Finite Group Databasebeta

Subgroups — #22611659

Order 9344.

220 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 89 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×#14556531	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#14556527	C₂	1	1	yes	maximal, characteristic, complemented
3	#17988394	C₂	1	4	yes	maximal, complemented
4	C₂×#135×C₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#11924012	—	2	1	no	—
6	#14556531	—	2	1	no	—
7	#14556531	—	2	1	no	—
8	C₂×#11924003	C₂²	1	2	yes	complemented
9	#14556527	—	2	1	no	—
10	#14556527	C₂²	1	2	yes	complemented
11	C₂×#41×C₇₃	—	2	1	no	—
12	#135×C₇₃	—	2	1	no	—
13	#135×C₇₃	—	2	1	no	—
14	#135×C₇₃	C₄	1	2	yes	—
15	#135×C₇₃	C₂²	1	2	yes	—
16	C₂×Q₁₆×C₇₃	C₄	1	1	yes	characteristic
17	C₂×SD₁₆×C₇₃	—	2	1	no	—
18	C₂×D₈×C₇₃	C₄	1	1	yes	characteristic
19	C₂²×C₈×C₇₃	C₂²	1	1	yes	characteristic
20	#11924012	—	2	2	no	—
21	#11924012	—	4	1	no	—
22	C₂×#10596831	—	4	1	no	—
23	#11924003	—	2	2	no	—
24	#41×C₇₃	—	2	2	no	—
25	#41×C₇₃	—	2	2	no	—
26	#41×C₇₃	—	4	1	no	—
27	#41×C₇₃	—	4	1	no	—
28	C₂×Q₈×C₇₃	—	2	1	no	—
29	C₂×D₄×C₇₃	—	2	1	no	—
30	C₂×D₄×C₇₃	—	4	1	no	—
31	C₂²×C₄×C₇₃	—	4	1	no	—
32	C₂²×C₄×C₇₃	D₄	1	1	yes	characteristic
33	Q₁₆×C₇₃	—	2	1	no	—
34	Q₁₆×C₇₃	C₂×C₄	1	2	yes	—
35	SD₁₆×C₇₃	—	2	2	no	—
36	SD₁₆×C₇₃	—	2	2	no	—
37	D₈×C₇₃	—	2	1	no	—
38	D₈×C₇₃	C₂×C₄	1	2	yes	—
39	C₂×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₈×C₇₃	D₄	1	1	yes	characteristic
42	C₂×C₈×C₇₃	D₄	1	1	yes	characteristic
43	C₂×C₈×C₇₃	D₄	1	1	yes	characteristic
44	C₂×C₈×C₇₃	C₂³	1	1	yes	characteristic
45	C₂³×C₇₃	—	4	1	no	—
46	C₂³×C₇₃	SD₁₆	1	1	yes	characteristic
47	Q₈×C₇₃	—	2	2	no	—
48	Q₈×C₇₃	—	4	1	no	—
49	D₄×C₇₃	—	2	2	no	—
50	D₄×C₇₃	—	4	2	no	—
51	D₄×C₇₃	—	4	2	no	—
52	D₄×C₇₃	—	4	1	no	—
53	C₂×C₄×C₇₃	—	4	1	no	—
54	C₂×C₄×C₇₃	—	4	2	no	—
55	C₂×C₄×C₇₃	—	4	2	no	—
56	C₂×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₇₃	D₈	1	1	yes	characteristic
59	C₂×C₄×C₇₃	D₈	1	1	yes	characteristic
60	C₂×C₄×C₇₃	SD₁₆	1	1	yes	characteristic
61	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
62	C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
63	C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
65	C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
66	#10596831	—	4	2	no	—
67	C₂²×C₇₃	—	4	1	no	—
68	C₂²×C₇₃	—	4	2	no	—
69	C₂²×C₇₃	—	8	2	no	—
70	C₂²×C₇₃	#102	1	1	yes	characteristic
71	C₂²×C₇₃	#102	1	1	yes	characteristic
72	C₂²×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
73	C₄×C₇₃	—	4	2	no	—
74	C₄×C₇₃	#102	1	1	yes	characteristic
75	C₄×C₇₃	#102	1	1	yes	characteristic
76	C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
78	C₂×C₇₃	—	2	1	no	—
79	C₂×C₇₃	—	2	1	no	—
80	C₂×C₇₃	—	2	1	no	—
81	C₂×C₇₃	—	8	2	no	—
82	C₂×C₇₃	C₂×#102	1	1	yes	characteristic
83	#2143	—	73	1	no	maximal, Sylow
84	C₇₃	#2143	1	1	yes	characteristic, complemented, Sylow
85	C₂×#135	D₇₃	1	1	yes	characteristic
86	C₂×M₃₂	—	73	1	no	—
