Subgroups — C₆₇×#20956 #30700207

Order 17152.

1902 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#535×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#3428	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#418×C₆₇	C₂	1	2	yes	maximal, complemented
4	C₂×#413×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1888	C₂	1	4	yes	maximal, complemented
6	C₆₇×#1887	C₂	1	4	yes	maximal, complemented
7	C₆₇×#1733	C₂	1	2	yes	maximal, complemented
8	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
9	C₂³×D₄×C₆₇	C₂²	1	1	yes	characteristic
10	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
11	#535×C₆₇	—	2	4	no	—
12	#535×C₆₇	C₄	1	8	yes	complemented
13	C₂×#127×C₆₇	C₂²	1	2	yes	—
14	C₂×#124×C₆₇	C₂²	1	2	yes	—
15	C₂×#121×C₆₇	—	2	2	no	—
16	C₂×#120×C₆₇	—	2	2	no	—
17	#520×C₆₇	—	2	4	no	—
18	#519×C₆₇	C₂²	1	2	yes	complemented
19	#518×C₆₇	—	2	2	no	—
20	#518×C₆₇	C₂²	1	2	yes	complemented
21	C₄×#41×C₆₇	—	2	4	no	—
22	C₂×#117×C₆₇	—	2	2	no	—
23	C₂×#117×C₆₇	C₂²	1	2	yes	—
24	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
25	#418×C₆₇	—	2	4	no	—
26	#418×C₆₇	C₂²	1	8	yes	—
27	C₂×#102×C₆₇	—	2	4	no	—
28	#413×C₆₇	—	2	4	no	—
29	C₂×#101×C₆₇	C₂²	1	2	yes	complemented
30	C₂×#100×C₆₇	C₂²	1	2	yes	complemented
31	#339×C₆₇	C₂²	1	8	yes	complemented
32	C₂⁵×C₆₇	—	2	1	no	—
33	C₂×#41×C₆₇	—	2	2	no	—
34	C₂×#41×C₆₇	—	2	2	no	—
35	C₂×#41×C₆₇	—	2	4	no	—
36	C₂×#41×C₆₇	C₂×C₄	1	4	yes	—
37	C₂×#41×C₆₇	D₄	1	4	yes	—
38	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
39	C₂²×D₄×C₆₇	—	2	4	no	—
40	C₂²×D₄×C₆₇	—	2	2	no	—
41	C₂²×D₄×C₆₇	—	2	2	no	—
42	C₂²×D₄×C₆₇	—	2	1	no	—
43	C₂²×D₄×C₆₇	—	2	2	no	—
44	C₂²×D₄×C₆₇	—	2	4	no	—
45	C₂²×D₄×C₆₇	C₂×C₄	1	4	yes	complemented
46	C₂²×D₄×C₆₇	D₄	1	2	yes	—
47	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
48	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
49	C₂³×C₄×C₆₇	D₄	1	2	yes	—
50	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
51	C₂×M₁₆×C₆₇	—	2	2	no	—
52	C₂×M₁₆×C₆₇	—	2	4	no	—
53	C₂×M₁₆×C₆₇	—	2	4	no	—
54	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
55	C₂²×C₈×C₆₇	—	2	2	no	—
56	#127×C₆₇	—	2	4	no	—
57	#127×C₆₇	C₂×C₄	1	8	yes	complemented
58	#124×C₆₇	—	2	4	no	—
59	#124×C₆₇	C₂×C₄	1	8	yes	complemented
60	#121×C₆₇	—	2	16	no	—
61	#120×C₆₇	—	2	16	no	—
62	C₄×Q₈×C₆₇	—	2	8	no	—
63	C₄×D₄×C₆₇	—	2	16	no	—
64	C₄×D₄×C₆₇	—	2	8	no	—
65	#117×C₆₇	—	2	4	no	—
66	#117×C₆₇	—	2	4	no	—
67	#117×C₆₇	—	2	2	no	—
68	#117×C₆₇	—	2	4	no	—
69	#117×C₆₇	—	2	2	no	—
70	#117×C₆₇	—	2	2	no	—
71	#117×C₆₇	C₂×C₄	1	4	yes	complemented
72	#117×C₆₇	D₄	1	4	yes	complemented
73	#117×C₆₇	C₂³	1	4	yes	—
74	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
75	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
76	C₂×C₄⋊C₄×C₆₇	D₄	1	4	yes	complemented
77	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
