Subgroups — C₆₇×#22663 #30701914

Order 17152.

1022 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#436×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#428×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₆₇×#2354	C₂	1	1	yes	maximal, characteristic
4	D₄×C₄⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1948	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#1947	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1854	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1853	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1844	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#1809	C₂	1	2	yes	maximal, complemented
11	C₄×#107×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
13	C₂×C₄×D₄×C₆₇	—	2	1	no	—
14	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×#117×C₆₇	C₂²	1	2	yes	—
16	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
17	#436×C₆₇	C₄	1	8	yes	complemented
18	#428×C₆₇	—	2	8	no	—
19	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×#107×C₆₇	C₂²	1	2	yes	—
21	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×#106×C₆₇	C₂²	1	2	yes	—
23	C₂×#106×C₆₇	C₂²	1	2	yes	—
24	C₂×#106×C₆₇	C₂²	1	2	yes	—
25	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
28	#390×C₆₇	—	2	1	no	—
29	#389×C₆₇	—	2	1	no	—
30	#388×C₆₇	—	2	2	no	—
31	#385×C₆₇	—	2	2	no	—
32	#384×C₆₇	C₂²	1	1	yes	characteristic
33	#384×C₆₇	C₂²	1	1	yes	characteristic
34	#380×C₆₇	C₂²	1	2	yes	—
35	#380×C₆₇	C₂²	1	2	yes	—
36	C₄×C₄⋊C₄×C₆₇	—	2	1	no	—
37	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
38	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
39	#340×C₆₇	C₂²	1	1	yes	characteristic
40	#340×C₆₇	C₂²	1	1	yes	characteristic
41	#340×C₆₇	C₂²	1	2	yes	—
42	#340×C₆₇	C₂²	1	2	yes	—
43	C₂²×D₄×C₆₇	D₄	1	1	yes	characteristic
44	C₂³×C₄×C₆₇	—	2	2	no	—
45	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
46	C₂×M₁₆×C₆₇	—	2	4	no	—
47	C₂×M₁₆×C₆₇	—	2	4	no	—
48	C₂×M₁₆×C₆₇	C₂×C₄	1	8	yes	complemented
49	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
50	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
51	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
52	C₄×D₄×C₆₇	—	4	4	no	—
53	C₄×D₄×C₆₇	C₂×C₄	1	4	yes	—
54	C₄×D₄×C₆₇	Q₈	1	4	yes	—
55	#117×C₆₇	—	2	4	no	—
56	#117×C₆₇	—	4	2	no	—
57	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
58	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
59	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
60	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
61	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
62	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
63	C₂×C₄⋊C₄×C₆₇	—	4	2	no	—
64	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
65	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
66	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
67	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
68	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
69	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
