Subgroups — #22569285

Order 8576.

402 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16863474	C₂	1	2	yes	maximal, complemented
2	#16863468	C₂	1	4	yes	maximal, complemented
3	D₄×C₆₇⋊C₈	C₂	1	2	yes	maximal, complemented
4	#16863453	C₂	1	1	yes	maximal, characteristic, complemented
5	#16863449	C₂	1	1	yes	maximal, characteristic, complemented
6	#16863363	C₂	1	4	yes	maximal, complemented
7	#532×C₆₇	C₂	1	1	yes	maximal, characteristic
8	C₂×#11761612	C₂²	1	4	yes	—
9	C₂²×C₆₇⋊C₈	C₂²	1	4	yes	—
10	C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
11	C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
12	C₂²⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
13	C₂²⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
14	C₂²⋊(C₆₇⋊C₈)	C₂²	1	4	yes	—
15	C₂²⋊(C₆₇⋊C₈)	C₂²	1	4	yes	—
16	#14523954	C₂²	1	2	yes	—
17	C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
18	C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
19	C₂×#41×C₆₇	C₂²	1	2	yes	—
20	C₄⋊Q₈×C₆₇	C₄	1	1	yes	characteristic
21	#127×C₆₇	C₄	1	1	yes	characteristic
22	#124×C₆₇	C₄	1	2	yes	—
23	#121×C₆₇	C₄	1	2	yes	—
24	#121×C₆₇	C₄	1	2	yes	—
25	C₄×D₄×C₆₇	C₂²	1	2	yes	—
26	C₄×D₄×C₆₇	C₂²	1	2	yes	—
27	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
28	#11761612	—	4	4	no	—
29	C₂×C₆₇⋊C₈	—	4	4	no	—
30	C₂×C₆₇⋊C₈	C₂³	1	2	yes	—
31	C₂×C₆₇⋊C₈	C₂³	1	2	yes	—
32	C₂×C₆₇⋊C₈	C₂³	1	4	yes	—
33	#41×C₆₇	—	4	4	no	—
34	C₂×Q₈×C₆₇	C₂×C₄	1	2	yes	—
35	C₂×D₄×C₆₇	C₂×C₄	1	2	yes	—
36	C₂×D₄×C₆₇	C₂×C₄	1	2	yes	—
37	C₂×D₄×C₆₇	C₂×C₄	1	4	yes	—
38	C₂×D₄×C₆₇	C₂×C₄	1	2	yes	—
39	C₂²×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₆₇	C₂³	1	4	yes	—
43	C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
44	C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
45	C₂²⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
46	C₂²⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
47	C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
48	C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
49	C₄²×C₆₇	C₂³	1	1	yes	characteristic
50	C₄²×C₆₇	C₂³	1	1	yes	characteristic
51	C₂³×C₆₇	C₂²×C₄	1	4	yes	—
52	C₂³×C₆₇	C₂²×C₄	1	1	yes	characteristic
53	Q₈×C₆₇	—	4	2	no	—
54	D₄×C₆₇	—	2	4	no	—
55	D₄×C₆₇	—	4	2	no	—
56	D₄×C₆₇	—	4	2	no	—
57	D₄×C₆₇	—	4	4	no	—
58	C₂×C₄×C₆₇	—	2	2	no	—
59	C₂×C₄×C₆₇	—	2	2	no	—
60	C₂×C₄×C₆₇	—	2	4	no	—
61	C₂×C₄×C₆₇	—	2	4	no	—
62	C₂×C₄×C₆₇	—	4	1	no	—
63	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₄×C₆₇	C₂²×C₄	1	4	yes	—
68	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
70	C₆₇⋊C₈	—	2	2	no	—
71	C₆₇⋊C₈	—	2	4	no	—
72	C₆₇⋊C₈	—	2	2	no	—
73	C₂²×C₆₇	—	2	1	no	—
74	C₂²×C₆₇	—	2	4	no	—
75	C₂²×C₆₇	—	2	1	no	—
76	C₂²×C₆₇	—	2	4	no	—
77	C₂²×C₆₇	—	2	1	no	—
78	C₂²×C₆₇	—	2	4	no	—
79	C₂²×C₆₇	C₂³×C₄	1	1	yes	characteristic
80	C₄×C₆₇	—	2	1	no	—
81	C₄×C₆₇	—	2	4	no	—
82	C₄×C₆₇	—	2	1	no	—
83	C₄×C₆₇	—	2	1	no	—
84	C₄×C₆₇	—	2	1	no	—
85	C₄×C₆₇	#131	1	2	yes	—
86	C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
87	C₂×C₆₇	—	4	1	no	—
88	C₂×C₆₇	—	4	4	no	—
89	C₂×C₆₇	#518	1	1	yes	characteristic
90	C₂×C₆₇	C₂×#131	1	1	yes	characteristic
91	C₂×C₆₇	#568	1	1	yes	characteristic
92	#2981	—	67	1	no	maximal, Sylow
93	C₆₇	#2981	1	1	yes	characteristic, complemented, Sylow
