Subgroups — #22572612

Order 8576.

366 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16864092	C₂	1	2	yes	maximal, complemented
2	#16863798	C₂	1	1	yes	maximal, characteristic, complemented
3	#16863721	C₂	1	2	yes	maximal, complemented
4	#16863715	C₂	1	2	yes	maximal, complemented
5	#16863697	C₂	1	2	yes	maximal, complemented
6	#16863632	C₂	1	1	yes	maximal, characteristic, complemented
7	#16863632	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×D₅₃₆	C₂	1	1	yes	maximal, characteristic, complemented
9	#16863434	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#14523977	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×D₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×D₅₃₆	C₂²	1	1	yes	characteristic
13	#14524038	—	2	2	no	—
14	#14524035	C₂²	1	2	yes	—
15	#14524025	—	2	2	no	—
16	#14524017	C₂²	1	2	yes	—
17	#14524017	C₂²	1	2	yes	—
18	#14524015	—	2	2	no	—
19	C₄×D₂₆₈	C₂²	1	2	yes	—
20	(C₂²⋊D₆₇)⋊C₄	—	2	2	no	—
21	#14524002	C₂²	1	2	yes	—
22	C₂×C₄⋊Dic₆₇	C₂²	1	2	yes	—
23	#14523977	C₂²	1	1	yes	characteristic
24	#14523977	C₂²	1	1	yes	characteristic
25	#14523977	C₂²	1	1	yes	characteristic
26	#14523977	C₂²	1	2	yes	—
27	#14523975	C₂²	1	2	yes	—
28	#14523975	C₂²	1	2	yes	—
29	#14523965	C₂²	1	2	yes	—
30	#14523965	C₂²	1	2	yes	—
31	#14523965	C₂²	1	2	yes	—
32	C₂²×C₈×C₆₇	C₂²	1	1	yes	characteristic
33	C₂²×C₈×C₆₇	C₂²	1	1	yes	characteristic
34	C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
35	C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
36	C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
37	C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
38	C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
39	C₂×D₂₆₈	C₂³	1	2	yes	—
40	C₂×C₂²⋊D₆₇	—	2	2	no	—
41	C₂×C₂²⋊D₆₇	—	2	2	no	—
42	C₂²×Dic₆₇	—	2	2	no	—
43	C₂×C₄×D₆₇	—	2	2	no	—
44	D₅₃₆	—	2	2	no	—
45	C₄⋊Dic₆₇	—	2	2	no	—
46	C₄⋊Dic₆₇	C₂³	1	2	yes	—
47	C₄⋊Dic₆₇	C₂³	1	2	yes	—
48	C₄⋊Dic₆₇	C₂³	1	2	yes	—
49	Dic₆₇⋊C₄	—	2	2	no	—
50	Dic₆₇⋊C₄	—	2	2	no	—
51	C₂²⋊Dic₆₇	—	2	2	no	—
52	(C₂×D₆₇)⋊C₄	—	2	2	no	—
53	(C₂×D₆₇)⋊C₄	—	2	2	no	—
54	(C₂×D₆₇)⋊C₄	—	2	2	no	—
55	C₄×Dic₆₇	—	2	2	no	—
56	C₂×D₄×C₆₇	D₄	1	1	yes	characteristic
57	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
58	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
59	C₂×C₈×C₆₇	—	2	2	no	—
60	C₂×C₈×C₆₇	—	2	2	no	—
61	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
62	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
63	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
64	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
65	C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
66	C₂²⋊C₄×C₆₇	D₄	1	1	yes	characteristic
67	C₂²⋊C₄×C₆₇	D₄	1	1	yes	characteristic
68	C₄²×C₆₇	C₂³	1	1	yes	characteristic
69	C₂³×C₆₇	C₂×D₄	1	1	yes	characteristic
