Finite Group Databasebeta

Subgroups — #22611329

Order 9344.

216 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 65 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	#17988636	C₂	1	1	yes	maximal, characteristic, complemented
2	#17988635	C₂	1	1	yes	maximal, characteristic, complemented
3	#17988635	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#14556508	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556508	C₂	1	1	yes	maximal, characteristic
6	#17988345	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#100×C₇₃	C₂	1	1	yes	maximal, characteristic
8	C₂×#11924012	C₂²	1	1	yes	characteristic
9	C₂×#11924012	C₂²	1	1	yes	characteristic
10	C₂×#11924012	C₂²	1	1	yes	characteristic
11	C₂×#11924012	C₂²	1	1	yes	characteristic
12	#14556527	—	2	2	no	—
13	#14556526	—	2	2	no	—
14	#14556526	—	2	2	no	—
15	#14556508	—	2	2	no	—
16	#14556508	—	2	2	no	—
17	C₂²×D₄×C₇₃	C₂²	1	1	yes	characteristic
18	C₂×M₁₆×C₇₃	C₂²	1	1	yes	characteristic
19	C₂×M₁₆×C₇₃	C₂²	1	1	yes	characteristic
20	#100×C₇₃	—	2	2	no	—
21	#11924012	—	2	1	no	—
22	#11924012	—	2	1	no	—
23	#11924012	—	2	1	no	—
24	#11924012	—	2	1	no	—
25	#11924012	—	2	1	no	—
26	#11924012	—	2	1	no	—
27	#11924012	D₄	1	2	yes	—
28	#11924012	D₄	1	2	yes	—
29	C₂×#10596831	—	2	1	no	—
30	C₂×#10596831	—	2	1	no	—
31	C₂×#10596831	—	2	1	no	—
32	C₂×#10596831	—	2	1	no	—
33	C₂⁴×C₇₃	Q₈	1	2	yes	—
34	C₂×D₄×C₇₃	—	2	1	no	—
35	C₂×D₄×C₇₃	—	2	1	no	—
36	C₂×D₄×C₇₃	—	2	1	no	—
37	C₂×D₄×C₇₃	—	2	1	no	—
38	C₂×D₄×C₇₃	—	2	1	no	—
39	C₂×D₄×C₇₃	—	2	1	no	—
40	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
41	M₁₆×C₇₃	—	2	1	no	—
42	M₁₆×C₇₃	—	2	1	no	—
43	M₁₆×C₇₃	—	2	1	no	—
44	M₁₆×C₇₃	D₄	1	2	yes	—
45	C₂×C₈×C₇₃	—	2	1	no	—
46	C₂×C₈×C₇₃	—	2	1	no	—
47	C₂³×C₇₃	—	2	2	no	—
48	C₂³×C₇₃	—	2	2	no	—
49	C₂³×C₇₃	—	2	2	no	—
50	C₂³×C₇₃	—	8	2	no	—
51	C₂³×C₇₃	C₂×Q₈	1	1	yes	characteristic
52	D₄×C₇₃	—	4	1	no	—
53	D₄×C₇₃	—	4	1	no	—
54	D₄×C₇₃	—	4	1	no	—
55	D₄×C₇₃	—	4	1	no	—
56	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
60	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
61	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
62	C₈×C₇₃	—	2	2	no	—
63	C₈×C₇₃	—	4	1	no	—
64	#10596831	—	2	2	no	—
65	#10596831	—	2	2	no	—
66	#10596831	—	4	1	no	—
67	#10596831	—	4	1	no	—
68	C₂²×C₇₃	—	4	1	no	—
69	C₂²×C₇₃	—	4	2	no	—
70	C₂²×C₇₃	—	8	2	no	—
71	C₂²×C₇₃	—	8	2	no	—
72	C₂²×C₇₃	—	8	2	no	—
73	C₂²×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
74	C₂²×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
75	C₂²×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
76	C₄×C₇₃	#121	1	1	yes	characteristic
77	C₄×C₇₃	#121	1	1	yes	characteristic
78	C₄×C₇₃	#121	1	1	yes	characteristic
79	C₄×C₇₃	#126	1	1	yes	characteristic
80	C₂×C₇₃	—	2	1	no	—
81	C₂×C₇₃	—	2	1	no	—
82	C₂×C₇₃	—	2	1	no	—
83	C₂×C₇₃	—	8	2	no	—
84	C₂×C₇₃	#396	1	1	yes	characteristic
