Finite Group Databasebeta

Subgroups — #15675059

Order 2944.

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

normal subgroups only

This group has 258 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	#13092965	C₂	1	1	yes	maximal, characteristic
2	#13092389	C₂	1	2	yes	maximal, complemented
3	#13092211	C₂	1	2	yes	maximal, complemented
4	#13092210	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#353	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#67290	C₄	1	1	yes	characteristic, complemented
7	C₂×#67290	C₂²	1	1	yes	characteristic
8	C₂₃×#127	C₂²	1	1	yes	characteristic
9	#10625109	—	2	2	no	—
10	#10625107	C₄	1	1	yes	characteristic, complemented
11	#10625107	C₂²	1	1	yes	characteristic
12	D₄×Dic₂₃	—	2	2	no	—
13	#10625048	—	2	2	no	—
14	C₄²×D₂₃	C₄	1	1	yes	characteristic, complemented
15	#10625040	C₄	1	1	yes	characteristic, complemented
16	C₂₃×#99	C₂²	1	2	yes	—
17	#10625016	C₂²	1	2	yes	—
18	#10624988	—	2	2	no	—
19	C₂×D₄×C₂₃	—	2	2	no	—
20	C₂×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
21	C₂×D₄×C₂₃	C₂³	1	1	yes	characteristic
22	C₂×C₂²⋊D₂₃	—	2	1	no	—
23	C₂×C₂²⋊D₂₃	D₄	1	2	yes	—
24	C₂²×Dic₂₃	—	2	1	no	—
25	C₂²×Dic₂₃	D₄	1	2	yes	—
26	#67290	—	4	1	no	—
27	#67290	—	4	1	no	—
28	#67290	—	4	1	no	—
29	#67290	—	4	1	no	—
30	C₂×C₄×D₂₃	—	2	1	no	—
31	C₂×C₄×D₂₃	C₂×C₄	1	1	yes	characteristic
32	C₂×Dic₄₆	C₂×C₄	1	1	yes	characteristic
33	C₂×Dic₄₆	C₂×C₄	1	1	yes	characteristic
34	C₂²⋊C₄×C₂₃	—	2	2	no	—
35	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
36	C₂²⋊Dic₂₃	—	2	2	no	—
37	C₂²⋊Dic₂₃	—	2	2	no	—
38	C₂²⋊Dic₂₃	—	2	2	no	—
39	C₄⋊Dic₂₃	—	2	2	no	—
40	Dic₂₃⋊C₄	—	4	2	no	—
41	C₄×Dic₂₃	—	2	1	no	—
42	C₄×Dic₂₃	—	4	2	no	—
43	C₄×Dic₂₃	C₂×C₄	1	1	yes	characteristic
44	C₂³×C₂₃	—	2	1	no	—
45	C₂³×C₂₃	C₂×D₄	1	2	yes	—
46	C₂²×D₂₃	C₂²⋊C₄	1	1	yes	characteristic
47	D₄×C₂₃	—	4	1	no	—
48	D₄×C₂₃	—	4	1	no	—
49	D₄×C₂₃	—	4	4	no	—
50	C₂×C₄×C₂₃	—	2	1	no	—
51	C₂×C₄×C₂₃	—	4	2	no	—
52	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
53	C₂²⋊D₂₃	—	4	2	no	—
54	C₂²⋊D₂₃	—	4	2	no	—
55	C₂×Dic₂₃	—	2	2	no	—
56	C₂×Dic₂₃	—	2	2	no	—
57	C₂×Dic₂₃	—	2	1	no	—
58	C₂×Dic₂₃	—	2	1	no	—
59	C₂×Dic₂₃	—	4	1	no	—
60	C₂×Dic₂₃	—	4	1	no	—
61	C₂×Dic₂₃	—	4	2	no	—
62	C₂×Dic₂₃	C₂²⋊C₄	1	2	yes	—
63	C₂×Dic₂₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₄×D₂₃	—	2	1	no	—
65	C₄×D₂₃	—	2	1	no	—
66	C₄×D₂₃	—	4	2	no	—
67	Dic₄₆	—	4	1	no	—
68	Dic₄₆	—	4	1	no	—
69	#2126	—	23	1	no	maximal, Sylow
70	C₂²×C₂₃	—	2	2	no	—
71	C₂²×C₂₃	—	4	2	no	—
72	C₂²×C₂₃	—	4	2	no	—
73	C₂²×C₂₃	—	4	1	no	—
74	C₂²×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₂×D₂₃	—	2	1	no	—
76	C₂×D₂₃	—	4	1	no	—
77	C₄×C₂₃	—	2	2	no	—
78	C₄×C₂₃	—	2	1	no	—
79	C₄×C₂₃	—	8	2	no	—
80	Dic₂₃	—	2	2	no	—
81	Dic₂₃	—	4	1	no	—
82	Dic₂₃	—	4	1	no	—
83	Dic₂₃	—	8	2	no	—
84	Dic₂₃	#99	1	1	yes	characteristic
85	Dic₂₃	#99	1	1	yes	characteristic
