Subgroups — C₆₇×#20932 #30700183

Order 17152.

1926 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×AGL(1,ℤ/8)×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#520×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#416×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×#102×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#2309	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#1880	C₂	1	4	yes	maximal, complemented
7	C₆₇×#1879	C₂	1	4	yes	maximal, complemented
8	C₆₇×#1736	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#1729	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
11	C₂³×D₄×C₆₇	C₂²	1	1	yes	characteristic
12	C₂×AGL(1,ℤ/8)×C₆₇	—	2	4	no	—
13	C₂×AGL(1,ℤ/8)×C₆₇	—	2	4	no	—
14	C₂×AGL(1,ℤ/8)×C₆₇	C₄	1	8	yes	complemented
15	C₂²×SD₁₆×C₆₇	C₂²	1	2	yes	—
16	C₂²×D₈×C₆₇	C₂²	1	2	yes	—
17	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
18	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
19	#520×C₆₇	—	2	4	no	—
20	#520×C₆₇	—	2	4	no	—
21	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	complemented
22	C₂×C₄×D₄×C₆₇	—	2	2	no	—
23	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	complemented
24	C₂×#117×C₆₇	—	2	1	no	—
25	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
27	C₂²×C₄⋊C₄×C₆₇	—	2	1	no	—
28	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
29	#416×C₆₇	—	2	4	no	—
30	C₂×#103×C₆₇	C₂²	1	2	yes	complemented
31	C₂×#102×C₆₇	—	2	4	no	—
32	C₂×#102×C₆₇	C₂²	1	4	yes	—
33	C₂×#102×C₆₇	C₂²	1	2	yes	complemented
34	#390×C₆₇	—	2	1	no	—
35	#390×C₆₇	—	2	1	no	—
36	#390×C₆₇	C₂²	1	4	yes	—
37	#379×C₆₇	—	2	2	no	—
38	#340×C₆₇	C₂²	1	4	yes	complemented
39	#336×C₆₇	C₂²	1	4	yes	complemented
40	C₂⁵×C₆₇	—	2	1	no	—
41	C₂×#41×C₆₇	—	2	2	no	—
42	C₂×#41×C₆₇	—	2	2	no	—
43	C₂×#41×C₆₇	—	2	2	no	—
44	C₂×#41×C₆₇	C₂×C₄	1	4	yes	—
45	C₂×#41×C₆₇	D₄	1	4	yes	—
46	C₂×#41×C₆₇	D₄	1	4	yes	—
47	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
48	C₂²×D₄×C₆₇	—	2	1	no	—
49	C₂²×D₄×C₆₇	—	2	2	no	—
50	C₂²×D₄×C₆₇	—	2	2	no	—
51	C₂²×D₄×C₆₇	—	2	2	no	—
52	C₂²×D₄×C₆₇	—	2	1	no	—
53	C₂²×D₄×C₆₇	—	2	2	no	—
54	C₂²×D₄×C₆₇	—	2	2	no	—
55	C₂²×D₄×C₆₇	C₂×C₄	1	4	yes	complemented
56	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
57	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
58	C₂³×C₄×C₆₇	—	2	1	no	—
59	C₂³×C₄×C₆₇	—	2	1	no	—
60	C₂³×C₄×C₆₇	—	2	2	no	—
61	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
62	AGL(1,ℤ/8)×C₆₇	—	2	32	no	—
63	C₂×SD₁₆×C₆₇	—	2	4	no	—
64	C₂×SD₁₆×C₆₇	—	2	4	no	—
65	C₂×SD₁₆×C₆₇	C₂×C₄	1	8	yes	complemented
66	C₂×D₈×C₆₇	—	2	4	no	—
67	C₂×D₈×C₆₇	—	2	4	no	—
68	C₂×D₈×C₆₇	C₂×C₄	1	8	yes	complemented
69	C₂×M₁₆×C₆₇	—	2	2	no	—
