Finite Group Databasebeta

Subgroups — #22627107

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	C₂×#14556687	C₂	1	2	yes	maximal, complemented
2	#17989686	C₂	1	2	yes	maximal, complemented
3	#17989685	C₂	1	2	yes	maximal, complemented
4	C₂×#14556678	C₂	1	1	yes	maximal, characteristic
5	#14556687	—	2	2	no	—
6	#14556687	C₂²	1	4	yes	—
7	C₂×#11924040	—	2	2	no	—
8	#14556678	—	2	2	no	—
9	#14556678	—	2	2	no	—
10	C₂×#11924038	C₂²	1	1	yes	characteristic
11	C₂×#11924037	C₄	1	1	yes	characteristic
12	C₂×#11924037	C₂²	1	1	yes	characteristic
13	C₂×#11924036	C₄	1	1	yes	characteristic
14	C₂²×#10596833	C₄	1	1	yes	characteristic
15	C₂²×#10596833	C₂²	1	1	yes	characteristic
16	C₂×#11924027	C₄	1	1	yes	characteristic
17	#11924040	—	2	4	no	—
18	#11924038	—	2	2	no	—
19	#11924037	—	2	2	no	—
20	#11924037	—	2	2	no	—
21	#11924036	—	2	1	no	—
22	#11924036	—	2	1	no	—
23	C₂×#10596833	—	2	1	no	—
24	C₂×#10596833	—	2	1	no	—
25	C₂×#10596833	C₂×C₄	1	1	yes	characteristic
26	C₂×#10596833	C₂×C₄	1	1	yes	characteristic
27	C₂×#10596833	C₂×C₄	1	1	yes	characteristic
28	C₂×#10596833	D₄	1	2	yes	—
29	C₂×#10596833	D₄	1	2	yes	—
30	C₂×#10596833	C₂³	1	1	yes	characteristic
31	#11924027	—	2	2	no	—
32	#11924027	—	2	2	no	—
33	C₂×Dic₁₄₆	C₈	1	1	yes	characteristic
34	C₂×C₂²⋊D₇₃	C₈	1	1	yes	characteristic
35	C₂×C₂²⋊D₇₃	C₈	1	1	yes	characteristic
36	C₂²×Dic₇₃	C₂×C₄	1	1	yes	characteristic
37	C₂²×Dic₇₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
39	C₂×D₄×C₇₃	C₈	1	1	yes	characteristic
40	C₂³×C₇₃	C₂×C₈	1	1	yes	characteristic
41	C₂³×C₇₃	C₂×C₈	1	1	yes	characteristic
42	C₂²×D₇₃	C₂×C₈	1	1	yes	characteristic
43	D₄×C₇₃	—	2	2	no	—
44	C₂×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
45	C₂²⋊D₇₃	—	2	2	no	—
46	C₂²⋊D₇₃	—	2	2	no	—
47	C₂×Dic₇₃	—	2	1	no	—
48	C₂×Dic₇₃	—	2	1	no	—
49	C₂×Dic₇₃	C₂²⋊C₄	1	2	yes	—
50	C₂×Dic₇₃	C₂×C₈	1	1	yes	characteristic
51	C₂×Dic₇₃	C₂×C₈	1	1	yes	characteristic
52	C₂×Dic₇₃	M₁₆	1	2	yes	—
53	C₂×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
54	C₄×D₇₃	—	2	1	no	—
55	C₄×D₇₃	—	2	1	no	—
56	Dic₁₄₆	—	2	2	no	—
57	#10596833	—	2	1	no	—
58	#10596833	—	2	1	no	—
59	#10596833	C₂²⋊C₄	1	1	yes	characteristic
60	#10596833	C₂²⋊C₄	1	1	yes	characteristic
61	#10596833	C₂×D₄	1	2	yes	—
62	C₂²×C₇₃	—	2	1	no	—
63	C₂²×C₇₃	—	4	1	no	—
64	C₂²×C₇₃	—	4	1	no	—
65	C₂²×C₇₃	C₂²⋊C₈	1	2	yes	—
66	C₂²×C₇₃	C₂²×C₈	1	1	yes	characteristic
67	C₂×D₇₃	—	2	1	no	—
68	C₂×D₇₃	—	4	1	no	—
69	C₄×C₇₃	—	2	1	no	—
70	Dic₇₃	—	2	1	no	—
71	Dic₇₃	C₂²⋊C₈	1	2	yes	—
72	Dic₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
