Subgroups — #22569371

Order 8576.

364 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16863930	C₂	1	1	yes	maximal, characteristic
2	#16863526	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈×Dic₁₃₄	C₂	1	1	yes	maximal, characteristic, complemented
4	#16863475	C₂	1	1	yes	maximal, characteristic, complemented
5	#16863464	C₂	1	2	yes	maximal, complemented
6	C₈×D₂₆₈	C₂	1	1	yes	maximal, characteristic, complemented
7	(C₂²⋊D₆₇)⋊C₈	C₂	1	2	yes	maximal, complemented
8	#16863456	C₂	1	2	yes	maximal, complemented
9	#16863421	C₂	1	2	yes	maximal, complemented
10	C₂×C₄⋊(C₆₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
11	#431×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#11761611	C₂²	1	2	yes	—
13	C₂²×C₆₇⋊C₈	C₂²	1	2	yes	—
14	C₂×C₈×D₆₇	C₂²	1	2	yes	—
15	#14524037	C₄	1	2	yes	—
16	#14524035	C₄	1	1	yes	characteristic
17	#14524031	C₄	1	1	yes	characteristic
18	#14524026	C₄	1	2	yes	—
19	#14524022	C₄	1	1	yes	characteristic
20	#14524017	C₄	1	1	yes	characteristic
21	C₄×Dic₁₃₄	C₂²	1	1	yes	characteristic
22	C₄×D₂₆₈	C₂²	1	1	yes	characteristic
23	C₄×C₂²⋊D₆₇	C₂²	1	2	yes	—
24	#14524000	C₂²	1	2	yes	—
25	C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
26	C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
27	C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
28	Dic₆₇⋊C₈	C₂²	1	2	yes	—
29	Dic₆₇⋊C₈	C₂²	1	2	yes	—
30	(C₂×D₆₇)⋊C₈	C₂²	1	2	yes	—
31	(C₂×D₆₇)⋊C₈	C₂²	1	2	yes	—
32	#14523953	C₂²	1	2	yes	—
33	C₈×Dic₆₇	C₂²	1	2	yes	—
34	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
35	C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
36	C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
37	C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
38	C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
39	C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
40	C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
41	C₂×Dic₁₃₄	C₂×C₄	1	1	yes	characteristic
42	C₂×D₂₆₈	C₂×C₄	1	1	yes	characteristic
43	C₂×C₂²⋊D₆₇	C₂×C₄	1	2	yes	—
44	C₂×C₄×D₆₇	C₂³	1	2	yes	—
45	#11761611	—	4	2	no	—
46	C₂×C₆₇⋊C₈	—	4	2	no	—
47	C₂×C₆₇⋊C₈	C₂³	1	2	yes	—
48	C₂×C₆₇⋊C₈	C₂³	1	2	yes	—
49	C₈×D₆₇	—	4	2	no	—
50	C₄⋊Dic₆₇	C₂×C₄	1	1	yes	characteristic
51	C₄⋊Dic₆₇	C₂×C₄	1	1	yes	characteristic
52	Dic₆₇⋊C₄	C₂×C₄	1	2	yes	—
53	Dic₆₇⋊C₄	C₂×C₄	1	2	yes	—
54	Dic₆₇⋊C₄	C₂×C₄	1	2	yes	—
55	Dic₆₇⋊C₄	C₂×C₄	1	2	yes	—
56	C₂²⋊Dic₆₇	C₂×C₄	1	2	yes	—
57	(C₂×D₆₇)⋊C₄	C₂×C₄	1	2	yes	—
58	(C₂×D₆₇)⋊C₄	C₂×C₄	1	2	yes	—
59	(C₂×D₆₇)⋊C₄	C₂×C₄	1	2	yes	—
60	(C₂×D₆₇)⋊C₄	C₂×C₄	1	2	yes	—
61	C₄×Dic₆₇	C₂³	1	2	yes	—
62	C₂²×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
65	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
66	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
67	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
68	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
