Finite Group Databasebeta

Subgroups — #15675371

Order 2944.

216 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 47 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	#13093218	C₂	1	1	yes	maximal, characteristic
2	#13092860	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#10625009	C₂	1	1	yes	maximal, characteristic, complemented
4	#13092693	C₂	1	1	yes	maximal, characteristic, complemented
5	#13092223	C₂	1	1	yes	maximal, characteristic, complemented
6	#13092202	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#361	C₂	1	1	yes	maximal, characteristic, complemented
8	#10625155	—	2	1	no	—
9	C₂×D₈×C₂₃	C₂²	1	1	yes	characteristic
10	#10625103	—	2	1	no	—
11	#10625083	—	2	1	no	—
12	#10625082	—	2	2	no	—
13	D₈×D₂₃	—	2	1	no	—
14	#10625080	—	2	1	no	—
15	C₂×Dic₉₂	C₂²	1	1	yes	characteristic, complemented
16	#10625075	C₂²	1	1	yes	characteristic
17	#10625072	C₂²	1	1	yes	characteristic
18	C₂₃×M₃₂	C₂²	1	1	yes	characteristic, complemented
19	C₂₃×#108	C₂²	1	1	yes	characteristic, complemented
20	#10625009	—	2	2	no	—
21	#10625006	—	2	1	no	—
22	C₂×C₂₃⋊C₁₆	C₂²	1	1	yes	characteristic, complemented
23	Dic₁₈₄	—	2	1	no	—
24	#10624982	—	2	1	no	—
25	C₂×D₄×C₂₃	—	2	1	no	—
26	C₂×C₂²⋊D₂₃	—	4	2	no	—
27	#67290	—	2	2	no	—
28	#67290	—	4	1	no	—
29	D₄×D₂₃	—	2	2	no	—
30	D₄×D₂₃	—	4	1	no	—
31	#67288	—	2	1	no	—
32	#67288	D₄	1	1	yes	characteristic
33	C₂×Dic₄₆	—	2	1	no	—
34	D₈×C₂₃	—	2	1	no	—
35	D₈×C₂₃	D₄	1	2	yes	—
36	M₁₆×C₂₃	—	2	1	no	—
37	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
38	#67273	—	2	2	no	—
39	#67272	—	2	2	no	—
40	#67267	D₄	1	1	yes	characteristic
41	Dic₉₂	—	2	1	no	—
42	Dic₉₂	—	2	1	no	—
43	Dic₉₂	—	4	1	no	—
44	Dic₉₂	D₄	1	1	yes	characteristic
45	D₁₈₄	D₄	1	1	yes	characteristic
46	#67263	—	2	1	no	—
47	#67263	—	4	1	no	—
48	#67262	—	2	1	no	—
49	C₈×D₂₃	—	2	1	no	—
50	C₁₆×C₂₃	—	2	1	no	—
51	C₂₃⋊C₁₆	—	2	1	no	—
52	C₂³×C₂₃	—	4	1	no	—
53	C₂²×D₂₃	—	4	2	no	—
54	D₄×C₂₃	—	2	2	no	—
55	D₄×C₂₃	—	4	1	no	—
56	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
57	C₂²⋊D₂₃	—	2	1	no	—
58	C₂²⋊D₂₃	—	4	2	no	—
59	C₂²⋊D₂₃	—	4	2	no	—
60	C₂²⋊D₂₃	—	4	1	no	—
61	C₂×Dic₂₃	—	4	2	no	—
62	C₂×Dic₂₃	—	4	1	no	—
63	D₉₂	—	2	1	no	—
64	D₉₂	D₈	1	1	yes	characteristic
65	C₄×D₂₃	—	2	1	no	—
66	C₄×D₂₃	—	4	1	no	—
67	Dic₄₆	—	2	1	no	—
68	Dic₄₆	—	4	1	no	—
69	Dic₄₆	—	4	1	no	—
70	Dic₄₆	D₈	1	1	yes	characteristic
71	C₈×C₂₃	—	4	1	no	—
72	C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
73	C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
74	C₂₃⋊C₈	—	2	1	no	—
75	#2194	—	23	1	no	maximal, Sylow
76	C₂²×C₂₃	—	4	2	no	—
77	C₂²×C₂₃	—	16	1	no	—
78	C₂²×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
79	C₂×D₂₃	—	2	1	no	—
80	C₂×D₂₃	—	4	1	no	—
81	C₂×D₂₃	—	16	2	no	—
82	C₄×C₂₃	#120	1	1	yes	characteristic
83	C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
84	Dic₂₃	—	2	1	no	—
85	Dic₂₃	—	4	1	no	—
86	Dic₂₃	—	8	1	no	—
87	#576	—	23	1	no	—
88	#510	—	23	1	no	—
89	C₂×SD₃₂	—	23	1	no	—
90	#473	—	23	1	no	—
91	#361	D₂₃	1	1	yes	characteristic, complemented
92	#359	—	23	1	no	—
93	#350	—	23	1	no	—
94	C₂×C₂₃	—	2	1	no	—
95	C₂×C₂₃	—	8	2	no	—
