Finite Group Databasebeta

Subgroups — #22611898

Order 9344.

210 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 47 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	#17989400	C₂	1	1	yes	maximal, characteristic, complemented
2	#17989025	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#14556534	C₂	1	1	yes	maximal, characteristic, complemented
4	#17988858	C₂	1	1	yes	maximal, characteristic, complemented
5	#17988393	C₂	1	1	yes	maximal, characteristic, complemented
6	#17988372	C₂	1	1	yes	maximal, characteristic, complemented
7	#362×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
8	#14556639	—	2	1	no	—
9	#14556623	—	2	2	no	—
10	#14556622	—	2	1	no	—
11	#14556619	—	2	1	no	—
12	#14556614	—	2	1	no	—
13	#14556613	C₂²	1	1	yes	characteristic
14	Q₁₆×D₇₃	—	2	1	no	—
15	C₂×D₅₈₄	C₂²	1	1	yes	characteristic
16	#14556597	C₂²	1	1	yes	characteristic
17	#14556536	—	2	1	no	—
18	#14556534	—	2	2	no	—
19	D₁₁₆₈	—	2	1	no	—
20	C₂×#11924003	C₂²	1	1	yes	characteristic
21	#14556507	—	2	1	no	—
22	C₂×Q₁₆×C₇₃	C₂²	1	1	yes	characteristic
23	M₃₂×C₇₃	C₂²	1	1	yes	characteristic
24	#108×C₇₃	C₂²	1	1	yes	characteristic
25	#11924029	—	2	2	no	—
26	#11924029	—	4	1	no	—
27	#11924028	—	2	1	no	—
28	#11924028	—	4	2	no	—
29	#11924028	D₄	1	1	yes	characteristic
30	Q₈×D₇₃	—	2	2	no	—
31	Q₈×D₇₃	—	4	1	no	—
32	C₂×D₂₉₂	—	2	1	no	—
33	Dic₂₉₂	D₄	1	1	yes	characteristic
34	#11924018	—	2	2	no	—
35	#11924017	—	2	1	no	—
36	#11924017	—	4	1	no	—
37	#11924015	—	2	2	no	—
38	D₅₈₄	—	2	1	no	—
39	D₅₈₄	—	2	1	no	—
40	D₅₈₄	—	4	1	no	—
41	D₅₈₄	D₄	1	1	yes	characteristic
42	#11924012	D₄	1	1	yes	characteristic
43	#11924011	—	2	1	no	—
44	C₈×D₇₃	—	2	1	no	—
45	#11924003	—	2	1	no	—
46	C₂×Q₈×C₇₃	—	2	1	no	—
47	Q₁₆×C₇₃	—	2	1	no	—
48	Q₁₆×C₇₃	D₄	1	2	yes	—
49	M₁₆×C₇₃	—	2	1	no	—
50	C₂×C₈×C₇₃	C₂³	1	1	yes	characteristic
51	C₁₆×C₇₃	—	2	1	no	—
52	C₂²×D₇₃	—	4	1	no	—
53	Q₈×C₇₃	—	2	2	no	—
54	Q₈×C₇₃	—	4	1	no	—
55	C₂×C₄×C₇₃	—	4	1	no	—
56	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
57	C₂²⋊D₇₃	—	2	1	no	—
58	C₂²⋊D₇₃	—	4	1	no	—
59	D₂₉₂	—	2	1	no	—
60	D₂₉₂	—	4	2	no	—
61	D₂₉₂	—	4	1	no	—
62	D₂₉₂	—	4	1	no	—
63	D₂₉₂	D₈	1	1	yes	characteristic
64	C₄×D₇₃	—	2	1	no	—
65	C₄×D₇₃	—	4	2	no	—
66	C₄×D₇₃	—	4	2	no	—
67	C₄×D₇₃	—	4	1	no	—
68	Dic₁₄₆	—	2	1	no	—
69	Dic₁₄₆	—	4	2	no	—
70	Dic₁₄₆	D₈	1	1	yes	characteristic
71	C₈×C₇₃	—	4	1	no	—
72	C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
73	C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
74	#10596831	—	2	1	no	—
75	C₂²×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
76	C₂×D₇₃	—	2	1	no	—
77	C₂×D₇₃	—	4	1	no	—
78	C₂×D₇₃	—	8	1	no	—
79	C₂×D₇₃	—	16	1	no	—
80	C₄×C₇₃	—	4	2	no	—
81	C₄×C₇₃	#120	1	1	yes	characteristic
82	C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
83	Dic₇₃	—	2	1	no	—
84	Dic₇₃	—	4	1	no	—
85	C₂×C₇₃	—	2	1	no	—
86	C₂×C₇₃	#447	1	1	yes	characteristic
87	D₇₃	—	4	1	no	—
