Finite Group Databasebeta

Subgroups — #15675145

Order 2944.

220 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 65 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	C₂×#10625090	C₂	1	1	yes	maximal, characteristic
2	M₃₂×D₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#108×D₂₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#13092233	C₂	1	1	yes	maximal, characteristic, complemented
5	#13092231	C₂	1	1	yes	maximal, characteristic, complemented
6	#13092229	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#362	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#67292	—	2	1	no	—
9	C₂×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic
10	C₂×#67274	—	2	1	no	—
11	#10625090	—	2	1	no	—
12	#10625090	—	2	1	no	—
13	#10625090	C₄	1	2	yes	—
14	#10625090	C₄	1	2	yes	—
15	M₁₆×D₂₃	—	2	1	no	—
16	C₂×D₁₈₄	C₂²	1	1	yes	characteristic
17	C₂×C₈×D₂₃	C₂²	1	1	yes	characteristic
18	C₂₃×M₃₂	C₂²	1	1	yes	characteristic
19	C₂₃×#108	C₂²	1	1	yes	characteristic
20	#10625004	—	2	1	no	—
21	#10625001	C₂²	1	1	yes	characteristic
22	#10624994	C₂²	1	1	yes	characteristic
23	#10624980	—	2	1	no	—
24	C₁₆×D₂₃	—	2	1	no	—
25	C₂×Q₈×C₂₃	—	2	1	no	—
26	#67292	—	2	2	no	—
27	#67292	—	2	2	no	—
28	#67292	—	4	1	no	—
29	#67292	—	4	1	no	—
30	C₂×D₉₂	—	2	1	no	—
31	C₂×D₉₂	—	4	1	no	—
32	C₂×C₄×D₂₃	—	4	1	no	—
33	C₂×C₄×D₂₃	D₄	1	1	yes	characteristic
34	Q₁₆×C₂₃	—	2	1	no	—
35	Q₁₆×C₂₃	C₂×C₄	1	2	yes	—
36	M₁₆×C₂₃	—	2	1	no	—
37	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
38	#67274	—	2	2	no	—
39	#67274	—	2	2	no	—
40	#67267	—	2	1	no	—
41	C₂×C₂₃⋊C₈	D₄	1	1	yes	characteristic
42	D₁₈₄	—	2	1	no	—
43	D₁₈₄	C₂×C₄	1	2	yes	—
44	#67262	—	4	1	no	—
45	C₈×D₂₃	—	4	1	no	—
46	C₈×D₂₃	C₂×C₄	1	1	yes	characteristic
47	C₈×D₂₃	C₂×C₄	1	1	yes	characteristic
48	C₈×D₂₃	D₄	1	1	yes	characteristic
49	C₈×D₂₃	D₄	1	1	yes	characteristic
50	C₁₆×C₂₃	—	2	1	no	—
51	C₂₃⋊C₁₆	—	2	1	no	—
52	C₂²×D₂₃	—	4	1	no	—
53	C₂²×D₂₃	SD₁₆	1	1	yes	characteristic
54	Q₈×C₂₃	—	2	2	no	—
55	Q₈×C₂₃	—	4	1	no	—
56	C₂×C₄×C₂₃	—	4	1	no	—
57	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
58	C₂×Dic₂₃	SD₁₆	1	1	yes	characteristic
59	D₉₂	—	2	2	no	—
60	D₉₂	—	4	2	no	—
61	D₉₂	—	4	2	no	—
62	D₉₂	—	4	1	no	—
63	C₄×D₂₃	—	4	2	no	—
64	C₄×D₂₃	—	4	2	no	—
65	C₄×D₂₃	C₂²⋊C₄	1	1	yes	characteristic
66	C₄×D₂₃	C₂²⋊C₄	1	1	yes	characteristic
67	C₄×D₂₃	D₈	1	1	yes	characteristic
68	C₄×D₂₃	D₈	1	1	yes	characteristic
69	C₈×C₂₃	—	4	1	no	—
70	C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
71	C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
72	C₂₃⋊C₈	—	4	1	no	—
73	C₂₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
74	C₂₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
75	#2148	—	23	1	no	maximal, Sylow
76	C₂²×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
77	C₂×D₂₃	—	4	1	no	—
78	C₂×D₂₃	—	4	2	no	—
79	C₂×D₂₃	—	8	2	no	—
80	C₂×D₂₃	#102	1	1	yes	characteristic
81	C₂×D₂₃	#102	1	1	yes	characteristic
82	C₄×C₂₃	—	4	2	no	—
83	C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
85	Dic₂₃	#102	1	1	yes	characteristic
86	Dic₂₃	#102	1	1	yes	characteristic
87	C₂×#135	—	23	1	no	—
