Subgroups — C₆₇×#36718 #30715969

Order 17152.

2842 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#543×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#121×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2848	C₂	1	2	yes	maximal, complemented
4	C₆₇×#2681	C₂	1	8	yes	maximal, complemented
5	C₆₇×#2672	C₂	1	8	yes	maximal, complemented
6	C₆₇×#2626	C₂	1	2	yes	maximal, complemented
7	C₆₇×#2583	C₂	1	2	yes	maximal, complemented
8	C₆₇×#2435	C₂	1	2	yes	maximal, complemented
9	C₆₇×#2405	C₂	1	2	yes	maximal, complemented
10	C₆₇×#2405	C₂	1	2	yes	maximal, complemented
11	C₆₇×#2312	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂³×D₄×C₆₇	C₂²	1	1	yes	characteristic
13	C₂³×D₄×C₆₇	C₂²	1	1	yes	characteristic
14	C₂³×D₄×C₆₇	C₂²	1	1	yes	characteristic
15	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
16	#543×C₆₇	—	2	8	no	—
17	C₂×#124×C₆₇	C₂²	1	4	yes	complemented
18	C₂×#123×C₆₇	C₂²	1	4	yes	complemented
19	C₂×#123×C₆₇	C₂²	1	4	yes	complemented
20	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
21	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
22	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	4	yes	complemented
23	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	4	yes	complemented
24	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	4	yes	complemented
25	C₂×#121×C₆₇	—	2	8	no	—
26	C₂×#121×C₆₇	C₂²	1	8	yes	—
27	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
28	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
29	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
30	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
31	#399×C₆₇	C₂²	1	4	yes	complemented
32	#398×C₆₇	C₂²	1	4	yes	complemented
33	#396×C₆₇	C₂²	1	8	yes	complemented
34	#395×C₆₇	C₂²	1	4	yes	complemented
35	#394×C₆₇	C₂²	1	8	yes	complemented
36	#394×C₆₇	C₂²	1	8	yes	complemented
37	#394×C₆₇	C₂²	1	8	yes	complemented
38	#393×C₆₇	C₂²	1	4	yes	complemented
39	#390×C₆₇	C₂²	1	4	yes	complemented
40	#389×C₆₇	C₂²	1	8	yes	complemented
41	#388×C₆₇	C₂²	1	8	yes	complemented
42	#386×C₆₇	C₂²	1	4	yes	complemented
43	#386×C₆₇	C₂²	1	4	yes	complemented
44	#385×C₆₇	C₂²	1	4	yes	—
45	#385×C₆₇	C₂²	1	4	yes	—
46	#385×C₆₇	C₂²	1	8	yes	complemented
47	#380×C₆₇	C₂²	1	2	yes	—
48	#380×C₆₇	C₂²	1	4	yes	complemented
49	#379×C₆₇	C₂²	1	2	yes	—
50	#379×C₆₇	C₂²	1	4	yes	—
51	#379×C₆₇	C₂²	1	4	yes	—
52	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
53	C₂⁵×C₆₇	C₂³	1	1	yes	characteristic
54	C₂⁵×C₆₇	C₂³	1	1	yes	characteristic
55	C₂⁵×C₆₇	C₂³	1	1	yes	characteristic
56	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
57	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
58	C₂²×D₄×C₆₇	—	2	4	no	—
59	C₂²×D₄×C₆₇	—	2	4	no	—
60	C₂²×D₄×C₆₇	—	2	4	no	—
61	C₂²×D₄×C₆₇	—	4	4	no	—
