Finite Group Databasebeta

Subgroups — #22627115

Order 9344.

220 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 67 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	#17989706	C₂	1	1	yes	maximal, characteristic, complemented
2	#17989706	C₂	1	1	yes	maximal, characteristic, complemented
3	#17989688	C₂	1	4	yes	maximal, complemented
4	C₂×#14556679	C₂	1	1	yes	maximal, characteristic
5	#14556688	—	2	1	no	—
6	#14556688	C₂²	1	2	yes	complemented
7	#14556687	—	2	1	no	—
8	#14556687	—	2	1	no	—
9	#14556686	—	2	1	no	—
10	#14556686	C₂²	1	2	yes	complemented
11	#14556679	—	2	1	no	—
12	#14556679	—	2	1	no	—
13	#14556679	C₄	1	2	yes	—
14	#14556679	C₂²	1	2	yes	—
15	C₂×#11924038	—	2	1	no	—
16	C₂×#11924038	C₄	1	1	yes	characteristic
17	C₂×#11924036	—	2	1	no	—
18	C₂×#11924036	C₂²	1	1	yes	characteristic
19	C₂×#11924029	C₄	1	1	yes	characteristic
20	#11924040	—	2	2	no	—
21	#11924040	—	2	2	no	—
22	#11924038	—	2	1	no	—
23	#11924038	—	2	2	no	—
24	#11924038	—	2	2	no	—
25	#11924038	C₂×C₄	1	2	yes	—
26	#11924036	—	2	2	no	—
27	#11924036	—	2	2	no	—
28	#11924036	C₂×C₄	1	1	yes	characteristic
29	#11924036	D₄	1	1	yes	characteristic
30	#11924036	D₄	1	1	yes	characteristic
31	#11924036	C₂³	1	1	yes	characteristic
32	C₂×#10596833	—	2	1	no	—
33	C₂×#10596833	D₄	1	1	yes	characteristic
34	C₂×#10596833	D₄	1	1	yes	characteristic
35	#11924029	—	2	1	no	—
36	#11924029	—	2	1	no	—
37	#11924029	C₈	1	2	yes	—
38	#11924029	C₂×C₄	1	2	yes	—
39	C₂×D₂₉₂	—	2	1	no	—
40	C₂×D₂₉₂	C₈	1	1	yes	characteristic
41	C₂×C₄×D₇₃	—	2	1	no	—
42	C₂×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
43	C₂×Q₈×C₇₃	C₈	1	1	yes	characteristic
44	C₂²×D₇₃	—	2	1	no	—
45	C₂²×D₇₃	M₁₆	1	1	yes	characteristic
46	Q₈×C₇₃	—	2	1	no	—
47	Q₈×C₇₃	C₂×C₈	1	2	yes	—
48	C₂×C₄×C₇₃	—	2	1	no	—
49	C₂×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
50	C₂×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
51	D₂₉₂	—	2	2	no	—
52	D₂₉₂	—	2	2	no	—
53	D₂₉₂	—	2	1	no	—
54	D₂₉₂	C₂×C₈	1	2	yes	—
55	C₄×D₇₃	—	2	2	no	—
56	C₄×D₇₃	—	2	2	no	—
57	C₄×D₇₃	C₂²⋊C₄	1	1	yes	characteristic
58	C₄×D₇₃	C₂×C₈	1	1	yes	characteristic
59	C₄×D₇₃	M₁₆	1	1	yes	characteristic
60	C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
61	#10596833	—	2	2	no	—
62	#10596833	C₂²⋊C₄	1	1	yes	characteristic
63	#10596833	C₂²⋊C₄	1	1	yes	characteristic
64	#10596833	C₂×D₄	1	1	yes	characteristic
65	#10596833	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
67	C₂×D₇₃	—	2	2	no	—
68	C₂×D₇₃	—	4	1	no	—
69	C₂×D₇₃	—	4	2	no	—
70	C₂×D₇₃	C₂²⋊C₈	1	1	yes	characteristic
71	C₂×D₇₃	C₂×M₁₆	1	1	yes	characteristic
72	C₄×C₇₃	—	2	2	no	—
73	C₄×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
74	C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
75	Dic₇₃	C₂²⋊C₈	1	1	yes	characteristic