70	C₂×M₁₆×C₆₇	—	2	2	no	—
71	C₂×M₁₆×C₆₇	C₂×C₄	1	4	yes	complemented
72	C₂²×C₈×C₆₇	—	2	2	no	—
73	C₂²×C₈×C₆₇	—	2	2	no	—
74	C₂²×C₈×C₆₇	—	2	2	no	—
75	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
76	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
77	C₄×Q₈×C₆₇	—	2	4	no	—
78	C₄×Q₈×C₆₇	—	2	4	no	—
79	C₄×D₄×C₆₇	—	2	16	no	—
80	C₄×D₄×C₆₇	—	2	4	no	—
81	C₄×D₄×C₆₇	—	2	4	no	—
82	#117×C₆₇	—	2	8	no	—
83	#117×C₆₇	—	2	2	no	—
84	#117×C₆₇	—	2	2	no	—
85	#117×C₆₇	—	2	2	no	—
86	#117×C₆₇	—	2	2	no	—
87	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
88	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
89	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
90	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
91	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
92	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
93	C₂×C₄⋊C₄×C₆₇	D₄	1	4	yes	—
94	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
95	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
96	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
97	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
98	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
99	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
100	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
101	C₂×C₄²×C₆₇	—	2	2	no	—
102	C₂×C₄²×C₆₇	—	2	1	no	—
103	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
104	#103×C₆₇	—	2	4	no	—
105	#102×C₆₇	—	2	4	no	—
106	#102×C₆₇	—	2	8	no	—
107	C₂⁴×C₆₇	—	2	1	no	—
108	C₂⁴×C₆₇	—	2	2	no	—
109	C₂⁴×C₆₇	—	2	4	no	—
110	C₂⁴×C₆₇	—	4	2	no	—
111	C₂⁴×C₆₇	—	4	2	no	—
112	C₂⁴×C₆₇	—	8	2	no	—
113	C₂⁴×C₆₇	—	8	2	no	—
114	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
115	#41×C₆₇	—	2	8	no	—
116	#41×C₆₇	—	2	8	no	—
117	#41×C₆₇	—	2	8	no	—
118	#41×C₆₇	C₂²⋊C₄	1	16	yes	complemented
119	C₂×Q₈×C₆₇	—	2	1	no	—
120	C₂×Q₈×C₆₇	—	2	1	no	—
121	C₂×Q₈×C₆₇	—	2	1	no	—
122	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
123	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
124	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
125	C₂×D₄×C₆₇	—	2	4	no	—
126	C₂×D₄×C₆₇	—	2	4	no	—
127	C₂×D₄×C₆₇	—	2	4	no	—
128	C₂×D₄×C₆₇	—	2	8	no	—
129	C₂×D₄×C₆₇	—	2	8	no	—
130	C₂×D₄×C₆₇	—	2	4	no	—
131	C₂×D₄×C₆₇	—	2	1	no	—
132	C₂×D₄×C₆₇	—	2	2	no	—
133	C₂×D₄×C₆₇	—	2	2	no	—
134	C₂×D₄×C₆₇	—	2	2	no	—
135	C₂×D₄×C₆₇	—	2	1	no	—
136	C₂×D₄×C₆₇	—	2	1	no	—
137	C₂×D₄×C₆₇	—	2	2	no	—
138	C₂×D₄×C₆₇	—	2	2	no	—
139	C₂×D₄×C₆₇	—	4	2	no	—
140	C₂×D₄×C₆₇	—	4	4	no	—
141	C₂×D₄×C₆₇	—	4	2	no	—
142	C₂×D₄×C₆₇	—	4	4	no	—
143	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
144	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
