Finite Group Databasebeta

Subgroups — #22610053

Order 9344.

172 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 73 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×#14556526	C₂	1	1	yes	maximal, characteristic
2	#17988623	C₂	1	1	yes	maximal, characteristic
3	#17988543	C₂	1	1	yes	maximal, characteristic
4	#17988359	C₂	1	1	yes	maximal, characteristic
5	#17988357	C₂	1	1	yes	maximal, characteristic
6	#17988343	C₂	1	1	yes	maximal, characteristic
7	H(3,ℤ/4)×C₇₃	C₂	1	1	yes	maximal, characteristic
8	C₂×#11924012	C₂²	1	1	yes	characteristic
9	C₂×#11924012	C₂²	1	1	yes	characteristic
10	C₂²×#10596831	C₂²	1	1	yes	characteristic
11	#14556550	C₂²	1	1	yes	characteristic
12	#14556526	—	2	1	no	—
13	#14556526	—	2	1	no	—
14	#14556524	—	2	1	no	—
15	#14556522	—	2	1	no	—
16	#14556502	C₄	1	2	yes	complemented
17	C₄×#10596831	C₄	1	2	yes	complemented
18	C₂×M₁₆×C₇₃	C₂²	1	1	yes	characteristic
19	#117×C₇₃	C₂²	1	1	yes	characteristic
20	#117×C₇₃	C₂²	1	1	yes	characteristic
21	#11924012	—	2	1	no	—
22	#11924012	—	2	1	no	—
23	#11924012	—	2	1	no	—
24	#11924012	—	2	1	no	—
25	C₂×#10596831	—	2	1	no	—
26	C₂×#10596831	C₂×C₄	1	2	yes	—
27	C₂×#10596831	C₂×C₄	1	1	yes	characteristic
28	C₂×#10596831	C₂×C₄	1	1	yes	characteristic
29	C₂×#10596831	D₄	1	1	yes	characteristic
30	C₂×#10596831	D₄	1	1	yes	characteristic
31	C₂×#10596831	Q₈	1	1	yes	characteristic
32	C₂×#10596831	Q₈	1	1	yes	characteristic
33	C₄⋊Dic₇₃	—	2	1	no	—
34	C₂²⋊Dic₇₃	—	2	1	no	—
35	C₄×Dic₇₃	—	2	1	no	—
36	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
37	M₁₆×C₇₃	—	2	1	no	—
38	M₁₆×C₇₃	—	2	1	no	—
39	C₂×C₈×C₇₃	—	2	1	no	—
40	C₄⋊C₄×C₇₃	—	2	1	no	—
41	C₄⋊C₄×C₇₃	—	2	1	no	—
42	C₂²⋊C₄×C₇₃	—	2	1	no	—
43	C₂²⋊C₄×C₇₃	—	2	1	no	—
44	C₄²×C₇₃	—	2	1	no	—
45	C₄²×C₇₃	C₂×C₄	1	2	yes	—
46	C₂³×C₇₃	#41	1	1	yes	characteristic
47	C₂×C₄×C₇₃	—	2	2	no	—
48	C₂×C₄×C₇₃	—	4	1	no	—
49	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
53	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
54	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
55	C₂×Dic₇₃	—	4	1	no	—
56	C₈×C₇₃	—	4	1	no	—
57	#10596831	—	2	2	no	—
58	#10596831	—	4	1	no	—
59	#10596831	C₄⋊C₄	1	1	yes	characteristic
60	#10596831	C₄⋊C₄	1	1	yes	characteristic
61	#10596831	C₄⋊C₄	1	1	yes	characteristic
62	#10596831	C₄⋊C₄	1	1	yes	characteristic
63	C₂²×C₇₃	—	4	1	no	—
64	C₂²×C₇₃	#117	1	1	yes	characteristic
65	C₂²×C₇₃	#124	1	1	yes	characteristic
66	C₂²×C₇₃	#125	1	1	yes	characteristic
67	C₄×C₇₃	—	4	2	no	—
68	C₄×C₇₃	—	8	1	no	—
69	C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₄×C₇₃	#117	1	1	yes	characteristic
71	C₄×C₇₃	#124	1	1	yes	characteristic
72	C₄×C₇₃	#125	1	1	yes	characteristic
73	Dic₇₃	—	8	1	no	—
74	C₂×C₇₃	—	2	1	no	—
75	C₂×C₇₃	—	2	1	no	—
76	C₂×C₇₃	—	2	1	no	—