62	C₂²×D₄×C₆₇	—	4	4	no	—
63	C₂²×D₄×C₆₇	—	4	4	no	—
64	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
65	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
66	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
67	C₂³×C₄×C₆₇	—	2	4	no	—
68	C₂³×C₄×C₆₇	—	4	4	no	—
69	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
70	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
71	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
72	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
73	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
74	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
75	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
76	#124×C₆₇	—	4	8	no	—
77	#123×C₆₇	—	4	8	no	—
78	#123×C₆₇	—	4	8	no	—
79	C₂²⋊Q₈×C₆₇	—	2	8	no	—
80	C₂²⋊Q₈×C₆₇	—	2	8	no	—
81	C₂²⋊Q₈×C₆₇	—	2	16	no	—
82	C₂²⋊Q₈×C₆₇	—	2	16	no	—
83	C₂²⋊Q₈×C₆₇	—	4	8	no	—
84	#121×C₆₇	—	2	32	no	—
85	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
86	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
87	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
88	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
89	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
90	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
91	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
92	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
93	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
94	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
95	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
96	C₂×C₂²⋊C₄×C₆₇	D₄	1	8	yes	—
97	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
98	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
99	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
100	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
101	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
102	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
103	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
104	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
105	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
106	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
107	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
108	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
109	#95×C₆₇	C₂³	1	4	yes	—
110	#95×C₆₇	C₂³	1	4	yes	—
111	#95×C₆₇	C₂³	1	8	yes	—
112	#95×C₆₇	C₂³	1	8	yes	—
113	C₂⁴×C₆₇	—	2	2	no	—
114	C₂⁴×C₆₇	—	2	4	no	—
115	C₂⁴×C₆₇	—	2	4	no	—
116	C₂⁴×C₆₇	—	2	2	no	—
117	C₂⁴×C₆₇	—	2	4	no	—
118	C₂⁴×C₆₇	—	2	4	no	—
119	C₂⁴×C₆₇	—	4	2	no	—
120	C₂⁴×C₆₇	—	4	2	no	—
121	C₂⁴×C₆₇	—	4	2	no	—
122	C₂⁴×C₆₇	—	4	2	no	—
123	C₂⁴×C₆₇	—	4	2	no	—
124	C₂⁴×C₆₇	C₂×D₄	1	4	yes	—
125	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