88	C₂×M₃₂	—	23	1	no	—
89	C₂×#108	—	23	1	no	—
90	#362	—	23	1	no	—
91	#362	D₂₃	1	1	yes	characteristic, complemented
92	#361	—	23	1	no	—
93	#361	—	23	1	no	—
94	C₂×C₂₃	—	2	1	no	—
95	C₂×C₂₃	C₂×#102	1	1	yes	characteristic
96	D₂₃	—	2	1	no	—
97	D₂₃	—	2	1	no	—
98	D₂₃	—	8	2	no	—
99	C₂×#41	—	46	1	no	—
100	#135	—	23	2	no	—
101	#135	—	23	2	no	—
102	#135	—	46	1	no	—
103	#135	—	46	1	no	—
104	C₂×Q₁₆	C₂×D₂₃	1	1	yes	characteristic
105	C₂×SD₁₆	—	46	1	no	—
106	C₂×D₈	—	23	1	no	—
107	C₂×M₁₆	—	46	1	no	—
108	C₂²×C₈	—	23	1	no	—
109	M₃₂	—	23	1	no	—
110	M₃₂	—	46	1	no	—
111	M₃₂	C₂×D₂₃	1	1	yes	characteristic
112	C₂×C₁₆	—	46	1	no	—
113	#108	—	23	1	no	—
114	#108	—	46	1	no	—
115	#108	C₂×D₂₃	1	1	yes	characteristic
116	C₂₃	#2148	1	1	yes	characteristic, complemented, Sylow
117	#41	—	46	2	no	—
118	#41	—	46	2	no	—
119	#41	—	92	1	no	—
120	#41	—	92	1	no	—
121	C₂×Q₈	—	2	1	no	—
122	C₂×D₄	—	46	1	no	—
123	C₂×D₄	—	92	1	no	—
124	C₂²×C₄	—	23	1	no	—
125	C₂²×C₄	—	92	1	no	—
126	Q₁₆	—	2	1	no	—
127	Q₁₆	C₄×D₂₃	1	2	yes	—
128	SD₁₆	—	46	2	no	—
129	SD₁₆	—	46	2	no	—
130	D₈	—	23	2	no	—
131	D₈	—	46	1	no	—
132	M₁₆	—	2	1	no	—
133	M₁₆	—	46	1	no	—
134	M₁₆	—	92	1	no	—
135	C₂×C₈	—	23	1	no	—
136	C₂×C₈	—	23	1	no	—
137	C₂×C₈	—	23	1	no	—
138	C₂×C₈	—	23	1	no	—
139	C₂×C₈	—	23	1	no	—
140	C₂×C₈	—	92	1	no	—
141	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
142	C₁₆	—	2	1	no	—
143	C₁₆	—	46	1	no	—
144	C₂³	—	23	1	no	—
145	C₂³	—	92	1	no	—
146	Q₈	—	2	2	no	—
147	Q₈	—	4	1	no	—
148	D₄	—	46	2	no	—
149	D₄	—	92	2	no	—
150	D₄	—	92	2	no	—
151	D₄	—	92	1	no	—
152	C₂×C₄	—	4	1	no	—
153	C₂×C₄	—	23	1	no	—
154	C₂×C₄	—	23	1	no	—
155	C₂×C₄	—	23	1	no	—
156	C₂×C₄	—	23	1	no	—
157	C₂×C₄	—	23	1	no	—
158	C₂×C₄	—	92	2	no	—
159	C₂×C₄	—	92	2	no	—
160	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
161	C₈	—	4	1	no	—
162	C₈	—	23	1	no	—
163	C₈	—	23	1	no	—
164	C₈	—	92	1	no	—
165	C₈	C₂×C₄×D₂₃	1	1	yes	characteristic
166	C₈	D₄×D₂₃	1	1	yes	characteristic
167	C₂²	—	23	1	no	—
168	C₂²	—	23	1	no	—
169	C₂²	—	92	1	no	—
170	C₂²	—	92	2	no	—
171	C₂²	—	184	2	no	—
172	C₂²	SD₁₆×D₂₃	1	1	yes	characteristic
173	C₄	—	4	2	no	—
174	C₄	—	23	1	no	—
175	C₄	—	23	1	no	—
176	C₄	C₂²⋊C₄×D₂₃	1	1	yes	characteristic
177	C₄	D₈×D₂₃	1	1	yes	characteristic
178	C₂	—	2	1	no	—
179	C₂	—	46	1	no	—
180	C₂	—	46	1	no	—
181	C₂	—	184	2	no	—
182	C₂	#102×D₂₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#102×D₂₃
commutator	C₈×C₂₃	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₂₃
Fitting	C₂₃×#362	C₂
socle	C₂×C₂₃	C₂×#102
prime radical	C₂×#10625090	C₂
cosocle	C₂×C₈×C₂₃	C₂³
hypercenter	#362	D₂₃
hypocenter	C₂₃	#2148
LCS	G ▷ C₈×C₂₃ ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #362

Complements

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

N	H ≅ G/N	type
2. M₃₂×D₂₃	C₂	semidirect
3. #108×D₂₃	C₂	semidirect
4. #13092233	C₂	semidirect
5. #13092231	C₂	semidirect
6. #13092229	C₂	semidirect
7. C₂₃×#362	C₂	semidirect
91. #362	D₂₃	semidirect
116. C₂₃	#2148	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10625090	2
M₃₂×D₂₃	2
#108×D₂₃	2
#13092233	2
#13092231	2
#13092229	2
C₂₃×#362	2
#2148	23