Finite Group Databasebeta

Subgroups — #15675067

Order 2944.

270 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 55 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	#13092980	C₂	1	1	yes	maximal, characteristic
2	#13092389	C₂	1	2	yes	maximal, complemented
3	#13092210	C₂	1	1	yes	maximal, characteristic, complemented
4	#13092209	C₂	1	2	yes	maximal, complemented
5	C₂₃×#353	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#67290	C₂²	1	1	yes	characteristic
7	C₂×D₄×D₂₃	C₄	1	1	yes	characteristic, complemented
8	C₂₃×#127	C₂²	1	1	yes	characteristic
9	#10625111	C₂²	1	1	yes	characteristic
10	#10625110	—	2	2	no	—
11	#10625108	—	2	2	no	—
12	#10625107	C₄	1	1	yes	characteristic, complemented
13	#10625048	—	2	2	no	—
14	#10625046	C₄	1	1	yes	characteristic, complemented
15	#10625043	C₄	1	1	yes	characteristic, complemented
16	C₂₃×#99	C₂²	1	2	yes	complemented
17	#10625016	C₂²	1	2	yes	complemented
18	#10624988	—	2	2	no	—
19	C₂³×D₂₃	—	2	1	no	—
20	C₂×D₄×C₂₃	—	2	2	no	—
21	C₂×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
22	C₂×D₄×C₂₃	C₂³	1	1	yes	characteristic
23	C₂×C₂²⋊D₂₃	—	2	2	no	—
24	C₂×C₂²⋊D₂₃	—	2	1	no	—
25	C₂×C₂²⋊D₂₃	D₄	1	2	yes	—
26	C₂²×Dic₂₃	D₄	1	2	yes	—
27	#67290	—	4	2	no	—
28	D₄×D₂₃	—	4	2	no	—
29	C₂×D₉₂	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×D₂₃	C₂×C₄	1	1	yes	characteristic
31	C₂×Dic₄₆	C₂×C₄	1	1	yes	characteristic
32	C₂²⋊C₄×C₂₃	—	2	2	no	—
33	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
34	C₂²⋊Dic₂₃	—	2	2	no	—
35	C₂²⋊Dic₂₃	—	2	2	no	—
36	(C₂×D₂₃)⋊C₄	—	2	1	no	—
37	(C₂×D₂₃)⋊C₄	—	2	2	no	—
38	Dic₂₃⋊C₄	—	2	1	no	—
39	Dic₂₃⋊C₄	—	4	2	no	—
40	C₄×Dic₂₃	—	4	2	no	—
41	C₄×Dic₂₃	C₂×C₄	1	1	yes	characteristic
42	C₂³×C₂₃	—	2	1	no	—
43	C₂³×C₂₃	C₂×D₄	1	2	yes	—
44	C₂²×D₂₃	—	2	1	no	—
45	C₂²×D₂₃	—	4	2	no	—
46	C₂²×D₂₃	—	8	2	no	—
47	C₂²×D₂₃	C₂²⋊C₄	1	1	yes	characteristic
48	D₄×C₂₃	—	4	1	no	—
49	D₄×C₂₃	—	4	1	no	—
50	D₄×C₂₃	—	8	2	no	—
51	C₂×C₄×C₂₃	—	2	1	no	—
52	C₂×C₄×C₂₃	—	4	2	no	—
53	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
54	C₂²⋊D₂₃	—	4	2	no	—
55	C₂²⋊D₂₃	—	4	2	no	—
56	C₂²⋊D₂₃	—	8	2	no	—
57	C₂×Dic₂₃	—	2	2	no	—
58	C₂×Dic₂₃	—	2	2	no	—
59	C₂×Dic₂₃	—	2	1	no	—
60	C₂×Dic₂₃	—	4	2	no	—
61	C₂×Dic₂₃	C₂²⋊C₄	1	2	yes	—
62	C₂×Dic₂₃	C₂²⋊C₄	1	1	yes	characteristic
63	D₉₂	—	4	1	no	—
64	C₄×D₂₃	—	2	2	no	—
65	Dic₄₆	—	4	1	no	—
66	#2128	—	23	1	no	maximal, Sylow
67	C₂²×C₂₃	—	2	2	no	—
68	C₂²×C₂₃	—	4	2	no	—
69	C₂²×C₂₃	—	4	2	no	—
70	C₂²×C₂₃	—	4	1	no	—
71	C₂²×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₂×D₂₃	—	2	1	no	—
73	C₂×D₂₃	—	4	1	no	—
74	C₂×D₂₃	—	4	2	no	—
75	C₂×D₂₃	—	4	1	no	—
76	C₂×D₂₃	—	8	4	no	—
77	C₄×C₂₃	—	2	1	no	—
78	C₄×C₂₃	—	4	1	no	—
79	C₄×C₂₃	—	8	2	no	—
80	Dic₂₃	—	2	2	no	—
81	Dic₂₃	—	4	1	no	—
82	Dic₂₃	—	8	2	no	—
83	Dic₂₃	#99	1	2	yes	—
84	#535	—	23	1	no	—
85	#410	—	23	2	no	—
86	#353	—	23	1	no	—
87	#353	—	23	2	no	—
88	#353	D₂₃	1	1	yes	characteristic, complemented
89	C₂×C₂₃	—	2	1	no	—
90	C₂×C₂₃	—	4	2	no	—
91	C₂×C₂₃	—	8	1	no	—