126	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
127	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
128	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
129	C₂×Q₈×C₆₇	—	2	4	no	—
130	C₂×Q₈×C₆₇	—	2	4	no	—
131	C₂×Q₈×C₆₇	—	4	4	no	—
132	C₂×Q₈×C₆₇	—	4	4	no	—
133	C₂×D₄×C₆₇	—	2	8	no	—
134	C₂×D₄×C₆₇	—	2	8	no	—
135	C₂×D₄×C₆₇	—	2	8	no	—
136	C₂×D₄×C₆₇	—	2	8	no	—
137	C₂×D₄×C₆₇	—	2	8	no	—
138	C₂×D₄×C₆₇	—	2	8	no	—
139	C₂×D₄×C₆₇	—	4	4	no	—
140	C₂×D₄×C₆₇	—	4	4	no	—
141	C₂×D₄×C₆₇	—	4	4	no	—
142	C₂×D₄×C₆₇	—	4	16	no	—
143	C₂×D₄×C₆₇	—	4	16	no	—
144	C₂×D₄×C₆₇	—	4	16	no	—
145	C₂²×C₄×C₆₇	—	2	2	no	—
146	C₂²×C₄×C₆₇	—	2	4	no	—
147	C₂²×C₄×C₆₇	—	2	2	no	—
148	C₂²×C₄×C₆₇	—	2	4	no	—
149	C₂²×C₄×C₆₇	—	2	4	no	—
150	C₂²×C₄×C₆₇	—	2	8	no	—
151	C₂²×C₄×C₆₇	—	2	8	no	—
152	C₂²×C₄×C₆₇	—	4	2	no	—
153	C₂²×C₄×C₆₇	—	4	4	no	—
154	C₂²×C₄×C₆₇	—	4	2	no	—
155	C₂²×C₄×C₆₇	—	4	2	no	—
156	C₂²×C₄×C₆₇	—	4	4	no	—
157	C₂²×C₄×C₆₇	—	4	4	no	—
158	C₂²×C₄×C₆₇	—	4	16	no	—
159	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
160	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
161	C₂²×C₄×C₆₇	C₂⁴	1	8	yes	—
162	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
163	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
164	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
165	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
166	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
167	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
168	C₄⋊C₄×C₆₇	—	2	8	no	—
169	C₄⋊C₄×C₆₇	—	2	4	no	—
170	C₄⋊C₄×C₆₇	—	2	8	no	—
171	C₄⋊C₄×C₆₇	—	2	8	no	—
172	C₄⋊C₄×C₆₇	—	2	4	no	—
173	C₄⋊C₄×C₆₇	—	2	16	no	—
174	C₄⋊C₄×C₆₇	—	4	4	no	—
175	C₄⋊C₄×C₆₇	—	4	8	no	—
176	C₂²⋊C₄×C₆₇	—	2	8	no	—
177	C₂²⋊C₄×C₆₇	—	2	8	no	—
178	C₂²⋊C₄×C₆₇	—	2	16	no	—
179	C₂²⋊C₄×C₆₇	—	2	8	no	—
180	C₂²⋊C₄×C₆₇	—	2	8	no	—
181	C₂²⋊C₄×C₆₇	—	4	8	no	—
182	C₂²⋊C₄×C₆₇	—	4	4	no	—
183	C₂²⋊C₄×C₆₇	—	4	4	no	—
184	C₂²⋊C₄×C₆₇	—	4	8	no	—
185	C₂²⋊C₄×C₆₇	—	4	8	no	—
186	C₂²⋊C₄×C₆₇	C₂×D₄	1	16	yes	—
187	C₄²×C₆₇	—	4	4	no	—
188	C₂³×C₆₇	—	2	2	no	—
189	C₂³×C₆₇	—	2	2	no	—
190	C₂³×C₆₇	—	2	8	no	—
191	C₂³×C₆₇	—	2	2	no	—
192	C₂³×C₆₇	—	2	4	no	—
193	C₂³×C₆₇	—	2	4	no	—
194	C₂³×C₆₇	—	2	4	no	—
195	C₂³×C₆₇	—	2	2	no	—
196	C₂³×C₆₇	—	2	4	no	—
197	C₂³×C₆₇	—	2	4	no	—
198	C₂³×C₆₇	—	2	8	no	—
199	C₂³×C₆₇	—	2	2	no	—
200	C₂³×C₆₇	—	2	4	no	—
201	C₂³×C₆₇	—	2	4	no	—
202	C₂³×C₆₇	—	2	4	no	—
203	C₂³×C₆₇	—	4	4	no	—
204	C₂³×C₆₇	—	4	4	no	—
205	C₂³×C₆₇	—	4	16	no	—
206	C₂³×C₆₇	—	4	4	no	—
207	C₂³×C₆₇	—	4	4	no	—
208	C₂³×C₆₇	—	4	4	no	—
209	C₂³×C₆₇	—	4	4	no	—
210	C₂³×C₆₇	—	4	4	no	—
211	C₂³×C₆₇	—	8	8	no	—
212	C₂³×C₆₇	—	8	8	no	—
213	C₂³×C₆₇	C₂²×D₄	1	4	yes	—
214	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
215	C₂³×C₆₇	H(5,𝔽₂)	1	4	yes	—