145	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
146	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
147	C₂²×C₄×C₆₇	—	2	1	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	2	no	—
152	C₂²×C₄×C₆₇	—	2	8	no	—
153	C₂²×C₄×C₆₇	—	2	4	no	—
154	C₂²×C₄×C₆₇	—	2	2	no	—
155	C₂²×C₄×C₆₇	—	8	4	no	—
156	C₂²×C₄×C₆₇	—	8	2	no	—
157	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
158	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
159	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
160	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
161	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
162	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
163	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
164	C₂²×C₄×C₆₇	#41	1	2	yes	—
165	C₂²×C₄×C₆₇	#41	1	2	yes	—
166	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
167	SD₁₆×C₆₇	—	2	16	no	—
168	D₈×C₆₇	—	2	16	no	—
169	M₁₆×C₆₇	—	2	8	no	—
170	C₂×C₈×C₆₇	—	2	2	no	—
171	C₂×C₈×C₆₇	—	2	2	no	—
172	C₂×C₈×C₆₇	—	2	2	no	—
173	C₂×C₈×C₆₇	—	2	2	no	—
174	C₂×C₈×C₆₇	—	2	2	no	—
175	C₂×C₈×C₆₇	—	4	4	no	—
176	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
177	C₄⋊C₄×C₆₇	—	2	8	no	—
178	C₄⋊C₄×C₆₇	—	2	2	no	—
179	C₄⋊C₄×C₆₇	—	2	4	no	—
180	C₄⋊C₄×C₆₇	—	2	2	no	—
181	C₄⋊C₄×C₆₇	—	2	2	no	—
182	C₄⋊C₄×C₆₇	—	2	2	no	—
183	C₄⋊C₄×C₆₇	—	2	2	no	—
184	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
185	C₂²⋊C₄×C₆₇	—	2	8	no	—
186	C₂²⋊C₄×C₆₇	—	2	4	no	—
187	C₂²⋊C₄×C₆₇	—	4	2	no	—
188	C₂²⋊C₄×C₆₇	—	8	4	no	—
189	C₂²⋊C₄×C₆₇	—	8	4	no	—
190	C₄²×C₆₇	—	2	8	no	—
191	C₄²×C₆₇	—	2	2	no	—
192	C₄²×C₆₇	—	2	2	no	—
193	C₄²×C₆₇	—	4	2	no	—
194	C₂³×C₆₇	—	2	2	no	—
195	C₂³×C₆₇	—	2	2	no	—
196	C₂³×C₆₇	—	2	2	no	—
197	C₂³×C₆₇	—	2	2	no	—
198	C₂³×C₆₇	—	2	2	no	—
199	C₂³×C₆₇	—	2	4	no	—
200	C₂³×C₆₇	—	4	2	no	—
201	C₂³×C₆₇	—	4	2	no	—
202	C₂³×C₆₇	—	4	2	no	—
203	C₂³×C₆₇	—	4	2	no	—
204	C₂³×C₆₇	—	4	4	no	—
205	C₂³×C₆₇	—	4	4	no	—
206	C₂³×C₆₇	—	4	8	no	—
207	C₂³×C₆₇	—	8	8	no	—
208	C₂³×C₆₇	—	8	4	no	—
209	C₂³×C₆₇	—	8	4	no	—
210	C₂³×C₆₇	—	8	8	no	—
211	C₂³×C₆₇	C₄×D₄	1	2	yes	—
212	C₂³×C₆₇	#120	1	2	yes	—
213	C₂³×C₆₇	#121	1	2	yes	—
214	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
215	Q₈×C₆₇	—	2	2	no	—
216	Q₈×C₆₇	—	2	2	no	—
217	Q₈×C₆₇	—	2	2	no	—
218	Q₈×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
219	D₄×C₆₇	—	2	16	no	—
220	D₄×C₆₇	—	2	4	no	—
221	D₄×C₆₇	—	2	2	no	—
222	D₄×C₆₇	—	2	4	no	—
223	D₄×C₆₇	—	2	2	no	—
224	D₄×C₆₇	—	2	2	no	—
225	D₄×C₆₇	—	4	2	no	—
226	D₄×C₆₇	—	4	2	no	—
227	D₄×C₆₇	—	4	8	no	—
228	D₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
229	C₂×C₄×C₆₇	—	2	2	no	—
230	C₂×C₄×C₆₇	—	2	2	no	—
231	C₂×C₄×C₆₇	—	2	2	no	—