76	Dic₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₇₃	—	2	1	no	—
78	C₂×C₇₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
79	D₇₃	—	2	1	no	—
80	D₇₃	—	2	1	no	—
81	D₇₃	—	4	2	no	—
82	#2117	—	73	1	no	maximal, Sylow
83	C₇₃	#2117	1	1	yes	characteristic, complemented, Sylow
84	C₂×#131	—	73	1	no	—
85	C₂×M₃₂	—	73	1	no	—
86	C₂×M₃₂	—	73	1	no	—
87	#350	—	73	4	no	—
88	C₂×#41	—	73	1	no	—
89	#131	—	73	2	no	—
90	#131	—	73	2	no	—
91	#131	—	146	1	no	—
92	#131	—	146	1	no	—
93	C₂×M₁₆	—	73	1	no	—
94	C₂×M₁₆	—	146	1	no	—
95	C₂²×C₈	—	73	1	no	—
96	C₂²×C₈	—	146	1	no	—
97	M₃₂	—	73	2	no	—
98	M₃₂	—	146	1	no	—
99	M₃₂	—	146	1	no	—
100	M₃₂	—	146	1	no	—
101	C₂×C₁₆	—	73	2	no	—
102	C₂×C₁₆	—	146	1	no	—
103	#41	—	73	2	no	—
104	#41	—	73	2	no	—
105	#41	—	146	1	no	—
106	#41	—	146	1	no	—
107	C₂×Q₈	#10596842	1	1	yes	characteristic
108	C₂×D₄	—	73	1	no	—
109	C₂×D₄	—	146	1	no	—
110	C₂²×C₄	—	73	1	no	—
111	C₂²×C₄	—	146	1	no	—
112	M₁₆	—	73	2	no	—
113	M₁₆	—	146	2	no	—
114	M₁₆	—	146	2	no	—
115	M₁₆	—	146	1	no	—
116	C₂×C₈	—	73	1	no	—
117	C₂×C₈	—	73	1	no	—
118	C₂×C₈	—	73	1	no	—
119	C₂×C₈	—	73	1	no	—
120	C₂×C₈	—	73	1	no	—
121	C₂×C₈	—	73	1	no	—
122	C₂×C₈	—	146	2	no	—
123	C₂×C₈	—	146	2	no	—
124	C₂×C₈	—	146	1	no	—
125	C₁₆	—	146	2	no	—
126	C₁₆	—	146	2	no	—
127	C₂³	—	73	1	no	—
128	C₂³	—	146	1	no	—
129	Q₈	—	2	1	no	—
130	Q₈	C₂×#10596842	1	2	yes	—
131	D₄	—	73	2	no	—
132	D₄	—	146	2	no	—
133	D₄	—	146	2	no	—
134	D₄	—	146	1	no	—
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	—	73	1	no	—
137	C₂×C₄	—	73	1	no	—
138	C₂×C₄	—	73	1	no	—
139	C₂×C₄	—	73	1	no	—
140	C₂×C₄	—	73	1	no	—
141	C₂×C₄	—	146	2	no	—
142	C₂×C₄	—	146	2	no	—
143	C₂×C₄	C₂×#10596842	1	1	yes	characteristic
144	C₈	—	73	1	no	—
145	C₈	—	73	1	no	—
146	C₈	—	73	1	no	—
147	C₈	—	73	1	no	—
148	C₈	—	146	2	no	—
149	C₂²	—	292	2	no	—
150	C₂²	—	292	1	no	—
151	C₂²	—	73	1	no	—
152	C₂²	—	73	1	no	—
153	C₂²	—	146	2	no	—
154	C₂²	C₂²⋊#10596842	1	1	yes	characteristic
155	C₄	—	2	2	no	—
156	C₄	—	73	1	no	—
157	C₄	—	73	1	no	—
158	C₄	C₂²⋊#10596842	1	1	yes	characteristic
159	C₄	C₂²×#10596842	1	1	yes	characteristic
160	C₂	—	2	1	no	—
161	C₂	—	292	2	no	—
162	C₂	—	146	1	no	—
163	C₂	—	146	1	no	—
164	C₂	C₂×C₂²⋊#10596842	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×C₂²⋊#10596842
commutator	C₄×C₇₃	C₂²×C₈
Frattini	C₄	C₂²×#10596842
Fitting	C₂×Q₈×C₇₃	C₈
socle	C₂×C₇₃	C₂×C₂²⋊C₈
hypercenter	C₂×Q₈	#10596842
hypocenter	C₇₃	#2117
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×Q₈

Complements

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

N	H ≅ G/N	type
1. #17989706	C₂	semidirect
2. #17989706	C₂	semidirect
3. #17989688	C₂	semidirect
6. #14556688	C₂²	semidirect
10. #14556686	C₂²	semidirect
83. C₇₃	#2117	semidirect