Finite Group Databasebeta

Subgroups — #22626252

Order 9344.

224 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 62 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	#17989643	C₂	1	1	yes	maximal, characteristic, complemented
2	#17989597	C₂	1	1	yes	maximal, characteristic, complemented
3	#17989503	C₂	1	4	yes	maximal, complemented
4	C₂×#14556610	C₂	1	1	yes	maximal, characteristic
5	#14556670	—	2	1	no	—
6	#14556666	C₂²	1	2	yes	complemented
7	#14556665	—	2	1	no	—
8	#14556664	—	2	1	no	—
9	#14556660	—	2	1	no	—
10	#14556659	C₂²	1	2	yes	complemented
11	C₂×#11924027	—	2	1	no	—
12	#14556610	—	2	1	no	—
13	#14556610	—	2	1	no	—
14	#14556610	C₄	1	2	yes	complemented
15	#14556610	C₂²	1	2	yes	—
16	C₂×Dic₂₉₂	C₄	1	1	yes	characteristic, complemented
17	C₂×#11924016	—	2	1	no	—
18	C₂×C₈×D₇₃	C₂²	1	1	yes	characteristic
19	C₂×D₈×C₇₃	C₄	1	1	yes	characteristic, complemented
20	C₄⋊#64408	—	2	2	no	—
21	C₂²⋊#64408	—	4	1	no	—
22	C₄×#64408	—	4	1	no	—
23	#11924031	—	2	2	no	—
24	#11924027	—	2	2	no	—
25	#11924027	—	2	2	no	—
26	#11924027	—	4	1	no	—
27	#11924027	—	4	1	no	—
28	C₂×Dic₁₄₆	—	2	1	no	—
29	C₂×C₂²⋊D₇₃	—	4	1	no	—
30	C₂²×Dic₇₃	—	4	1	no	—
31	C₂×C₄×D₇₃	D₄	1	1	yes	characteristic
32	Dic₂₉₂	—	2	1	no	—
33	Dic₂₉₂	C₂×C₄	1	2	yes	—
34	#11924016	—	2	2	no	—
35	#11924016	—	2	2	no	—
36	C₂×#10596831	D₄	1	1	yes	characteristic
37	C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
38	C₈×D₇₃	D₄	1	1	yes	characteristic
39	C₈×D₇₃	D₄	1	1	yes	characteristic
40	C₈×D₇₃	C₂³	1	1	yes	characteristic
41	C₂×D₄×C₇₃	—	2	1	no	—
42	D₈×C₇₃	—	2	1	no	—
43	D₈×C₇₃	C₂×C₄	1	2	yes	—
44	C₂×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
45	C₂³×C₇₃	—	4	1	no	—
46	C₂²×D₇₃	D₈	1	1	yes	characteristic
47	C₂×#64408	—	4	2	no	—
48	D₄×C₇₃	—	2	2	no	—
49	D₄×C₇₃	—	4	1	no	—
50	C₂×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
51	C₂²⋊D₇₃	—	4	2	no	—
52	C₂²⋊D₇₃	—	4	2	no	—
53	C₂×Dic₇₃	—	4	1	no	—
54	C₂×Dic₇₃	—	4	2	no	—
55	C₂×Dic₇₃	—	4	2	no	—
56	C₂×Dic₇₃	SD₁₆	1	1	yes	characteristic
57	C₄×D₇₃	C₂²⋊C₄	1	1	yes	characteristic
58	C₄×D₇₃	D₈	1	1	yes	characteristic
59	C₄×D₇₃	SD₁₆	1	1	yes	characteristic
60	C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
61	Dic₁₄₆	—	2	2	no	—
62	Dic₁₄₆	—	4	1	no	—
63	C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
65	#10596831	C₂²⋊C₄	1	1	yes	characteristic
66	#10596831	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₇₃	—	4	2	no	—
68	C₂²×C₇₃	—	8	2	no	—
69	C₂²×C₇₃	#102	1	1	yes	characteristic
70	C₂×D₇₃	—	4	1	no	—
71	C₂×D₇₃	#102	1	1	yes	characteristic
72	C₂×D₇₃	C₂×D₈	1	1	yes	characteristic
73	#64408	—	8	2	no	—
74	C₄×C₇₃	#102	1	1	yes	characteristic
75	C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
76	Dic₇₃	—	4	2	no	—
77	Dic₇₃	#102	1	1	yes	characteristic
78	Dic₇₃	C₂×SD₁₆	1	1	yes	characteristic
79	C₂×C₇₃	—	2	1	no	—
