Subgroups — #22574026

Order 8576.

1060 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	Q₈×Dic₁₃₄	C₂	1	2	yes	maximal, complemented
2	#16864125	C₂	1	1	yes	maximal, characteristic, complemented
3	#16864102	C₂	1	6	yes	maximal, complemented
4	Q₈×D₂₆₈	C₂	1	3	yes	maximal, complemented
5	#16864061	C₂	1	3	yes	maximal, complemented
6	#16864026	C₂	1	6	yes	maximal, complemented
7	#16863950	C₂	1	3	yes	maximal, complemented
8	C₄⋊Q₈×D₆₇	C₂	1	3	yes	maximal, complemented
9	#16863878	C₂	1	2	yes	maximal
10	C₄×#11761629	C₂	1	1	yes	maximal, characteristic
11	#557×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#11761629	C₂²	1	1	yes	characteristic
13	C₂×Q₈×D₆₇	C₂²	1	6	yes	—
14	#14524043	C₂²	1	6	yes	—
15	#14524043	C₂²	1	3	yes	—
16	#14524042	C₂²	1	6	yes	—
17	#14524042	C₂²	1	6	yes	—
18	#14524042	C₂²	1	6	yes	—
19	#14524042	C₂²	1	1	yes	characteristic
20	#14524041	C₂²	1	3	yes	—
21	#14524034	C₂²	1	6	yes	—
22	#14524033	C₂²	1	6	yes	—
23	#14524023	C₂²	1	12	yes	—
24	#14524023	C₂²	1	3	yes	—
25	#14524021	C₂²	1	12	yes	—
26	#14524021	C₂²	1	6	yes	—
27	#14524019	C₂²	1	3	yes	—
28	Q₈×Dic₆₇	C₂²	1	1	yes	characteristic
29	Q₈×Dic₆₇	C₂²	1	2	yes	—
30	C₄×Dic₁₃₄	C₂²	1	3	yes	—
31	C₄×Dic₁₃₄	C₂²	1	6	yes	—
32	C₄×D₂₆₈	C₂²	1	3	yes	—
33	C₄×D₂₆₈	C₂²	1	6	yes	—
34	#14524005	C₂²	1	6	yes	—
35	#14524005	C₂²	1	3	yes	—
36	#14524001	C₂²	1	3	yes	—
37	#14524000	C₂²	1	6	yes	—
38	C₄⋊C₄×D₆₇	C₂²	1	6	yes	—
39	C₄⋊C₄×D₆₇	C₂²	1	6	yes	—
40	C₄²×D₆₇	C₂²	1	3	yes	—
41	C₄⋊Q₈×C₆₇	C₂²	1	3	yes	—
42	#125×C₆₇	C₂²	1	6	yes	—
43	C₄×Q₈×C₆₇	C₂²	1	3	yes	—
44	C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
45	C₄×Q₈×C₆₇	C₂²	1	2	yes	—
46	#11761629	Q₈	1	8	yes	complemented
47	C₂×Dic₁₃₄	C₂³	1	2	yes	—
48	C₂×Dic₁₃₄	C₂³	1	3	yes	—
49	C₂×Dic₁₃₄	C₂³	1	12	yes	—
50	Q₈×D₆₇	—	4	12	no	—
51	C₂×D₂₆₈	C₂³	1	3	yes	—
52	C₂×C₄×D₆₇	C₂³	1	3	yes	—
53	C₂×C₄×D₆₇	C₂³	1	3	yes	—
54	C₂×C₄×D₆₇	C₂³	1	12	yes	—
55	C₂×C₄×D₆₇	C₂³	1	3	yes	—