78	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
79	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
80	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
81	C₂×C₄²×C₆₇	—	2	4	no	—
82	C₂×C₄²×C₆₇	—	2	2	no	—
83	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
84	#102×C₆₇	—	2	16	no	—
85	#101×C₆₇	—	2	4	no	—
86	#100×C₆₇	—	2	4	no	—
87	C₂⁴×C₆₇	—	2	2	no	—
88	C₂⁴×C₆₇	—	2	4	no	—
89	C₂⁴×C₆₇	—	4	2	no	—
90	C₂⁴×C₆₇	—	4	2	no	—
91	C₂⁴×C₆₇	—	8	4	no	—
92	C₂⁴×C₆₇	C₂×D₄	1	2	yes	—
93	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
94	#41×C₆₇	—	2	8	no	—
95	#41×C₆₇	—	2	8	no	—
96	#41×C₆₇	—	2	8	no	—
97	#41×C₆₇	—	2	8	no	—
98	C₂×Q₈×C₆₇	—	2	2	no	—
99	C₂×Q₈×C₆₇	—	2	2	no	—
100	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
101	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
102	C₂×D₄×C₆₇	—	2	4	no	—
103	C₂×D₄×C₆₇	—	2	4	no	—
104	C₂×D₄×C₆₇	—	2	16	no	—
105	C₂×D₄×C₆₇	—	2	4	no	—
106	C₂×D₄×C₆₇	—	2	8	no	—
107	C₂×D₄×C₆₇	—	2	8	no	—
108	C₂×D₄×C₆₇	—	2	2	no	—
109	C₂×D₄×C₆₇	—	2	4	no	—
110	C₂×D₄×C₆₇	—	2	2	no	—
111	C₂×D₄×C₆₇	—	2	2	no	—
112	C₂×D₄×C₆₇	—	2	4	no	—
113	C₂×D₄×C₆₇	—	4	1	no	—
114	C₂×D₄×C₆₇	—	4	1	no	—
115	C₂×D₄×C₆₇	—	4	4	no	—
116	C₂×D₄×C₆₇	—	4	4	no	—
117	C₂×D₄×C₆₇	—	4	2	no	—
118	C₂×D₄×C₆₇	—	8	8	no	—
119	C₂×D₄×C₆₇	C₂²⋊C₄	1	8	yes	—
120	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
121	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
122	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
123	C₂²×C₄×C₆₇	—	2	2	no	—
124	C₂²×C₄×C₆₇	—	2	2	no	—
125	C₂²×C₄×C₆₇	—	2	4	no	—
126	C₂²×C₄×C₆₇	—	2	2	no	—
127	C₂²×C₄×C₆₇	—	2	8	no	—
128	C₂²×C₄×C₆₇	—	2	4	no	—
129	C₂²×C₄×C₆₇	C₂²⋊C₄	1	8	yes	—
130	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
131	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
132	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
133	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
134	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
135	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
136	C₂²×C₄×C₆₇	#41	1	4	yes	—
137	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
138	M₁₆×C₆₇	—	2	4	no	—
139	M₁₆×C₆₇	—	2	4	no	—
140	M₁₆×C₆₇	—	4	4	no	—
141	C₂×C₈×C₆₇	—	2	4	no	—
142	C₂×C₈×C₆₇	—	4	4	no	—
143	C₄⋊C₄×C₆₇	—	2	8	no	—
144	C₄⋊C₄×C₆₇	—	2	4	no	—
145	C₄⋊C₄×C₆₇	—	2	4	no	—
146	C₄⋊C₄×C₆₇	C₂×D₄	1	8	yes	—
147	C₂²⋊C₄×C₆₇	—	2	8	no	—
148	C₂²⋊C₄×C₆₇	—	2	4	no	—
149	C₂²⋊C₄×C₆₇	—	2	4	no	—
150	C₂²⋊C₄×C₆₇	—	2	4	no	—
151	C₂²⋊C₄×C₆₇	—	2	16	no	—
152	C₄²×C₆₇	—	2	8	no	—
153	C₄²×C₆₇	—	2	4	no	—
154	C₄²×C₆₇	—	2	4	no	—
155	C₄²×C₆₇	—	2	2	no	—
156	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
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	4	no	—
162	C₂³×C₆₇	—	4	8	no	—
163	C₂³×C₆₇	—	4	2	no	—
164	C₂³×C₆₇	—	4	2	no	—
165	C₂³×C₆₇	—	4	8	no	—