85	#2069	—	73	1	no	maximal, Sylow
86	C₇₃	#2069	1	1	yes	characteristic, complemented, Sylow
87	#430	—	73	1	no	—
88	#430	—	73	1	no	—
89	#430	—	73	1	no	—
90	C₂×#100	—	73	1	no	—
91	C₂×#100	—	73	1	no	—
92	C₂×#100	D₇₃	1	1	yes	characteristic
93	#338	—	73	1	no	—
94	C₂²×D₄	C₂×D₇₃	1	1	yes	characteristic
95	C₂×M₁₆	—	73	1	no	—
96	C₂×M₁₆	—	73	1	no	—
97	C₂×M₁₆	—	73	1	no	—
98	C₂×M₁₆	—	73	1	no	—
99	C₂×M₁₆	C₂×D₇₃	1	1	yes	characteristic
100	C₂×M₁₆	C₂×D₇₃	1	1	yes	characteristic
101	#108	—	146	2	no	—
102	#108	—	146	2	no	—
103	#108	—	146	2	no	—
104	#100	—	2	2	no	—
105	#100	—	146	2	no	—
106	#100	—	146	2	no	—
107	C₂⁴	Dic₁₄₆	1	2	yes	—
108	C₂×D₄	—	2	1	no	—
109	C₂×D₄	—	2	1	no	—
110	C₂×D₄	—	2	1	no	—
111	C₂×D₄	—	2	1	no	—
112	C₂×D₄	—	2	1	no	—
113	C₂×D₄	—	2	1	no	—
114	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
115	M₁₆	—	2	1	no	—
116	M₁₆	—	2	1	no	—
117	M₁₆	—	2	1	no	—
118	M₁₆	—	73	2	no	—
119	M₁₆	—	73	2	no	—
120	M₁₆	—	146	1	no	—
121	M₁₆	—	146	1	no	—
122	M₁₆	—	146	1	no	—
123	M₁₆	—	146	1	no	—
124	M₁₆	—	146	1	no	—
125	M₁₆	—	146	1	no	—
126	M₁₆	C₂²⋊D₇₃	1	2	yes	—
127	C₂×C₈	—	2	1	no	—
128	C₂×C₈	—	2	1	no	—
129	C₂×C₈	—	146	1	no	—
130	C₂×C₈	—	146	1	no	—
131	C₂×C₈	—	146	1	no	—
132	C₂×C₈	—	146	1	no	—
133	C₂³	—	2	2	no	—
134	C₂³	—	2	2	no	—
135	C₂³	—	2	2	no	—
136	C₂³	—	8	2	no	—
137	C₂³	C₂×Dic₁₄₆	1	1	yes	characteristic
138	D₄	—	4	1	no	—
139	D₄	—	4	1	no	—
140	D₄	—	4	1	no	—
141	D₄	—	4	1	no	—
142	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
143	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
144	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
145	C₂×C₄	#11924027	1	1	yes	characteristic
146	C₂×C₄	#11924027	1	1	yes	characteristic
147	C₂×C₄	#11924028	1	1	yes	characteristic
148	C₈	—	2	2	no	—
149	C₈	—	4	1	no	—
150	C₈	—	292	1	no	—
151	C₈	—	292	1	no	—
152	C₈	—	146	2	no	—
153	C₈	—	146	2	no	—
154	C₂²	—	4	1	no	—
155	C₂²	—	4	2	no	—
156	C₂²	—	8	2	no	—
157	C₂²	—	8	2	no	—
158	C₂²	—	8	2	no	—
159	C₂²	#14556568	1	1	yes	characteristic
160	C₂²	#14556568	1	1	yes	characteristic
161	C₂²	#14556570	1	1	yes	characteristic
162	C₄	#14556562	1	1	yes	characteristic
163	C₄	#14556563	1	1	yes	characteristic
164	C₄	#14556564	1	1	yes	characteristic
165	C₄	#14556584	1	1	yes	characteristic
166	C₂	—	2	1	no	—
167	C₂	—	2	1	no	—
168	C₂	—	2	1	no	—
169	C₂	—	8	2	no	—
170	C₂	#17988510	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#17988510
commutator	C₂²×C₄×C₇₃	C₂³
Frattini	C₂²×C₄	C₂²×D₇₃
Fitting	C₂×#100×C₇₃	C₂
socle	C₂×C₇₃	#396
hypercenter	C₂×#100	D₇₃
hypocenter	C₇₃	#2069
LCS	G ▷ C₂²×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#100

Complements

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

N	H ≅ G/N	type
1. #17988636	C₂	semidirect
2. #17988635	C₂	semidirect
3. #17988635	C₂	semidirect
6. #17988345	C₂	semidirect
86. C₇₃	#2069	semidirect