94	#532	D₆₇	1	1	yes	characteristic
95	#436	—	67	2	no	—
96	#435	—	67	4	no	—
97	C₈×D₄	—	67	2	no	—
98	#432	—	67	1	no	—
99	#431	—	67	1	no	—
100	#408	—	67	4	no	—
101	C₂×#41	C₂×D₆₇	1	2	yes	—
102	C₂×M₁₆	—	67	4	no	—
103	C₂²×C₈	—	67	4	no	—
104	C₄⋊Q₈	Dic₆₇	1	1	yes	characteristic
105	#127	Dic₆₇	1	1	yes	characteristic
106	#124	Dic₆₇	1	2	yes	—
107	#121	Dic₆₇	1	2	yes	—
108	#121	Dic₆₇	1	2	yes	—
109	C₄×D₄	C₂×D₆₇	1	2	yes	—
110	C₄×D₄	C₂×D₆₇	1	2	yes	—
111	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
112	C₄⋊C₈	—	67	2	no	—
113	C₄⋊C₈	—	67	2	no	—
114	C₂²⋊C₈	—	67	2	no	—
115	C₂²⋊C₈	—	67	2	no	—
116	C₂²⋊C₈	—	67	4	no	—
117	C₂²⋊C₈	—	67	4	no	—
118	#97	—	67	2	no	—
119	C₄×C₈	—	67	1	no	—
120	C₄×C₈	—	67	1	no	—
121	#41	—	4	4	no	—
122	C₂×Q₈	C₂×Dic₆₇	1	2	yes	—
123	C₂×D₄	C₂×Dic₆₇	1	2	yes	—
124	C₂×D₄	C₂×Dic₆₇	1	2	yes	—
125	C₂×D₄	C₂×Dic₆₇	1	4	yes	—
126	C₂×D₄	C₂×Dic₆₇	1	2	yes	—
127	C₂²×C₄	C₂×Dic₆₇	1	1	yes	characteristic
128	C₂²×C₄	C₂×Dic₆₇	1	1	yes	characteristic
129	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
130	C₂²×C₄	C₂²×D₆₇	1	4	yes	—
131	M₁₆	—	268	4	no	—
132	C₂×C₈	—	268	4	no	—
133	C₂×C₈	—	67	2	no	—
134	C₂×C₈	—	67	2	no	—
135	C₂×C₈	—	67	4	no	—
136	C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
137	C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
138	C₂²⋊C₄	C₂×Dic₆₇	1	4	yes	—
139	C₂²⋊C₄	C₂×Dic₆₇	1	4	yes	—
140	C₄²	C₂×Dic₆₇	1	1	yes	characteristic
141	C₄²	C₂×Dic₆₇	1	1	yes	characteristic
142	C₄²	C₂²×D₆₇	1	1	yes	characteristic
143	C₄²	C₂²×D₆₇	1	1	yes	characteristic
144	C₂³	C₂²×Dic₆₇	1	4	yes	—
145	C₂³	C₂²×Dic₆₇	1	1	yes	characteristic
146	Q₈	—	4	2	no	—
147	D₄	—	2	4	no	—
148	D₄	—	4	2	no	—
149	D₄	—	4	4	no	—
150	D₄	—	4	2	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	4	no	—
153	C₂×C₄	—	2	4	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	4	1	no	—
156	C₂×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
157	C₂×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
158	C₂×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
159	C₂×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
160	C₂×C₄	C₂²×Dic₆₇	1	4	yes	—
161	C₂×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
162	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
163	C₈	—	134	2	no	—
164	C₈	—	134	4	no	—
165	C₈	—	134	2	no	—
166	C₂²	—	2	1	no	—
167	C₂²	—	2	4	no	—
168	C₂²	—	2	1	no	—
169	C₂²	—	2	4	no	—
170	C₂²	—	2	1	no	—
171	C₂²	—	2	4	no	—
172	C₂²	C₂³×Dic₆₇	1	1	yes	characteristic
173	C₄	—	2	1	no	—
174	C₄	—	2	4	no	—
175	C₄	—	2	1	no	—
176	C₄	—	2	1	no	—
177	C₄	—	2	1	no	—
178	C₄	#14524050	1	2	yes	—
179	C₄	#14524088	1	2	yes	—
180	C₂	—	4	1	no	—
181	C₂	—	4	4	no	—
182	C₂	#16863871	1	1	yes	characteristic
183	C₂	C₂×#14524050	1	1	yes	characteristic
184	C₂	#16864163	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₆₇
commutator, socle	C₂²×C₆₇	C₂³×C₄
Fitting	#532×C₆₇	C₂
hypercenter	#532	D₆₇
hypocenter	C₆₇	#2981
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ #532

Complements

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

N	H ≅ G/N	type
1. #16863474	C₂	semidirect
2. #16863468	C₂	semidirect
3. D₄×C₆₇⋊C₈	C₂	semidirect
4. #16863453	C₂	semidirect
5. #16863449	C₂	semidirect
6. #16863363	C₂	semidirect
93. C₆₇	#2981	semidirect