92	C₂×C₂₃	C₂×#99	1	1	yes	characteristic
93	D₂₃	—	4	1	no	—
94	D₂₃	—	8	1	no	—
95	C₂×#41	—	23	1	no	—
96	C₂²×D₄	—	23	1	no	—
97	#127	—	23	1	no	—
98	#127	C₂×D₂₃	1	1	yes	characteristic
99	#124	—	23	1	no	—
100	#124	—	23	1	no	—
101	#121	—	46	2	no	—
102	#120	—	46	2	no	—
103	#117	—	23	1	no	—
104	#117	—	46	2	no	—
105	#99	—	23	2	no	—
106	#99	—	46	2	no	—
107	#99	C₂×D₂₃	1	2	yes	complemented
108	C₂₃	#2128	1	1	yes	characteristic, complemented, Sylow
109	C₂⁴	—	46	1	no	—
110	#41	—	92	2	no	—
111	C₂×Q₈	—	23	1	no	—
112	C₂×D₄	—	2	2	no	—
113	C₂×D₄	—	23	2	no	—
114	C₂×D₄	—	23	1	no	—
115	C₂×D₄	—	46	1	no	—
116	C₂×D₄	—	46	2	no	—
117	C₂×D₄	—	92	2	no	—
118	C₂×D₄	C₄×D₂₃	1	1	yes	characteristic
119	C₂×D₄	C₂²×D₂₃	1	1	yes	characteristic
120	C₂²×C₄	—	23	2	no	—
121	C₂²×C₄	—	23	1	no	—
122	C₄⋊C₄	—	46	1	no	—
123	C₄⋊C₄	—	92	2	no	—
124	C₂²⋊C₄	—	2	2	no	—
125	C₂²⋊C₄	—	46	2	no	—
126	C₂²⋊C₄	—	46	1	no	—
127	C₂²⋊C₄	—	46	2	no	—
128	C₂²⋊C₄	—	46	2	no	—
129	C₄²	—	23	1	no	—
130	C₄²	—	92	2	no	—
131	C₄²	C₄×D₂₃	1	1	yes	characteristic
132	C₂³	—	2	1	no	—
133	C₂³	—	23	1	no	—
134	C₂³	—	46	1	no	—
135	C₂³	—	92	2	no	—
136	C₂³	—	184	2	no	—
137	C₂³	D₄×D₂₃	1	2	yes	—
138	Q₈	—	92	1	no	—
139	D₄	—	4	1	no	—
140	D₄	—	4	1	no	—
141	D₄	—	8	2	no	—
142	D₄	—	92	1	no	—
143	D₄	—	92	2	no	—
144	D₄	—	92	2	no	—
145	D₄	—	184	2	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	4	2	no	—
148	C₂×C₄	—	23	1	no	—
149	C₂×C₄	—	23	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	—	92	2	no	—
155	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
156	C₂²	—	2	2	no	—
157	C₂²	—	4	2	no	—
158	C₂²	—	4	2	no	—
159	C₂²	—	4	1	no	—
160	C₂²	—	46	1	no	—
161	C₂²	—	92	1	no	—
162	C₂²	—	92	2	no	—
163	C₂²	—	92	1	no	—
164	C₂²	—	184	4	no	—
165	C₂²	C₂²⋊C₄×D₂₃	1	1	yes	characteristic
166	C₄	—	2	1	no	—
167	C₄	—	4	1	no	—
168	C₄	—	8	2	no	—
169	C₄	—	23	2	no	—
170	C₄	—	46	2	no	—
171	C₄	—	92	1	no	—
172	C₄	—	184	2	no	—
173	C₂	—	2	1	no	—
174	C₂	—	4	2	no	—
175	C₂	—	8	1	no	—
176	C₂	—	92	1	no	—
177	C₂	—	184	1	no	—
178	C₂	#99×D₂₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#99×D₂₃
commutator	C₂×C₄×C₂₃	C₂²×C₄
Frattini	C₂×D₄	C₂²×D₂₃
Fitting	C₂₃×#353	C₂
socle	C₂×C₂₃	C₂×#99
prime radical	#13092980	C₂
cosocle	C₂×D₄×C₂₃	C₂³
hypercenter	#353	D₂₃
hypocenter	C₂₃	#2128
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂²×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×D₄ ◁ #353

Complements

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

N	H ≅ G/N	type
2. #13092389	C₂	semidirect
3. #13092210	C₂	semidirect
4. #13092209	C₂	semidirect
5. C₂₃×#353	C₂	semidirect
7. C₂×D₄×D₂₃	C₄	semidirect
12. #10625107	C₄	semidirect
14. #10625046	C₄	semidirect
15. #10625043	C₄	semidirect
16. C₂₃×#99	C₂²	semidirect
17. #10625016	C₂²	semidirect
88. #353	D₂₃	semidirect
107. #99	C₂×D₂₃	semidirect
108. C₂₃	#2128	semidirect

Maximal subgroups

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

subgroup	index
#13092980	2
#13092389	2
#13092210	2
#13092209	2
C₂₃×#353	2
#2128	23