216	C₂³×C₆₇	2₋¹⁺⁴	1	4	yes	—
217	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
218	Q₈×C₆₇	—	8	4	no	—
219	Q₈×C₆₇	—	8	4	no	—
220	D₄×C₆₇	—	4	16	no	—
221	D₄×C₆₇	—	4	16	no	—
222	D₄×C₆₇	—	4	16	no	—
223	C₂×C₄×C₆₇	—	2	8	no	—
224	C₂×C₄×C₆₇	—	2	2	no	—
225	C₂×C₄×C₆₇	—	2	4	no	—
226	C₂×C₄×C₆₇	—	2	4	no	—
227	C₂×C₄×C₆₇	—	2	2	no	—
228	C₂×C₄×C₆₇	—	2	4	no	—
229	C₂×C₄×C₆₇	—	2	4	no	—
230	C₂×C₄×C₆₇	—	2	2	no	—
231	C₂×C₄×C₆₇	—	2	8	no	—
232	C₂×C₄×C₆₇	—	2	8	no	—
233	C₂×C₄×C₆₇	—	2	8	no	—
234	C₂×C₄×C₆₇	—	2	8	no	—
235	C₂×C₄×C₆₇	—	2	4	no	—
236	C₂×C₄×C₆₇	—	2	8	no	—
237	C₂×C₄×C₆₇	—	4	4	no	—
238	C₂×C₄×C₆₇	—	4	4	no	—
239	C₂×C₄×C₆₇	—	4	16	no	—
240	C₂×C₄×C₆₇	—	4	4	no	—
241	C₂×C₄×C₆₇	—	8	4	no	—
242	C₂×C₄×C₆₇	—	8	4	no	—
243	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
244	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
245	C₂×C₄×C₆₇	H(5,𝔽₂)	1	4	yes	—
246	C₂²×C₆₇	—	2	2	no	—
247	C₂²×C₆₇	—	2	2	no	—
248	C₂²×C₆₇	—	2	8	no	—
249	C₂²×C₆₇	—	2	4	no	—
250	C₂²×C₆₇	—	2	4	no	—
251	C₂²×C₆₇	—	2	4	no	—
252	C₂²×C₆₇	—	2	4	no	—
253	C₂²×C₆₇	—	2	8	no	—
254	C₂²×C₆₇	—	4	16	no	—
255	C₂²×C₆₇	—	4	2	no	—
256	C₂²×C₆₇	—	4	2	no	—
257	C₂²×C₆₇	—	4	4	no	—
258	C₂²×C₆₇	—	4	8	no	—
259	C₂²×C₆₇	—	4	2	no	—
260	C₂²×C₆₇	—	4	2	no	—
261	C₂²×C₆₇	—	4	2	no	—
262	C₂²×C₆₇	—	4	8	no	—
263	C₂²×C₆₇	—	8	8	no	—
264	C₂²×C₆₇	—	8	8	no	—
265	C₂²×C₆₇	#536	1	4	yes	—
266	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
267	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	2	yes	—
268	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	2	yes	—
269	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	2	yes	—
270	C₄×C₆₇	—	2	4	no	—
271	C₄×C₆₇	—	2	4	no	—
272	C₄×C₆₇	—	2	4	no	—
273	C₄×C₆₇	—	4	2	no	—
274	C₄×C₆₇	—	4	4	no	—
275	C₄×C₆₇	—	4	4	no	—
276	C₄×C₆₇	—	4	8	no	—
277	#36718	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
278	C₂×C₆₇	—	2	4	no	—
279	C₂×C₆₇	—	4	4	no	—
280	C₂×C₆₇	—	4	4	no	—
281	C₂×C₆₇	—	8	2	no	—
282	C₂×C₆₇	C₂×#536	1	1	yes	characteristic
283	C₂×C₆₇	#3484	1	1	yes	characteristic
284	C₂×C₆₇	#3484	1	1	yes	characteristic
285	C₂×C₆₇	#3536	1	4	yes	—
286	C₂×#543	C₂×C₆₇	1	1	yes	characteristic, complemented
287	C₂²×#121	C₂×C₆₇	1	1	yes	characteristic
288	#2848	C₂×C₆₇	1	2	yes	complemented
289	#2681	C₂×C₆₇	1	8	yes	complemented
290	#2672	C₂×C₆₇	1	8	yes	complemented
291	#2626	C₂×C₆₇	1	2	yes	complemented
292	#2583	C₂×C₆₇	1	2	yes	complemented
293	#2435	C₂×C₆₇	1	2	yes	complemented
294	#2405	C₂×C₆₇	1	2	yes	complemented
295	#2405	C₂×C₆₇	1	2	yes	complemented
296	#2312	C₂×C₆₇	1	1	yes	characteristic, complemented
297	C₆₇	#36718	1	1	yes	characteristic, direct product factor, Sylow
298	C₂³×D₄	C₂²×C₆₇	1	1	yes	characteristic
299	C₂³×D₄	C₂²×C₆₇	1	1	yes	characteristic
300	C₂³×D₄	C₂²×C₆₇	1	1	yes	characteristic