70	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
71	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
72	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
73	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
74	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
75	C₂×C₄²×C₆₇	—	2	2	no	—
76	C₂×C₄²×C₆₇	—	2	1	no	—
77	C₂×C₄²×C₆₇	—	2	1	no	—
78	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
79	#107×C₆₇	—	2	2	no	—
80	#107×C₆₇	—	2	4	no	—
81	#107×C₆₇	D₄	1	4	yes	complemented
82	#106×C₆₇	—	2	4	no	—
83	#106×C₆₇	—	2	4	no	—
84	#106×C₆₇	—	4	2	no	—
85	C₄⋊C₈×C₆₇	C₂×C₄	1	4	yes	complemented
86	C₂²⋊C₈×C₆₇	C₂×C₄	1	8	yes	complemented
87	C₄×C₈×C₆₇	C₂×C₄	1	4	yes	complemented
88	#95×C₆₇	—	2	2	no	—
89	#95×C₆₇	—	2	1	no	—
90	#95×C₆₇	—	2	1	no	—
91	#95×C₆₇	—	2	2	no	—
92	C₂⁴×C₆₇	C₂×D₄	1	2	yes	—
93	C₂×D₄×C₆₇	—	2	2	no	—
94	C₂×D₄×C₆₇	—	2	2	no	—
95	C₂×D₄×C₆₇	C₄⋊C₄	1	4	yes	—
96	C₂²×C₄×C₆₇	—	2	2	no	—
97	C₂²×C₄×C₆₇	—	2	2	no	—
98	C₂²×C₄×C₆₇	—	2	2	no	—
99	C₂²×C₄×C₆₇	—	2	2	no	—
100	C₂²×C₄×C₆₇	—	2	1	no	—
101	C₂²×C₄×C₆₇	—	2	1	no	—
102	C₂²×C₄×C₆₇	—	2	2	no	—
103	C₂²×C₄×C₆₇	—	2	1	no	—
104	C₂²×C₄×C₆₇	—	2	1	no	—
105	C₂²×C₄×C₆₇	—	2	2	no	—
106	C₂²×C₄×C₆₇	—	4	4	no	—
107	C₂²×C₄×C₆₇	—	4	4	no	—
108	C₂²×C₄×C₆₇	—	8	2	no	—
109	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
110	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
111	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
112	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
113	C₂²×C₄×C₆₇	C₂×Q₈	1	4	yes	—
114	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
115	M₁₆×C₆₇	—	2	16	no	—
116	C₂×C₈×C₆₇	—	2	2	no	—
117	C₂×C₈×C₆₇	—	2	2	no	—
118	C₂×C₈×C₆₇	—	2	1	no	—
119	C₂×C₈×C₆₇	—	2	1	no	—
120	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
121	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
122	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
123	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
124	C₂×C₈×C₆₇	#41	1	2	yes	—
125	C₄⋊C₄×C₆₇	—	2	1	no	—
126	C₄⋊C₄×C₆₇	—	2	1	no	—
127	C₄⋊C₄×C₆₇	—	2	2	no	—
128	C₄⋊C₄×C₆₇	—	2	2	no	—
129	C₄⋊C₄×C₆₇	—	2	1	no	—
130	C₄⋊C₄×C₆₇	—	2	1	no	—
131	C₄⋊C₄×C₆₇	—	2	2	no	—
132	C₄⋊C₄×C₆₇	—	2	2	no	—
133	C₄⋊C₄×C₆₇	—	2	1	no	—
134	C₄⋊C₄×C₆₇	—	2	1	no	—
135	C₄⋊C₄×C₆₇	—	4	4	no	—
136	C₄⋊C₄×C₆₇	—	4	2	no	—
137	C₄⋊C₄×C₆₇	—	4	2	no	—
138	C₄⋊C₄×C₆₇	—	8	1	no	—
139	C₄⋊C₄×C₆₇	—	8	1	no	—
140	C₄⋊C₄×C₆₇	C₄⋊C₄	1	4	yes	—
141	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
142	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
143	C₂²⋊C₄×C₆₇	—	4	4	no	—
144	C₂²⋊C₄×C₆₇	—	4	4	no	—
145	C₂²⋊C₄×C₆₇	—	8	2	no	—
146	C₂²⋊C₄×C₆₇	C₄⋊C₄	1	8	yes	—
147	C₄²×C₆₇	—	4	4	no	—
148	C₄²×C₆₇	—	4	2	no	—
149	C₄²×C₆₇	—	4	2	no	—
150	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
151	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
152	C₂³×C₆₇	—	2	2	no	—
153	C₂³×C₆₇	—	2	2	no	—
154	C₂³×C₆₇	—	2	2	no	—
155	C₂³×C₆₇	—	2	2	no	—
156	C₂³×C₆₇	—	2	2	no	—
157	C₂³×C₆₇	—	2	2	no	—
158	C₂³×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