88	D₇₃	—	8	1	no	—
89	D₇₃	—	16	1	no	—
90	#2193	—	73	1	no	maximal, Sylow
91	C₇₃	#2193	1	1	yes	characteristic, complemented, Sylow
92	#578	—	73	1	no	—
93	#509	—	73	1	no	—
94	C₂×SD₃₂	—	73	1	no	—
95	#472	—	73	1	no	—
96	#362	D₇₃	1	1	yes	characteristic, complemented
97	#359	—	73	1	no	—
98	#350	—	73	1	no	—
99	2₋¹⁺⁴	—	146	1	no	—
100	#137	—	146	1	no	—
101	#137	—	146	2	no	—
102	AGL(1,ℤ/8)	—	146	1	no	—
103	#135	—	73	1	no	—
104	#135	—	146	1	no	—
105	C₂×Q₁₆	—	146	1	no	—
106	C₂×Q₁₆	C₂×D₇₃	1	1	yes	characteristic
107	C₂×D₈	—	73	1	no	—
108	#131	—	73	1	no	—
109	SD₃₂	—	146	1	no	—
110	SD₃₂	—	146	2	no	—
111	D₁₆	—	146	1	no	—
112	M₃₂	C₂×D₇₃	1	1	yes	characteristic
113	C₂×C₁₆	—	73	1	no	—
114	#108	C₂×D₇₃	1	1	yes	characteristic
115	#100	—	146	1	no	—
116	#41	—	292	2	no	—
117	#41	—	292	1	no	—
118	#41	—	73	1	no	—
119	#41	—	146	2	no	—
120	#41	—	146	1	no	—
121	C₂×Q₈	—	2	1	no	—
122	C₂×Q₈	—	292	1	no	—
123	C₂×Q₈	—	146	2	no	—
124	C₂×D₄	—	146	1	no	—
125	Q₁₆	—	2	1	no	—
126	Q₁₆	—	73	1	no	—
127	Q₁₆	—	146	2	no	—
128	Q₁₆	D₂₉₂	1	2	yes	—
129	SD₁₆	—	292	1	no	—
130	SD₁₆	—	146	2	no	—
131	SD₁₆	—	146	1	no	—
132	D₈	—	292	1	no	—
133	D₈	—	73	1	no	—
134	D₈	—	146	1	no	—
135	D₈	—	146	1	no	—
136	M₁₆	—	2	1	no	—
137	M₁₆	—	73	1	no	—
138	M₁₆	—	146	1	no	—
139	C₂×C₈	—	146	1	no	—
140	C₂×C₈	C₂²×D₇₃	1	1	yes	characteristic
141	C₁₆	—	2	1	no	—
142	C₁₆	—	146	1	no	—
143	C₂³	—	292	1	no	—
144	Q₈	—	2	2	no	—
145	Q₈	—	4	1	no	—
146	Q₈	—	292	2	no	—
147	Q₈	—	73	1	no	—
148	Q₈	—	146	1	no	—
149	D₄	—	292	2	no	—
150	D₄	—	292	1	no	—
151	D₄	—	292	1	no	—
152	D₄	—	292	1	no	—
153	D₄	—	73	1	no	—
154	D₄	—	146	1	no	—
155	D₄	—	146	1	no	—
156	C₂×C₄	—	4	1	no	—
157	C₂×C₄	—	292	2	no	—
158	C₂×C₄	—	292	2	no	—
159	C₂×C₄	—	292	1	no	—
160	C₂×C₄	—	146	1	no	—
161	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
162	C₈	—	4	1	no	—
163	C₈	—	146	1	no	—
164	C₈	D₄×D₇₃	1	1	yes	characteristic
165	C₈	C₂×D₂₉₂	1	1	yes	characteristic
166	C₂²	—	292	1	no	—
167	C₂²	—	584	1	no	—
168	C₂²	—	1168	1	no	—
169	C₂²	—	146	1	no	—
170	C₂²	#14556618	1	1	yes	characteristic
171	C₄	—	4	2	no	—
172	C₄	—	292	1	no	—
173	C₄	—	146	1	no	—
174	C₄	#14556561	1	1	yes	characteristic
175	C₄	D₈×D₇₃	1	1	yes	characteristic
176	C₂	—	2	1	no	—
177	C₂	—	292	1	no	—
178	C₂	—	584	1	no	—
179	C₂	—	1168	1	no	—
180	C₂	#17988722	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#17988722
commutator	C₂×C₈×C₇₃	C₂³
Frattini	C₂×C₈	C₂²×D₇₃
Fitting	#362×C₇₃	C₂
socle	C₂×C₇₃	#447
hypercenter	#362	D₇₃
hypocenter	C₇₃	#2193
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #362

Complements

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

N	H ≅ G/N	type
1. #17989400	C₂	semidirect
2. #17989025	C₂	semidirect
3. C₂×#14556534	C₂	semidirect
4. #17988858	C₂	semidirect
5. #17988393	C₂	semidirect
6. #17988372	C₂	semidirect
7. #362×C₇₃	C₂	semidirect
91. C₇₃	#2193	semidirect
96. #362	D₇₃	semidirect