77	C₂×C₇₃	#382	1	1	yes	characteristic
78	#1833	—	73	1	no	maximal, Sylow
79	C₇₃	#1833	1	1	yes	characteristic, complemented, Sylow
80	C₂×#108	—	73	1	no	—
81	#427	—	73	1	no	—
82	#405	—	73	1	no	—
83	#344	—	73	1	no	—
84	#344	—	73	1	no	—
85	H(3,ℤ/4)	—	73	1	no	—
86	H(3,ℤ/4)	D₇₃	1	1	yes	characteristic
87	C₂×M₁₆	—	73	1	no	—
88	C₂×M₁₆	—	73	1	no	—
89	C₂×M₁₆	C₂×D₇₃	1	1	yes	characteristic
90	C₂²×C₈	—	73	1	no	—
91	#117	—	73	1	no	—
92	#117	C₂×D₇₃	1	1	yes	characteristic
93	#117	C₂×D₇₃	1	1	yes	characteristic
94	#108	—	146	1	no	—
95	#108	—	146	1	no	—
96	#107	—	146	1	no	—
97	#106	—	146	1	no	—
98	#97	—	73	2	no	—
99	C₄×C₈	—	73	2	no	—
100	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
101	M₁₆	—	2	1	no	—
102	M₁₆	—	2	1	no	—
103	M₁₆	—	146	1	no	—
104	M₁₆	—	146	1	no	—
105	M₁₆	—	146	1	no	—
106	M₁₆	—	146	1	no	—
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	—	73	1	no	—
109	C₂×C₈	—	73	1	no	—
110	C₂×C₈	—	73	1	no	—
111	C₂×C₈	—	73	1	no	—
112	C₂×C₈	—	73	1	no	—
113	C₂×C₈	—	73	1	no	—
114	C₂×C₈	—	73	2	no	—
115	C₂×C₈	—	146	1	no	—
116	C₄⋊C₄	—	2	1	no	—
117	C₄⋊C₄	—	2	1	no	—
118	C₄⋊C₄	—	146	1	no	—
119	C₂²⋊C₄	—	2	1	no	—
120	C₂²⋊C₄	—	2	1	no	—
121	C₂²⋊C₄	—	146	1	no	—
122	C₄²	—	2	1	no	—
123	C₄²	—	146	1	no	—
124	C₄²	C₄×D₇₃	1	2	yes	—
125	C₂³	#11924027	1	1	yes	characteristic
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	4	1	no	—
128	C₂×C₄	—	292	1	no	—
129	C₂×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
130	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
131	C₂×C₄	Q₈×D₇₃	1	1	yes	characteristic
132	C₂×C₄	#11924027	1	1	yes	characteristic
133	C₂×C₄	#11924028	1	1	yes	characteristic
134	C₂×C₄	#11924028	1	1	yes	characteristic
135	C₈	—	4	1	no	—
136	C₈	—	292	1	no	—
137	C₈	—	73	1	no	—
138	C₈	—	73	1	no	—
139	C₈	—	73	1	no	—
140	C₈	—	73	1	no	—
141	C₈	—	146	2	no	—
142	C₂²	—	4	1	no	—
143	C₂²	#14556547	1	1	yes	characteristic
144	C₂²	#14556577	1	1	yes	characteristic
145	C₂²	#14556581	1	1	yes	characteristic
146	C₄	—	4	2	no	—
147	C₄	—	8	1	no	—
148	C₄	—	584	1	no	—
149	C₄	C₄⋊C₄×D₇₃	1	1	yes	characteristic
150	C₄	#14556548	1	1	yes	characteristic
151	C₄	#14556577	1	1	yes	characteristic
152	C₄	#14556581	1	1	yes	characteristic
153	C₂	—	2	1	no	—
154	C₂	—	2	1	no	—
155	C₂	—	2	1	no	—
156	C₂	#17988451	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#14556548
commutator	C₂×C₄×C₇₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇₃
Fitting	H(3,ℤ/4)×C₇₃	C₂
socle	C₂×C₇₃	#382
hypercenter	H(3,ℤ/4)	D₇₃
hypocenter	C₇₃	#1833
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄ ◁ #117 ◁ H(3,ℤ/4)

Complements

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

N	H ≅ G/N	type
16. #14556502	C₄	semidirect
17. C₄×#10596831	C₄	semidirect
79. C₇₃	#1833	semidirect