70	C₂³×C₆₇	C₂×D₄	1	1	yes	characteristic
71	C₂²×D₆₇	—	2	2	no	—
72	D₄×C₆₇	—	2	2	no	—
73	C₂×C₄×C₆₇	—	2	1	no	—
74	C₂×C₄×C₆₇	—	2	1	no	—
75	C₂×C₄×C₆₇	—	2	1	no	—
76	C₂×C₄×C₆₇	—	2	1	no	—
77	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
78	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
79	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
80	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
81	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
82	C₂²⋊D₆₇	—	4	4	no	—
83	C₂²⋊D₆₇	—	8	2	no	—
84	C₂×Dic₆₇	—	2	2	no	—
85	C₂×Dic₆₇	—	2	2	no	—
86	C₂×Dic₆₇	—	2	2	no	—
87	C₂×Dic₆₇	—	8	2	no	—
88	D₂₆₈	—	2	2	no	—
89	D₂₆₈	—	2	2	no	—
90	C₄×D₆₇	—	8	2	no	—
91	C₈×C₆₇	—	2	1	no	—
92	C₈×C₆₇	—	2	1	no	—
93	C₈×C₆₇	—	2	1	no	—
94	C₈×C₆₇	#41	1	2	yes	—
95	C₂²×C₆₇	—	2	1	no	—
96	C₂²×C₆₇	—	2	1	no	—
97	C₂²×C₆₇	—	2	1	no	—
98	C₂²×C₆₇	—	2	1	no	—
99	C₂²×C₆₇	—	2	1	no	—
100	C₂²×C₆₇	#135	1	2	yes	—
101	C₂²×C₆₇	C₂²×D₄	1	1	yes	characteristic
102	C₂×D₆₇	—	4	2	no	—
103	C₂×D₆₇	—	4	2	no	—
104	C₂×D₆₇	—	4	2	no	—
105	C₄×C₆₇	—	2	1	no	—
106	C₄×C₆₇	—	2	1	no	—
107	C₄×C₆₇	—	2	1	no	—
108	C₄×C₆₇	—	2	1	no	—
109	C₄×C₆₇	C₂×#41	1	1	yes	characteristic
110	C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
111	Dic₆₇	—	4	2	no	—
112	Dic₆₇	—	4	2	no	—
113	Dic₆₇	—	4	2	no	—
114	C₂×C₆₇	—	2	2	no	—
115	C₂×C₆₇	—	4	1	no	—
116	C₂×C₆₇	#547	1	1	yes	characteristic
117	C₂×C₆₇	C₂×#135	1	1	yes	characteristic
118	C₂×C₆₇	#576	1	1	yes	characteristic
119	D₆₇	—	8	2	no	—
120	#3339	—	67	1	no	maximal, Sylow
121	C₆₇	#3339	1	1	yes	characteristic, complemented, Sylow
122	#553	—	67	2	no	—
123	#499	—	67	1	no	—
124	#482	—	67	2	no	—
125	#481	—	67	2	no	—
126	#478	—	67	2	no	—
127	#466	—	67	1	no	—
128	#466	—	67	1	no	—
129	C₄×D₈	—	67	1	no	—
130	C₈×D₄	D₆₇	1	1	yes	characteristic, complemented
131	#427	—	67	1	no	—
132	C₂×#107	—	67	1	no	—
133	C₂×D₈	—	67	1	no	—
134	C₂²×C₈	C₂×D₆₇	1	1	yes	characteristic
135	C₂²×C₈	C₂×D₆₇	1	1	yes	characteristic
136	#126	—	134	2	no	—
137	#125	—	67	2	no	—
138	#123	—	134	2	no	—
139	#121	—	67	2	no	—
140	#121	—	67	2	no	—
141	#121	—	134	2	no	—
142	C₄×D₄	—	67	2	no	—
143	C₄×D₄	—	134	2	no	—
144	C₄×D₄	C₂×D₆₇	1	1	yes	characteristic
145	#117	—	67	2	no	—
146	C₂×C₄⋊C₄	—	67	2	no	—
147	#107	—	67	1	no	—
148	#107	—	67	1	no	—
149	#107	—	67	2	no	—
150	#107	—	67	1	no	—
151	#106	—	67	2	no	—
152	#106	—	67	2	no	—
153	C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
154	#102	—	67	2	no	—
155	#102	—	67	2	no	—
156	#102	—	67	2	no	—
157	C₂²⋊C₈	C₂×D₆₇	1	1	yes	characteristic
158	C₂²⋊C₈	C₂×D₆₇	1	1	yes	characteristic
159	C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
160	C₂×D₄	—	67	2	no	—
161	C₂×D₄	—	134	2	no	—
162	C₂×D₄	—	134	2	no	—
163	C₂×D₄	D₂₆₈	1	1	yes	characteristic
164	C₂²×C₄	—	134	2	no	—