232	C₂×C₄×C₆₇	—	2	16	no	—
233	C₂×C₄×C₆₇	—	2	2	no	—
234	C₂×C₄×C₆₇	—	2	2	no	—
235	C₂×C₄×C₆₇	—	2	4	no	—
236	C₂×C₄×C₆₇	—	2	4	no	—
237	C₂×C₄×C₆₇	—	2	4	no	—
238	C₂×C₄×C₆₇	—	4	4	no	—
239	C₂×C₄×C₆₇	—	4	2	no	—
240	C₂×C₄×C₆₇	—	4	2	no	—
241	C₂×C₄×C₆₇	—	4	2	no	—
242	C₂×C₄×C₆₇	—	8	4	no	—
243	C₂×C₄×C₆₇	—	8	8	no	—
244	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
245	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
246	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
247	C₂×C₄×C₆₇	#120	1	2	yes	—
248	C₂×C₄×C₆₇	#121	1	4	yes	—
249	C₂×C₄×C₆₇	#121	1	2	yes	—
250	C₂×C₄×C₆₇	#123	1	4	yes	—
251	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
252	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
253	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
254	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
255	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
256	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
257	C₈×C₆₇	—	2	4	no	—
258	C₈×C₆₇	—	4	2	no	—
259	C₂²×C₆₇	—	2	4	no	—
260	C₂²×C₆₇	—	2	2	no	—
261	C₂²×C₆₇	—	2	2	no	—
262	C₂²×C₆₇	—	2	4	no	—
263	C₂²×C₆₇	—	2	4	no	—
264	C₂²×C₆₇	—	4	4	no	—
265	C₂²×C₆₇	—	4	2	no	—
266	C₂²×C₆₇	—	4	2	no	—
267	C₂²×C₆₇	—	4	4	no	—
268	C₂²×C₆₇	—	4	8	no	—
269	C₂²×C₆₇	—	4	4	no	—
270	C₂²×C₆₇	—	8	16	no	—
271	C₂²×C₆₇	—	8	2	no	—
272	C₂²×C₆₇	—	8	2	no	—
273	C₂²×C₆₇	—	8	4	no	—
274	C₂²×C₆₇	#386	1	4	yes	—
275	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
276	C₂²×C₆₇	C₂×#120	1	1	yes	characteristic
277	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
278	C₂²×C₆₇	#576	1	2	yes	—
279	C₂²×C₆₇	#577	1	2	yes	—
280	C₄×C₆₇	—	2	4	no	—
281	C₄×C₆₇	—	4	2	no	—
282	C₄×C₆₇	—	4	4	no	—
283	C₄×C₆₇	—	4	2	no	—
284	C₄×C₆₇	—	8	2	no	—
285	C₄×C₆₇	#386	1	4	yes	—
286	C₄×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
287	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
288	C₄×C₆₇	C₂×#121	1	1	yes	characteristic
289	C₄×C₆₇	C₂×#123	1	1	yes	characteristic
290	#20932	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
291	C₂×C₆₇	—	2	4	no	—
292	C₂×C₆₇	—	4	4	no	—
293	C₂×C₆₇	—	8	4	no	—
294	C₂×C₆₇	C₂×#386	1	1	yes	characteristic
295	C₂×C₆₇	#2946	1	2	yes	—
296	C₂×C₆₇	#3008	1	1	yes	characteristic
297	C₂×C₆₇	#3010	1	1	yes	characteristic
298	C₂×C₆₇	#3045	1	2	yes	—
299	C₂²×AGL(1,ℤ/8)	C₂×C₆₇	1	1	yes	characteristic
300	C₂×#520	C₂×C₆₇	1	1	yes	characteristic, complemented
301	C₂×#416	C₂×C₆₇	1	1	yes	characteristic, complemented
302	C₂²×#102	C₂×C₆₇	1	1	yes	characteristic, complemented
303	#2309	C₂×C₆₇	1	1	yes	characteristic, complemented
304	#1880	C₂×C₆₇	1	4	yes	complemented
305	#1879	C₂×C₆₇	1	4	yes	complemented
