Finite Group Databasebeta

Subgroups — #22559740

Order 8576.

170 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 51 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16863742	C₂	1	1	yes	maximal, characteristic, complemented
2	#16863686	C₂	1	1	yes	maximal, characteristic, complemented
3	#16863474	C₂	1	1	yes	maximal, characteristic, complemented
4	#16863148	C₂	1	1	yes	maximal, characteristic, complemented
5	#16863144	C₂	1	1	yes	maximal, characteristic, complemented
6	#16863140	C₂	1	1	yes	maximal, characteristic, complemented
7	#459×C₆₇	C₂	1	1	yes	maximal, characteristic
8	C₂×#11761612	—	2	1	no	—
9	#14524043	C₂²	1	1	yes	characteristic
10	#14523974	—	2	1	no	—
11	C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
12	C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
13	#14523964	—	2	1	no	—
14	#14523964	—	2	1	no	—
15	#14523964	—	2	1	no	—
16	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
17	C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×D₈×C₆₇	—	2	1	no	—
19	#127×C₆₇	C₂²	1	1	yes	characteristic
20	C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
22	#102×C₆₇	—	2	1	no	—
23	C₂×Dic₁₃₄	—	2	1	no	—
24	#11761612	—	4	2	no	—
25	C₂×C₆₇⋊C₈	—	2	1	no	—
26	C₂×C₆₇⋊C₈	—	2	1	no	—
27	C₂×C₆₇⋊C₈	D₄	1	1	yes	characteristic
28	C₂×C₆₇⋊C₈	D₄	1	1	yes	characteristic
29	C₄⋊Dic₆₇	—	2	1	no	—
30	C₄⋊Dic₆₇	—	2	1	no	—
31	C₂×D₄×C₆₇	—	2	1	no	—
32	C₂×D₄×C₆₇	—	2	1	no	—
33	C₂×D₄×C₆₇	—	2	1	no	—
34	C₂×D₄×C₆₇	D₄	1	1	yes	characteristic
35	C₂²×C₄×C₆₇	—	2	1	no	—
36	D₈×C₆₇	—	4	2	no	—
37	C₂×C₈×C₆₇	—	2	1	no	—
38	C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
39	C₂²⋊C₄×C₆₇	—	2	1	no	—
40	C₄²×C₆₇	C₂³	1	1	yes	characteristic
41	C₂³×C₆₇	—	2	1	no	—
42	C₂³×C₆₇	—	4	1	no	—
43	D₄×C₆₇	—	2	1	no	—
44	D₄×C₆₇	—	2	1	no	—
45	D₄×C₆₇	—	4	1	no	—
46	D₄×C₆₇	—	4	1	no	—
47	D₄×C₆₇	—	4	1	no	—
48	D₄×C₆₇	—	4	1	no	—
49	D₄×C₆₇	—	8	1	no	—
50	C₂×C₄×C₆₇	—	2	1	no	—
51	C₂×C₄×C₆₇	—	4	1	no	—
52	C₂×C₄×C₆₇	—	4	1	no	—
53	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
56	C₂×Dic₆₇	—	4	1	no	—
57	Dic₁₃₄	—	8	1	no	—
58	C₈×C₆₇	—	4	1	no	—
59	C₆₇⋊C₈	—	2	1	no	—
60	C₆₇⋊C₈	—	2	1	no	—
61	C₆₇⋊C₈	—	4	1	no	—
62	C₆₇⋊C₈	—	4	1	no	—
63	C₂²×C₆₇	—	4	1	no	—
64	C₂²×C₆₇	—	4	1	no	—
65	C₂²×C₆₇	—	4	1	no	—
66	C₂²×C₆₇	—	8	1	no	—
67	C₂²×C₆₇	—	8	1	no	—
68	C₂²×C₆₇	—	8	1	no	—
69	C₂²×C₆₇	#121	1	1	yes	characteristic
70	C₄×C₆₇	—	2	1	no	—
71	C₄×C₆₇	—	4	1	no	—
72	C₄×C₆₇	#135	1	1	yes	characteristic
73	C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
74	C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
75	C₄×C₆₇	#137	1	1	yes	characteristic
76	Dic₆₇	—	8	1	no	—
77	C₂×C₆₇	—	8	1	no	—
78	C₂×C₆₇	—	16	1	no	—
79	C₂×C₆₇	#463	1	1	yes	characteristic
80	C₂×C₆₇	#468	1	1	yes	characteristic
81	C₂×C₆₇	#472	1	1	yes	characteristic
