Finite Group Databasebeta

Subgroups — #15675359

Order 2944.

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

normal subgroups only

This group has 210 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	#13093234	C₂	1	1	yes	maximal, characteristic, complemented
2	#13092853	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#10625009	C₂	1	1	yes	maximal, characteristic, complemented
4	#13092232	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	#10625161	—	2	1	no	—
9	C₂₃×AGL(1,ℤ/8)	—	2	1	no	—
10	C₂×D₈×C₂₃	C₂²	1	1	yes	characteristic
11	C₂₃×#131	C₂²	1	1	yes	characteristic
12	#10625080	—	2	1	no	—
13	#10625080	—	2	2	no	—
14	C₂×Dic₉₂	—	2	1	no	—
15	C₂×Dic₉₂	C₂²	1	1	yes	characteristic
16	#10625075	—	2	1	no	—
17	#10625075	C₂²	1	1	yes	characteristic
18	C₂₃×#108	C₂²	1	1	yes	characteristic
19	C₂₃×#100	—	2	1	no	—
20	#10625009	—	2	2	no	—
21	#10625009	—	2	1	no	—
22	#10625008	—	2	1	no	—
23	#10624994	C₂²	1	1	yes	characteristic
24	C₂×C₂₃⋊C₁₆	C₂²	1	1	yes	characteristic
25	#41×C₂₃	D₄	1	1	yes	characteristic
26	C₂×D₄×C₂₃	—	2	1	no	—
27	Q₈×D₂₃	—	4	1	no	—
28	#67290	—	4	2	no	—
29	#67290	—	4	1	no	—
30	#67288	—	2	2	no	—
31	#67288	—	2	1	no	—
32	C₂×Dic₄₆	—	2	2	no	—
33	C₂×Dic₄₆	—	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	—	2	2	no	—
44	Dic₉₂	—	2	1	no	—
45	Dic₉₂	D₄	1	1	yes	characteristic
46	Dic₉₂	D₄	1	2	yes	—
47	D₁₈₄	D₄	1	1	yes	characteristic
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	Q₈×C₂₃	D₈	1	1	yes	characteristic
54	D₄×C₂₃	—	2	1	no	—
55	D₄×C₂₃	—	4	1	no	—
56	D₄×C₂₃	—	4	1	no	—
57	D₄×C₂₃	D₈	1	1	yes	characteristic
58	C₂×C₄×C₂₃	—	2	1	no	—
59	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
60	C₂²⋊D₂₃	—	4	2	no	—
61	C₂²⋊D₂₃	—	4	1	no	—
62	C₂×Dic₂₃	—	4	2	no	—
63	C₂×Dic₂₃	—	4	1	no	—
64	D₉₂	—	2	1	no	—
65	C₄×D₂₃	—	4	2	no	—
66	C₄×D₂₃	—	4	1	no	—
67	Dic₄₆	—	2	2	no	—
68	Dic₄₆	—	2	1	no	—
69	Dic₄₆	—	4	2	no	—
70	Dic₄₆	—	4	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	#2193	—	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	1	no	—
81	C₄×C₂₃	—	2	1	no	—
82	C₄×C₂₃	#120	1	1	yes	characteristic
83	C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
84	Dic₂₃	—	4	1	no	—
85	Dic₂₃	—	4	2	no	—
86	#578	—	23	1	no	—
87	#509	—	23	1	no	—
88	C₂×SD₃₂	—	23	1	no	—
89	#472	D₂₃	1	1	yes	characteristic, complemented
90	#362	—	23	1	no	—
91	#359	—	23	1	no	—
92	#350	—	23	1	no	—
93	C₂×C₂₃	—	2	1	no	—
94	C₂×C₂₃	—	4	1	no	—
95	C₂×C₂₃	—	16	1	no	—
96	C₂×C₂₃	#447	1	1	yes	characteristic
97	D₂₃	—	8	1	no	—
