Finite Group Databasebeta

Subgroups — #22627101

Order 9344.

228 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 71 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	C₂×#14556687	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#11924040	C₂	1	1	yes	maximal, characteristic, complemented
3	#17989684	C₂	1	2	yes	maximal, complemented
4	C₂²⋊#11924040	C₂	1	2	yes	maximal, complemented
5	C₂×#14556678	C₂	1	1	yes	maximal, characteristic
6	#14556687	—	2	2	no	—
7	C₂×#11924040	—	2	2	no	—
8	C₂×#11924040	C₂²	1	2	yes	—
9	C₂×#11924040	C₂²	1	2	yes	—
10	#14556678	—	2	2	no	—
11	#14556678	—	2	2	no	—
12	C₂×#11924038	C₄	1	1	yes	characteristic
13	C₂×#11924037	C₄	1	1	yes	characteristic
14	C₂×#11924037	C₄	1	1	yes	characteristic
15	C₂×#11924036	C₂²	1	1	yes	characteristic
16	C₂²×#10596833	C₂²	1	1	yes	characteristic
17	C₂²×#10596833	C₂²	1	1	yes	characteristic
18	C₂×#11924027	C₄	1	1	yes	characteristic
19	#11924040	—	2	2	no	—
20	#11924040	—	2	2	no	—
21	#11924038	—	2	2	no	—
22	#11924037	—	2	2	no	—
23	#11924037	—	2	2	no	—
24	#11924036	—	2	1	no	—
25	#11924036	—	2	1	no	—
26	C₂×#10596833	—	2	1	no	—
27	C₂×#10596833	—	2	1	no	—
28	C₂×#10596833	C₂×C₄	1	1	yes	characteristic
29	C₂×#10596833	C₂×C₄	1	1	yes	characteristic
30	C₂×#10596833	C₂×C₄	1	1	yes	characteristic
31	C₂×#10596833	D₄	1	2	yes	—
32	C₂×#10596833	D₄	1	2	yes	—
33	C₂×#10596833	C₂³	1	1	yes	characteristic
34	#11924027	—	2	2	no	—
35	#11924027	—	2	2	no	—
36	C₂×Dic₁₄₆	C₈	1	1	yes	characteristic
37	C₂×C₂²⋊D₇₃	C₈	1	1	yes	characteristic
38	C₂×C₂²⋊D₇₃	C₈	1	1	yes	characteristic
39	C₂²×Dic₇₃	C₂×C₄	1	1	yes	characteristic
40	C₂²×Dic₇₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
42	C₂×D₄×C₇₃	C₈	1	1	yes	characteristic
43	C₂³×C₇₃	C₂×C₈	1	1	yes	characteristic
44	C₂³×C₇₃	C₂×C₈	1	1	yes	characteristic
45	C₂²×D₇₃	C₂×C₈	1	1	yes	characteristic
46	D₄×C₇₃	—	2	2	no	—
47	C₂×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
48	C₂²⋊D₇₃	—	2	2	no	—
49	C₂²⋊D₇₃	—	2	2	no	—
50	C₂×Dic₇₃	—	2	1	no	—
51	C₂×Dic₇₃	—	2	1	no	—
52	C₂×Dic₇₃	C₂²⋊C₄	1	2	yes	—
53	C₂×Dic₇₃	C₂×C₈	1	1	yes	characteristic
54	C₂×Dic₇₃	C₂×C₈	1	1	yes	characteristic
55	C₂×Dic₇₃	M₁₆	1	2	yes	—
56	C₂×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
57	C₄×D₇₃	—	2	1	no	—
58	C₄×D₇₃	—	2	1	no	—
59	Dic₁₄₆	—	2	2	no	—
60	#10596833	—	2	1	no	—
61	#10596833	—	2	1	no	—
62	#10596833	C₂²⋊C₄	1	2	yes	—
63	#10596833	C₂×D₄	1	1	yes	characteristic
64	#10596833	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₇₃	—	2	2	no	—
66	C₂²×C₇₃	—	2	1	no	—
67	C₂²×C₇₃	—	2	2	no	—
68	C₂²×C₇₃	C₂²⋊C₈	1	2	yes	—
69	C₂²×C₇₃	C₂²×C₈	1	1	yes	characteristic
70	C₂×D₇₃	—	2	1	no	—
71	C₂×D₇₃	—	2	2	no	—
72	C₄×C₇₃	—	2	1	no	—