73	Dic₇₃	C₂×M₁₆	1	1	yes	characteristic
74	C₂×C₇₃	—	2	1	no	—
75	C₂×C₇₃	—	2	2	no	—
76	C₂×C₇₃	—	4	1	no	—
77	C₂×C₇₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
78	D₇₃	—	4	1	no	—
79	#2115	—	73	1	no	maximal, Sylow
80	C₇₃	#2115	1	1	yes	characteristic, complemented, Sylow
81	C₂×#131	—	73	1	no	—
82	C₂×M₃₂	—	73	2	no	—
83	#349	—	73	2	no	—
84	#349	—	73	2	no	—
85	C₂×#41	—	73	1	no	—
86	#131	—	146	2	no	—
87	#131	—	146	2	no	—
88	C₂×M₁₆	—	73	1	no	—
89	C₂×M₁₆	—	73	1	no	—
90	C₂×M₁₆	—	73	1	no	—
91	C₂²×C₈	—	73	1	no	—
92	C₂²×C₈	—	73	1	no	—
93	C₂²×C₈	—	73	1	no	—
94	M₃₂	—	73	4	no	—
95	M₃₂	—	146	2	no	—
96	C₂×C₁₆	—	146	2	no	—
97	#41	—	146	2	no	—
98	#41	—	146	2	no	—
99	C₂×Q₈	—	73	1	no	—
100	C₂×D₄	—	73	1	no	—
101	C₂×D₄	—	73	1	no	—
102	C₂×D₄	#10596842	1	1	yes	characteristic
103	C₂²×C₄	—	73	1	no	—
104	C₂²×C₄	—	73	1	no	—
105	C₂²×C₄	—	73	1	no	—
106	M₁₆	—	146	2	no	—
107	M₁₆	—	146	2	no	—
108	M₁₆	—	146	2	no	—
109	C₂×C₈	—	73	1	no	—
110	C₂×C₈	—	73	2	no	—
111	C₂×C₈	—	73	2	no	—
112	C₂×C₈	—	73	1	no	—
113	C₂×C₈	—	73	1	no	—
114	C₂×C₈	—	73	1	no	—
115	C₂×C₈	—	146	1	no	—
116	C₂×C₈	—	146	1	no	—
117	C₂×C₈	—	146	1	no	—
118	C₂×C₈	—	146	1	no	—
119	C₁₆	—	146	4	no	—
120	C₂³	—	73	1	no	—
121	C₂³	C₂×#10596842	1	1	yes	characteristic
122	C₂³	C₂×#10596842	1	1	yes	characteristic
123	Q₈	—	146	2	no	—
124	D₄	—	2	2	no	—
125	D₄	—	146	2	no	—
126	D₄	—	146	2	no	—
127	C₂×C₄	—	73	2	no	—
128	C₂×C₄	—	73	2	no	—
129	C₂×C₄	—	73	1	no	—
130	C₂×C₄	—	73	1	no	—
131	C₂×C₄	—	73	1	no	—
132	C₂×C₄	—	146	1	no	—
133	C₂×C₄	—	146	1	no	—
134	C₂×C₄	—	146	1	no	—
135	C₂×C₄	—	146	1	no	—
136	C₂×C₄	C₂×#10596842	1	1	yes	characteristic
137	C₈	—	73	2	no	—
138	C₈	—	73	1	no	—
139	C₈	—	73	1	no	—
140	C₈	—	146	1	no	—
141	C₈	—	146	1	no	—
142	C₂²	—	2	1	no	—
143	C₂²	—	4	1	no	—
144	C₂²	—	4	1	no	—
145	C₂²	—	292	1	no	—
146	C₂²	—	146	1	no	—
147	C₂²	C₂²⋊#10596842	1	2	yes	—
148	C₂²	C₂²×#10596842	1	1	yes	characteristic
149	C₄	—	2	1	no	—
150	C₄	—	73	1	no	—
151	C₄	—	73	1	no	—
152	C₄	—	73	2	no	—
153	C₄	—	146	1	no	—
154	C₂	—	2	1	no	—
155	C₂	—	2	2	no	—
156	C₂	—	4	1	no	—
157	C₂	—	292	1	no	—
158	C₂	C₂×C₂²⋊#10596842	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×C₂²⋊#10596842
commutator	C₂²×C₇₃	C₂²×C₈
Frattini	C₂²	C₂²×#10596842
Fitting	C₂×D₄×C₇₃	C₈
socle	C₂×C₇₃	C₂×C₂²⋊C₈
hypercenter	C₂×D₄	#10596842
hypocenter	C₇₃	#2115
LCS	G ▷ C₂²×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×#14556687	C₂	semidirect
2. #17989686	C₂	semidirect
3. #17989685	C₂	semidirect
80. C₇₃	#2115	semidirect