301	C₂⁴×C₄	C₂²×C₆₇	1	1	yes	characteristic
302	#543	—	2	8	no	—
303	C₂×#124	C₂²×C₆₇	1	4	yes	complemented
304	C₂×#123	C₂²×C₆₇	1	4	yes	complemented
305	C₂×#123	C₂²×C₆₇	1	4	yes	complemented
306	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
307	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
308	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	4	yes	complemented
309	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	4	yes	complemented
310	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	4	yes	complemented
311	C₂×#121	—	2	8	no	—
312	C₂×#121	C₂²×C₆₇	1	8	yes	—
313	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
314	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
315	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
316	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
317	#399	C₂²×C₆₇	1	4	yes	complemented
318	#398	C₂²×C₆₇	1	4	yes	complemented
319	#396	C₂²×C₆₇	1	8	yes	complemented
320	#395	C₂²×C₆₇	1	4	yes	complemented
321	#394	C₂²×C₆₇	1	8	yes	complemented
322	#394	C₂²×C₆₇	1	8	yes	complemented
323	#394	C₂²×C₆₇	1	8	yes	complemented
324	#393	C₂²×C₆₇	1	4	yes	complemented
325	#390	C₂²×C₆₇	1	4	yes	complemented
326	#389	C₂²×C₆₇	1	8	yes	complemented
327	#388	C₂²×C₆₇	1	8	yes	complemented
328	#386	C₂²×C₆₇	1	4	yes	complemented
329	#386	C₂²×C₆₇	1	4	yes	complemented
330	#385	C₂²×C₆₇	1	4	yes	—
331	#385	C₂²×C₆₇	1	4	yes	—
332	#385	C₂²×C₆₇	1	8	yes	complemented
333	#380	C₂²×C₆₇	1	2	yes	—
334	#380	C₂²×C₆₇	1	4	yes	complemented
335	#379	C₂²×C₆₇	1	2	yes	—
336	#379	C₂²×C₆₇	1	4	yes	—
337	#379	C₂²×C₆₇	1	4	yes	—
338	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
339	C₂⁵	C₂³×C₆₇	1	1	yes	characteristic
340	C₂⁵	C₂³×C₆₇	1	1	yes	characteristic
341	C₂⁵	C₂³×C₆₇	1	1	yes	characteristic
342	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
343	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
344	C₂²×D₄	—	2	4	no	—
345	C₂²×D₄	—	2	4	no	—
346	C₂²×D₄	—	2	4	no	—
347	C₂²×D₄	—	4	4	no	—
348	C₂²×D₄	—	4	4	no	—
349	C₂²×D₄	—	4	4	no	—
350	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
351	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
352	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
353	C₂³×C₄	—	2	4	no	—
354	C₂³×C₄	—	4	4	no	—
355	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
356	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
357	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
358	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
359	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
360	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
361	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
362	#124	—	4	8	no	—
363	#123	—	4	8	no	—
364	#123	—	4	8	no	—
