Finite Group Databasebeta

Subgroups — #13093233

Order 1472.

248 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 99 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	#10625161	C₂	1	1	yes	maximal, characteristic, complemented
2	#10625158	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#137	C₂	1	1	yes	maximal, characteristic, complemented
4	#10625121	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#67275	C₂	1	1	yes	maximal, characteristic, complemented
6	#10625089	C₂	1	2	yes	maximal, complemented
7	Q₁₆×D₂₃	C₂	1	2	yes	maximal, complemented
8	#10625087	C₂	1	2	yes	maximal, complemented
9	#10625086	C₂	1	2	yes	maximal, complemented
10	#10625080	C₂	1	1	yes	maximal, characteristic, complemented
11	#10625078	C₂	1	1	yes	maximal, characteristic, complemented
12	#41×C₂₃	C₂²	1	1	yes	characteristic
13	C₂×Q₈×C₂₃	C₂²	1	1	yes	characteristic
14	#67292	—	2	1	no	—
15	#67292	C₂²	1	2	yes	—
16	Q₈×D₂₃	—	2	1	no	—
17	Q₈×D₂₃	C₂²	1	2	yes	—
18	Q₈×D₂₃	C₂²	1	2	yes	—
19	#67290	—	2	2	no	—
20	#67290	C₂²	1	2	yes	—
21	#67288	—	2	2	no	—
22	#67288	—	2	1	no	—
23	#67288	C₂²	1	1	yes	characteristic
24	#67288	C₂²	1	1	yes	characteristic
25	C₂×Dic₄₆	—	2	1	no	—
26	C₂×Dic₄₆	C₂²	1	1	yes	characteristic
27	Q₁₆×C₂₃	C₂²	1	2	yes	—
28	SD₁₆×C₂₃	C₂²	1	2	yes	—
29	M₁₆×C₂₃	C₂²	1	1	yes	characteristic
30	#67275	C₂²	1	2	yes	—
31	#67275	C₂²	1	2	yes	—
32	#67275	C₂²	1	1	yes	characteristic
33	#67274	C₂²	1	1	yes	characteristic
34	#67273	C₂²	1	1	yes	characteristic
35	#67272	C₂²	1	1	yes	characteristic
36	C₂×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
37	Dic₉₂	C₂²	1	2	yes	—
38	#67263	C₂²	1	2	yes	—
39	#67262	C₂²	1	2	yes	—
40	C₈×D₂₃	C₂²	1	2	yes	—
41	Q₈×C₂₃	—	2	1	no	—
42	Q₈×C₂₃	C₂³	1	1	yes	characteristic
43	Q₈×C₂₃	C₂³	1	2	yes	—
44	D₄×C₂₃	—	2	1	no	—
45	D₄×C₂₃	C₂³	1	1	yes	characteristic
46	C₂×C₄×C₂₃	—	2	1	no	—
47	C₂×C₄×C₂₃	—	2	1	no	—
48	C₂×C₄×C₂₃	C₂³	1	1	yes	characteristic
49	C₂²⋊D₂₃	—	2	1	no	—
50	C₂²⋊D₂₃	—	2	2	no	—
51	C₂²⋊D₂₃	D₄	1	2	yes	—
52	C₂×Dic₂₃	—	2	1	no	—
53	C₂×Dic₂₃	—	2	2	no	—
54	D₉₂	—	2	2	no	—
55	D₉₂	D₄	1	1	yes	characteristic
56	D₉₂	C₂³	1	1	yes	characteristic
57	C₄×D₂₃	—	2	2	no	—
58	C₄×D₂₃	—	2	2	no	—
59	C₄×D₂₃	—	2	1	no	—
60	C₄×D₂₃	—	2	2	no	—
61	C₄×D₂₃	C₂³	1	2	yes	—
62	Dic₄₆	—	2	1	no	—
63	Dic₄₆	—	2	2	no	—
64	Dic₄₆	—	2	2	no	—
65	Dic₄₆	D₄	1	1	yes	characteristic
66	Dic₄₆	C₂³	1	2	yes	—
67	Dic₄₆	C₂³	1	1	yes	characteristic
68	C₈×C₂₃	C₂³	1	2	yes	—
69	C₂₃⋊C₈	C₂³	1	1	yes	characteristic
70	C₂₃⋊C₈	C₂³	1	1	yes	characteristic
71	C₂²×C₂₃	—	2	1	no	—
72	C₂²×C₂₃	C₂×D₄	1	1	yes	characteristic
73	C₂×D₂₃	—	2	1	no	—
74	C₂×D₂₃	C₂×D₄	1	2	yes	—
75	C₄×C₂₃	—	2	1	no	—
76	C₄×C₂₃	—	2	2	no	—
77	C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
78	C₄×C₂₃	C₂⁴	1	1	yes	characteristic
79	Dic₂₃	—	2	1	no	—
80	Dic₂₃	—	2	2	no	—
81	Dic₂₃	C₂×D₄	1	2	yes	—
82	#578	—	23	1	no	maximal, Sylow
83	C₂×C₂₃	—	2	1	no	—
84	C₂×C₂₃	—	4	1	no	—
85	C₂×C₂₃	C₂²×D₄	1	1	yes	characteristic
86	D₂₃	—	2	2	no	—