159	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
160	D₄×C₆₇	—	2	8	no	—
161	C₂×C₄×C₆₇	—	2	2	no	—
162	C₂×C₄×C₆₇	—	2	2	no	—
163	C₂×C₄×C₆₇	—	2	1	no	—
164	C₂×C₄×C₆₇	—	2	1	no	—
165	C₂×C₄×C₆₇	—	2	4	no	—
166	C₂×C₄×C₆₇	—	2	4	no	—
167	C₂×C₄×C₆₇	—	2	4	no	—
168	C₂×C₄×C₆₇	—	2	4	no	—
169	C₂×C₄×C₆₇	—	2	2	no	—
170	C₂×C₄×C₆₇	—	2	2	no	—
171	C₂×C₄×C₆₇	—	4	1	no	—
172	C₂×C₄×C₆₇	—	4	1	no	—
173	C₂×C₄×C₆₇	—	4	1	no	—
174	C₂×C₄×C₆₇	—	4	1	no	—
175	C₂×C₄×C₆₇	—	4	2	no	—
176	C₂×C₄×C₆₇	—	4	2	no	—
177	C₂×C₄×C₆₇	—	4	1	no	—
178	C₂×C₄×C₆₇	—	4	1	no	—
179	C₂×C₄×C₆₇	—	4	1	no	—
180	C₂×C₄×C₆₇	—	4	1	no	—
181	C₂×C₄×C₆₇	—	4	2	no	—
182	C₂×C₄×C₆₇	—	4	2	no	—
183	C₂×C₄×C₆₇	—	4	1	no	—
184	C₂×C₄×C₆₇	—	4	2	no	—
185	C₂×C₄×C₆₇	—	4	2	no	—
186	C₂×C₄×C₆₇	—	4	1	no	—
187	C₂×C₄×C₆₇	—	8	8	no	—
188	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
189	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
190	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
191	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
192	C₂×C₄×C₆₇	#137	1	2	yes	—
193	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
194	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
195	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
196	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
197	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
198	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
199	C₈×C₆₇	—	2	4	no	—
200	C₈×C₆₇	—	2	2	no	—
201	C₈×C₆₇	C₄×D₄	1	4	yes	complemented
202	C₂²×C₆₇	—	2	4	no	—
203	C₂²×C₆₇	—	2	4	no	—
204	C₂²×C₆₇	—	2	4	no	—
205	C₂²×C₆₇	—	4	2	no	—
206	C₂²×C₆₇	—	4	2	no	—
207	C₂²×C₆₇	—	4	2	no	—
208	C₂²×C₆₇	—	4	2	no	—
209	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
210	C₂²×C₆₇	D₄²	1	1	yes	characteristic
211	C₂²×C₆₇	#547	1	1	yes	characteristic
212	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
213	C₂²×C₆₇	C₂×#137	1	1	yes	characteristic
214	C₂²×C₆₇	#577	1	1	yes	characteristic
215	C₂²×C₆₇	#577	1	1	yes	characteristic
216	C₄×C₆₇	—	2	4	no	—
217	C₄×C₆₇	—	2	2	no	—
218	C₄×C₆₇	—	4	2	no	—
219	C₄×C₆₇	—	4	2	no	—
220	C₄×C₆₇	—	8	1	no	—
221	C₄×C₆₇	—	8	1	no	—
222	C₄×C₆₇	—	8	2	no	—
223	C₄×C₆₇	—	8	2	no	—
224	C₄×C₆₇	#428	1	4	yes	complemented
225	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
226	C₄×C₆₇	#520	1	1	yes	characteristic
227	C₄×C₆₇	D₄×Q₈	1	1	yes	characteristic
228	C₄×C₆₇	#554	1	1	yes	characteristic
229	#22663	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
230	C₂×C₆₇	—	4	4	no	—
231	C₂×C₆₇	D₄×C₄⋊C₄	1	1	yes	characteristic
232	C₂×C₆₇	C₂×#428	1	1	yes	characteristic
233	C₂×C₆₇	#2912	1	1	yes	characteristic
234	C₂×C₆₇	#3272	1	1	yes	characteristic
235	C₂×C₆₇	#3278	1	1	yes	characteristic
236	C₂×C₆₇	#3354	1	1	yes	characteristic
237	C₂×C₆₇	#3356	1	1	yes	characteristic
238	C₂×#436	C₂×C₆₇	1	1	yes	characteristic
239	C₂×#428	C₂×C₆₇	1	2	yes	complemented
240	#2354	C₂×C₆₇	1	1	yes	characteristic
241	D₄×C₄⋊C₄	C₂×C₆₇	1	1	yes	characteristic
242	#1948	C₂×C₆₇	1	1	yes	characteristic, complemented
243	#1947	C₂×C₆₇	1	1	yes	characteristic, complemented