80	C₂×C₇₃	—	8	2	no	—
81	C₂×C₇₃	C₂×#102	1	1	yes	characteristic
82	D₇₃	—	2	1	no	—
83	D₇₃	—	2	1	no	—
84	#2145	—	73	1	no	maximal, Sylow
85	C₇₃	#2145	1	1	yes	characteristic, complemented, Sylow
86	C₂×#135	—	73	1	no	—
87	C₂×M₃₂	—	73	1	no	—
88	#427	—	73	1	no	—
89	#360	—	73	4	no	—
90	C₂×#41	—	146	1	no	—
91	#135	—	73	2	no	—
92	#135	—	73	2	no	—
93	#135	—	146	1	no	—
94	#135	—	146	1	no	—
95	C₂×Q₁₆	—	73	1	no	—
96	C₂×SD₁₆	—	146	1	no	—
97	C₂×D₈	#64408	1	1	yes	characteristic, complemented
98	C₂²×C₈	—	73	1	no	—
99	#117	—	146	1	no	—
100	M₃₂	—	73	2	no	—
101	M₃₂	—	146	1	no	—
102	C₂×C₁₆	—	146	1	no	—
103	#107	—	146	1	no	—
104	#106	—	73	2	no	—
105	#41	—	292	1	no	—
106	#41	—	292	1	no	—
107	#41	—	146	2	no	—
108	#41	—	146	2	no	—
109	C₂×Q₈	—	146	1	no	—
110	C₂×D₄	—	2	1	no	—
111	C₂×D₄	—	292	1	no	—
112	C₂²×C₄	—	292	1	no	—
113	C₂²×C₄	—	73	1	no	—
114	Q₁₆	—	73	2	no	—
115	Q₁₆	—	146	1	no	—
116	SD₁₆	—	146	2	no	—
117	SD₁₆	—	146	2	no	—
118	D₈	—	2	1	no	—
119	D₈	C₂×#64408	1	2	yes	—
120	C₂×C₈	—	73	1	no	—
121	C₂×C₈	—	73	1	no	—
122	C₂×C₈	—	73	1	no	—
123	C₂×C₈	—	73	1	no	—
124	C₂×C₈	—	73	1	no	—
125	C₂×C₈	C₂×#64408	1	1	yes	characteristic
126	C₄⋊C₄	—	146	2	no	—
127	C₂²⋊C₄	—	292	1	no	—
128	C₄²	—	292	1	no	—
129	C₁₆	—	146	2	no	—
130	C₂³	—	4	1	no	—
131	C₂³	—	73	1	no	—
132	Q₈	—	292	1	no	—
133	Q₈	—	146	2	no	—
134	D₄	—	2	2	no	—
135	D₄	—	4	1	no	—
136	D₄	—	292	2	no	—
137	D₄	—	292	2	no	—
138	C₂×C₄	—	292	1	no	—
139	C₂×C₄	—	292	2	no	—
140	C₂×C₄	—	292	2	no	—
141	C₂×C₄	—	292	2	no	—
142	C₂×C₄	—	73	1	no	—
143	C₂×C₄	—	73	1	no	—
144	C₂×C₄	—	73	1	no	—
145	C₂×C₄	—	73	1	no	—
146	C₂×C₄	—	73	1	no	—
147	C₂×C₄	C₂²⋊#64408	1	1	yes	characteristic
148	C₈	—	73	1	no	—
149	C₈	—	73	1	no	—
150	C₈	C₂²⋊#64408	1	1	yes	characteristic
151	C₈	C₂²×#64408	1	1	yes	characteristic
152	C₂²	—	4	2	no	—
153	C₂²	—	8	2	no	—
154	C₂²	—	292	1	no	—
155	C₂²	—	73	1	no	—
156	C₂²	—	73	1	no	—
157	C₂²	#14556653	1	1	yes	characteristic
158	C₄	—	292	2	no	—
159	C₄	—	584	2	no	—
160	C₄	—	73	1	no	—
161	C₄	—	73	1	no	—
162	C₄	#14556653	1	1	yes	characteristic
163	C₄	C₂×C₂²⋊#64408	1	1	yes	characteristic
164	C₂	—	2	1	no	—
165	C₂	—	8	2	no	—
166	C₂	—	146	1	no	—
167	C₂	—	146	1	no	—
168	C₂	C₂×#14556653	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#14556653
commutator	C₈×C₇₃	C₂²×C₄
Frattini	C₈	C₂²×#64408
Fitting	C₂×D₈×C₇₃	C₄
socle	C₂×C₇₃	C₂×#102
hypercenter	C₂×D₈	#64408
hypocenter	C₇₃	#2145
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×D₈

Complements

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

N	H ≅ G/N	type
1. #17989643	C₂	semidirect
2. #17989597	C₂	semidirect
3. #17989503	C₂	semidirect
6. #14556666	C₂²	semidirect
10. #14556659	C₂²	semidirect
14. #14556610	C₄	semidirect
16. C₂×Dic₂₉₂	C₄	semidirect
19. C₂×D₈×C₇₃	C₄	semidirect
85. C₇₃	#2145	semidirect
97. C₂×D₈	#64408	semidirect