Finite Group Databasebeta

Subgroups — #15675352

Order 2944.

220 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 49 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	#13093222	C₂	1	1	yes	maximal, characteristic, complemented
2	#13092853	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#10625008	C₂	1	1	yes	maximal, characteristic, complemented
4	#13092229	C₂	1	1	yes	maximal, characteristic, complemented
5	#13092223	C₂	1	1	yes	maximal, characteristic, complemented
6	#13092201	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#472	C₂	1	1	yes	maximal, characteristic, complemented
8	#10625160	—	2	1	no	—
9	C₂₃×AGL(1,ℤ/8)	—	2	1	no	—
10	C₂×D₈×C₂₃	C₂²	1	1	yes	characteristic, complemented
11	C₂₃×#131	C₂²	1	1	yes	characteristic
12	#10625079	—	2	1	no	—
13	#10625079	—	2	2	no	—
14	#10625075	—	2	1	no	—
15	#10625075	C₂²	1	1	yes	characteristic
16	C₂×D₁₈₄	—	2	1	no	—
17	C₂×D₁₈₄	C₂²	1	1	yes	characteristic
18	C₂₃×#108	C₂²	1	1	yes	characteristic, complemented
19	C₂₃×#100	—	2	1	no	—
20	#10625009	—	2	1	no	—
21	#10625008	—	2	2	no	—
22	#10625008	—	2	1	no	—
23	#10624994	C₂²	1	1	yes	characteristic, complemented
24	C₂×C₂₃⋊C₁₆	C₂²	1	1	yes	characteristic, complemented
25	#41×C₂₃	D₄	1	1	yes	characteristic
26	C₂×D₄×C₂₃	—	2	1	no	—
27	#67292	—	4	1	no	—
28	D₄×D₂₃	—	4	2	no	—
29	D₄×D₂₃	—	4	1	no	—
30	#67288	—	2	2	no	—
31	#67288	—	2	1	no	—
32	C₂×D₉₂	—	2	2	no	—
33	C₂×D₉₂	—	2	1	no	—
34	SD₁₆×C₂₃	—	4	1	no	—
35	D₈×C₂₃	—	2	1	no	—
36	D₈×C₂₃	—	2	1	no	—
37	D₈×C₂₃	—	4	1	no	—
38	M₁₆×C₂₃	—	2	1	no	—
39	M₁₆×C₂₃	—	2	1	no	—
40	M₁₆×C₂₃	D₄	1	1	yes	characteristic
41	C₂×C₈×C₂₃	—	2	1	no	—
42	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
43	Dic₉₂	D₄	1	1	yes	characteristic
44	D₁₈₄	—	2	2	no	—
45	D₁₈₄	—	2	1	no	—
46	D₁₈₄	D₄	1	1	yes	characteristic
47	D₁₈₄	D₄	1	2	yes	—
48	#67263	—	2	2	no	—
49	#67263	—	2	1	no	—
50	C₂₃⋊C₁₆	—	2	1	no	—
51	C₂₃⋊C₁₆	—	2	1	no	—
52	C₂³×C₂₃	—	4	1	no	—
53	C₂²×D₂₃	—	4	2	no	—
54	C₂²×D₂₃	—	4	1	no	—
55	Q₈×C₂₃	D₈	1	1	yes	characteristic
56	D₄×C₂₃	—	2	1	no	—
57	D₄×C₂₃	—	4	1	no	—
58	D₄×C₂₃	—	4	1	no	—
59	D₄×C₂₃	D₈	1	1	yes	characteristic
60	C₂×C₄×C₂₃	—	2	1	no	—
61	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
62	C₂²⋊D₂₃	—	4	2	no	—
63	C₂²⋊D₂₃	—	4	1	no	—
64	D₉₂	—	2	2	no	—
65	D₉₂	—	2	1	no	—
66	D₉₂	—	4	2	no	—
67	D₉₂	—	4	1	no	—
68	C₄×D₂₃	—	4	2	no	—
69	C₄×D₂₃	—	4	1	no	—
70	Dic₄₆	—	2	1	no	—
71	C₈×C₂₃	—	2	1	no	—
72	C₈×C₂₃	—	4	1	no	—
73	C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
74	C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
75	#2192	—	23	1	no	maximal, Sylow
76	C₂²×C₂₃	—	2	1	no	—
77	C₂²×C₂₃	—	8	1	no	—
78	C₂²×C₂₃	—	16	1	no	—
79	C₂²×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
80	C₂×D₂₃	—	4	2	no	—
81	C₂×D₂₃	—	4	1	no	—
82	C₂×D₂₃	—	16	1	no	—
83	C₂×D₂₃	—	16	2	no	—
84	C₄×C₂₃	—	2	1	no	—
85	C₄×C₂₃	#120	1	1	yes	characteristic
86	C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
87	Dic₂₃	—	4	1	no	—
88	#576	—	23	1	no	—
89	#509	—	23	1	no	—
90	C₂×D₁₆	—	23	1	no	—
91	#472	D₂₃	1	1	yes	characteristic, complemented
92	#361	—	23	1	no	—
93	#359	—	23	1	no	—
94	#350	—	23	1	no	—
95	C₂×C₂₃	—	2	1	no	—
96	C₂×C₂₃	—	4	1	no	—
97	C₂×C₂₃	—	16	1	no	—
98	C₂×C₂₃	#447	1	1	yes	characteristic