166	C₂³×C₆₇	—	8	8	no	—
167	C₂³×C₆₇	—	8	8	no	—
168	C₂³×C₆₇	—	8	8	no	—
169	C₂³×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
170	C₂³×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
171	C₂³×C₆₇	#120	1	2	yes	—
172	C₂³×C₆₇	#123	1	2	yes	—
173	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
174	Q₈×C₆₇	—	2	4	no	—
175	Q₈×C₆₇	—	2	4	no	—
176	D₄×C₆₇	—	2	16	no	—
177	D₄×C₆₇	—	2	4	no	—
178	D₄×C₆₇	—	2	8	no	—
179	D₄×C₆₇	—	2	4	no	—
180	D₄×C₆₇	—	4	4	no	—
181	D₄×C₆₇	—	4	4	no	—
182	D₄×C₆₇	—	4	4	no	—
183	D₄×C₆₇	—	8	16	no	—
184	C₂×C₄×C₆₇	—	2	2	no	—
185	C₂×C₄×C₆₇	—	2	2	no	—
186	C₂×C₄×C₆₇	—	2	8	no	—
187	C₂×C₄×C₆₇	—	2	8	no	—
188	C₂×C₄×C₆₇	—	2	4	no	—
189	C₂×C₄×C₆₇	—	2	8	no	—
190	C₂×C₄×C₆₇	—	4	4	no	—
191	C₂×C₄×C₆₇	—	8	4	no	—
192	C₂×C₄×C₆₇	—	8	8	no	—
193	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
194	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
195	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
196	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
197	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
198	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
199	C₂×C₄×C₆₇	#121	1	4	yes	—
200	C₂×C₄×C₆₇	#121	1	4	yes	—
201	C₂×C₄×C₆₇	#123	1	2	yes	—
202	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
203	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
204	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
205	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
206	C₈×C₆₇	—	4	4	no	—
207	C₂²×C₆₇	—	2	2	no	—
208	C₂²×C₆₇	—	2	4	no	—
209	C₂²×C₆₇	—	2	4	no	—
210	C₂²×C₆₇	—	4	2	no	—
211	C₂²×C₆₇	—	4	2	no	—
212	C₂²×C₆₇	—	4	4	no	—
213	C₂²×C₆₇	—	4	8	no	—
214	C₂²×C₆₇	—	4	4	no	—
215	C₂²×C₆₇	—	4	4	no	—
216	C₂²×C₆₇	—	8	2	no	—
217	C₂²×C₆₇	—	8	4	no	—
218	C₂²×C₆₇	—	8	16	no	—
219	C₂²×C₆₇	—	8	4	no	—
220	C₂²×C₆₇	#386	1	4	yes	—
221	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
222	C₂²×C₆₇	C₂×#120	1	1	yes	characteristic
223	C₂²×C₆₇	C₂×#123	1	1	yes	characteristic
224	C₄×C₆₇	—	2	4	no	—
225	C₄×C₆₇	—	4	4	no	—
226	C₄×C₆₇	—	4	8	no	—
227	C₄×C₆₇	#386	1	2	yes	—
228	C₄×C₆₇	#386	1	2	yes	—
229	C₄×C₆₇	C₂×C₄×D₄	1	2	yes	—
230	C₄×C₆₇	C₂×#121	1	2	yes	—
231	#20956	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
232	C₂×C₆₇	—	2	1	no	—
233	C₂×C₆₇	—	2	1	no	—
234	C₂×C₆₇	—	2	4	no	—
235	C₂×C₆₇	—	4	4	no	—
236	C₂×C₆₇	—	8	4	no	—
237	C₂×C₆₇	C₂×#386	1	1	yes	characteristic
238	C₂×C₆₇	#3021	1	2	yes	—
239	C₂×#535	C₂×C₆₇	1	1	yes	characteristic
240	#3428	C₂×C₆₇	1	1	yes	characteristic, complemented
241	C₂×#418	C₂×C₆₇	1	2	yes	complemented
242	C₂×#413	C₂×C₆₇	1	1	yes	characteristic, complemented
243	#1888	C₂×C₆₇	1	4	yes	complemented
244	#1887	C₂×C₆₇	1	4	yes	complemented
245	#1733	C₂×C₆₇	1	2	yes	complemented
246	C₆₇	#20956	1	1	yes	characteristic, direct product factor, Sylow