165	C₂²×C₄	—	134	2	no	—
166	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
167	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
168	D₈	—	134	2	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
172	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
173	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
174	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
175	C₄⋊C₄	—	67	2	no	—
176	C₄⋊C₄	—	67	2	no	—
177	C₄⋊C₄	—	67	2	no	—
178	C₄⋊C₄	—	134	2	no	—
179	C₄⋊C₄	—	134	2	no	—
180	C₄⋊C₄	—	134	2	no	—
181	C₄⋊C₄	D₂₆₈	1	1	yes	characteristic
182	C₂²⋊C₄	—	134	2	no	—
183	C₂²⋊C₄	—	134	2	no	—
184	C₂²⋊C₄	—	134	2	no	—
185	C₂²⋊C₄	—	134	2	no	—
186	C₂²⋊C₄	D₂₆₈	1	1	yes	characteristic
187	C₂²⋊C₄	D₂₆₈	1	1	yes	characteristic
188	C₄²	—	134	2	no	—
189	C₄²	C₂²×D₆₇	1	1	yes	characteristic
190	C₂³	—	134	2	no	—
191	C₂³	C₂×D₂₆₈	1	1	yes	characteristic
192	C₂³	C₂×D₂₆₈	1	1	yes	characteristic
193	D₄	—	2	2	no	—
194	D₄	—	268	4	no	—
195	D₄	—	536	2	no	—
196	D₄	—	134	2	no	—
197	D₄	—	134	2	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	536	2	no	—
203	C₂×C₄	—	536	2	no	—
204	C₂×C₄	—	134	2	no	—
205	C₂×C₄	—	134	2	no	—
206	C₂×C₄	—	134	2	no	—
207	C₂×C₄	C₂×D₂₆₈	1	1	yes	characteristic
208	C₂×C₄	C₂×D₂₆₈	1	1	yes	characteristic
209	C₂×C₄	C₂×D₂₆₈	1	1	yes	characteristic
210	C₂×C₄	C₂×D₂₆₈	1	1	yes	characteristic
211	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
212	C₈	—	2	1	no	—
213	C₈	—	2	1	no	—
214	C₈	—	2	1	no	—
215	C₈	#11761627	1	2	yes	—
216	C₂²	—	2	1	no	—
217	C₂²	—	2	1	no	—
218	C₂²	—	2	1	no	—
219	C₂²	—	2	1	no	—
220	C₂²	—	2	1	no	—
221	C₂²	—	268	2	no	—
222	C₂²	—	268	2	no	—
223	C₂²	—	268	2	no	—
224	C₂²	#14524065	1	2	yes	—
225	C₂²	C₂²×D₂₆₈	1	1	yes	characteristic
226	C₄	—	2	1	no	—
227	C₄	—	2	1	no	—
228	C₄	—	2	1	no	—
229	C₄	—	2	1	no	—
230	C₄	—	268	2	no	—
231	C₄	—	268	2	no	—
232	C₄	—	268	2	no	—
233	C₄	C₂×#11761627	1	1	yes	characteristic
234	C₄	#14524090	1	1	yes	characteristic
235	C₂	—	2	2	no	—
236	C₂	—	4	1	no	—
237	C₂	—	536	2	no	—
238	C₂	#16864050	1	1	yes	characteristic
239	C₂	C₂×#14524065	1	1	yes	characteristic
240	C₂	#16864201	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₆₇
Fitting	C₈×D₄×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×D₄
hypercenter	C₈×D₄	D₆₇
hypocenter	C₆₇	#3339
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄

Complements

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

N	H ≅ G/N	type
1. #16864092	C₂	semidirect
2. #16863798	C₂	semidirect
3. #16863721	C₂	semidirect
4. #16863715	C₂	semidirect
5. #16863697	C₂	semidirect
6. #16863632	C₂	semidirect
7. #16863632	C₂	semidirect
8. C₄×D₅₃₆	C₂	semidirect
9. #16863434	C₂	semidirect
10. C₂×#14523977	C₂	semidirect
11. C₈×D₄×C₆₇	C₂	semidirect
121. C₆₇	#3339	semidirect
130. C₈×D₄	D₆₇	semidirect