Finite Group Databasebeta

Subgroups — #22611906

Order 9344.

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

normal subgroups only

This group has 206 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	#17989401	C₂	1	1	yes	maximal, characteristic
2	C₂×#14556537	C₂	1	1	yes	maximal, characteristic, complemented
3	#17988859	C₂	1	1	yes	maximal, characteristic, complemented
4	#17988404	C₂	1	1	yes	maximal, characteristic, complemented
5	#17988394	C₂	1	1	yes	maximal, characteristic, complemented
6	#17988372	C₂	1	1	yes	maximal, characteristic, complemented
7	#510×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
8	#14556638	—	2	1	no	—
9	#14556622	—	2	1	no	—
10	#14556621	—	2	1	no	—
11	#14556620	—	2	2	no	—
12	#14556610	—	2	1	no	—
13	C₂×Dic₂₉₂	C₂²	1	1	yes	characteristic
14	Q₁₆×D₇₃	—	2	1	no	—
15	#14556597	C₂²	1	1	yes	characteristic
16	#14556537	—	2	2	no	—
17	C₂×#11924003	C₂²	1	1	yes	characteristic
18	#14556527	C₂²	1	1	yes	characteristic
19	#14556510	—	2	1	no	—
20	#135×C₇₃	C₂²	1	1	yes	characteristic
21	C₂×Q₁₆×C₇₃	C₂²	1	1	yes	characteristic
22	Q₃₂×C₇₃	—	2	1	no	—
23	SD₃₂×C₇₃	—	2	1	no	—
24	M₃₂×C₇₃	C₂²	1	1	yes	characteristic
25	#11924028	—	4	1	no	—
26	#11924028	D₄	1	1	yes	characteristic
27	#11924027	—	2	2	no	—
28	#11924027	—	4	2	no	—
29	C₂×Dic₁₄₆	—	2	1	no	—
30	C₂×Dic₁₄₆	—	4	1	no	—
31	Q₈×D₇₃	—	2	2	no	—
32	Dic₂₉₂	—	2	1	no	—
33	Dic₂₉₂	D₄	1	2	yes	—
34	#11924018	—	2	2	no	—
35	#11924018	—	4	1	no	—
36	#11924016	—	2	2	no	—
37	#11924015	—	4	1	no	—
38	#11924012	—	2	1	no	—
39	#11924012	D₄	1	1	yes	characteristic
40	#11924011	—	2	1	no	—
41	C₈×D₇₃	—	2	1	no	—
42	#11924003	—	2	1	no	—
43	#41×C₇₃	—	2	1	no	—
44	C₂×Q₈×C₇₃	—	2	1	no	—
45	Q₁₆×C₇₃	—	2	1	no	—
46	Q₁₆×C₇₃	—	2	1	no	—
47	Q₁₆×C₇₃	D₄	1	1	yes	characteristic
48	SD₁₆×C₇₃	—	2	1	no	—
49	D₈×C₇₃	D₄	1	1	yes	characteristic
50	C₂×C₈×C₇₃	C₂³	1	1	yes	characteristic
51	C₁₆×C₇₃	—	2	1	no	—
52	Q₈×C₇₃	—	2	1	no	—
53	Q₈×C₇₃	—	4	1	no	—
54	Q₈×C₇₃	—	4	1	no	—
55	D₄×C₇₃	—	2	1	no	—
56	D₄×C₇₃	—	4	1	no	—
57	C₂×C₄×C₇₃	—	4	1	no	—
58	C₂×C₄×C₇₃	—	4	1	no	—
59	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
60	C₂²⋊D₇₃	—	2	1	no	—
61	C₂²⋊D₇₃	—	4	2	no	—
62	C₂×Dic₇₃	—	4	2	no	—
63	C₂×Dic₇₃	—	4	1	no	—
64	D₂₉₂	D₈	1	1	yes	characteristic
65	C₄×D₇₃	—	2	1	no	—
66	C₄×D₇₃	—	4	2	no	—
67	Dic₁₄₆	—	2	2	no	—
68	Dic₁₄₆	—	4	2	no	—
69	Dic₁₄₆	—	4	1	no	—
70	Dic₁₄₆	D₈	1	1	yes	characteristic
71	C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
72	C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
73	#10596831	—	2	1	no	—
74	#10596831	—	4	1	no	—
75	C₂²×C₇₃	—	4	1	no	—
76	C₂²×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
77	C₂×D₇₃	—	2	1	no	—
78	C₄×C₇₃	—	4	1	no	—
79	C₄×C₇₃	—	8	1	no	—
80	C₄×C₇₃	#120	1	1	yes	characteristic
81	C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
82	Dic₇₃	—	2	1	no	—
83	Dic₇₃	—	4	2	no	—
84	C₂×C₇₃	—	2	1	no	—