365	C₂²⋊Q₈	—	2	8	no	—
366	C₂²⋊Q₈	—	2	8	no	—
367	C₂²⋊Q₈	—	2	16	no	—
368	C₂²⋊Q₈	—	2	16	no	—
369	C₂²⋊Q₈	—	4	8	no	—
370	#121	—	2	32	no	—
371	C₂×C₄⋊C₄	—	2	4	no	—
372	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
373	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
374	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
375	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
376	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
377	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
378	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
379	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
380	C₂×C₂²⋊C₄	—	2	8	no	—
381	C₂×C₂²⋊C₄	—	2	4	no	—
382	C₂×C₂²⋊C₄	D₄×C₆₇	1	8	yes	—
383	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
384	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
385	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
386	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
387	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
388	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
389	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
390	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
391	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
392	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
393	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
394	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
395	#95	C₂³×C₆₇	1	4	yes	—
396	#95	C₂³×C₆₇	1	4	yes	—
397	#95	C₂³×C₆₇	1	8	yes	—
398	#95	C₂³×C₆₇	1	8	yes	—
399	C₂⁴	—	2	2	no	—
400	C₂⁴	—	2	4	no	—
401	C₂⁴	—	2	4	no	—
402	C₂⁴	—	2	2	no	—
403	C₂⁴	—	2	4	no	—
404	C₂⁴	—	2	4	no	—
405	C₂⁴	—	4	2	no	—
406	C₂⁴	—	4	2	no	—
407	C₂⁴	—	4	2	no	—
408	C₂⁴	—	4	2	no	—
409	C₂⁴	—	4	2	no	—
410	C₂⁴	C₂×D₄×C₆₇	1	4	yes	—
411	C₂⁴	C₂⁴×C₆₇	1	2	yes	—
412	C₂⁴	C₂⁴×C₆₇	1	2	yes	—
413	C₂⁴	C₂⁴×C₆₇	1	2	yes	—
414	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
415	C₂×Q₈	—	2	4	no	—
416	C₂×Q₈	—	2	4	no	—
417	C₂×Q₈	—	4	4	no	—
418	C₂×Q₈	—	4	4	no	—
419	C₂×D₄	—	2	8	no	—
420	C₂×D₄	—	2	8	no	—
421	C₂×D₄	—	2	8	no	—
422	C₂×D₄	—	2	8	no	—
423	C₂×D₄	—	2	8	no	—
424	C₂×D₄	—	2	8	no	—
425	C₂×D₄	—	4	4	no	—
426	C₂×D₄	—	4	4	no	—
427	C₂×D₄	—	4	4	no	—
428	C₂×D₄	—	4	16	no	—
429	C₂×D₄	—	4	16	no	—
430	C₂×D₄	—	4	16	no	—
431	C₂²×C₄	—	2	2	no	—
432	C₂²×C₄	—	2	4	no	—
433	C₂²×C₄	—	2	2	no	—
434	C₂²×C₄	—	2	4	no	—
435	C₂²×C₄	—	2	4	no	—
436	C₂²×C₄	—	2	8	no	—
437	C₂²×C₄	—	2	8	no	—
438	C₂²×C₄	—	4	2	no	—
439	C₂²×C₄	—	4	4	no	—
440	C₂²×C₄	—	4	2	no	—
441	C₂²×C₄	—	4	2	no	—
442	C₂²×C₄	—	4	4	no	—
443	C₂²×C₄	—	4	4	no	—
444	C₂²×C₄	—	4	16	no	—
445	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
446	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
447	C₂²×C₄	C₂⁴×C₆₇	1	8	yes	—
