Finite Group Databasebeta

Subgroups — #22626314

Order 9344.

164 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 164 conjugacy classes of subgroups — too many to draw. Only its 64 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×#14556666	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#14556664	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄⋊#11924031	C₂	1	1	yes	maximal, characteristic, complemented
4	#17989482	C₂	1	1	yes	maximal, characteristic, complemented
5	#17989482	C₂	1	1	yes	maximal, characteristic, complemented
6	#17989475	C₂	1	1	yes	maximal, characteristic, complemented
7	#17988652	C₂	1	1	yes	maximal, characteristic
8	#14556666	—	2	2	no	—
9	#14556664	—	2	2	no	—
10	C₂×#11924031	C₂²	1	1	yes	characteristic
11	C₂×#11924031	C₂²	1	1	yes	characteristic
12	C₂×#11924031	C₂²	1	1	yes	characteristic
13	C₂×#11924031	C₂²	1	1	yes	characteristic
14	C₂×C₈×D₇₃	C₂²	1	1	yes	characteristic
15	C₂×C₈×D₇₃	C₂²	1	1	yes	characteristic
16	C₄×D₂₉₂	C₄	1	1	yes	characteristic
17	#14556503	C₄	1	1	yes	characteristic
18	#14556503	C₄	1	1	yes	characteristic
19	C₄×#10596831	C₂²	1	1	yes	characteristic
20	C₄⋊C₈×C₇₃	C₄	1	1	yes	characteristic
21	#11924031	—	2	1	no	—
22	#11924031	—	2	1	no	—
23	#11924031	—	2	1	no	—
24	#11924031	D₄	1	2	yes	complemented
25	C₂×D₂₉₂	C₈	1	1	yes	characteristic
26	C₂×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
28	C₂×#10596831	C₂×C₄	1	1	yes	characteristic
29	C₂×#10596831	C₂³	1	1	yes	characteristic
30	C₈×D₇₃	—	2	1	no	—
31	C₈×D₇₃	—	2	1	no	—
32	C₈×D₇₃	—	2	1	no	—
33	C₈×D₇₃	—	2	1	no	—
34	C₄⋊Dic₇₃	C₈	1	1	yes	characteristic
35	#11924005	C₈	1	1	yes	characteristic
36	#11924005	C₈	1	1	yes	characteristic
37	C₂×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
39	C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
40	C₂²×D₇₃	C₂×C₈	1	1	yes	characteristic
41	C₂²×D₇₃	C₂×C₈	1	1	yes	characteristic
42	C₂×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
43	C₂×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
44	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
45	C₂×Dic₇₃	C₂×C₈	1	1	yes	characteristic
46	C₂×Dic₇₃	C₂×C₈	1	1	yes	characteristic
47	D₂₉₂	—	2	2	no	—
48	C₄×D₇₃	—	2	1	no	—
49	C₄×D₇₃	—	2	1	no	—
50	C₄×D₇₃	—	2	1	no	—
51	C₄×D₇₃	—	2	1	no	—
52	C₈×C₇₃	—	2	1	no	—
53	C₈×C₇₃	—	2	1	no	—
54	#10596831	—	2	1	no	—
55	#10596831	C₂×D₄	1	1	yes	characteristic
56	#10596831	#41	1	1	yes	characteristic
57	C₂²×C₇₃	C₂²×C₈	1	1	yes	characteristic
58	C₂×D₇₃	—	2	1	no	—
59	C₂×D₇₃	—	2	1	no	—
60	C₂×D₇₃	—	2	1	no	—
61	C₂×D₇₃	—	2	1	no	—
62	C₂×D₇₃	—	2	1	no	—
63	C₂×D₇₃	M₃₂	1	2	yes	complemented
64	C₄×C₇₃	—	2	1	no	—
65	C₄×C₇₃	—	2	1	no	—
66	C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