82	#1697	—	67	1	no	maximal, Sylow
83	C₆₇	#1697	1	1	yes	characteristic, complemented, Sylow
84	#486	—	67	1	no	—
85	#476	—	67	1	no	—
86	#459	D₆₇	1	1	yes	characteristic
87	#436	—	67	1	no	—
88	#335	—	67	1	no	—
89	#333	—	67	1	no	—
90	#331	—	67	1	no	—
91	C₂×D₈	—	2	1	no	—
92	C₂×M₁₆	—	134	1	no	—
93	C₄⋊Q₈	—	67	1	no	—
94	#127	C₂×D₆₇	1	1	yes	characteristic
95	C₄×D₄	C₂×D₆₇	1	1	yes	characteristic
96	#106	—	134	1	no	—
97	C₄⋊C₈	—	67	1	no	—
98	C₄⋊C₈	—	67	1	no	—
99	C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
100	#102	—	2	1	no	—
101	#102	—	134	1	no	—
102	#102	—	134	1	no	—
103	#102	—	134	1	no	—
104	C₂²⋊C₈	—	134	1	no	—
105	C₄×C₈	—	67	1	no	—
106	C₂×Q₈	—	134	1	no	—
107	C₂×D₄	—	2	1	no	—
108	C₂×D₄	—	2	1	no	—
109	C₂×D₄	—	2	1	no	—
110	C₂×D₄	C₂²⋊D₆₇	1	1	yes	characteristic
111	C₂²×C₄	—	2	1	no	—
112	D₈	—	4	2	no	—
113	M₁₆	—	268	2	no	—
114	C₂×C₈	—	2	1	no	—
115	C₂×C₈	—	67	1	no	—
116	C₂×C₈	—	67	1	no	—
117	C₂×C₈	—	134	1	no	—
118	C₂×C₈	—	134	1	no	—
119	C₄⋊C₄	—	134	1	no	—
120	C₄⋊C₄	—	134	1	no	—
121	C₄⋊C₄	C₂²⋊D₆₇	1	1	yes	characteristic
122	C₂²⋊C₄	—	2	1	no	—
123	C₄²	C₂²×D₆₇	1	1	yes	characteristic
124	C₂³	—	2	1	no	—
125	C₂³	—	4	1	no	—
126	Q₈	—	536	1	no	—
127	D₄	—	2	1	no	—
128	D₄	—	2	1	no	—
129	D₄	—	4	1	no	—
130	D₄	—	4	1	no	—
131	D₄	—	4	1	no	—
132	D₄	—	4	1	no	—
133	D₄	—	8	1	no	—
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	—	4	1	no	—
136	C₂×C₄	—	4	1	no	—
137	C₂×C₄	—	268	1	no	—
138	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
139	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
140	C₂×C₄	#11761627	1	1	yes	characteristic
141	C₈	—	4	1	no	—
142	C₈	—	268	1	no	—
143	C₈	—	268	1	no	—
144	C₈	—	134	1	no	—
145	C₈	—	134	1	no	—
146	C₂²	—	4	1	no	—
147	C₂²	—	4	1	no	—
148	C₂²	—	4	1	no	—
149	C₂²	—	8	1	no	—
150	C₂²	—	8	1	no	—
151	C₂²	—	8	1	no	—
152	C₂²	#14524015	1	1	yes	characteristic
153	C₄	—	2	1	no	—
154	C₄	—	4	1	no	—
155	C₄	—	536	1	no	—
156	C₄	#14524062	1	1	yes	characteristic
157	C₄	#14524067	1	1	yes	characteristic
158	C₄	#14524069	1	1	yes	characteristic
159	C₄	#14524073	1	1	yes	characteristic
160	C₂	—	8	1	no	—
161	C₂	—	16	1	no	—
162	C₂	#16863609	1	1	yes	characteristic
163	C₂	#16863640	1	1	yes	characteristic
164	C₂	#16863664	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#14524015
commutator	C₄²×C₆₇	C₂³
Frattini	C₄²	C₂²×D₆₇
Fitting	#459×C₆₇	C₂
socle	C₂²×C₆₇	#121
hypercenter	#459	D₆₇
hypocenter	C₆₇	#1697
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #459

Complements

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

N	H ≅ G/N	type
1. #16863742	C₂	semidirect
2. #16863686	C₂	semidirect
3. #16863474	C₂	semidirect
4. #16863148	C₂	semidirect
5. #16863144	C₂	semidirect
6. #16863140	C₂	semidirect
83. C₆₇	#1697	semidirect