448	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
449	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
450	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
451	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
452	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
453	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
454	C₄⋊C₄	—	2	8	no	—
455	C₄⋊C₄	—	2	4	no	—
456	C₄⋊C₄	—	2	8	no	—
457	C₄⋊C₄	—	2	8	no	—
458	C₄⋊C₄	—	2	4	no	—
459	C₄⋊C₄	—	2	16	no	—
460	C₄⋊C₄	—	4	4	no	—
461	C₄⋊C₄	—	4	8	no	—
462	C₂²⋊C₄	—	2	8	no	—
463	C₂²⋊C₄	—	2	8	no	—
464	C₂²⋊C₄	—	2	16	no	—
465	C₂²⋊C₄	—	2	8	no	—
466	C₂²⋊C₄	—	2	8	no	—
467	C₂²⋊C₄	—	4	8	no	—
468	C₂²⋊C₄	—	4	4	no	—
469	C₂²⋊C₄	—	4	4	no	—
470	C₂²⋊C₄	—	4	8	no	—
471	C₂²⋊C₄	—	4	8	no	—
472	C₂²⋊C₄	C₂×D₄×C₆₇	1	16	yes	—
473	C₄²	—	4	4	no	—
474	C₂³	—	2	2	no	—
475	C₂³	—	2	2	no	—
476	C₂³	—	2	8	no	—
477	C₂³	—	2	2	no	—
478	C₂³	—	2	4	no	—
479	C₂³	—	2	4	no	—
480	C₂³	—	2	4	no	—
481	C₂³	—	2	2	no	—
482	C₂³	—	2	4	no	—
483	C₂³	—	2	4	no	—
484	C₂³	—	2	8	no	—
485	C₂³	—	2	2	no	—
486	C₂³	—	2	4	no	—
487	C₂³	—	2	4	no	—
488	C₂³	—	2	4	no	—
489	C₂³	—	4	4	no	—
490	C₂³	—	4	4	no	—
491	C₂³	—	4	16	no	—
492	C₂³	—	4	4	no	—
493	C₂³	—	4	4	no	—
494	C₂³	—	4	4	no	—
495	C₂³	—	4	4	no	—
496	C₂³	—	4	4	no	—
497	C₂³	—	8	8	no	—
498	C₂³	—	8	8	no	—
499	C₂³	C₂²×D₄×C₆₇	1	4	yes	—
500	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
501	C₂³	H(5,𝔽₂)×C₆₇	1	4	yes	—
502	C₂³	2₋¹⁺⁴×C₆₇	1	4	yes	—
503	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
504	Q₈	—	8	4	no	—
505	Q₈	—	8	4	no	—
506	D₄	—	4	16	no	—
507	D₄	—	4	16	no	—
508	D₄	—	4	16	no	—
509	C₂×C₄	—	2	8	no	—
510	C₂×C₄	—	2	2	no	—
511	C₂×C₄	—	2	4	no	—
512	C₂×C₄	—	2	4	no	—
513	C₂×C₄	—	2	2	no	—
514	C₂×C₄	—	2	4	no	—
515	C₂×C₄	—	2	4	no	—
516	C₂×C₄	—	2	2	no	—
517	C₂×C₄	—	2	8	no	—
518	C₂×C₄	—	2	8	no	—
519	C₂×C₄	—	2	8	no	—
520	C₂×C₄	—	2	8	no	—
521	C₂×C₄	—	2	4	no	—
522	C₂×C₄	—	2	8	no	—
523	C₂×C₄	—	4	4	no	—
524	C₂×C₄	—	4	4	no	—
525	C₂×C₄	—	4	16	no	—
526	C₂×C₄	—	4	4	no	—
527	C₂×C₄	—	8	4	no	—
528	C₂×C₄	—	8	4	no	—
529	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
530	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
531	C₂×C₄	H(5,𝔽₂)×C₆₇	1	4	yes	—
532	C₂²	—	2	8	no	—
533	C₂²	—	2	2	no	—
534	C₂²	—	2	2	no	—
535	C₂²	—	2	8	no	—
536	C₂²	—	2	4	no	—
537	C₂²	—	2	4	no	—
538	C₂²	—	2	4	no	—
539	C₂²	—	2	4	no	—
540	C₂²	—	4	16	no	—
541	C₂²	—	4	4	no	—
542	C₂²	—	4	8	no	—
543	C₂²	—	4	2	no	—
544	C₂²	—	4	2	no	—
545	C₂²	—	4	8	no	—
546	C₂²	—	4	2	no	—
547	C₂²	—	4	2	no	—
548	C₂²	—	4	2	no	—
549	C₂²	—	8	8	no	—
550	C₂²	—	8	8	no	—
551	C₂²	#536×C₆₇	1	4	yes	—
552	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
553	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	2	yes	—
554	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	2	yes	—
