Finite Group Databasebeta

Subgroups — #22609461

Order 9344.

170 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 51 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	#17988874	C₂	1	1	yes	maximal, characteristic, complemented
2	#17988779	C₂	1	1	yes	maximal, characteristic
3	#17988666	C₂	1	1	yes	maximal, characteristic, complemented
4	#17988340	C₂	1	1	yes	maximal, characteristic, complemented
5	#17988336	C₂	1	1	yes	maximal, characteristic, complemented
6	#17988331	C₂	1	1	yes	maximal, characteristic, complemented
7	#486×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11924013	—	2	1	no	—
9	C₂×#11924011	—	2	1	no	—
10	C₄×D₂₉₂	C₂²	1	1	yes	characteristic
11	#14556522	—	2	1	no	—
12	C₄⋊#10596831	C₂²	1	1	yes	characteristic
13	C₄⋊#10596831	C₂²	1	1	yes	characteristic
14	C₄⋊#10596831	C₂²	1	1	yes	characteristic
15	#14556512	—	2	1	no	—
16	#14556511	—	2	1	no	—
17	#14556503	—	2	1	no	—
18	C₄⋊Q₈×C₇₃	C₂²	1	1	yes	characteristic
19	#127×C₇₃	C₂²	1	1	yes	characteristic
20	#102×C₇₃	—	2	1	no	—
21	#102×C₇₃	—	2	1	no	—
22	C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
23	C₂×D₂₉₂	D₄	1	1	yes	characteristic
24	C₂×C₄×D₇₃	—	2	1	no	—
25	#11924013	—	4	2	no	—
26	#11924011	—	4	2	no	—
27	C₂×#10596831	—	2	1	no	—
28	C₂×#10596831	—	2	1	no	—
29	C₂×#10596831	—	2	1	no	—
30	C₄⋊Dic₇₃	D₄	1	1	yes	characteristic
31	#11924005	—	2	1	no	—
32	C₂×Q₈×C₇₃	—	2	1	no	—
33	C₂×D₄×C₇₃	—	2	1	no	—
34	C₂×D₄×C₇₃	—	2	1	no	—
35	C₂×D₄×C₇₃	—	2	1	no	—
36	C₂×C₈×C₇₃	D₄	1	1	yes	characteristic
37	C₂×C₈×C₇₃	D₄	1	1	yes	characteristic
38	C₄⋊C₄×C₇₃	—	2	1	no	—
39	C₄⋊C₄×C₇₃	—	2	1	no	—
40	C₄²×C₇₃	C₂³	1	1	yes	characteristic
41	C₂³×C₇₃	—	4	1	no	—
42	C₂²×D₇₃	—	2	1	no	—
43	Q₈×C₇₃	—	8	1	no	—
44	D₄×C₇₃	—	4	1	no	—
45	D₄×C₇₃	—	4	1	no	—
46	D₄×C₇₃	—	4	1	no	—
47	D₄×C₇₃	—	4	1	no	—
48	D₄×C₇₃	—	8	1	no	—
49	C₂×C₄×C₇₃	—	4	1	no	—
50	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
53	C₂×Dic₇₃	—	2	1	no	—
54	D₂₉₂	—	2	1	no	—
55	D₂₉₂	—	2	1	no	—
56	C₄×D₇₃	—	4	1	no	—
57	C₄×D₇₃	—	4	1	no	—
58	C₈×C₇₃	—	2	1	no	—
59	C₈×C₇₃	—	2	1	no	—
60	#10596831	—	4	1	no	—
61	#10596831	—	4	1	no	—
62	#10596831	—	4	1	no	—
63	C₂²×C₇₃	—	8	1	no	—
64	C₂²×C₇₃	—	8	1	no	—
65	C₂²×C₇₃	—	8	1	no	—
66	C₂²×C₇₃	#121	1	1	yes	characteristic
67	C₂×D₇₃	—	4	1	no	—
68	C₂×D₇₃	—	4	1	no	—
69	C₂×D₇₃	—	4	1	no	—
70	C₄×C₇₃	—	2	1	no	—
71	C₄×C₇₃	—	8	1	no	—
72	C₄×C₇₃	#135	1	1	yes	characteristic
73	C₄×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
74	C₄×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
75	C₄×C₇₃	#137	1	1	yes	characteristic
76	Dic₇₃	—	4	1	no	—
77	C₂×C₇₃	—	16	1	no	—
78	C₂×C₇₃	#463	1	1	yes	characteristic
79	C₂×C₇₃	#468	1	1	yes	characteristic
80	C₂×C₇₃	#472	1	1	yes	characteristic
81	D₇₃	—	8	1	no	—
82	#1697	—	73	1	no	maximal, Sylow