56	C₄⋊Dic₆₇	C₂³	1	6	yes	—
57	C₄⋊Dic₆₇	C₂³	1	6	yes	—
58	C₄⋊Dic₆₇	C₂³	1	3	yes	—
59	C₄⋊Dic₆₇	C₂³	1	1	yes	characteristic
60	C₄⋊Dic₆₇	C₂³	1	6	yes	—
61	Dic₆₇⋊C₄	C₂³	1	6	yes	—
62	Dic₆₇⋊C₄	C₂³	1	6	yes	—
63	Dic₆₇⋊C₄	C₂³	1	2	yes	—
64	Dic₆₇⋊C₄	C₂³	1	6	yes	—
65	Dic₆₇⋊C₄	C₂³	1	6	yes	—
66	(C₂×D₆₇)⋊C₄	C₂³	1	6	yes	—
67	(C₂×D₆₇)⋊C₄	C₂³	1	6	yes	—
68	(C₂×D₆₇)⋊C₄	C₂³	1	6	yes	—
69	C₄×Dic₆₇	C₂³	1	1	yes	characteristic
70	C₄×Dic₆₇	C₂³	1	3	yes	—
71	C₄×Dic₆₇	C₂³	1	3	yes	—
72	C₄×Dic₆₇	C₂³	1	6	yes	—
73	C₂×Q₈×C₆₇	C₂³	1	3	yes	—
74	C₂×Q₈×C₆₇	C₂³	1	1	yes	characteristic
75	C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
76	C₄⋊C₄×C₆₇	C₂³	1	3	yes	—
77	C₄⋊C₄×C₆₇	C₂³	1	6	yes	—
78	C₄⋊C₄×C₆₇	C₂³	1	6	yes	—
79	C₄⋊C₄×C₆₇	C₂³	1	6	yes	—
80	C₄²×C₆₇	C₂³	1	6	yes	—
81	C₄²×C₆₇	C₂³	1	3	yes	—
82	C₂²×D₆₇	C₂⁴	1	3	yes	—
83	Q₈×C₆₇	—	2	6	no	—
84	Q₈×C₆₇	C₂×Q₈	1	4	yes	complemented
85	C₂×C₄×C₆₇	C₂⁴	1	2	yes	—
86	C₂×C₄×C₆₇	C₂⁴	1	3	yes	—
87	C₂×C₄×C₆₇	C₂⁴	1	6	yes	—
88	C₂×C₄×C₆₇	C₂⁴	1	3	yes	—
89	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
90	C₂×Dic₆₇	C₂⁴	1	2	yes	—
91	C₂×Dic₆₇	C₂⁴	1	3	yes	—
92	C₂×Dic₆₇	C₂⁴	1	6	yes	—
93	C₂×Dic₆₇	C₂⁴	1	1	yes	characteristic
94	C₂×Dic₆₇	C₂⁴	1	1	yes	characteristic
95	D₂₆₈	C₂×Q₈	1	12	yes	complemented
96	C₄×D₆₇	—	2	6	no	—
97	C₄×D₆₇	—	2	6	no	—
98	C₄×D₆₇	—	4	12	no	—
99	C₄×D₆₇	C₂×Q₈	1	12	yes	complemented
100	Dic₁₃₄	—	2	6	no	—
101	Dic₁₃₄	—	2	4	no	—
102	Dic₁₃₄	—	4	12	no	—
103	C₂²×C₆₇	C₂⁵	1	1	yes	characteristic
104	C₂×D₆₇	—	2	3	no	—
105	C₂×D₆₇	—	2	3	no	—
106	C₂×D₆₇	C₂²×Q₈	1	6	yes	—
107	C₄×C₆₇	—	2	6	no	—
108	C₄×C₆₇	—	2	2	no	—
109	C₄×C₆₇	—	2	3	no	—
110	C₄×C₆₇	C₂²×Q₈	1	6	yes	—
111	C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
112	Dic₆₇	—	2	6	no	—
113	Dic₆₇	—	2	3	no	—
114	Dic₆₇	—	2	2	no	—
115	Dic₆₇	C₂²×Q₈	1	2	yes	—
116	Dic₆₇	2₋¹⁺⁴	1	2	yes	—
117	C₂×C₆₇	C₂³×Q₈	1	1	yes	characteristic
118	C₂×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
119	C₂×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