98	2₋¹⁺⁴	—	46	1	no	—
99	#137	—	46	2	no	—
100	#137	—	46	1	no	—
101	AGL(1,ℤ/8)	—	2	1	no	—
102	#135	—	23	1	no	—
103	#135	—	46	1	no	—
104	C₂×Q₁₆	—	23	1	no	—
105	C₂×Q₁₆	—	46	1	no	—
106	C₂×D₈	C₂×D₂₃	1	1	yes	characteristic
107	#131	C₂×D₂₃	1	1	yes	characteristic
108	SD₃₂	—	46	2	no	—
109	SD₃₂	—	46	1	no	—
110	D₁₆	—	46	1	no	—
111	M₃₂	—	23	1	no	—
112	C₂×C₁₆	—	23	1	no	—
113	#108	C₂×D₂₃	1	1	yes	characteristic
114	#100	—	2	1	no	—
115	C₂₃	#2193	1	1	yes	characteristic, complemented, Sylow
116	#41	—	46	2	no	—
117	#41	—	46	1	no	—
118	#41	—	92	2	no	—
119	#41	—	92	1	no	—
120	#41	C₂²⋊D₂₃	1	1	yes	characteristic
121	C₂×Q₈	—	46	2	no	—
122	C₂×Q₈	—	46	1	no	—
123	C₂×Q₈	—	92	1	no	—
124	C₂×D₄	—	2	1	no	—
125	Q₁₆	—	23	2	no	—
126	Q₁₆	—	23	1	no	—
127	Q₁₆	—	46	2	no	—
128	Q₁₆	—	46	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	1	no	—
136	M₁₆	—	2	1	no	—
137	M₁₆	—	2	1	no	—
138	M₁₆	C₂²⋊D₂₃	1	1	yes	characteristic
139	C₂×C₈	—	2	1	no	—
140	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
141	C₁₆	—	46	1	no	—
142	C₁₆	—	46	1	no	—
143	C₂³	—	4	1	no	—
144	Q₈	—	46	2	no	—
145	Q₈	—	46	1	no	—
146	Q₈	—	92	2	no	—
147	Q₈	—	92	1	no	—
148	Q₈	#67272	1	1	yes	characteristic
149	D₄	—	2	1	no	—
150	D₄	—	4	1	no	—
151	D₄	—	4	1	no	—
152	D₄	—	46	1	no	—
153	D₄	—	92	2	no	—
154	D₄	—	92	1	no	—
155	D₄	#67272	1	1	yes	characteristic
156	C₂×C₄	—	2	1	no	—
157	C₂×C₄	—	92	2	no	—
158	C₂×C₄	—	92	2	no	—
159	C₂×C₄	—	92	1	no	—
160	C₂×C₄	—	92	1	no	—
161	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
162	C₈	—	2	1	no	—
163	C₈	—	4	1	no	—
164	C₈	D₄×D₂₃	1	1	yes	characteristic
165	C₈	D₄×D₂₃	1	1	yes	characteristic
166	C₂²	—	2	1	no	—
167	C₂²	—	8	1	no	—
168	C₂²	—	16	1	no	—
169	C₂²	—	92	1	no	—
170	C₂²	#10625120	1	1	yes	characteristic
171	C₄	—	2	1	no	—
172	C₄	—	92	2	no	—
173	C₄	—	92	1	no	—
174	C₄	C₂×#67272	1	1	yes	characteristic
175	C₄	#10625108	1	1	yes	characteristic
176	C₂	—	2	1	no	—
177	C₂	—	4	1	no	—
178	C₂	—	16	1	no	—
179	C₂	—	184	1	no	—
180	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	#2193
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. #13093234	C₂	semidirect
2. #13092853	C₂	semidirect
3. C₂×#10625009	C₂	semidirect
4. #13092232	C₂	semidirect
5. #13092223	C₂	semidirect
6. #13092201	C₂	semidirect
7. C₂₃×#472	C₂	semidirect
89. #472	D₂₃	semidirect
115. C₂₃	#2193	semidirect

Maximal subgroups

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

subgroup	index
#13093234	2
#13092853	2
C₂×#10625009	2
#13092232	2
#13092223	2
#13092201	2
C₂₃×#472	2
#2193	23