87	C₂×#108	—	73	1	no	—
88	#359	—	73	4	no	—
89	C₂×#41	—	2	1	no	—
90	#135	—	2	1	no	—
91	#135	—	2	1	no	—
92	#135	Dic₇₃	1	2	yes	—
93	#135	C₂×D₇₃	1	2	yes	—
94	C₂×Q₁₆	Dic₇₃	1	1	yes	characteristic
95	C₂×SD₁₆	—	2	1	no	—
96	C₂×D₈	Dic₇₃	1	1	yes	characteristic
97	C₂×M₁₆	—	146	1	no	—
98	C₂²×C₈	C₂×D₇₃	1	1	yes	characteristic
99	M₃₂	—	146	1	no	—
100	M₃₂	—	146	1	no	—
101	C₂×C₁₆	—	73	2	no	—
102	#108	—	73	2	no	—
103	#108	—	146	1	no	—
104	#41	—	2	2	no	—
105	#41	—	2	2	no	—
106	#41	—	4	1	no	—
107	#41	—	4	1	no	—
108	C₂×Q₈	—	2	1	no	—
109	C₂×D₄	—	2	1	no	—
110	C₂×D₄	—	4	1	no	—
111	C₂²×C₄	—	4	1	no	—
112	C₂²×C₄	C₂²⋊D₇₃	1	1	yes	characteristic
113	Q₁₆	—	2	1	no	—
114	Q₁₆	C₂×Dic₇₃	1	2	yes	—
115	SD₁₆	—	2	2	no	—
116	SD₁₆	—	2	2	no	—
117	D₈	—	2	1	no	—
118	D₈	C₂×Dic₇₃	1	2	yes	—
119	M₁₆	—	292	1	no	—
120	M₁₆	—	146	2	no	—
121	C₂×C₈	—	292	1	no	—
122	C₂×C₈	C₂×Dic₇₃	1	1	yes	characteristic
123	C₂×C₈	C₂×Dic₇₃	1	1	yes	characteristic
124	C₂×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
125	C₂×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
126	C₂×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
127	C₂×C₈	C₂²×D₇₃	1	1	yes	characteristic
128	C₁₆	—	146	2	no	—
129	C₂³	—	4	1	no	—
130	C₂³	#11924016	1	1	yes	characteristic
131	Q₈	—	2	2	no	—
132	Q₈	—	4	1	no	—
133	D₄	—	2	2	no	—
134	D₄	—	4	2	no	—
135	D₄	—	4	2	no	—
136	D₄	—	4	1	no	—
137	C₂×C₄	—	4	1	no	—
138	C₂×C₄	—	4	2	no	—
139	C₂×C₄	—	4	2	no	—
140	C₂×C₄	C₂²⋊Dic₇₃	1	1	yes	characteristic
141	C₂×C₄	C₂²⋊Dic₇₃	1	1	yes	characteristic
142	C₂×C₄	#11924013	1	1	yes	characteristic
143	C₂×C₄	#11924013	1	1	yes	characteristic
144	C₂×C₄	#11924016	1	1	yes	characteristic
145	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
146	C₈	—	292	2	no	—
147	C₈	C₂²⋊Dic₇₃	1	1	yes	characteristic
148	C₈	C₂²⋊Dic₇₃	1	1	yes	characteristic
149	C₈	C₂²×Dic₇₃	1	1	yes	characteristic
150	C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
151	C₂²	—	4	1	no	—
152	C₂²	—	4	2	no	—
153	C₂²	—	8	2	no	—
154	C₂²	#14556512	1	1	yes	characteristic
155	C₂²	#14556512	1	1	yes	characteristic
156	C₂²	C₂×#11924016	1	1	yes	characteristic
157	C₄	—	4	2	no	—
158	C₄	#14556512	1	1	yes	characteristic
159	C₄	#14556512	1	1	yes	characteristic
160	C₄	C₂×C₂²⋊Dic₇₃	1	1	yes	characteristic
161	C₄	C₂×#11924013	1	1	yes	characteristic
162	C₂	—	2	1	no	—
163	C₂	—	2	1	no	—
164	C₂	—	2	1	no	—
165	C₂	—	8	2	no	—
166	C₂	C₂×#14556512	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#11924013
commutator	C₈×C₇₃	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₇₃
Fitting	C₂×#135×C₇₃	C₂
socle	C₂×C₇₃	C₂×#102
hypercenter	C₂×#135	D₇₃
hypocenter	C₇₃	#2143
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#135

Complements

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

N	H ≅ G/N	type
1. C₂×#14556531	C₂	semidirect
2. C₂×#14556527	C₂	semidirect
3. #17988394	C₂	semidirect
8. C₂×#11924003	C₂²	semidirect
10. #14556527	C₂²	semidirect
84. C₇₃	#2143	semidirect