120	D₆₇	—	2	6	no	—
121	#3503	—	67	1	no	maximal, Sylow
122	C₆₇	#3503	1	1	yes	characteristic, complemented, Sylow
123	Q₈²	—	67	2	no	—
124	#557	—	67	1	no	—
125	#557	D₆₇	1	1	yes	characteristic, complemented
126	#554	—	67	6	no	—
127	D₄×Q₈	—	67	3	no	—
128	D₄×Q₈	—	67	3	no	—
129	#544	—	67	6	no	—
130	#533	—	67	3	no	—
131	C₂×C₄⋊Q₈	—	67	3	no	—
132	#520	—	67	2	no	—
133	C₄×#41	—	67	1	no	—
134	C₂×#41	—	67	1	no	—
135	C₂²×Q₈	—	67	6	no	—
136	C₄⋊Q₈	—	67	3	no	—
137	C₄⋊Q₈	—	67	6	no	—
138	C₄⋊Q₈	—	67	6	no	—
139	C₄⋊Q₈	—	67	1	no	—
140	C₄⋊Q₈	—	67	6	no	—
141	C₄⋊Q₈	—	67	3	no	—
142	C₄⋊Q₈	—	67	6	no	—
143	C₄⋊Q₈	C₂×D₆₇	1	3	yes	—
144	#125	—	67	6	no	—
145	#125	—	67	6	no	—
146	#125	C₂×D₆₇	1	6	yes	—
147	C₂²⋊Q₈	—	67	3	no	—
148	C₂²⋊Q₈	—	67	3	no	—
149	C₂²⋊Q₈	—	67	6	no	—
150	C₂²⋊Q₈	—	67	12	no	—
151	C₂²⋊Q₈	—	67	12	no	—
152	C₄×Q₈	—	67	1	no	—
153	C₄×Q₈	—	67	2	no	—
154	C₄×Q₈	—	67	3	no	—
155	C₄×Q₈	—	67	6	no	—
156	C₄×Q₈	C₂×D₆₇	1	1	yes	characteristic
157	C₄×Q₈	C₂×D₆₇	1	2	yes	—
158	C₄×Q₈	C₂×D₆₇	1	3	yes	—
159	C₄×D₄	—	67	3	no	—
160	C₄×D₄	—	67	6	no	—
161	C₄×D₄	—	67	3	no	—
162	C₄×D₄	—	67	6	no	—
163	#117	—	67	3	no	—
164	#117	—	67	6	no	—
165	C₂×C₄⋊C₄	—	67	6	no	—
166	C₂×C₄⋊C₄	—	67	6	no	—
167	C₂×C₄²	—	67	3	no	—
168	#41	—	67	8	no	—
169	C₂×Q₈	—	268	12	no	—
170	C₂×Q₈	—	67	12	no	—
171	C₂×Q₈	—	67	2	no	—
172	C₂×Q₈	—	67	3	no	—
173	C₂×Q₈	C₂²×D₆₇	1	3	yes	—
174	C₂×Q₈	C₂²×D₆₇	1	1	yes	characteristic
175	C₂×D₄	—	67	3	no	—
176	C₂²×C₄	—	67	12	no	—
177	C₂²×C₄	—	67	3	no	—
178	C₂²×C₄	—	67	3	no	—
179	C₂²×C₄	—	67	3	no	—
180	C₄⋊C₄	—	67	6	no	—
181	C₄⋊C₄	—	67	6	no	—
182	C₄⋊C₄	—	67	6	no	—
183	C₄⋊C₄	—	67	6	no	—
184	C₄⋊C₄	—	67	2	no	—
185	C₄⋊C₄	—	67	3	no	—
186	C₄⋊C₄	—	67	1	no	—
187	C₄⋊C₄	—	67	6	no	—
188	C₄⋊C₄	—	67	6	no	—
189	C₄⋊C₄	—	67	6	no	—
190	C₄⋊C₄	C₂²×D₆₇	1	6	yes	—
191	C₄⋊C₄	C₂²×D₆₇	1	6	yes	—
192	C₄⋊C₄	C₂²×D₆₇	1	3	yes	—
193	C₄⋊C₄	C₂²×D₆₇	1	1	yes	characteristic
194	C₄⋊C₄	C₂²×D₆₇	1	6	yes	—