247	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
248	C₂³×D₄	C₂²×C₆₇	1	1	yes	characteristic
249	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
250	#535	—	2	4	no	—
251	#535	C₄×C₆₇	1	8	yes	complemented
252	C₂×#127	C₂²×C₆₇	1	2	yes	—
253	C₂×#124	C₂²×C₆₇	1	2	yes	—
254	C₂×#121	—	2	2	no	—
255	C₂×#120	—	2	2	no	—
256	#520	—	2	4	no	—
257	#519	C₂²×C₆₇	1	2	yes	complemented
258	#518	—	2	2	no	—
259	#518	C₂²×C₆₇	1	2	yes	complemented
260	C₄×#41	—	2	4	no	—
261	C₂×#117	—	2	2	no	—
262	C₂×#117	C₂²×C₆₇	1	2	yes	—
263	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
264	#418	—	2	4	no	—
265	#418	C₂²×C₆₇	1	8	yes	—
266	C₂×#102	—	2	4	no	—
267	#413	—	2	4	no	—
268	C₂×#101	C₂²×C₆₇	1	2	yes	complemented
269	C₂×#100	C₂²×C₆₇	1	2	yes	complemented
270	#339	C₂²×C₆₇	1	8	yes	complemented
271	C₂⁵	—	2	1	no	—
272	C₂×#41	—	2	2	no	—
273	C₂×#41	—	2	2	no	—
274	C₂×#41	—	2	4	no	—
275	C₂×#41	C₂×C₄×C₆₇	1	4	yes	—
276	C₂×#41	D₄×C₆₇	1	4	yes	—
277	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
278	C₂²×D₄	—	2	4	no	—
279	C₂²×D₄	—	2	2	no	—
280	C₂²×D₄	—	2	2	no	—
281	C₂²×D₄	—	2	1	no	—
282	C₂²×D₄	—	2	2	no	—
283	C₂²×D₄	—	2	4	no	—
284	C₂²×D₄	C₂×C₄×C₆₇	1	4	yes	complemented
285	C₂²×D₄	D₄×C₆₇	1	2	yes	—
286	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
287	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
288	C₂³×C₄	D₄×C₆₇	1	2	yes	—
289	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
290	C₂×M₁₆	—	2	2	no	—
291	C₂×M₁₆	—	2	4	no	—
292	C₂×M₁₆	—	2	4	no	—
293	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
294	C₂²×C₈	—	2	2	no	—
295	#127	—	2	4	no	—
296	#127	C₂×C₄×C₆₇	1	8	yes	complemented
297	#124	—	2	4	no	—
298	#124	C₂×C₄×C₆₇	1	8	yes	complemented
299	#121	—	2	16	no	—
300	#120	—	2	16	no	—
301	C₄×Q₈	—	2	8	no	—
302	C₄×D₄	—	2	16	no	—
303	C₄×D₄	—	2	8	no	—
304	#117	—	2	4	no	—
305	#117	—	2	4	no	—
306	#117	—	2	2	no	—
307	#117	—	2	4	no	—
308	#117	—	2	2	no	—
309	#117	—	2	2	no	—
310	#117	C₂×C₄×C₆₇	1	4	yes	complemented
311	#117	D₄×C₆₇	1	4	yes	complemented
312	#117	C₂³×C₆₇	1	4	yes	—
313	C₂×C₄⋊C₄	—	2	4	no	—
314	C₂×C₄⋊C₄	—	2	1	no	—
315	C₂×C₄⋊C₄	D₄×C₆₇	1	4	yes	complemented
316	C₂×C₂²⋊C₄	—	2	4	no	—
317	C₂×C₂²⋊C₄	—	2	2	no	—
318	C₂×C₂²⋊C₄	—	2	1	no	—
319	C₂×C₂²⋊C₄	—	2	4	no	—
320	C₂×C₄²	—	2	4	no	—
321	C₂×C₄²	—	2	2	no	—
322	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
323	#102	—	2	16	no	—
324	#101	—	2	4	no	—
325	#100	—	2	4	no	—
326	C₂⁴	—	2	2	no	—
327	C₂⁴	—	2	4	no	—
328	C₂⁴	—	4	2	no	—
329	C₂⁴	—	4	2	no	—
330	C₂⁴	—	8	4	no	—
331	C₂⁴	C₂×D₄×C₆₇	1	2	yes	—
332	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
333	#41	—	2	8	no	—
334	#41	—	2	8	no	—
335	#41	—	2	8	no	—
336	#41	—	2	8	no	—
337	C₂×Q₈	—	2	2	no	—
338	C₂×Q₈	—	2	2	no	—
339	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