244	#1854	C₂×C₆₇	1	2	yes	complemented
245	#1853	C₂×C₆₇	1	2	yes	complemented
246	#1844	C₂×C₆₇	1	1	yes	characteristic, complemented
247	#1809	C₂×C₆₇	1	2	yes	complemented
248	C₄×#107	C₂×C₆₇	1	1	yes	characteristic, complemented
249	C₆₇	#22663	1	1	yes	characteristic, direct product factor, Sylow
250	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
251	C₂×C₄×D₄	—	2	1	no	—
252	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
253	C₂×#117	C₂²×C₆₇	1	2	yes	—
254	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
255	#436	C₄×C₆₇	1	8	yes	complemented
256	#428	—	2	8	no	—
257	C₂×#107	C₂²×C₆₇	1	1	yes	characteristic
258	C₂×#107	C₂²×C₆₇	1	2	yes	—
259	C₂×#107	C₂²×C₆₇	1	1	yes	characteristic
260	C₂×#106	C₂²×C₆₇	1	2	yes	—
261	C₂×#106	C₂²×C₆₇	1	2	yes	—
262	C₂×#106	C₂²×C₆₇	1	2	yes	—
263	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
264	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
265	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
266	#390	—	2	1	no	—
267	#389	—	2	1	no	—
268	#388	—	2	2	no	—
269	#385	—	2	2	no	—
270	#384	C₂²×C₆₇	1	1	yes	characteristic
271	#384	C₂²×C₆₇	1	1	yes	characteristic
272	#380	C₂²×C₆₇	1	2	yes	—
273	#380	C₂²×C₆₇	1	2	yes	—
274	C₄×C₄⋊C₄	—	2	1	no	—
275	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
276	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
277	#340	C₂²×C₆₇	1	2	yes	—
278	#340	C₂²×C₆₇	1	2	yes	—
279	#340	C₂²×C₆₇	1	1	yes	characteristic
280	#340	C₂²×C₆₇	1	1	yes	characteristic
281	C₂²×D₄	D₄×C₆₇	1	1	yes	characteristic
282	C₂³×C₄	—	2	2	no	—
283	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
284	C₂×M₁₆	—	2	4	no	—
285	C₂×M₁₆	—	2	4	no	—
286	C₂×M₁₆	C₂×C₄×C₆₇	1	8	yes	complemented
287	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
288	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
289	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
290	C₄×D₄	—	4	4	no	—
291	C₄×D₄	C₂×C₄×C₆₇	1	4	yes	—
292	C₄×D₄	Q₈×C₆₇	1	4	yes	—
293	#117	—	2	4	no	—
294	#117	—	4	2	no	—
295	C₂×C₄⋊C₄	—	2	4	no	—
296	C₂×C₄⋊C₄	—	2	2	no	—
297	C₂×C₄⋊C₄	—	2	1	no	—
298	C₂×C₄⋊C₄	—	2	1	no	—
299	C₂×C₄⋊C₄	—	2	1	no	—
300	C₂×C₄⋊C₄	—	2	1	no	—
301	C₂×C₄⋊C₄	—	4	2	no	—
302	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
303	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
304	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
305	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
306	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
307	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
308	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
309	C₂×C₂²⋊C₄	—	2	2	no	—
310	C₂×C₂²⋊C₄	—	2	2	no	—
311	C₂×C₂²⋊C₄	—	2	2	no	—
312	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
313	C₂×C₄²	—	2	2	no	—
314	C₂×C₄²	—	2	1	no	—
315	C₂×C₄²	—	2	1	no	—
316	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
317	#107	—	2	2	no	—
318	#107	—	2	4	no	—
319	#107	D₄×C₆₇	1	4	yes	complemented
320	#106	—	2	4	no	—
321	#106	—	2	4	no	—
322	#106	—	4	2	no	—