306	#1736	C₂×C₆₇	1	1	yes	characteristic, complemented
307	#1729	C₂×C₆₇	1	1	yes	characteristic, complemented
308	C₆₇	#20932	1	1	yes	characteristic, direct product factor, Sylow
309	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
310	C₂³×D₄	C₂²×C₆₇	1	1	yes	characteristic
311	C₂×AGL(1,ℤ/8)	—	2	4	no	—
312	C₂×AGL(1,ℤ/8)	—	2	4	no	—
313	C₂×AGL(1,ℤ/8)	C₄×C₆₇	1	8	yes	complemented
314	C₂²×SD₁₆	C₂²×C₆₇	1	2	yes	—
315	C₂²×D₈	C₂²×C₆₇	1	2	yes	—
316	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
317	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
318	#520	—	2	4	no	—
319	#520	—	2	4	no	—
320	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	complemented
321	C₂×C₄×D₄	—	2	2	no	—
322	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	complemented
323	C₂×#117	—	2	1	no	—
324	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
325	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
326	C₂²×C₄⋊C₄	—	2	1	no	—
327	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
328	#416	—	2	4	no	—
329	C₂×#103	C₂²×C₆₇	1	2	yes	complemented
330	C₂×#102	—	2	4	no	—
331	C₂×#102	C₂²×C₆₇	1	4	yes	—
332	C₂×#102	C₂²×C₆₇	1	2	yes	complemented
333	#390	—	2	1	no	—
334	#390	—	2	1	no	—
335	#390	C₂²×C₆₇	1	4	yes	—
336	#379	—	2	2	no	—
337	#340	C₂²×C₆₇	1	4	yes	complemented
338	#336	C₂²×C₆₇	1	4	yes	complemented
339	C₂⁵	—	2	1	no	—
340	C₂×#41	—	2	2	no	—
341	C₂×#41	—	2	2	no	—
342	C₂×#41	—	2	2	no	—
343	C₂×#41	C₂×C₄×C₆₇	1	4	yes	—
344	C₂×#41	D₄×C₆₇	1	4	yes	—
345	C₂×#41	D₄×C₆₇	1	4	yes	—
346	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
347	C₂²×D₄	—	2	1	no	—
348	C₂²×D₄	—	2	2	no	—
349	C₂²×D₄	—	2	2	no	—
350	C₂²×D₄	—	2	2	no	—
351	C₂²×D₄	—	2	1	no	—
352	C₂²×D₄	—	2	2	no	—
353	C₂²×D₄	—	2	2	no	—
354	C₂²×D₄	C₂×C₄×C₆₇	1	4	yes	complemented
355	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
356	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
357	C₂³×C₄	—	2	1	no	—
358	C₂³×C₄	—	2	1	no	—
359	C₂³×C₄	—	2	2	no	—
360	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
361	AGL(1,ℤ/8)	—	2	32	no	—
362	C₂×SD₁₆	—	2	4	no	—
363	C₂×SD₁₆	—	2	4	no	—
364	C₂×SD₁₆	C₂×C₄×C₆₇	1	8	yes	complemented
365	C₂×D₈	—	2	4	no	—
366	C₂×D₈	—	2	4	no	—
367	C₂×D₈	C₂×C₄×C₆₇	1	8	yes	complemented
368	C₂×M₁₆	—	2	2	no	—
369	C₂×M₁₆	—	2	2	no	—
370	C₂×M₁₆	C₂×C₄×C₆₇	1	4	yes	complemented
371	C₂²×C₈	—	2	2	no	—
372	C₂²×C₈	—	2	2	no	—
373	C₂²×C₈	—	2	2	no	—
374	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
375	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
376	C₄×Q₈	—	2	4	no	—
377	C₄×Q₈	—	2	4	no	—
378	C₄×D₄	—	2	16	no	—
379	C₄×D₄	—	2	4	no	—
380	C₄×D₄	—	2	4	no	—
381	#117	—	2	8	no	—
382	#117	—	2	2	no	—
383	#117	—	2	2	no	—