87	D₂₃	—	4	1	no	—
88	2₋¹⁺⁴	—	23	1	no	—
89	2₋¹⁺⁴	—	23	1	no	—
90	#137	—	23	2	no	—
91	#137	—	23	2	no	—
92	#137	—	23	1	no	—
93	#137	D₂₃	1	1	yes	characteristic, complemented
94	#135	—	23	1	no	—
95	#135	—	23	2	no	—
96	C₂×Q₁₆	—	23	1	no	—
97	C₂×Q₁₆	—	23	2	no	—
98	#131	—	23	1	no	—
99	C₂₃	#578	1	1	yes	characteristic, complemented, Sylow
100	#41	—	23	1	no	—
101	#41	—	23	2	no	—
102	#41	—	23	2	no	—
103	#41	—	23	1	no	—
104	#41	—	46	2	no	—
105	#41	—	46	1	no	—
106	#41	—	46	2	no	—
107	#41	—	46	1	no	—
108	#41	C₂×D₂₃	1	1	yes	characteristic
109	C₂×Q₈	—	23	2	no	—
110	C₂×Q₈	—	23	2	no	—
111	C₂×Q₈	—	23	1	no	—
112	C₂×Q₈	—	46	1	no	—
113	C₂×Q₈	—	46	1	no	—
114	C₂×Q₈	C₂×D₂₃	1	1	yes	characteristic
115	Q₁₆	—	23	1	no	—
116	Q₁₆	—	23	2	no	—
117	Q₁₆	—	23	2	no	—
118	Q₁₆	—	23	2	no	—
119	Q₁₆	C₂×D₂₃	1	2	yes	—
120	SD₁₆	—	23	1	no	—
121	SD₁₆	—	23	1	no	—
122	SD₁₆	—	23	2	no	—
123	SD₁₆	C₂×D₂₃	1	2	yes	—
124	D₈	—	23	1	no	—
125	M₁₆	—	23	2	no	—
126	M₁₆	C₂×D₂₃	1	1	yes	characteristic
127	C₂×C₈	—	23	1	no	—
128	C₂×C₈	—	23	2	no	—
129	Q₈	—	2	1	no	—
130	Q₈	—	23	1	no	—
131	Q₈	—	23	1	no	—
132	Q₈	—	23	2	no	—
133	Q₈	—	46	2	no	—
134	Q₈	—	46	2	no	—
135	Q₈	—	46	1	no	—
136	Q₈	C₂²×D₂₃	1	1	yes	characteristic
137	Q₈	C₂²×D₂₃	1	2	yes	—
138	D₄	—	2	1	no	—
139	D₄	—	23	2	no	—
140	D₄	—	23	1	no	—
141	D₄	—	23	1	no	—
142	D₄	—	46	2	no	—
143	D₄	—	46	2	no	—
144	D₄	—	46	1	no	—
145	D₄	C₂²×D₂₃	1	1	yes	characteristic
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	23	2	no	—
149	C₂×C₄	—	46	2	no	—
150	C₂×C₄	—	46	2	no	—
151	C₂×C₄	—	46	2	no	—
152	C₂×C₄	—	46	2	no	—
153	C₂×C₄	—	46	1	no	—
154	C₂×C₄	—	46	1	no	—
155	C₂×C₄	C₂²×D₂₃	1	1	yes	characteristic
156	C₈	—	23	1	no	—
157	C₈	—	23	1	no	—
158	C₈	C₂²×D₂₃	1	2	yes	—
159	C₂²	—	2	1	no	—
160	C₂²	—	23	2	no	—
161	C₂²	—	46	1	no	—
162	C₂²	D₄×D₂₃	1	1	yes	characteristic
163	C₄	—	2	1	no	—
164	C₄	—	2	2	no	—
165	C₄	—	23	2	no	—
166	C₄	—	46	1	no	—
167	C₄	—	46	2	no	—
168	C₄	D₄×D₂₃	1	1	yes	characteristic
169	C₄	C₂³×D₂₃	1	1	yes	characteristic
170	C₂	—	2	1	no	—
171	C₂	—	4	1	no	—
172	C₂	—	46	2	no	—
173	C₂	—	92	1	no	—
174	C₂	C₂×D₄×D₂₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₄×D₂₃
commutator, cosocle	C₄×C₂₃	C₂⁴
Frattini	C₄	C₂³×D₂₃
Fitting	C₂₃×#137	C₂
socle	C₂×C₂₃	C₂²×D₄
hypercenter	#137	D₂₃
hypocenter	C₂₃	#578
LCS	G ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #137

Complements

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

N	H ≅ G/N	type
1. #10625161	C₂	semidirect
2. #10625158	C₂	semidirect
3. C₂₃×#137	C₂	semidirect
4. #10625121	C₂	semidirect
5. C₂×#67275	C₂	semidirect
6. #10625089	C₂	semidirect
7. Q₁₆×D₂₃	C₂	semidirect
8. #10625087	C₂	semidirect
9. #10625086	C₂	semidirect
10. #10625080	C₂	semidirect
11. #10625078	C₂	semidirect
93. #137	D₂₃	semidirect
99. C₂₃	#578	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#10625161	2
#10625158	2
C₂₃×#137	2
#10625121	2
C₂×#67275	2
#10625089	2
Q₁₆×D₂₃	2
#10625087	2
#10625086	2
#10625080	2
#10625078	2
#578	23