69	C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
70	C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
71	C₄²×C₆₇	C₂³	1	1	yes	characteristic
72	C₄²×C₆₇	C₂³	1	1	yes	characteristic
73	C₂³×C₆₇	C₂²×C₄	1	1	yes	characteristic
74	C₂²×D₆₇	C₂²×C₄	1	2	yes	—
75	C₂×C₄×C₆₇	—	2	2	no	—
76	C₂×C₄×C₆₇	—	2	2	no	—
77	C₂×C₄×C₆₇	—	2	1	no	—
78	C₂×C₄×C₆₇	—	2	1	no	—
79	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
85	C₂²⋊D₆₇	—	2	4	no	—
86	C₂×Dic₆₇	C₂²×C₄	1	2	yes	—
87	C₂×Dic₆₇	C₂²×C₄	1	2	yes	—
88	C₂×Dic₆₇	C₂²×C₄	1	2	yes	—
89	D₂₆₈	—	2	2	no	—
90	C₄×D₆₇	—	2	2	no	—
91	C₄×D₆₇	—	2	2	no	—
92	Dic₁₃₄	—	2	2	no	—
93	C₈×C₆₇	—	2	1	no	—
94	C₈×C₆₇	—	2	1	no	—
95	C₈×C₆₇	—	2	1	no	—
96	C₈×C₆₇	—	2	1	no	—
97	C₆₇⋊C₈	—	2	2	no	—
98	C₆₇⋊C₈	—	2	2	no	—
99	C₂²×C₆₇	—	2	1	no	—
100	C₂²×C₆₇	—	2	1	no	—
101	C₂²×C₆₇	#131	1	2	yes	—
102	C₂²×C₆₇	C₂³×C₄	1	1	yes	characteristic
103	C₂×D₆₇	—	2	2	no	—
104	C₂×D₆₇	—	2	2	no	—
105	C₂×D₆₇	—	2	2	no	—
106	C₄×C₆₇	—	2	1	no	—
107	C₄×C₆₇	—	2	1	no	—
108	C₄×C₆₇	—	2	1	no	—
109	C₄×C₆₇	—	2	1	no	—
110	C₄×C₆₇	#131	1	2	yes	—
111	C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
112	C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
113	Dic₆₇	—	2	2	no	—
114	Dic₆₇	—	2	2	no	—
115	Dic₆₇	—	2	2	no	—
116	C₂×C₆₇	—	2	2	no	—
117	C₂×C₆₇	#520	1	1	yes	characteristic
118	C₂×C₆₇	C₂×#131	1	1	yes	characteristic
119	C₂×C₆₇	C₂×#131	1	1	yes	characteristic
120	D₆₇	—	4	2	no	—
121	#2994	—	67	1	no	maximal, Sylow
122	C₆₇	#2994	1	1	yes	characteristic, complemented, Sylow
123	#529	—	67	1	no	—
124	#446	—	67	1	no	—
125	C₈×Q₈	—	67	1	no	—
126	#436	—	67	1	no	—
127	#435	—	67	2	no	—
128	C₈×D₄	—	67	2	no	—
129	C₈×D₄	—	67	1	no	—
130	#433	—	67	2	no	—
131	#431	D₆₇	1	1	yes	characteristic, complemented
132	#424	—	67	2	no	—
133	C₂×C₄⋊C₈	—	67	1	no	—
134	C₂×M₁₆	—	67	2	no	—
135	C₂²×C₈	—	67	2	no	—
136	C₂²×C₈	—	67	2	no	—
137	#126	—	67	2	no	—
138	#125	—	67	1	no	—
139	#124	—	67	1	no	—
140	#123	—	67	2	no	—
141	C₂²⋊Q₈	—	67	1	no	—
142	#121	—	67	1	no	—
143	C₄×Q₈	—	67	1	no	—
144	C₄×D₄	—	67	2	no	—
145	C₄×D₄	—	67	1	no	—
146	#117	—	67	2	no	—
147	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
148	C₄⋊C₈	—	67	2	no	—
149	C₄⋊C₈	—	67	2	no	—
150	C₄⋊C₈	—	67	2	no	—
151	C₄⋊C₈	—	67	1	no	—
152	C₄⋊C₈	—	67	1	no	—
153	C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
154	C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
155	C₂²⋊C₈	—	67	2	no	—
156	C₂²⋊C₈	—	67	2	no	—
157	C₂²⋊C₈	C₂×D₆₇	1	1	yes	characteristic
158	C₂²⋊C₈	C₂×D₆₇	1	1	yes	characteristic
159	#97	—	67	2	no	—
160	C₄×C₈	—	67	2	no	—
161	C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
162	C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
163	C₂×Q₈	—	67	1	no	—
164	C₂×D₄	—	67	2	no	—
165	C₂×D₄	—	67	1	no	—
166	C₂²×C₄	—	67	2	no	—
167	C₂²×C₄	C₄×D₆₇	1	1	yes	characteristic