384	#117	—	2	2	no	—
385	#117	—	2	2	no	—
386	C₂×C₄⋊C₄	—	2	8	no	—
387	C₂×C₄⋊C₄	—	2	1	no	—
388	C₂×C₄⋊C₄	—	2	2	no	—
389	C₂×C₄⋊C₄	—	2	2	no	—
390	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
391	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
392	C₂×C₄⋊C₄	D₄×C₆₇	1	4	yes	—
393	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
394	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
395	C₂×C₂²⋊C₄	—	2	2	no	—
396	C₂×C₂²⋊C₄	—	2	1	no	—
397	C₂×C₂²⋊C₄	—	2	4	no	—
398	C₂×C₂²⋊C₄	—	2	4	no	—
399	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
400	C₂×C₄²	—	2	2	no	—
401	C₂×C₄²	—	2	1	no	—
402	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
403	#103	—	2	4	no	—
404	#102	—	2	4	no	—
405	#102	—	2	8	no	—
406	C₂⁴	—	2	1	no	—
407	C₂⁴	—	2	2	no	—
408	C₂⁴	—	2	4	no	—
409	C₂⁴	—	4	2	no	—
410	C₂⁴	—	4	2	no	—
411	C₂⁴	—	8	2	no	—
412	C₂⁴	—	8	2	no	—
413	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
414	#41	—	2	8	no	—
415	#41	—	2	8	no	—
416	#41	—	2	8	no	—
417	#41	C₂²⋊C₄×C₆₇	1	16	yes	complemented
418	C₂×Q₈	—	2	1	no	—
419	C₂×Q₈	—	2	1	no	—
420	C₂×Q₈	—	2	1	no	—
421	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
422	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
423	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
424	C₂×D₄	—	2	4	no	—
425	C₂×D₄	—	2	4	no	—
426	C₂×D₄	—	2	4	no	—
427	C₂×D₄	—	2	8	no	—
428	C₂×D₄	—	2	8	no	—
429	C₂×D₄	—	2	4	no	—
430	C₂×D₄	—	2	1	no	—
431	C₂×D₄	—	2	2	no	—
432	C₂×D₄	—	2	2	no	—
433	C₂×D₄	—	2	2	no	—
434	C₂×D₄	—	2	1	no	—
435	C₂×D₄	—	2	1	no	—
436	C₂×D₄	—	2	2	no	—
437	C₂×D₄	—	2	2	no	—
438	C₂×D₄	—	4	2	no	—
439	C₂×D₄	—	4	4	no	—
440	C₂×D₄	—	4	2	no	—
441	C₂×D₄	—	4	4	no	—
442	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
443	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
444	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
445	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
446	C₂²×C₄	—	2	1	no	—
447	C₂²×C₄	—	2	4	no	—
448	C₂²×C₄	—	2	4	no	—
449	C₂²×C₄	—	2	4	no	—
450	C₂²×C₄	—	2	2	no	—
451	C₂²×C₄	—	2	8	no	—
452	C₂²×C₄	—	2	4	no	—
453	C₂²×C₄	—	2	2	no	—
454	C₂²×C₄	—	8	4	no	—
455	C₂²×C₄	—	8	2	no	—
456	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
457	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
458	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
459	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
460	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
461	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
462	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
463	C₂²×C₄	#41×C₆₇	1	2	yes	—
464	C₂²×C₄	#41×C₆₇	1	2	yes	—
465	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
