Finite Group Databasebeta

Subgroups — #15675355

Order 2944.

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

normal subgroups only

This group has 220 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	#13093219	C₂	1	1	yes	maximal, characteristic, complemented
2	#13092852	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×D₃₆₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#13092688	C₂	1	1	yes	maximal, characteristic, complemented
5	#13092229	C₂	1	1	yes	maximal, characteristic, complemented
6	#13092200	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#359	C₂	1	1	yes	maximal, characteristic, complemented
8	#10625160	—	2	1	no	—
9	C₂₃×#135	C₂²	1	1	yes	characteristic
10	#10625120	—	2	1	no	—
11	#10625090	—	2	1	no	—
12	#10625085	—	2	2	no	—
13	D₈×D₂₃	—	2	1	no	—
14	#10625079	—	2	1	no	—
15	C₂×D₁₈₄	C₂²	1	1	yes	characteristic
16	C₂×D₁₈₄	C₂²	1	1	yes	characteristic, complemented
17	#10625072	C₂²	1	1	yes	characteristic
18	C₂×C₁₆×C₂₃	C₂²	1	1	yes	characteristic, complemented
19	C₂₃×#108	C₂²	1	1	yes	characteristic, complemented
20	#10625010	—	2	1	no	—
21	#10625008	—	2	1	no	—
22	#10625005	—	2	1	no	—
23	#10624994	C₂²	1	1	yes	characteristic, complemented
24	D₃₆₈	—	2	2	no	—
25	#41×C₂₃	—	2	1	no	—
26	#67292	—	2	2	no	—
27	D₄×D₂₃	—	2	2	no	—
28	D₄×D₂₃	—	4	2	no	—
29	#67288	—	4	1	no	—
30	#67288	D₄	1	1	yes	characteristic
31	C₂×D₉₂	—	2	1	no	—
32	C₂×D₉₂	—	2	1	no	—
33	C₂×D₉₂	—	4	1	no	—
34	Q₁₆×C₂₃	D₄	1	1	yes	characteristic
35	SD₁₆×C₂₃	—	2	1	no	—
36	D₈×C₂₃	D₄	1	1	yes	characteristic
37	M₁₆×C₂₃	—	2	1	no	—
38	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
39	#67274	—	2	2	no	—
40	#67272	—	2	2	no	—
41	#67267	D₄	1	1	yes	characteristic
42	D₁₈₄	—	2	1	no	—
43	D₁₈₄	—	2	1	no	—
44	D₁₈₄	—	2	1	no	—
45	D₁₈₄	—	4	1	no	—
46	D₁₈₄	D₄	1	2	yes	—
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	C₂²×D₂₃	—	4	2	no	—
53	C₂²×D₂₃	—	4	1	no	—
54	C₂²×D₂₃	—	4	1	no	—
55	Q₈×C₂₃	—	2	1	no	—
56	D₄×C₂₃	—	2	1	no	—
57	D₄×C₂₃	—	4	1	no	—
58	C₂×C₄×C₂₃	—	4	1	no	—
59	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
60	C₂²⋊D₂₃	—	2	1	no	—
61	C₂²⋊D₂₃	—	4	2	no	—
62	D₉₂	—	2	2	no	—
63	D₉₂	—	4	2	no	—
64	D₉₂	—	4	1	no	—
65	D₉₂	—	4	1	no	—
66	D₉₂	—	4	1	no	—
67	D₉₂	D₈	1	1	yes	characteristic
68	C₄×D₂₃	—	2	1	no	—
69	C₄×D₂₃	—	4	2	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	#2192	—	23	1	no	maximal, Sylow
76	C₂²×C₂₃	—	4	1	no	—
77	C₂²×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
78	C₂×D₂₃	—	2	1	no	—
79	C₂×D₂₃	—	4	2	no	—
80	C₂×D₂₃	—	8	1	no	—
81	C₂×D₂₃	—	16	2	no	—
82	C₂×D₂₃	—	16	1	no	—
83	C₂×D₂₃	—	16	1	no	—
84	C₄×C₂₃	—	4	1	no	—
85	C₄×C₂₃	#120	1	1	yes	characteristic
86	C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
87	Dic₂₃	—	2	1	no	—
88	#576	—	23	1	no	—
89	#509	—	23	1	no	—
90	C₂×D₁₆	—	23	1	no	—
91	#472	—	23	1	no	—
92	#361	—	23	1	no	—
93	#359	D₂₃	1	1	yes	characteristic, complemented
94	#350	—	23	1	no	—
95	C₂×C₂₃	—	2	1	no	—
96	C₂×C₂₃	—	8	1	no	—
97	C₂×C₂₃	#447	1	1	yes	characteristic