340	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
341	C₂×D₄	—	2	4	no	—
342	C₂×D₄	—	2	4	no	—
343	C₂×D₄	—	2	16	no	—
344	C₂×D₄	—	2	4	no	—
345	C₂×D₄	—	2	8	no	—
346	C₂×D₄	—	2	8	no	—
347	C₂×D₄	—	2	2	no	—
348	C₂×D₄	—	2	4	no	—
349	C₂×D₄	—	2	2	no	—
350	C₂×D₄	—	2	2	no	—
351	C₂×D₄	—	2	4	no	—
352	C₂×D₄	—	4	1	no	—
353	C₂×D₄	—	4	1	no	—
354	C₂×D₄	—	4	4	no	—
355	C₂×D₄	—	4	4	no	—
356	C₂×D₄	—	4	2	no	—
357	C₂×D₄	—	8	8	no	—
358	C₂×D₄	C₂²⋊C₄×C₆₇	1	8	yes	—
359	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
360	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
361	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
362	C₂²×C₄	—	2	2	no	—
363	C₂²×C₄	—	2	2	no	—
364	C₂²×C₄	—	2	4	no	—
365	C₂²×C₄	—	2	2	no	—
366	C₂²×C₄	—	2	8	no	—
367	C₂²×C₄	—	2	4	no	—
368	C₂²×C₄	C₂²⋊C₄×C₆₇	1	8	yes	—
369	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
370	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
371	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
372	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
373	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
374	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
375	C₂²×C₄	#41×C₆₇	1	4	yes	—
376	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
377	M₁₆	—	2	4	no	—
378	M₁₆	—	2	4	no	—
379	M₁₆	—	4	4	no	—
380	C₂×C₈	—	2	4	no	—
381	C₂×C₈	—	4	4	no	—
382	C₄⋊C₄	—	2	8	no	—
383	C₄⋊C₄	—	2	4	no	—
384	C₄⋊C₄	—	2	4	no	—
385	C₄⋊C₄	C₂×D₄×C₆₇	1	8	yes	—
386	C₂²⋊C₄	—	2	8	no	—
387	C₂²⋊C₄	—	2	4	no	—
388	C₂²⋊C₄	—	2	4	no	—
389	C₂²⋊C₄	—	2	4	no	—
390	C₂²⋊C₄	—	2	16	no	—
391	C₄²	—	2	8	no	—
392	C₄²	—	2	4	no	—
393	C₄²	—	2	4	no	—
394	C₄²	—	2	2	no	—
395	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
396	C₂³	—	2	2	no	—
397	C₂³	—	2	2	no	—
398	C₂³	—	2	4	no	—
399	C₂³	—	4	2	no	—
400	C₂³	—	4	4	no	—
401	C₂³	—	4	8	no	—
402	C₂³	—	4	2	no	—
403	C₂³	—	4	2	no	—
404	C₂³	—	4	8	no	—
405	C₂³	—	8	8	no	—
406	C₂³	—	8	8	no	—
407	C₂³	—	8	8	no	—
408	C₂³	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
409	C₂³	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
410	C₂³	#120×C₆₇	1	2	yes	—
411	C₂³	#123×C₆₇	1	2	yes	—
412	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
413	Q₈	—	2	4	no	—
414	Q₈	—	2	4	no	—
415	D₄	—	2	16	no	—
416	D₄	—	2	4	no	—
417	D₄	—	2	8	no	—
418	D₄	—	2	4	no	—
419	D₄	—	4	4	no	—
420	D₄	—	4	4	no	—
421	D₄	—	4	4	no	—
422	D₄	—	8	16	no	—
423	C₂×C₄	—	2	2	no	—
424	C₂×C₄	—	2	2	no	—
425	C₂×C₄	—	2	8	no	—
426	C₂×C₄	—	2	8	no	—
427	C₂×C₄	—	2	4	no	—
428	C₂×C₄	—	2	8	no	—
429	C₂×C₄	—	4	4	no	—
430	C₂×C₄	—	8	4	no	—
431	C₂×C₄	—	8	8	no	—
432	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
433	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
434	C₂×C₄	C₄×D₄×C₆₇	1	4	yes	—
435	C₂×C₄	C₄×D₄×C₆₇	1	4	yes	—
436	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