86	#532	—	23	1	no	—
87	#410	—	23	2	no	—
88	#354	—	23	2	no	—
89	#353	—	23	1	no	—
90	#353	D₂₃	1	1	yes	characteristic, complemented
91	C₂×C₂₃	—	2	1	no	—
92	C₂×C₂₃	—	4	2	no	—
93	C₂×C₂₃	—	8	1	no	—
94	C₂×C₂₃	C₂×#99	1	1	yes	characteristic
95	D₂₃	—	4	1	no	—
96	C₂×#41	—	23	1	no	—
97	C₂×#41	—	23	1	no	—
98	C₄⋊Q₈	—	23	1	no	—
99	#127	C₂×D₂₃	1	1	yes	characteristic
100	#124	—	23	1	no	—
101	#124	—	23	1	no	—
102	#121	—	46	2	no	—
103	C₄×D₄	—	46	2	no	—
104	#117	—	46	2	no	—
105	C₂×C₄²	—	23	1	no	—
106	#99	—	23	2	no	—
107	#99	—	46	2	no	—
108	#99	C₂×D₂₃	1	2	yes	—
109	C₂₃	#2126	1	1	yes	characteristic, complemented, Sylow
110	#41	—	92	1	no	—
111	#41	—	92	1	no	—
112	#41	—	92	1	no	—
113	#41	—	92	1	no	—
114	C₂×Q₈	—	23	1	no	—
115	C₂×Q₈	—	23	1	no	—
116	C₂×D₄	—	2	2	no	—
117	C₂×D₄	—	23	2	no	—
118	C₂×D₄	—	46	1	no	—
119	C₂×D₄	C₄×D₂₃	1	1	yes	characteristic
120	C₂×D₄	C₂²×D₂₃	1	1	yes	characteristic
121	C₂²×C₄	—	23	2	no	—
122	C₂²×C₄	—	23	1	no	—
123	C₂²×C₄	—	46	1	no	—
124	C₂²×C₄	—	46	1	no	—
125	C₄⋊C₄	—	46	2	no	—
126	C₄⋊C₄	—	92	2	no	—
127	C₂²⋊C₄	—	2	2	no	—
128	C₂²⋊C₄	—	46	2	no	—
129	C₂²⋊C₄	—	46	2	no	—
130	C₂²⋊C₄	—	46	2	no	—
131	C₄²	—	23	1	no	—
132	C₄²	—	46	1	no	—
133	C₄²	—	92	2	no	—
134	C₄²	C₄×D₂₃	1	1	yes	characteristic
135	C₂³	—	2	1	no	—
136	C₂³	—	23	1	no	—
137	C₂³	D₄×D₂₃	1	2	yes	—
138	Q₈	—	92	1	no	—
139	Q₈	—	92	1	no	—
140	D₄	—	4	1	no	—
141	D₄	—	4	1	no	—
142	D₄	—	4	4	no	—
143	D₄	—	92	2	no	—
144	D₄	—	92	2	no	—
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	4	2	no	—
147	C₂×C₄	—	23	1	no	—
148	C₂×C₄	—	23	2	no	—
149	C₂×C₄	—	46	1	no	—
150	C₂×C₄	—	46	1	no	—
151	C₂×C₄	—	46	2	no	—
152	C₂×C₄	—	46	2	no	—
153	C₂×C₄	—	46	1	no	—
154	C₂×C₄	—	46	1	no	—
155	C₂×C₄	—	92	2	no	—
156	C₂×C₄	—	92	1	no	—
157	C₂×C₄	—	92	1	no	—
158	C₂×C₄	—	92	2	no	—
159	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
160	C₂²	—	2	2	no	—
161	C₂²	—	4	2	no	—
162	C₂²	—	4	2	no	—
163	C₂²	—	4	1	no	—
164	C₂²	—	46	1	no	—
165	C₂²	—	92	1	no	—
166	C₂²	C₂²⋊C₄×D₂₃	1	1	yes	characteristic
167	C₄	—	2	2	no	—
168	C₄	—	2	1	no	—
169	C₄	—	8	2	no	—
170	C₄	—	23	1	no	—
171	C₄	—	23	1	no	—
172	C₄	—	46	2	no	—
173	C₄	—	92	1	no	—
174	C₄	—	92	1	no	—
175	C₄	—	184	2	no	—
176	C₂	—	2	1	no	—
177	C₂	—	4	2	no	—
178	C₂	—	8	1	no	—
179	C₂	—	92	1	no	—
180	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	#13092965	C₂
cosocle	C₂×D₄×C₂₃	C₂³
hypercenter	#353	D₂₃
hypocenter	C₂₃	#2126
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. #13092211	C₂	semidirect
4. #13092210	C₂	semidirect
5. C₂₃×#353	C₂	semidirect
6. C₂×#67290	C₄	semidirect
10. #10625107	C₄	semidirect
14. C₄²×D₂₃	C₄	semidirect
15. #10625040	C₄	semidirect
90. #353	D₂₃	semidirect
109. C₂₃	#2126	semidirect

Maximal subgroups

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

subgroup	index
#13092965	2
#13092389	2
#13092211	2
#13092210	2
C₂₃×#353	2
#2126	23