98	D₂₃	—	4	1	no	—
99	D₂₃	—	8	2	no	—
100	D₂₃	—	16	1	no	—
101	H(5,𝔽₂)	—	46	1	no	—
102	AGL(1,ℤ/8)	—	46	1	no	—
103	AGL(1,ℤ/8)	—	46	1	no	—
104	AGL(1,ℤ/8)	—	46	2	no	—
105	#135	—	46	1	no	—
106	#135	C₂×D₂₃	1	1	yes	characteristic
107	C₂×D₈	—	23	1	no	—
108	C₂×D₈	—	23	1	no	—
109	C₂×D₈	—	46	1	no	—
110	#131	—	23	1	no	—
111	SD₃₂	—	46	1	no	—
112	D₁₆	—	46	2	no	—
113	D₁₆	—	46	1	no	—
114	M₃₂	—	23	1	no	—
115	C₂×C₁₆	C₂×D₂₃	1	1	yes	characteristic, complemented
116	#108	C₂×D₂₃	1	1	yes	characteristic, complemented
117	#100	—	46	1	no	—
118	C₂₃	#2192	1	1	yes	characteristic, complemented, Sylow
119	#41	—	2	1	no	—
120	#41	—	23	1	no	—
121	#41	—	46	2	no	—
122	#41	—	92	1	no	—
123	C₂×D₄	—	46	2	no	—
124	C₂×D₄	—	46	1	no	—
125	C₂×D₄	—	46	1	no	—
126	C₂×D₄	—	92	1	no	—
127	C₂×D₄	—	92	2	no	—
128	Q₁₆	D₉₂	1	1	yes	characteristic
129	SD₁₆	—	2	1	no	—
130	SD₁₆	—	46	2	no	—
131	SD₁₆	—	92	1	no	—
132	D₈	—	23	2	no	—
133	D₈	—	46	2	no	—
134	D₈	—	46	1	no	—
135	D₈	—	46	1	no	—
136	D₈	—	46	1	no	—
137	D₈	—	92	1	no	—
138	D₈	D₉₂	1	1	yes	characteristic
139	M₁₆	—	2	1	no	—
140	M₁₆	—	23	1	no	—
141	M₁₆	—	46	1	no	—
142	C₂×C₈	—	46	1	no	—
143	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
144	C₁₆	—	2	1	no	—
145	C₁₆	—	46	1	no	—
146	C₂³	—	92	2	no	—
147	C₂³	—	92	1	no	—
148	C₂³	—	92	1	no	—
149	Q₈	—	2	1	no	—
150	Q₈	—	23	1	no	—
151	D₄	—	2	1	no	—
152	D₄	—	4	1	no	—
153	D₄	—	23	1	no	—
154	D₄	—	46	1	no	—
155	D₄	—	46	2	no	—
156	D₄	—	92	2	no	—
157	D₄	—	92	2	no	—
158	D₄	—	92	1	no	—
159	D₄	—	92	1	no	—
160	D₄	—	92	1	no	—
161	C₂×C₄	—	4	1	no	—
162	C₂×C₄	—	46	1	no	—
163	C₂×C₄	—	92	2	no	—
164	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
165	C₈	—	4	1	no	—
166	C₈	—	46	1	no	—
167	C₈	C₂×D₉₂	1	1	yes	characteristic
168	C₈	D₄×D₂₃	1	1	yes	characteristic
169	C₂²	—	4	1	no	—
170	C₂²	—	46	1	no	—
171	C₂²	—	92	2	no	—
172	C₂²	—	368	1	no	—
173	C₂²	—	368	1	no	—
174	C₂²	—	368	2	no	—
175	C₂²	—	184	1	no	—
176	C₂²	#10625085	1	1	yes	characteristic
177	C₄	—	4	1	no	—
178	C₄	—	46	1	no	—
179	C₄	#10625053	1	1	yes	characteristic
180	C₄	D₈×D₂₃	1	1	yes	characteristic
181	C₂	—	2	1	no	—
182	C₂	—	8	1	no	—
183	C₂	—	92	1	no	—
184	C₂	—	368	1	no	—
185	C₂	—	184	2	no	—
186	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×#359	C₂
socle	C₂×C₂₃	#447
hypercenter	#359	D₂₃
hypocenter	C₂₃	#2192
LCS	G ▷ C₂×C₈×C₂₃ ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #359

Complements

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

N	H ≅ G/N	type
1. #13093219	C₂	semidirect
2. #13092852	C₂	semidirect
3. C₂×D₃₆₈	C₂	semidirect
4. #13092688	C₂	semidirect
5. #13092229	C₂	semidirect
6. #13092200	C₂	semidirect
7. C₂₃×#359	C₂	semidirect
16. C₂×D₁₈₄	C₂²	semidirect
18. C₂×C₁₆×C₂₃	C₂²	semidirect
19. C₂₃×#108	C₂²	semidirect
23. #10624994	C₂²	semidirect
93. #359	D₂₃	semidirect
115. C₂×C₁₆	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
#13093219	2
#13092852	2
C₂×D₃₆₈	2
#13092688	2
#13092229	2
#13092200	2
C₂₃×#359	2
#2192	23