555	C₂²	C₂×2₋¹⁺⁴×C₆₇	1	2	yes	—
556	C₄	—	2	4	no	—
557	C₄	—	2	4	no	—
558	C₄	—	2	4	no	—
559	C₄	—	4	8	no	—
560	C₄	—	4	2	no	—
561	C₄	—	4	4	no	—
562	C₄	—	4	4	no	—
563	C₂	—	2	4	no	—
564	C₂	—	4	4	no	—
565	C₂	—	4	4	no	—
566	C₂	—	8	2	no	—
567	C₂	C₂×#536×C₆₇	1	1	yes	characteristic
568	C₂	C₆₇×#3484	1	1	yes	characteristic
569	C₂	C₆₇×#3484	1	1	yes	characteristic
570	C₂	C₆₇×#3536	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₆₇
prime radical	C₂²×#121×C₆₇	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#543×C₆₇	C₂	semidirect
3. C₆₇×#2848	C₂	semidirect
4. C₆₇×#2681	C₂	semidirect
5. C₆₇×#2672	C₂	semidirect
6. C₆₇×#2626	C₂	semidirect
7. C₆₇×#2583	C₂	semidirect
8. C₆₇×#2435	C₂	semidirect
9. C₆₇×#2405	C₂	semidirect
10. C₆₇×#2405	C₂	semidirect
11. C₆₇×#2312	C₂	semidirect
17. C₂×#124×C₆₇	C₂²	semidirect
18. C₂×#123×C₆₇	C₂²	semidirect
19. C₂×#123×C₆₇	C₂²	semidirect
22. C₂×C₂²⋊Q₈×C₆₇	C₂²	semidirect
23. C₂×C₂²⋊Q₈×C₆₇	C₂²	semidirect
24. C₂×C₂²⋊Q₈×C₆₇	C₂²	semidirect
31. #399×C₆₇	C₂²	semidirect
32. #398×C₆₇	C₂²	semidirect
33. #396×C₆₇	C₂²	semidirect
34. #395×C₆₇	C₂²	semidirect
35. #394×C₆₇	C₂²	semidirect
36. #394×C₆₇	C₂²	semidirect
37. #394×C₆₇	C₂²	semidirect
38. #393×C₆₇	C₂²	semidirect
39. #390×C₆₇	C₂²	semidirect
40. #389×C₆₇	C₂²	semidirect
41. #388×C₆₇	C₂²	semidirect
42. #386×C₆₇	C₂²	semidirect
43. #386×C₆₇	C₂²	semidirect
46. #385×C₆₇	C₂²	semidirect
48. #380×C₆₇	C₂²	semidirect
277. #36718	C₆₇	direct (N × H)
286. C₂×#543	C₂×C₆₇	semidirect
288. #2848	C₂×C₆₇	semidirect
289. #2681	C₂×C₆₇	semidirect
290. #2672	C₂×C₆₇	semidirect
291. #2626	C₂×C₆₇	semidirect
292. #2583	C₂×C₆₇	semidirect
293. #2435	C₂×C₆₇	semidirect
294. #2405	C₂×C₆₇	semidirect
295. #2405	C₂×C₆₇	semidirect
296. #2312	C₂×C₆₇	semidirect
297. C₆₇	#36718	direct (N × H)
303. C₂×#124	C₂²×C₆₇	semidirect
304. C₂×#123	C₂²×C₆₇	semidirect
305. C₂×#123	C₂²×C₆₇	semidirect
308. C₂×C₂²⋊Q₈	C₂²×C₆₇	semidirect
309. C₂×C₂²⋊Q₈	C₂²×C₆₇	semidirect
310. C₂×C₂²⋊Q₈	C₂²×C₆₇	semidirect
317. #399	C₂²×C₆₇	semidirect
318. #398	C₂²×C₆₇	semidirect
319. #396	C₂²×C₆₇	semidirect
320. #395	C₂²×C₆₇	semidirect
321. #394	C₂²×C₆₇	semidirect
322. #394	C₂²×C₆₇	semidirect
323. #394	C₂²×C₆₇	semidirect
324. #393	C₂²×C₆₇	semidirect
325. #390	C₂²×C₆₇	semidirect
326. #389	C₂²×C₆₇	semidirect
327. #388	C₂²×C₆₇	semidirect
328. #386	C₂²×C₆₇	semidirect
329. #386	C₂²×C₆₇	semidirect
332. #385	C₂²×C₆₇	semidirect
334. #380	C₂²×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#543×C₆₇	2
C₂²×#121×C₆₇	2
C₆₇×#2848	2
C₆₇×#2681	2
C₆₇×#2672	2
C₆₇×#2626	2
C₆₇×#2583	2
C₆₇×#2435	2
C₆₇×#2405	2
C₆₇×#2312	2
#36718	67

Subgroups — C67×#36718 #30715969

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#36718 #30715969