73	Dic₇₃	—	2	1	no	—
74	Dic₇₃	C₂²⋊C₈	1	2	yes	—
75	Dic₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
76	Dic₇₃	C₂×M₁₆	1	1	yes	characteristic
77	C₂×C₇₃	—	2	2	no	—
78	C₂×C₇₃	—	4	1	no	—
79	C₂×C₇₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
80	C₂×C₇₃	#504	1	2	yes	—
81	D₇₃	—	4	1	no	—
82	#2113	—	73	1	no	maximal, Sylow
83	C₇₃	#2113	1	1	yes	characteristic, complemented, Sylow
84	C₂×#131	—	73	1	no	—
85	C₂×M₃₂	—	73	1	no	—
86	C₂²×C₁₆	—	73	1	no	—
87	C₂²⋊C₁₆	—	73	2	no	—
88	C₂²⋊C₁₆	—	73	2	no	—
89	C₂×#41	—	73	1	no	—
90	#131	—	146	2	no	—
91	#131	—	146	2	no	—
92	C₂×M₁₆	—	73	1	no	—
93	C₂×M₁₆	—	73	1	no	—
94	C₂×M₁₆	—	73	1	no	—
95	C₂²×C₈	—	73	1	no	—
96	C₂²×C₈	—	73	1	no	—
97	C₂²×C₈	—	73	1	no	—
98	M₃₂	—	146	2	no	—
99	C₂×C₁₆	—	73	2	no	—
100	C₂×C₁₆	—	73	2	no	—
101	C₂×C₁₆	—	146	2	no	—
102	#41	—	146	2	no	—
103	#41	—	146	2	no	—
104	C₂×Q₈	—	73	1	no	—
105	C₂×D₄	—	73	1	no	—
106	C₂×D₄	—	73	1	no	—
107	C₂×D₄	#10596842	1	1	yes	characteristic
108	C₂²×C₄	—	73	1	no	—
109	C₂²×C₄	—	73	1	no	—
110	C₂²×C₄	—	73	1	no	—
111	M₁₆	—	146	2	no	—
112	M₁₆	—	146	2	no	—
113	M₁₆	—	146	2	no	—
114	C₂×C₈	—	73	1	no	—
115	C₂×C₈	—	73	2	no	—
116	C₂×C₈	—	73	2	no	—
117	C₂×C₈	—	73	1	no	—
118	C₂×C₈	—	73	1	no	—
119	C₂×C₈	—	73	1	no	—
120	C₂×C₈	—	146	1	no	—
121	C₂×C₈	—	146	1	no	—
122	C₂×C₈	—	146	1	no	—
123	C₂×C₈	—	146	1	no	—
124	C₁₆	—	146	2	no	—
125	C₁₆	—	146	2	no	—
126	C₂³	—	73	1	no	—
127	C₂³	C₂×#10596842	1	1	yes	characteristic
128	C₂³	C₂×#10596842	1	1	yes	characteristic
129	Q₈	—	146	2	no	—
130	D₄	—	2	2	no	—
131	D₄	—	146	2	no	—
132	D₄	—	146	2	no	—
133	C₂×C₄	—	73	2	no	—
134	C₂×C₄	—	73	2	no	—
135	C₂×C₄	—	73	1	no	—
136	C₂×C₄	—	73	1	no	—
137	C₂×C₄	—	73	1	no	—
138	C₂×C₄	—	146	1	no	—
139	C₂×C₄	—	146	1	no	—
140	C₂×C₄	—	146	1	no	—
141	C₂×C₄	—	146	1	no	—
142	C₂×C₄	C₂×#10596842	1	1	yes	characteristic
143	C₈	—	73	1	no	—
144	C₈	—	73	1	no	—
145	C₈	—	73	2	no	—
146	C₈	—	146	1	no	—
147	C₈	—	146	1	no	—
148	C₂²	—	2	2	no	—
149	C₂²	—	2	1	no	—
150	C₂²	—	2	2	no	—
151	C₂²	—	146	1	no	—
152	C₂²	—	146	2	no	—
153	C₂²	C₂²⋊#10596842	1	2	yes	—
154	C₂²	C₂²×#10596842	1	1	yes	characteristic
155	C₄	—	2	1	no	—
156	C₄	—	73	1	no	—
157	C₄	—	73	1	no	—
158	C₄	—	73	2	no	—
159	C₄	—	146	1	no	—
160	C₂	—	2	2	no	—
161	C₂	—	4	1	no	—
162	C₂	—	292	1	no	—
163	C₂	C₂×C₂²⋊#10596842	1	1	yes	characteristic
164	C₂	#17989708	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂²×#10596842
commutator, socle	C₂²×C₇₃	C₂²×C₈
Fitting	C₂×D₄×C₇₃	C₈
hypercenter	C₂×D₄	#10596842
hypocenter	C₇₃	#2113
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×#14556687	C₂	semidirect
2. C₂²×#11924040	C₂	semidirect
3. #17989684	C₂	semidirect
4. C₂²⋊#11924040	C₂	semidirect
83. C₇₃	#2113	semidirect