67	C₄×C₇₃	#131	1	1	yes	characteristic
68	Dic₇₃	—	2	1	no	—
69	Dic₇₃	—	2	1	no	—
70	C₂×C₇₃	C₈×D₄	1	1	yes	characteristic
71	C₂×C₇₃	C₂×M₃₂	1	1	yes	characteristic
72	C₂×C₇₃	#504	1	1	yes	characteristic
73	D₇₃	—	2	2	no	—
74	D₇₃	—	4	1	no	—
75	#2168	—	73	1	no	maximal, Sylow
76	C₇₃	#2168	1	1	yes	characteristic, complemented, Sylow
77	C₂×M₃₂	—	73	1	no	—
78	C₂²×C₁₆	—	73	1	no	—
79	C₈×D₄	—	73	1	no	—
80	C₄⋊C₁₆	—	73	1	no	—
81	C₂²⋊C₁₆	—	73	1	no	—
82	C₂²⋊C₁₆	—	73	1	no	—
83	#346	—	73	1	no	—
84	C₂²×C₈	—	73	1	no	—
85	C₂²×C₈	—	73	1	no	—
86	C₄×D₄	—	73	1	no	—
87	M₃₂	—	146	2	no	—
88	C₂×C₁₆	—	73	1	no	—
89	C₂×C₁₆	—	73	1	no	—
90	C₂×C₁₆	—	73	1	no	—
91	C₂×C₁₆	—	73	1	no	—
92	C₂×C₁₆	—	146	2	no	—
93	C₄⋊C₈	#64408	1	1	yes	characteristic
94	C₂²⋊C₈	—	73	1	no	—
95	C₂²⋊C₈	—	73	1	no	—
96	C₄×C₈	—	73	1	no	—
97	C₂×D₄	—	73	1	no	—
98	C₂²×C₄	—	73	1	no	—
99	C₂²×C₄	—	73	1	no	—
100	C₂×C₈	—	73	1	no	—
101	C₂×C₈	—	73	1	no	—
102	C₂×C₈	—	146	1	no	—
103	C₂×C₈	—	146	1	no	—
104	C₂×C₈	—	146	1	no	—
105	C₂×C₈	—	146	1	no	—
106	C₂×C₈	C₂×#64408	1	1	yes	characteristic
107	C₂×C₈	C₂×#64408	1	1	yes	characteristic
108	C₄⋊C₄	—	73	1	no	—
109	C₂²⋊C₄	—	73	1	no	—
110	C₂²⋊C₄	—	73	1	no	—
111	C₄²	C₂×#64408	1	1	yes	characteristic
112	C₁₆	—	73	2	no	—
113	C₁₆	—	146	1	no	—
114	C₁₆	—	146	1	no	—
115	C₁₆	—	146	1	no	—
116	C₂³	—	73	1	no	—
117	C₂³	—	73	1	no	—
118	D₄	—	146	2	no	—
119	C₂×C₄	—	73	1	no	—
120	C₂×C₄	—	73	1	no	—
121	C₂×C₄	—	146	1	no	—
122	C₂×C₄	—	146	1	no	—
123	C₂×C₄	—	146	1	no	—
124	C₂×C₄	—	146	1	no	—
125	C₂×C₄	#11924036	1	1	yes	characteristic
126	C₂×C₄	#11924036	1	1	yes	characteristic
127	C₂×C₄	C₂²×#64408	1	1	yes	characteristic
128	C₈	—	2	1	no	—
129	C₈	—	2	1	no	—
130	C₈	—	73	1	no	—
131	C₈	—	73	1	no	—
132	C₈	—	146	1	no	—
133	C₂²	—	73	2	no	—
134	C₂²	—	146	1	no	—
135	C₂²	—	146	1	no	—
136	C₂²	—	146	1	no	—
137	C₂²	—	146	1	no	—
138	C₂²	—	146	1	no	—
139	C₂²	C₂×#11924036	1	1	yes	characteristic
140	C₄	—	2	1	no	—
141	C₄	—	2	1	no	—
142	C₄	—	146	1	no	—
143	C₄	—	146	1	no	—
144	C₄	D₄×#64408	1	1	yes	characteristic
145	C₄	#14556679	1	1	yes	characteristic
146	C₂	—	292	1	no	—
147	C₂	—	146	2	no	—
148	C₂	#17989615	1	1	yes	characteristic
149	C₂	#17989643	1	1	yes	characteristic
150	C₂	#17989646	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#64408
commutator, socle	C₂²×C₇₃	C₂²×C₈
Fitting	C₄⋊C₈×C₇₃	C₄
hypercenter	C₄⋊C₈	#64408
hypocenter	C₇₃	#2168
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
1. C₂×#14556666	C₂	semidirect
2. C₂×#14556664	C₂	semidirect
3. C₄⋊#11924031	C₂	semidirect
4. #17989482	C₂	semidirect
5. #17989482	C₂	semidirect
6. #17989475	C₂	semidirect
24. #11924031	D₄	semidirect
63. C₂×D₇₃	M₃₂	semidirect
76. C₇₃	#2168	semidirect