Finite Group Databasebeta

Subgroups — #15675061

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	#13092966	C₂	1	1	yes	maximal, characteristic
2	#13092389	C₂	1	2	yes	maximal, complemented
3	#13092212	C₂	1	1	yes	maximal, characteristic, complemented
4	#13092209	C₂	1	2	yes	maximal, complemented
5	C₂₃×#354	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#67292	C₄	1	1	yes	characteristic, complemented
7	C₂×#67290	C₂²	1	1	yes	characteristic
8	C₂₃×#124	C₂²	1	1	yes	characteristic
9	#10625115	C₄	1	1	yes	characteristic, complemented
10	#10625111	C₂²	1	1	yes	characteristic
11	#10625055	—	2	2	no	—
12	(C₂²⋊D₂₃)⋊C₄	—	2	2	no	—
13	#10625048	—	2	2	no	—
14	#10625046	C₄	1	1	yes	characteristic, complemented
15	C₄²×D₂₃	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×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
20	C₂×D₄×C₂₃	C₂³	1	1	yes	characteristic
21	C₂×C₂²⋊D₂₃	—	2	2	no	—
22	C₂×C₂²⋊D₂₃	D₄	1	2	yes	—
23	C₂²×Dic₂₃	D₄	1	2	yes	—
24	#67292	—	4	1	no	—
25	#67292	—	4	1	no	—
26	#67290	—	4	1	no	—
27	#67290	—	4	1	no	—
28	C₂×D₉₂	—	2	1	no	—
29	C₂×D₉₂	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×D₂₃	—	2	1	no	—
31	C₂×C₄×D₂₃	—	2	1	no	—
32	C₂×C₄×D₂₃	C₂×C₄	1	1	yes	characteristic
33	C₂×Dic₄₆	C₂×C₄	1	1	yes	characteristic
34	C₂²⋊C₄×C₂₃	—	2	2	no	—
35	C₂²⋊C₄×C₂₃	—	2	2	no	—
36	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
37	C₂²⋊Dic₂₃	—	2	2	no	—
38	(C₂×D₂₃)⋊C₄	—	2	2	no	—
39	Dic₂₃⋊C₄	—	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	C₂×D₄	1	2	yes	—
45	C₂²×D₂₃	—	2	1	no	—
46	C₂²×D₂₃	C₂²⋊C₄	1	1	yes	characteristic
47	Q₈×C₂₃	—	4	1	no	—
48	D₄×C₂₃	—	4	1	no	—
49	C₂×C₄×C₂₃	—	2	1	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	4	no	—
55	C₂×Dic₂₃	—	2	1	no	—
56	C₂×Dic₂₃	—	2	2	no	—
57	C₂×Dic₂₃	—	2	2	no	—
58	C₂×Dic₂₃	—	4	2	no	—
59	C₂×Dic₂₃	C₂²⋊C₄	1	2	yes	—
60	C₂×Dic₂₃	C₂²⋊C₄	1	1	yes	characteristic
61	D₉₂	—	4	1	no	—
62	D₉₂	—	4	2	no	—
63	C₄×D₂₃	—	2	1	no	—
64	C₄×D₂₃	—	2	1	no	—
65	C₄×D₂₃	—	4	2	no	—
66	C₄×D₂₃	—	4	1	no	—
67	C₄×D₂₃	—	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	C₂×C₂²⋊C₄	1	1	yes	characteristic
73	C₂×D₂₃	—	2	1	no	—
74	C₂×D₂₃	—	4	1	no	—
75	C₂×D₂₃	—	4	2	no	—
76	C₂×D₂₃	—	4	1	no	—
77	C₄×C₂₃	—	2	1	no	—
78	C₄×C₂₃	—	4	1	no	—
79	C₄×C₂₃	—	4	1	no	—
80	C₄×C₂₃	—	8	2	no	—
81	Dic₂₃	—	2	2	no	—
82	Dic₂₃	—	2	2	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	1	no	—
89	#354	D₂₃	1	1	yes	characteristic, complemented
90	#353	—	23	2	no	—
91	C₂×C₂₃	—	2	1	no	—
92	C₂×C₂₃	—	4	2	no	—
93	C₂×C₂₃	C₂×#99	1	1	yes	characteristic
94	D₂₃	—	4	1	no	—
95	D₂₃	—	8	1	no	—
96	C₂×#41	—	23	1	no	—
97	C₂×#41	—	23	1	no	—
98	C₄⋊Q₈	—	23	1	no	—
99	#127	—	23	1	no	—
100	#124	—	23	1	no	—
101	#124	C₂×D₂₃	1	1	yes	characteristic
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	C₄×D₂₃	1	1	yes	characteristic
116	C₂×D₄	—	23	2	no	—
117	C₂×D₄	—	23	1	no	—
118	C₂×D₄	—	46	1	no	—
119	C₂×D₄	—	46	2	no	—
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	—	2	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	—	23	1	no	—
136	C₂³	—	46	1	no	—
137	C₂³	D₄×D₂₃	1	2	yes	—
138	Q₈	—	4	1	no	—
139	Q₈	—	92	1	no	—
140	D₄	—	4	1	no	—
141	D₄	—	92	1	no	—
142	D₄	—	92	2	no	—
143	D₄	—	92	2	no	—
144	D₄	—	92	4	no	—
145	C₂×C₄	—	2	1	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	1	no	—
151	C₂×C₄	—	46	1	no	—
152	C₂×C₄	—	46	2	no	—
153	C₂×C₄	—	46	2	no	—
154	C₂×C₄	—	46	1	no	—
155	C₂×C₄	—	92	1	no	—
156	C₂×C₄	—	92	1	no	—
157	C₂×C₄	—	92	2	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	—	46	1	no	—
163	C₂²	—	92	1	no	—
164	C₂²	—	92	2	no	—
165	C₂²	—	92	1	no	—
166	C₂²	C₂²⋊C₄×D₂₃	1	1	yes	characteristic
167	C₄	—	2	1	no	—
168	C₄	—	4	1	no	—
169	C₄	—	4	1	no	—
170	C₄	—	8	2	no	—
171	C₄	—	23	1	no	—
172	C₄	—	23	1	no	—
173	C₄	—	46	2	no	—
174	C₄	—	46	2	no	—
175	C₄	—	184	2	no	—
176	C₂	—	2	1	no	—
177	C₂	—	4	2	no	—
178	C₂	—	92	1	no	—
179	C₂	—	184	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×#354	C₂
socle	C₂×C₂₃	C₂×#99
prime radical	#13092966	C₂
cosocle	C₂×D₄×C₂₃	C₂³
hypercenter	#354	D₂₃
hypocenter	C₂₃	#2126
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂²×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×D₄ ◁ #354

Complements

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

N	H ≅ G/N	type
2. #13092389	C₂	semidirect
3. #13092212	C₂	semidirect
4. #13092209	C₂	semidirect
5. C₂₃×#354	C₂	semidirect
6. C₂×#67292	C₄	semidirect
9. #10625115	C₄	semidirect
14. #10625046	C₄	semidirect
15. C₄²×D₂₃	C₄	semidirect
89. #354	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
#13092966	2
#13092389	2
#13092212	2
#13092209	2
C₂₃×#354	2
#2126	23