195	C₂²⋊C₄	—	67	6	no	—
196	C₂²⋊C₄	—	67	6	no	—
197	C₂²⋊C₄	—	67	6	no	—
198	C₄²	—	67	1	no	—
199	C₄²	—	67	3	no	—
200	C₄²	—	67	3	no	—
201	C₄²	—	67	6	no	—
202	C₄²	C₂²×D₆₇	1	3	yes	—
203	C₄²	C₂²×D₆₇	1	6	yes	—
204	C₂³	—	67	3	no	—
205	Q₈	—	2	6	no	—
206	Q₈	—	268	12	no	—
207	Q₈	—	134	4	no	—
208	Q₈	—	134	6	no	—
209	Q₈	Q₈×D₆₇	1	4	yes	complemented
210	D₄	—	67	12	no	—
211	C₂×C₄	—	268	12	no	—
212	C₂×C₄	—	67	1	no	—
213	C₂×C₄	—	67	1	no	—
214	C₂×C₄	—	67	12	no	—
215	C₂×C₄	—	67	2	no	—
216	C₂×C₄	—	67	3	no	—
217	C₂×C₄	—	67	6	no	—
218	C₂×C₄	—	134	6	no	—
219	C₂×C₄	—	134	6	no	—
220	C₂×C₄	C₂³×D₆₇	1	3	yes	—
221	C₂×C₄	C₂³×D₆₇	1	2	yes	—
222	C₂×C₄	C₂³×D₆₇	1	6	yes	—
223	C₂×C₄	C₂³×D₆₇	1	3	yes	—
224	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
225	C₂²	—	67	6	no	—
226	C₂²	—	134	3	no	—
227	C₂²	—	134	3	no	—
228	C₂²	C₂⁴×D₆₇	1	1	yes	characteristic
229	C₄	—	2	6	no	—
230	C₄	—	2	3	no	—
231	C₄	—	2	2	no	—
232	C₄	—	67	2	no	—
233	C₄	—	67	2	no	—
234	C₄	—	134	6	no	—
235	C₄	—	134	3	no	—
236	C₄	—	134	2	no	—
237	C₄	C₂×Q₈×D₆₇	1	6	yes	—
238	C₄	#14524089	1	2	yes	—
239	C₂	—	134	6	no	—
240	C₂	C₂²×Q₈×D₆₇	1	1	yes	characteristic
241	C₂	C₂×#14524089	1	1	yes	characteristic
242	C₂	2₋¹⁺⁴×D₆₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₆₇
commutator, socle	C₂²×C₆₇	C₂⁵
Fitting	#557×C₆₇	C₂
hypercenter	#557	D₆₇
hypocenter	C₆₇	#3503
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ #557

Complements

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

N	H ≅ G/N	type
1. Q₈×Dic₁₃₄	C₂	semidirect
2. #16864125	C₂	semidirect
3. #16864102	C₂	semidirect
4. Q₈×D₂₆₈	C₂	semidirect
5. #16864061	C₂	semidirect
6. #16864026	C₂	semidirect
7. #16863950	C₂	semidirect
8. C₄⋊Q₈×D₆₇	C₂	semidirect
11. #557×C₆₇	C₂	semidirect
46. #11761629	Q₈	semidirect
84. Q₈×C₆₇	C₂×Q₈	semidirect
95. D₂₆₈	C₂×Q₈	semidirect
99. C₄×D₆₇	C₂×Q₈	semidirect
122. C₆₇	#3503	semidirect
125. #557	D₆₇	semidirect
209. Q₈	Q₈×D₆₇	semidirect