466	SD₁₆	—	2	16	no	—
467	D₈	—	2	16	no	—
468	M₁₆	—	2	8	no	—
469	C₂×C₈	—	2	2	no	—
470	C₂×C₈	—	2	2	no	—
471	C₂×C₈	—	2	2	no	—
472	C₂×C₈	—	2	2	no	—
473	C₂×C₈	—	2	2	no	—
474	C₂×C₈	—	4	4	no	—
475	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
476	C₄⋊C₄	—	2	8	no	—
477	C₄⋊C₄	—	2	2	no	—
478	C₄⋊C₄	—	2	4	no	—
479	C₄⋊C₄	—	2	2	no	—
480	C₄⋊C₄	—	2	2	no	—
481	C₄⋊C₄	—	2	2	no	—
482	C₄⋊C₄	—	2	2	no	—
483	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
484	C₂²⋊C₄	—	2	8	no	—
485	C₂²⋊C₄	—	2	4	no	—
486	C₂²⋊C₄	—	4	2	no	—
487	C₂²⋊C₄	—	8	4	no	—
488	C₂²⋊C₄	—	8	4	no	—
489	C₄²	—	2	8	no	—
490	C₄²	—	2	2	no	—
491	C₄²	—	2	2	no	—
492	C₄²	—	4	2	no	—
493	C₂³	—	2	2	no	—
494	C₂³	—	2	2	no	—
495	C₂³	—	2	2	no	—
496	C₂³	—	2	2	no	—
497	C₂³	—	2	2	no	—
498	C₂³	—	2	4	no	—
499	C₂³	—	4	2	no	—
500	C₂³	—	4	2	no	—
501	C₂³	—	4	2	no	—
502	C₂³	—	4	2	no	—
503	C₂³	—	4	4	no	—
504	C₂³	—	4	4	no	—
505	C₂³	—	4	8	no	—
506	C₂³	—	8	8	no	—
507	C₂³	—	8	4	no	—
508	C₂³	—	8	8	no	—
509	C₂³	—	8	4	no	—
510	C₂³	C₄×D₄×C₆₇	1	2	yes	—
511	C₂³	#120×C₆₇	1	2	yes	—
512	C₂³	#121×C₆₇	1	2	yes	—
513	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
514	Q₈	—	2	2	no	—
515	Q₈	—	2	2	no	—
516	Q₈	—	2	2	no	—
517	Q₈	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
518	D₄	—	2	16	no	—
519	D₄	—	2	4	no	—
520	D₄	—	2	2	no	—
521	D₄	—	2	4	no	—
522	D₄	—	2	2	no	—
523	D₄	—	2	2	no	—
524	D₄	—	4	2	no	—
525	D₄	—	4	2	no	—
526	D₄	—	4	8	no	—
527	D₄	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
528	C₂×C₄	—	2	2	no	—
529	C₂×C₄	—	2	2	no	—
530	C₂×C₄	—	2	2	no	—
531	C₂×C₄	—	2	16	no	—
532	C₂×C₄	—	2	2	no	—
533	C₂×C₄	—	2	2	no	—
534	C₂×C₄	—	2	4	no	—
535	C₂×C₄	—	2	4	no	—
536	C₂×C₄	—	2	4	no	—
537	C₂×C₄	—	4	4	no	—
538	C₂×C₄	—	4	2	no	—
539	C₂×C₄	—	4	2	no	—
540	C₂×C₄	—	4	2	no	—
541	C₂×C₄	—	8	4	no	—
542	C₂×C₄	—	8	8	no	—
543	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
544	C₂×C₄	C₄×D₄×C₆₇	1	4	yes	—
545	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
546	C₂×C₄	#120×C₆₇	1	2	yes	—
547	C₂×C₄	#121×C₆₇	1	4	yes	—
548	C₂×C₄	#121×C₆₇	1	2	yes	—
549	C₂×C₄	#123×C₆₇	1	4	yes	—
550	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
551	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
552	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
553	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
554	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
555	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
556	C₈	—	2	4	no	—
557	C₈	—	4	2	no	—
558	C₂²	—	2	2	no	—
559	C₂²	—	2	4	no	—
560	C₂²	—	2	4	no	—
561	C₂²	—	2	2	no	—
562	C₂²	—	2	4	no	—
563	C₂²	—	4	2	no	—
564	C₂²	—	4	4	no	—
565	C₂²	—	4	4	no	—
566	C₂²	—	4	2	no	—