99	D₂₃	—	8	2	no	—
100	D₂₃	—	8	1	no	—
101	H(5,𝔽₂)	—	46	1	no	—
102	AGL(1,ℤ/8)	—	2	1	no	—
103	AGL(1,ℤ/8)	—	46	2	no	—
104	AGL(1,ℤ/8)	—	46	1	no	—
105	#135	—	23	1	no	—
106	#135	—	46	1	no	—
107	C₂×D₈	—	23	1	no	—
108	C₂×D₈	—	46	1	no	—
109	C₂×D₈	C₂×D₂₃	1	1	yes	characteristic, complemented
110	#131	C₂×D₂₃	1	1	yes	characteristic
111	SD₃₂	—	46	1	no	—
112	D₁₆	—	46	2	no	—
113	D₁₆	—	46	1	no	—
114	M₃₂	—	23	1	no	—
115	C₂×C₁₆	—	23	1	no	—
116	#108	C₂×D₂₃	1	1	yes	characteristic, complemented
117	#100	—	2	1	no	—
118	C₂₃	#2192	1	1	yes	characteristic, complemented, Sylow
119	#41	—	46	2	no	—
120	#41	—	46	1	no	—
121	#41	—	92	1	no	—
122	#41	C₂²⋊D₂₃	1	1	yes	characteristic
123	C₂×D₄	—	2	1	no	—
124	C₂×D₄	—	46	2	no	—
125	C₂×D₄	—	46	1	no	—
126	C₂×D₄	—	92	2	no	—
127	C₂×D₄	—	92	1	no	—
128	Q₁₆	—	23	1	no	—
129	SD₁₆	—	4	1	no	—
130	SD₁₆	—	46	2	no	—
131	SD₁₆	—	46	1	no	—
132	D₈	—	2	1	no	—
133	D₈	—	2	1	no	—
134	D₈	—	4	1	no	—
135	D₈	—	23	2	no	—
136	D₈	—	23	1	no	—
137	D₈	—	46	2	no	—
138	D₈	—	46	1	no	—
139	M₁₆	—	2	1	no	—
140	M₁₆	—	2	1	no	—
141	M₁₆	C₂²⋊D₂₃	1	1	yes	characteristic
142	C₂×C₈	—	2	1	no	—
143	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
144	C₁₆	—	46	1	no	—
145	C₁₆	—	46	1	no	—
146	C₂³	—	4	1	no	—
147	C₂³	—	92	2	no	—
148	C₂³	—	92	1	no	—
149	Q₈	—	46	1	no	—
150	Q₈	#67272	1	1	yes	characteristic
151	D₄	—	2	1	no	—
152	D₄	—	4	1	no	—
153	D₄	—	4	1	no	—
154	D₄	—	46	2	no	—
155	D₄	—	46	1	no	—
156	D₄	—	92	2	no	—
157	D₄	—	92	2	no	—
158	D₄	—	92	1	no	—
159	D₄	—	92	1	no	—
160	D₄	#67272	1	1	yes	characteristic
161	C₂×C₄	—	2	1	no	—
162	C₂×C₄	—	92	2	no	—
163	C₂×C₄	—	92	1	no	—
164	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
165	C₈	—	2	1	no	—
166	C₈	—	4	1	no	—
167	C₈	D₄×D₂₃	1	1	yes	characteristic
168	C₈	D₄×D₂₃	1	1	yes	characteristic
169	C₂²	—	2	1	no	—
170	C₂²	—	8	1	no	—
171	C₂²	—	16	1	no	—
172	C₂²	—	92	2	no	—
173	C₂²	—	92	1	no	—
174	C₂²	—	368	1	no	—
175	C₂²	—	368	2	no	—
176	C₂²	#10625120	1	1	yes	characteristic
177	C₄	—	2	1	no	—
178	C₄	—	92	1	no	—
179	C₄	C₂×#67272	1	1	yes	characteristic
180	C₄	#10625108	1	1	yes	characteristic
181	C₂	—	2	1	no	—
182	C₂	—	4	1	no	—
183	C₂	—	16	1	no	—
184	C₂	—	184	2	no	—
185	C₂	—	184	1	no	—
186	C₂	#13092550	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#13092550
commutator, cosocle	C₂×C₈×C₂₃	C₂³
Frattini	C₂×C₈	C₂²×D₂₃
Fitting	C₂₃×#472	C₂
socle	C₂×C₂₃	#447
hypercenter	#472	D₂₃
hypocenter	C₂₃	#2192
LCS	G ▷ C₂×C₈×C₂₃ ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #131 ◁ #472

Complements

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

N	H ≅ G/N	type
1. #13093222	C₂	semidirect
2. #13092853	C₂	semidirect
3. C₂×#10625008	C₂	semidirect
4. #13092229	C₂	semidirect
5. #13092223	C₂	semidirect
6. #13092201	C₂	semidirect
7. C₂₃×#472	C₂	semidirect
10. C₂×D₈×C₂₃	C₂²	semidirect
18. C₂₃×#108	C₂²	semidirect
23. #10624994	C₂²	semidirect
24. C₂×C₂₃⋊C₁₆	C₂²	semidirect
91. #472	D₂₃	semidirect
109. C₂×D₈	C₂×D₂₃	semidirect
116. #108	C₂×D₂₃	semidirect
118. C₂₃	#2192	semidirect

Maximal subgroups

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

subgroup	index
#13093222	2
#13092853	2
C₂×#10625008	2
#13092229	2
#13092223	2
#13092201	2
C₂₃×#472	2
#2192	23