85	C₂×C₇₃	—	8	1	no	—
86	C₂×C₇₃	#447	1	1	yes	characteristic
87	D₇₃	—	4	1	no	—
88	#2195	—	73	1	no	maximal, Sylow
89	C₇₃	#2195	1	1	yes	characteristic, complemented, Sylow
90	#578	—	73	1	no	—
91	#510	D₇₃	1	1	yes	characteristic, complemented
92	C₂×Q₃₂	—	73	1	no	—
93	#473	—	73	1	no	—
94	#362	—	73	1	no	—
95	#359	—	73	1	no	—
96	#350	—	73	1	no	—
97	2₋¹⁺⁴	—	146	1	no	—
98	#137	—	146	1	no	—
99	#137	—	146	2	no	—
100	#137	—	146	1	no	—
101	#135	—	146	1	no	—
102	#135	C₂×D₇₃	1	1	yes	characteristic
103	C₂×Q₁₆	—	73	1	no	—
104	C₂×Q₁₆	—	146	1	no	—
105	C₂×Q₁₆	C₂×D₇₃	1	1	yes	characteristic
106	#131	—	73	1	no	—
107	Q₃₂	—	2	1	no	—
108	Q₃₂	—	146	2	no	—
109	SD₃₂	—	2	1	no	—
110	M₃₂	C₂×D₇₃	1	1	yes	characteristic
111	C₂×C₁₆	—	73	1	no	—
112	#108	—	73	1	no	—
113	#101	—	146	1	no	—
114	#41	—	2	1	no	—
115	#41	—	292	1	no	—
116	#41	—	292	2	no	—
117	#41	—	73	1	no	—
118	#41	—	146	2	no	—
119	C₂×Q₈	—	2	1	no	—
120	C₂×Q₈	—	292	1	no	—
121	C₂×Q₈	—	146	2	no	—
122	C₂×Q₈	—	146	1	no	—
123	Q₁₆	—	2	1	no	—
124	Q₁₆	—	2	1	no	—
125	Q₁₆	—	292	1	no	—
126	Q₁₆	—	73	2	no	—
127	Q₁₆	—	146	2	no	—
128	Q₁₆	—	146	1	no	—
129	Q₁₆	C₂²⋊D₇₃	1	1	yes	characteristic
130	SD₁₆	—	2	1	no	—
131	SD₁₆	—	292	1	no	—
132	SD₁₆	—	146	2	no	—
133	D₈	C₂²⋊D₇₃	1	1	yes	characteristic
134	M₁₆	—	73	1	no	—
135	M₁₆	—	146	1	no	—
136	M₁₆	—	146	1	no	—
137	C₂×C₈	—	146	1	no	—
138	C₂×C₈	C₂²×D₇₃	1	1	yes	characteristic
139	C₁₆	—	2	1	no	—
140	C₁₆	—	146	1	no	—
141	Q₈	—	2	1	no	—
142	Q₈	—	4	1	no	—
143	Q₈	—	4	1	no	—
144	Q₈	—	292	2	no	—
145	Q₈	—	292	1	no	—
146	Q₈	—	73	1	no	—
147	Q₈	—	146	2	no	—
148	D₄	—	2	1	no	—
149	D₄	—	4	1	no	—
150	D₄	—	292	2	no	—
151	D₄	—	73	1	no	—
152	D₄	—	146	1	no	—
153	C₂×C₄	—	4	1	no	—
154	C₂×C₄	—	4	1	no	—
155	C₂×C₄	—	292	2	no	—
156	C₂×C₄	—	292	1	no	—
157	C₂×C₄	—	292	2	no	—
158	C₂×C₄	—	146	1	no	—
159	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
160	C₈	—	292	1	no	—
161	C₈	—	146	1	no	—
162	C₈	D₄×D₇₃	1	1	yes	characteristic
163	C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
164	C₂²	—	4	1	no	—
165	C₂²	—	146	1	no	—
166	C₂²	#14556615	1	1	yes	characteristic
167	C₄	—	4	1	no	—
168	C₄	—	8	1	no	—
169	C₄	—	292	2	no	—
170	C₄	—	146	1	no	—
171	C₄	#14556559	1	1	yes	characteristic
172	C₄	D₈×D₇₃	1	1	yes	characteristic
173	C₂	—	2	1	no	—
174	C₂	—	8	1	no	—
175	C₂	—	292	1	no	—
176	C₂	#17988719	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#17988719
commutator	C₂×C₈×C₇₃	C₂³
Frattini	C₂×C₈	C₂²×D₇₃
Fitting	#510×C₇₃	C₂
socle	C₂×C₇₃	#447
hypercenter	#510	D₇₃
hypocenter	C₇₃	#2195
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #510

Complements

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

N	H ≅ G/N	type
2. C₂×#14556537	C₂	semidirect
3. #17988859	C₂	semidirect
4. #17988404	C₂	semidirect
5. #17988394	C₂	semidirect
6. #17988372	C₂	semidirect
7. #510×C₇₃	C₂	semidirect
89. C₇₃	#2195	semidirect
91. #510	D₇₃	semidirect