437	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
438	C₂×C₄	#121×C₆₇	1	4	yes	—
439	C₂×C₄	#121×C₆₇	1	4	yes	—
440	C₂×C₄	#123×C₆₇	1	2	yes	—
441	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
442	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
443	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
444	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
445	C₈	—	4	4	no	—
446	C₂²	—	2	2	no	—
447	C₂²	—	2	4	no	—
448	C₂²	—	2	4	no	—
449	C₂²	—	4	2	no	—
450	C₂²	—	4	2	no	—
451	C₂²	—	4	4	no	—
452	C₂²	—	4	8	no	—
453	C₂²	—	4	4	no	—
454	C₂²	—	4	4	no	—
455	C₂²	—	8	2	no	—
456	C₂²	—	8	4	no	—
457	C₂²	—	8	16	no	—
458	C₂²	—	8	4	no	—
459	C₂²	#386×C₆₇	1	4	yes	—
460	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
461	C₂²	C₂×#120×C₆₇	1	1	yes	characteristic
462	C₂²	C₂×#123×C₆₇	1	1	yes	characteristic
463	C₄	—	2	4	no	—
464	C₄	—	4	4	no	—
465	C₄	—	4	8	no	—
466	C₄	#386×C₆₇	1	2	yes	—
467	C₄	#386×C₆₇	1	2	yes	—
468	C₄	C₂×C₄×D₄×C₆₇	1	2	yes	—
469	C₄	C₂×#121×C₆₇	1	2	yes	—
470	C₂	—	2	1	no	—
471	C₂	—	2	1	no	—
472	C₂	—	2	4	no	—
473	C₂	—	4	4	no	—
474	C₂	—	8	4	no	—
475	C₂	C₂×#386×C₆₇	1	1	yes	characteristic
476	C₂	C₆₇×#3021	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂×#120
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂×#535×C₆₇	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂³×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#3428	C₂	semidirect
3. C₂×#418×C₆₇	C₂	semidirect
4. C₂×#413×C₆₇	C₂	semidirect
5. C₆₇×#1888	C₂	semidirect
6. C₆₇×#1887	C₂	semidirect
7. C₆₇×#1733	C₂	semidirect
12. #535×C₆₇	C₄	semidirect
18. #519×C₆₇	C₂²	semidirect
20. #518×C₆₇	C₂²	semidirect
29. C₂×#101×C₆₇	C₂²	semidirect
30. C₂×#100×C₆₇	C₂²	semidirect
31. #339×C₆₇	C₂²	semidirect
45. C₂²×D₄×C₆₇	C₂×C₄	semidirect
57. #127×C₆₇	C₂×C₄	semidirect
59. #124×C₆₇	C₂×C₄	semidirect
71. #117×C₆₇	C₂×C₄	semidirect
72. #117×C₆₇	D₄	semidirect
76. C₂×C₄⋊C₄×C₆₇	D₄	semidirect
231. #20956	C₆₇	direct (N × H)
240. #3428	C₂×C₆₇	semidirect
241. C₂×#418	C₂×C₆₇	semidirect
242. C₂×#413	C₂×C₆₇	semidirect
243. #1888	C₂×C₆₇	semidirect
244. #1887	C₂×C₆₇	semidirect
245. #1733	C₂×C₆₇	semidirect
246. C₆₇	#20956	direct (N × H)
251. #535	C₄×C₆₇	semidirect
257. #519	C₂²×C₆₇	semidirect
259. #518	C₂²×C₆₇	semidirect
268. C₂×#101	C₂²×C₆₇	semidirect
269. C₂×#100	C₂²×C₆₇	semidirect
270. #339	C₂²×C₆₇	semidirect
284. C₂²×D₄	C₂×C₄×C₆₇	semidirect
296. #127	C₂×C₄×C₆₇	semidirect
298. #124	C₂×C₄×C₆₇	semidirect
310. #117	C₂×C₄×C₆₇	semidirect
311. #117	D₄×C₆₇	semidirect
315. C₂×C₄⋊C₄	D₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#535×C₆₇	2
C₆₇×#3428	2
C₂×#418×C₆₇	2
C₂×#413×C₆₇	2
C₆₇×#1888	2
C₆₇×#1887	2
C₆₇×#1733	2
#20956	67

Subgroups — C67×#20956 #30700207

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#20956 #30700207