	—
429	C₂×C₄	—	4	2	no	—
430	C₂×C₄	—	4	2	no	—
431	C₂×C₄	—	4	2	no	—
432	C₂×C₄	—	4	2	no	—
433	C₂×C₄	—	4	2	no	—
434	C₂×C₄	—	4	2	no	—
435	C₂×C₄	—	8	8	no	—
436	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
437	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
438	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
439	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
440	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
441	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
442	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
443	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
444	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
445	C₈	—	2	2	no	—
446	C₈	—	2	2	no	—
447	C₈	—	2	2	no	—
448	C₈	—	2	2	no	—
449	C₂²	—	2	2	no	—
450	C₂²	—	2	4	no	—
451	C₂²	—	4	1	no	—
452	C₂²	—	4	1	no	—
453	C₂²	—	4	2	no	—
454	C₂²	—	4	4	no	—
455	C₂²	—	4	4	no	—
456	C₂²	—	4	4	no	—
457	C₂²	—	8	2	no	—
458	C₂²	—	8	2	no	—
459	C₂²	—	8	2	no	—
460	C₂²	—	8	2	no	—
461	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
462	C₂²	#532×C₆₇	1	1	yes	characteristic
463	C₂²	#532×C₆₇	1	1	yes	characteristic
464	C₂²	#576×C₆₇	1	2	yes	—
465	C₂²	#577×C₆₇	1	2	yes	—
466	C₄	—	2	2	no	—
467	C₄	—	2	4	no	—
468	C₄	—	2	4	no	—
469	C₄	—	4	4	no	—
470	C₄	—	8	2	no	—
471	C₄	—	8	2	no	—
472	C₄	C₄×#41×C₆₇	1	1	yes	characteristic
473	C₄	C₄×#41×C₆₇	1	1	yes	characteristic
474	C₄	#525×C₆₇	1	1	yes	characteristic
475	C₄	#525×C₆₇	1	1	yes	characteristic
476	C₂	—	4	2	no	—
477	C₂	—	8	4	no	—
478	C₂	C₆₇×#2297	1	1	yes	characteristic
479	C₂	C₆₇×#2897	1	2	yes	—
480	C₂	C₆₇×#3174	1	1	yes	characteristic
481	C₂	C₆₇×#3176	1	1	yes	characteristic
482	C₂	C₆₇×#3179	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂×#540×C₆₇	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂×#117×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#405×C₆₇	C₂	semidirect
3. C₆₇×#2314	C₂	semidirect
4. C₆₇×#1992	C₂	semidirect
5. C₆₇×#1991	C₂	semidirect
6. C₆₇×#1961	C₂	semidirect
7. C₆₇×#1957	C₂	semidirect
8. C₆₇×#1895	C₂	semidirect
9. C₆₇×#1893	C₂	semidirect
10. C₆₇×#1762	C₂	semidirect
11. C₄×#102×C₆₇	C₂	semidirect
15. #540×C₆₇	C₄	semidirect
234. #21414	C₆₇	direct (N × H)
243. C₂×#405	C₂×C₆₇	semidirect
244. #2314	C₂×C₆₇	semidirect
245. #1992	C₂×C₆₇	semidirect
246. #1991	C₂×C₆₇	semidirect
247. #1961	C₂×C₆₇	semidirect
248. #1957	C₂×C₆₇	semidirect
249. #1895	C₂×C₆₇	semidirect
250. #1893	C₂×C₆₇	semidirect
251. #1762	C₂×C₆₇	semidirect
252. C₄×#102	C₂×C₆₇	semidirect
253. C₆₇	#21414	direct (N × H)
257. #540	C₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#540×C₆₇	2
C₂×#405×C₆₇	2
C₆₇×#2314	2
C₆₇×#1992	2
C₆₇×#1991	2
C₆₇×#1961	2
C₆₇×#1957	2
C₆₇×#1895	2
C₆₇×#1893	2
C₆₇×#1762	2
C₄×#102×C₆₇	2
#21414	67

Subgroups — C67×#21414 #30700665

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#21414 #30700665