323	C₄⋊C₈	C₂×C₄×C₆₇	1	4	yes	complemented
324	C₂²⋊C₈	C₂×C₄×C₆₇	1	8	yes	complemented
325	C₄×C₈	C₂×C₄×C₆₇	1	4	yes	complemented
326	#95	—	2	2	no	—
327	#95	—	2	1	no	—
328	#95	—	2	1	no	—
329	#95	—	2	2	no	—
330	C₂⁴	C₂×D₄×C₆₇	1	2	yes	—
331	C₂×D₄	—	2	2	no	—
332	C₂×D₄	—	2	2	no	—
333	C₂×D₄	C₄⋊C₄×C₆₇	1	4	yes	—
334	C₂²×C₄	—	2	2	no	—
335	C₂²×C₄	—	2	2	no	—
336	C₂²×C₄	—	2	2	no	—
337	C₂²×C₄	—	2	2	no	—
338	C₂²×C₄	—	2	1	no	—
339	C₂²×C₄	—	2	1	no	—
340	C₂²×C₄	—	2	1	no	—
341	C₂²×C₄	—	2	1	no	—
342	C₂²×C₄	—	2	2	no	—
343	C₂²×C₄	—	2	2	no	—
344	C₂²×C₄	—	4	4	no	—
345	C₂²×C₄	—	4	4	no	—
346	C₂²×C₄	—	8	2	no	—
347	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
348	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
349	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
350	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
351	C₂²×C₄	C₂×Q₈×C₆₇	1	4	yes	—
352	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
353	M₁₆	—	2	16	no	—
354	C₂×C₈	—	2	2	no	—
355	C₂×C₈	—	2	2	no	—
356	C₂×C₈	—	2	1	no	—
357	C₂×C₈	—	2	1	no	—
358	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
359	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
360	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
361	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
362	C₂×C₈	#41×C₆₇	1	2	yes	—
363	C₄⋊C₄	—	2	1	no	—
364	C₄⋊C₄	—	2	1	no	—
365	C₄⋊C₄	—	2	2	no	—
366	C₄⋊C₄	—	2	2	no	—
367	C₄⋊C₄	—	2	1	no	—
368	C₄⋊C₄	—	2	1	no	—
369	C₄⋊C₄	—	2	2	no	—
370	C₄⋊C₄	—	2	2	no	—
371	C₄⋊C₄	—	2	1	no	—
372	C₄⋊C₄	—	2	1	no	—
373	C₄⋊C₄	—	4	4	no	—
374	C₄⋊C₄	—	4	2	no	—
375	C₄⋊C₄	—	4	2	no	—
376	C₄⋊C₄	—	8	1	no	—
377	C₄⋊C₄	—	8	1	no	—
378	C₄⋊C₄	C₄⋊C₄×C₆₇	1	4	yes	—
379	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
380	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
381	C₂²⋊C₄	—	4	4	no	—
382	C₂²⋊C₄	—	4	4	no	—
383	C₂²⋊C₄	—	8	2	no	—
384	C₂²⋊C₄	C₄⋊C₄×C₆₇	1	8	yes	—
385	C₄²	—	4	4	no	—
386	C₄²	—	4	2	no	—
387	C₄²	—	4	2	no	—
388	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
389	C₄²	C₂×Q₈×C₆₇	1	2	yes	—
390	C₂³	—	2	2	no	—
391	C₂³	—	2	2	no	—
392	C₂³	—	2	2	no	—
393	C₂³	—	2	2	no	—
394	C₂³	—	2	2	no	—
395	C₂³	—	2	2	no	—
396	C₂³	C₂×C₄⋊C₄×C₆₇	1	4	yes	—
397	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
398	D₄	—	2	8	no	—
399	C₂×C₄	—	2	2	no	—
400	C₂×C₄	—	2	2	no	—
401	C₂×C₄	—	2	1	no	—
402	C₂×C₄	—	2	1	no	—
403	C₂×C₄	—	2	4	no	—
404	C₂×C₄	—	2	4	no	—
405	C₂×C₄	—	2	4	no	—
406	C₂×C₄	—	2	4	no	—
407	C₂×C₄	—	2	2	no	—
408	C₂×C₄	—	2	2	no	—
409	C₂×C₄	—	4	1	no	—
410	C₂×C₄	—	4	1	no	—
411	C₂×C₄	—	4	1	no	—
412	C₂×C₄	—	4	1	no	—
413	C₂×C₄	—	4	2	no	—
414	C₂×C₄	—	4	2	no	—
415	C₂×C₄	—	4	1	no	—
416	C₂×C₄	—	4	1	no	—
417	C₂×C₄	—	4	1	no	—
418	C₂×C₄	—	4	1	no	—
419	C₂×C₄	—	4	2	no	—
420	C₂×C₄	—	4	2	no	—
421	C₂×C₄	—	4	1	no	—
422	C₂×C₄	—	4	2	no	—
423	C₂×C₄	—	4	2	no	—
424	C₂×C₄	—	4	1	no	—
425	C₂×C₄	—	8	8	no	—
426	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
427	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	4	yes	—
428	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