96	C₂×C₂₃	#447	1	1	yes	characteristic
97	D₂₃	—	4	1	no	—
98	D₂₃	—	8	1	no	—
99	H(5,𝔽₂)	—	46	1	no	—
100	#137	—	46	1	no	—
101	AGL(1,ℤ/8)	—	46	1	no	—
102	AGL(1,ℤ/8)	—	46	2	no	—
103	#135	—	23	1	no	—
104	#135	—	46	1	no	—
105	C₂×Q₁₆	—	23	1	no	—
106	C₂×D₈	—	46	1	no	—
107	C₂×D₈	C₂×D₂₃	1	1	yes	characteristic
108	#131	—	23	1	no	—
109	Q₃₂	—	46	1	no	—
110	SD₃₂	—	46	1	no	—
111	SD₃₂	—	46	2	no	—
112	M₃₂	C₂×D₂₃	1	1	yes	characteristic, complemented
113	C₂×C₁₆	—	23	1	no	—
114	#108	C₂×D₂₃	1	1	yes	characteristic, complemented
115	#101	—	46	1	no	—
116	C₂₃	#2194	1	1	yes	characteristic, complemented, Sylow
117	#41	—	23	1	no	—
118	#41	—	46	2	no	—
119	#41	—	46	1	no	—
120	#41	—	92	1	no	—
121	C₂×Q₈	—	46	1	no	—
122	C₂×D₄	—	2	1	no	—
123	C₂×D₄	—	46	2	no	—
124	C₂×D₄	—	92	1	no	—
125	C₂×D₄	—	92	2	no	—
126	Q₁₆	—	23	1	no	—
127	Q₁₆	—	46	1	no	—
128	Q₁₆	—	46	1	no	—
129	Q₁₆	—	92	1	no	—
130	SD₁₆	—	46	2	no	—
131	SD₁₆	—	46	1	no	—
132	SD₁₆	—	92	1	no	—
133	D₈	—	2	1	no	—
134	D₈	—	23	1	no	—
135	D₈	—	46	2	no	—
136	D₈	D₉₂	1	2	yes	—
137	M₁₆	—	2	1	no	—
138	M₁₆	—	23	1	no	—
139	M₁₆	—	46	1	no	—
140	C₂×C₈	—	46	1	no	—
141	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
142	C₁₆	—	2	1	no	—
143	C₁₆	—	46	1	no	—
144	C₂³	—	4	1	no	—
145	C₂³	—	92	2	no	—
146	Q₈	—	23	1	no	—
147	Q₈	—	46	1	no	—
148	Q₈	—	92	1	no	—
149	Q₈	—	92	1	no	—
150	D₄	—	2	2	no	—
151	D₄	—	4	1	no	—
152	D₄	—	23	1	no	—
153	D₄	—	46	1	no	—
154	D₄	—	46	1	no	—
155	D₄	—	92	2	no	—
156	D₄	—	92	2	no	—
157	D₄	—	92	1	no	—
158	C₂×C₄	—	46	1	no	—
159	C₂×C₄	—	92	2	no	—
160	C₂×C₄	—	92	1	no	—
161	C₂×C₄	—	92	1	no	—
162	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
163	C₈	—	4	1	no	—
164	C₈	—	46	1	no	—
165	C₈	C₂×D₉₂	1	1	yes	characteristic
166	C₈	D₄×D₂₃	1	1	yes	characteristic
167	C₂²	—	4	2	no	—
168	C₂²	—	16	1	no	—
169	C₂²	—	46	1	no	—
170	C₂²	—	92	1	no	—
171	C₂²	—	368	2	no	—
172	C₂²	#10625085	1	1	yes	characteristic
173	C₄	—	46	1	no	—
174	C₄	—	92	1	no	—
175	C₄	—	184	1	no	—
176	C₄	#10625053	1	1	yes	characteristic
177	C₄	D₈×D₂₃	1	1	yes	characteristic
178	C₂	—	2	1	no	—
179	C₂	—	8	2	no	—
180	C₂	—	92	1	no	—
181	C₂	—	184	1	no	—
182	C₂	#13092552	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#13092552
commutator, cosocle	C₂×C₈×C₂₃	C₂³
Frattini	C₂×C₈	C₂²×D₂₃
Fitting	C₂₃×#361	C₂
socle	C₂×C₂₃	#447
prime radical	#13093218	C₂
hypercenter	#361	D₂₃
hypocenter	C₂₃	#2194
LCS	G ▷ C₂×C₈×C₂₃ ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #361

Complements

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

N	H ≅ G/N	type
2. #13092860	C₂	semidirect
3. C₂×#10625009	C₂	semidirect
4. #13092693	C₂	semidirect
5. #13092223	C₂	semidirect
6. #13092202	C₂	semidirect
7. C₂₃×#361	C₂	semidirect
15. C₂×Dic₉₂	C₂²	semidirect
18. C₂₃×M₃₂	C₂²	semidirect
19. C₂₃×#108	C₂²	semidirect
22. C₂×C₂₃⋊C₁₆	C₂²	semidirect
91. #361	D₂₃	semidirect
112. M₃₂	C₂×D₂₃	semidirect
114. #108	C₂×D₂₃	semidirect
116. C₂₃	#2194	semidirect

Maximal subgroups

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

subgroup	index
#13093218	2
#13092860	2
C₂×#10625009	2
#13092693	2
#13092223	2
#13092202	2
C₂₃×#361	2
#2194	23