567	C₂²	—	4	4	no	—
568	C₂²	—	4	8	no	—
569	C₂²	—	8	16	no	—
570	C₂²	—	8	2	no	—
571	C₂²	—	8	4	no	—
572	C₂²	—	8	2	no	—
573	C₂²	#386×C₆₇	1	4	yes	—
574	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
575	C₂²	C₂×#120×C₆₇	1	1	yes	characteristic
576	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
577	C₂²	#576×C₆₇	1	2	yes	—
578	C₂²	#577×C₆₇	1	2	yes	—
579	C₄	—	2	4	no	—
580	C₄	—	4	2	no	—
581	C₄	—	4	2	no	—
582	C₄	—	4	4	no	—
583	C₄	—	8	2	no	—
584	C₄	#386×C₆₇	1	4	yes	—
585	C₄	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
586	C₄	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
587	C₄	C₂×#121×C₆₇	1	1	yes	characteristic
588	C₄	C₂×#123×C₆₇	1	1	yes	characteristic
589	C₂	—	2	4	no	—
590	C₂	—	4	4	no	—
591	C₂	—	8	4	no	—
592	C₂	C₂×#386×C₆₇	1	1	yes	characteristic
593	C₂	C₆₇×#2946	1	2	yes	—
594	C₂	C₆₇×#3008	1	1	yes	characteristic
595	C₂	C₆₇×#3010	1	1	yes	characteristic
596	C₂	C₆₇×#3045	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	C₂²×AGL(1,ℤ/8)×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×#520×C₆₇	C₂	semidirect
3. C₂×#416×C₆₇	C₂	semidirect
4. C₂²×#102×C₆₇	C₂	semidirect
5. C₆₇×#2309	C₂	semidirect
6. C₆₇×#1880	C₂	semidirect
7. C₆₇×#1879	C₂	semidirect
8. C₆₇×#1736	C₂	semidirect
9. C₆₇×#1729	C₂	semidirect
14. C₂×AGL(1,ℤ/8)×C₆₇	C₄	semidirect
21. C₂×C₄×Q₈×C₆₇	C₂²	semidirect
23. C₂×C₄×D₄×C₆₇	C₂²	semidirect
30. C₂×#103×C₆₇	C₂²	semidirect
33. C₂×#102×C₆₇	C₂²	semidirect
38. #340×C₆₇	C₂²	semidirect
39. #336×C₆₇	C₂²	semidirect
55. C₂²×D₄×C₆₇	C₂×C₄	semidirect
65. C₂×SD₁₆×C₆₇	C₂×C₄	semidirect
68. C₂×D₈×C₆₇	C₂×C₄	semidirect
71. C₂×M₁₆×C₆₇	C₂×C₄	semidirect
118. #41×C₆₇	C₂²⋊C₄	semidirect
290. #20932	C₆₇	direct (N × H)
300. C₂×#520	C₂×C₆₇	semidirect
301. C₂×#416	C₂×C₆₇	semidirect
302. C₂²×#102	C₂×C₆₇	semidirect
303. #2309	C₂×C₆₇	semidirect
304. #1880	C₂×C₆₇	semidirect
305. #1879	C₂×C₆₇	semidirect
306. #1736	C₂×C₆₇	semidirect
307. #1729	C₂×C₆₇	semidirect
308. C₆₇	#20932	direct (N × H)
313. C₂×AGL(1,ℤ/8)	C₄×C₆₇	semidirect
320. C₂×C₄×Q₈	C₂²×C₆₇	semidirect
322. C₂×C₄×D₄	C₂²×C₆₇	semidirect
329. C₂×#103	C₂²×C₆₇	semidirect
332. C₂×#102	C₂²×C₆₇	semidirect
337. #340	C₂²×C₆₇	semidirect
338. #336	C₂²×C₆₇	semidirect
354. C₂²×D₄	C₂×C₄×C₆₇	semidirect
364. C₂×SD₁₆	C₂×C₄×C₆₇	semidirect
367. C₂×D₈	C₂×C₄×C₆₇	semidirect
370. C₂×M₁₆	C₂×C₄×C₆₇	semidirect
417. #41	C₂²⋊C₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×AGL(1,ℤ/8)×C₆₇	2
C₂×#520×C₆₇	2
C₂×#416×C₆₇	2
C₂²×#102×C₆₇	2
C₆₇×#2309	2
C₆₇×#1880	2
C₆₇×#1879	2
C₆₇×#1736	2
C₆₇×#1729	2
#20932	67

Subgroups — C67×#20932 #30700183

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#20932 #30700183