83	C₇₃	#1697	1	1	yes	characteristic, complemented, Sylow
84	#486	D₇₃	1	1	yes	characteristic, complemented
85	#476	—	73	1	no	—
86	#459	—	73	1	no	—
87	#436	—	73	1	no	—
88	#335	—	73	1	no	—
89	#333	—	73	1	no	—
90	#331	—	73	1	no	—
91	C₂×D₈	—	146	1	no	—
92	C₂×M₁₆	—	146	1	no	—
93	C₄⋊Q₈	C₂×D₇₃	1	1	yes	characteristic
94	#127	C₂×D₇₃	1	1	yes	characteristic
95	C₄×D₄	—	73	1	no	—
96	#106	—	146	1	no	—
97	C₄⋊C₈	—	73	1	no	—
98	C₄⋊C₈	—	73	1	no	—
99	C₄⋊C₈	—	73	1	no	—
100	#102	—	2	1	no	—
101	#102	—	2	1	no	—
102	#102	—	146	1	no	—
103	#102	—	146	1	no	—
104	C₂²⋊C₈	—	146	1	no	—
105	C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
106	C₂×Q₈	—	2	1	no	—
107	C₂×D₄	—	2	1	no	—
108	C₂×D₄	—	2	1	no	—
109	C₂×D₄	—	2	1	no	—
110	C₂×D₄	—	73	1	no	—
111	C₂²×C₄	—	146	1	no	—
112	D₈	—	292	2	no	—
113	M₁₆	—	292	2	no	—
114	C₂×C₈	—	146	1	no	—
115	C₂×C₈	—	146	1	no	—
116	C₂×C₈	—	146	1	no	—
117	C₂×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
118	C₂×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
119	C₄⋊C₄	—	2	1	no	—
120	C₄⋊C₄	—	2	1	no	—
121	C₄⋊C₄	—	73	1	no	—
122	C₂²⋊C₄	—	146	1	no	—
123	C₄²	C₂²×D₇₃	1	1	yes	characteristic
124	C₂³	—	4	1	no	—
125	C₂³	—	146	1	no	—
126	Q₈	—	8	1	no	—
127	D₄	—	4	1	no	—
128	D₄	—	4	1	no	—
129	D₄	—	4	1	no	—
130	D₄	—	4	1	no	—
131	D₄	—	8	1	no	—
132	D₄	—	146	1	no	—
133	D₄	—	146	1	no	—
134	C₂×C₄	—	4	1	no	—
135	C₂×C₄	—	292	1	no	—
136	C₂×C₄	—	292	1	no	—
137	C₂×C₄	—	146	1	no	—
138	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
139	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
140	C₂×C₄	#11924027	1	1	yes	characteristic
141	C₈	—	2	1	no	—
142	C₈	—	2	1	no	—
143	C₈	—	292	1	no	—
144	C₈	—	292	1	no	—
145	C₈	—	292	1	no	—
146	C₂²	—	8	1	no	—
147	C₂²	—	8	1	no	—
148	C₂²	—	8	1	no	—
149	C₂²	—	292	1	no	—
150	C₂²	—	292	1	no	—
151	C₂²	—	292	1	no	—
152	C₂²	#14556562	1	1	yes	characteristic
153	C₄	—	2	1	no	—
154	C₄	—	8	1	no	—
155	C₄	—	292	1	no	—
156	C₄	#14556611	1	1	yes	characteristic
157	C₄	#14556616	1	1	yes	characteristic
158	C₄	#14556618	1	1	yes	characteristic
159	C₄	#14556620	1	1	yes	characteristic
160	C₂	—	16	1	no	—
161	C₂	—	584	1	no	—
162	C₂	#17988799	1	1	yes	characteristic
163	C₂	#17988830	1	1	yes	characteristic
164	C₂	#17988854	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#14556562
commutator	C₄²×C₇₃	C₂³
Frattini	C₄²	C₂²×D₇₃
Fitting	#486×C₇₃	C₂
socle	C₂²×C₇₃	#121
hypercenter	#486	D₇₃
hypocenter	C₇₃	#1697
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #486

Complements

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

N	H ≅ G/N	type
1. #17988874	C₂	semidirect
3. #17988666	C₂	semidirect
4. #17988340	C₂	semidirect
5. #17988336	C₂	semidirect
6. #17988331	C₂	semidirect
7. #486×C₇₃	C₂	semidirect
83. C₇₃	#1697	semidirect
84. #486	D₇₃	semidirect