168	C₂²×C₄	C₄×D₆₇	1	1	yes	characteristic
169	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
170	M₁₆	—	268	2	no	—
171	C₂×C₈	—	268	2	no	—
172	C₂×C₈	—	268	2	no	—
173	C₂×C₈	—	67	2	no	—
174	C₂×C₈	—	67	2	no	—
175	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
176	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
177	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
178	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
179	C₄⋊C₄	—	67	2	no	—
180	C₄⋊C₄	—	67	2	no	—
181	C₄⋊C₄	—	67	1	no	—
182	C₄⋊C₄	—	67	2	no	—
183	C₄⋊C₄	—	67	1	no	—
184	C₄⋊C₄	—	67	2	no	—
185	C₂²⋊C₄	—	67	2	no	—
186	C₂²⋊C₄	—	67	2	no	—
187	C₂²⋊C₄	—	67	2	no	—
188	C₂²⋊C₄	—	67	2	no	—
189	C₂²⋊C₄	—	67	2	no	—
190	C₄²	—	67	2	no	—
191	C₄²	C₄×D₆₇	1	1	yes	characteristic
192	C₄²	C₄×D₆₇	1	1	yes	characteristic
193	C₄²	C₂²×D₆₇	1	1	yes	characteristic
194	C₄²	C₂²×D₆₇	1	1	yes	characteristic
195	C₂³	—	67	2	no	—
196	C₂³	C₂×C₄×D₆₇	1	1	yes	characteristic
197	Q₈	—	134	2	no	—
198	D₄	—	134	4	no	—
199	D₄	—	134	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	67	2	no	—
205	C₂×C₄	—	67	2	no	—
206	C₂×C₄	—	67	2	no	—
207	C₂×C₄	—	134	2	no	—
208	C₂×C₄	—	134	2	no	—
209	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
210	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
211	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
212	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
213	C₂×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
214	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
215	C₈	—	2	1	no	—
216	C₈	—	2	1	no	—
217	C₈	—	2	1	no	—
218	C₈	—	2	1	no	—
219	C₈	—	134	2	no	—
220	C₈	—	134	2	no	—
221	C₂²	—	2	1	no	—
222	C₂²	—	2	1	no	—
223	C₂²	—	134	2	no	—
224	C₂²	—	134	2	no	—
225	C₂²	—	134	2	no	—
226	C₂²	#14524049	1	2	yes	—
227	C₂²	C₂²×C₄×D₆₇	1	1	yes	characteristic
228	C₄	—	2	1	no	—
229	C₄	—	2	1	no	—
230	C₄	—	2	1	no	—
231	C₄	—	2	1	no	—
232	C₄	—	134	2	no	—
233	C₄	—	134	2	no	—
234	C₄	—	134	2	no	—
235	C₄	#14524051	1	2	yes	—
236	C₄	#14524089	1	1	yes	characteristic
237	C₄	#14524090	1	1	yes	characteristic
238	C₂	—	2	2	no	—
239	C₂	—	268	2	no	—
240	C₂	#16863876	1	1	yes	characteristic
241	C₂	C₂×#14524049	1	1	yes	characteristic
242	C₂	C₂×#14524051	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₆₇
commutator, socle	C₂²×C₆₇	C₂³×C₄
Fitting	#431×C₆₇	C₂
hypercenter	#431	D₆₇
hypocenter	C₆₇	#2994
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ #431

Complements

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

N	H ≅ G/N	type
2. #16863526	C₂	semidirect
3. C₈×Dic₁₃₄	C₂	semidirect
4. #16863475	C₂	semidirect
5. #16863464	C₂	semidirect
6. C₈×D₂₆₈	C₂	semidirect
7. (C₂²⋊D₆₇)⋊C₈	C₂	semidirect
8. #16863456	C₂	semidirect
9. #16863421	C₂	semidirect
10. C₂×C₄⋊(C₆₇⋊C₈)	C₂	semidirect
11. #431×C₆₇	C₂	semidirect
122. C₆₇	#2994	semidirect
131. #431	D₆₇	semidirect