429	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
430	C₂×C₄	#137×C₆₇	1	2	yes	—
431	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
432	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
433	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
434	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
435	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
436	C₂×C₄	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
437	C₈	—	2	4	no	—
438	C₈	—	2	2	no	—
439	C₈	C₄×D₄×C₆₇	1	4	yes	complemented
440	C₂²	—	2	4	no	—
441	C₂²	—	2	4	no	—
442	C₂²	—	2	4	no	—
443	C₂²	—	4	2	no	—
444	C₂²	—	4	2	no	—
445	C₂²	—	4	2	no	—
446	C₂²	—	4	2	no	—
447	C₂²	C₂²×C₄⋊C₄×C₆₇	1	1	yes	characteristic
448	C₂²	D₄²×C₆₇	1	1	yes	characteristic
449	C₂²	#547×C₆₇	1	1	yes	characteristic
450	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
451	C₂²	C₂×#137×C₆₇	1	1	yes	characteristic
452	C₂²	#577×C₆₇	1	1	yes	characteristic
453	C₂²	#577×C₆₇	1	1	yes	characteristic
454	C₄	—	2	4	no	—
455	C₄	—	2	2	no	—
456	C₄	—	4	2	no	—
457	C₄	—	4	2	no	—
458	C₄	—	8	2	no	—
459	C₄	—	8	2	no	—
460	C₄	—	8	1	no	—
461	C₄	—	8	1	no	—
462	C₄	#428×C₆₇	1	4	yes	complemented
463	C₄	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
464	C₄	#520×C₆₇	1	1	yes	characteristic
465	C₄	D₄×Q₈×C₆₇	1	1	yes	characteristic
466	C₄	#554×C₆₇	1	1	yes	characteristic
467	C₂	—	4	4	no	—
468	C₂	D₄×C₄⋊C₄×C₆₇	1	1	yes	characteristic
469	C₂	C₂×#428×C₆₇	1	1	yes	characteristic
470	C₂	C₆₇×#2912	1	1	yes	characteristic
471	C₂	C₆₇×#3272	1	1	yes	characteristic
472	C₂	C₆₇×#3278	1	1	yes	characteristic
473	C₂	C₆₇×#3354	1	1	yes	characteristic
474	C₂	C₆₇×#3356	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₂×#428×C₆₇	C₂	semidirect
5. C₆₇×#1948	C₂	semidirect
6. C₆₇×#1947	C₂	semidirect
7. C₆₇×#1854	C₂	semidirect
8. C₆₇×#1853	C₂	semidirect
9. C₆₇×#1844	C₂	semidirect
10. C₆₇×#1809	C₂	semidirect
11. C₄×#107×C₆₇	C₂	semidirect
17. #436×C₆₇	C₄	semidirect
48. C₂×M₁₆×C₆₇	C₂×C₄	semidirect
81. #107×C₆₇	D₄	semidirect
85. C₄⋊C₈×C₆₇	C₂×C₄	semidirect
86. C₂²⋊C₈×C₆₇	C₂×C₄	semidirect
87. C₄×C₈×C₆₇	C₂×C₄	semidirect
201. C₈×C₆₇	C₄×D₄	semidirect
224. C₄×C₆₇	#428	semidirect
229. #22663	C₆₇	direct (N × H)
239. C₂×#428	C₂×C₆₇	semidirect
242. #1948	C₂×C₆₇	semidirect
243. #1947	C₂×C₆₇	semidirect
244. #1854	C₂×C₆₇	semidirect
245. #1853	C₂×C₆₇	semidirect
246. #1844	C₂×C₆₇	semidirect
247. #1809	C₂×C₆₇	semidirect
248. C₄×#107	C₂×C₆₇	semidirect
249. C₆₇	#22663	direct (N × H)
255. #436	C₄×C₆₇	semidirect
286. C₂×M₁₆	C₂×C₄×C₆₇	semidirect
319. #107	D₄×C₆₇	semidirect
323. C₄⋊C₈	C₂×C₄×C₆₇	semidirect
324. C₂²⋊C₈	C₂×C₄×C₆₇	semidirect
325. C₄×C₈	C₂×C₄×C₆₇	semidirect
439. C₈	C₄×D₄×C₆₇	semidirect
462. C₄	#428×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#436×C₆₇	2
C₂×#428×C₆₇	2
C₆₇×#2354	2
D₄×C₄⋊C₄×C₆₇	2
C₆₇×#1948	2
C₆₇×#1947	2
C₆₇×#1854	2
C₆₇×#1853	2
C₆₇×#1844	2
C₆₇×#1809	2
C₄×#107×C₆₇	2
#22663	67

Subgroups — C67×#22663 #30701914

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#22663 #30701914