Subgroups — #15682092

Order 2944.

520 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10625119	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×C₂₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#13092489	C₂	1	4	yes	maximal, complemented
4	D₄×C₂₃⋊C₈	C₂	1	4	yes	maximal, complemented
5	#13092444	C₂	1	1	yes	maximal, characteristic, complemented
6	#13092384	C₂	1	1	yes	maximal, characteristic, complemented
7	#13092381	C₂	1	1	yes	maximal, characteristic, complemented
8	#13092373	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#525	C₂	1	1	yes	maximal, characteristic
10	C₂×#41×C₂₃	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
12	C₂₃×#123	C₄	1	2	yes	—
13	C₂₃×C₂²⋊Q₈	C₄	1	2	yes	—
14	C₂₃×#121	C₄	1	2	yes	—
15	C₂₃×#120	C₄	1	2	yes	—
16	C₄×D₄×C₂₃	C₂²	1	4	yes	—
17	C₂₃×#117	C₂²	1	1	yes	characteristic
18	#10625119	—	2	4	no	—
19	C₂×#67267	C₂²	1	1	yes	characteristic
20	C₂×#67267	C₂²	1	1	yes	characteristic
21	C₂×#67267	C₂²	1	2	yes	complemented
22	C₂²×C₂₃⋊C₈	—	2	4	no	—
23	C₂²×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
24	C₂²×C₂₃⋊C₈	C₂²	1	2	yes	—
25	C₂²×C₂₃⋊C₈	C₂²	1	2	yes	—
26	C₂²×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
27	C₂²×C₂₃⋊C₈	C₂²	1	2	yes	complemented
28	C₂²⋊(C₂₃⋊C₈)	C₂²	1	4	yes	—
29	C₂²⋊(C₂₃⋊C₈)	C₂²	1	4	yes	complemented
30	C₄⋊(C₂₃⋊C₈)	C₂²	1	4	yes	complemented
31	#10624985	C₂²	1	2	yes	complemented
32	C₄×C₂₃⋊C₈	C₂²	1	2	yes	complemented
33	C₂⁴×C₂₃	C₂×C₄	1	1	yes	characteristic
34	#41×C₂₃	—	2	2	no	—
35	#41×C₂₃	—	2	2	no	—
36	C₂×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
37	C₂×D₄×C₂₃	C₂×C₄	1	4	yes	—
38	C₂×D₄×C₂₃	C₂×C₄	1	2	yes	—
39	C₂×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₂₃	—	2	2	no	—
41	C₂²×C₄×C₂₃	—	2	2	no	—
42	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
43	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
45	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
46	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
47	C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
48	C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
49	C₂²⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
50	C₂²⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
51	C₂²⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
52	C₄²×C₂₃	C₂³	1	2	yes	—
53	#67267	—	2	2	no	—
54	#67267	—	2	2	no	—
55	#67267	—	2	4	no	—
56	C₂×C₂₃⋊C₈	—	2	2	no	—
57	C₂×C₂₃⋊C₈	—	2	4	no	—
58	C₂×C₂₃⋊C₈	—	2	4	no	—
59	C₂×C₂₃⋊C₈	—	2	4	no	—
60	C₂×C₂₃⋊C₈	D₄	1	4	yes	complemented
61	C₂×C₂₃⋊C₈	C₂³	1	2	yes	—
62	C₂×C₂₃⋊C₈	C₂³	1	2	yes	—
63	C₂×C₂₃⋊C₈	C₂³	1	2	yes	—
64	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
65	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
66	C₂³×C₂₃	—	2	2	no	—
67	C₂³×C₂₃	—	2	4	no	—
68	C₂³×C₂₃	C₂²×C₄	1	2	yes	—
69	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
70	C₂³×C₂₃	C₂²×C₄	1	2	yes	—
71	Q₈×C₂₃	—	2	2	no	—
72	D₄×C₂₃	—	2	4	no	—
73	D₄×C₂₃	—	2	8	no	—
74	D₄×C₂₃	—	2	2	no	—
75	C₂×C₄×C₂₃	—	2	2	no	—
76	C₂×C₄×C₂₃	—	2	4	no	—
77	C₂×C₄×C₂₃	—	2	2	no	—
78	C₂×C₄×C₂₃	—	2	2	no	—
79	C₂×C₄×C₂₃	—	2	2	no	—
80	C₂×C₄×C₂₃	—	2	2	no	—
81	C₂×C₄×C₂₃	C₂²×C₄	1	4	yes	—
82	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
83	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
84	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
86	C₂×C₄×C₂₃	#41	1	2	yes	—
87	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
88	C₂₃⋊C₈	—	2	2	no	—
89	C₂₃⋊C₈	—	2	2	no	—
90	C₂₃⋊C₈	—	2	2	no	—
91	C₂₃⋊C₈	C₂×D₄	1	2	yes	—
92	C₂₃⋊C₈	C₂×D₄	1	2	yes	—
93	#2929	—	23	1	no	maximal, Sylow
94	C₂²×C₂₃	—	2	8	no	—
95	C₂²×C₂₃	—	2	2	no	—
96	C₂²×C₂₃	—	2	2	no	—
97	C₂²×C₂₃	—	2	2	no	—
98	C₂²×C₂₃	—	2	2	no	—
99	C₂²×C₂₃	—	2	4	no	—
100	C₂²×C₂₃	C₄×D₄	1	2	yes	—
101	C₂²×C₂₃	#131	1	4	yes	complemented
102	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
103	C₄×C₂₃	—	2	2	no	—
104	C₄×C₂₃	—	2	2	no	—
105	C₄×C₂₃	—	2	4	no	—
106	C₄×C₂₃	C₄×D₄	1	2	yes	—
107	C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
108	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
109	C₂×#131	—	23	1	no	—
110	C₂³×C₈	—	23	1	no	—
111	#525	D₂₃	1	1	yes	characteristic
112	#435	—	23	4	no	—
113	C₈×D₄	—	23	4	no	—
114	#424	—	23	1	no	—
115	#408	—	23	1	no	—
116	#407	—	23	1	no	—
117	#405	—	23	1	no	—
118	C₂×C₂₃	—	2	2	no	—
119	C₂×C₂₃	—	2	4	no	—
120	C₂×C₂₃	—	4	2	no	—
121	C₂×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
122	C₂×C₂₃	C₂×#131	1	2	yes	—
123	C₂×#41	C₂×D₂₃	1	1	yes	characteristic
124	C₂³×C₄	C₂×D₂₃	1	1	yes	characteristic
125	#131	—	46	4	no	—
126	C₂×M₁₆	—	23	1	no	—
127	C₂×M₁₆	—	23	2	no	—
128	C₂×M₁₆	—	23	1	no	—
129	C₂²×C₈	—	23	1	no	—
130	C₂²×C₈	—	23	1	no	—
131	C₂²×C₈	—	23	2	no	—
132	C₂²×C₈	—	23	2	no	—
133	C₂²×C₈	—	23	2	no	—
134	C₂²×C₈	—	46	4	no	—
135	#123	Dic₂₃	1	2	yes	—
136	C₂²⋊Q₈	Dic₂₃	1	2	yes	—
137	#121	Dic₂₃	1	2	yes	—
138	#120	Dic₂₃	1	2	yes	—
139	C₄×D₄	C₂×D₂₃	1	4	yes	—
140	#117	C₂×D₂₃	1	1	yes	characteristic
141	C₄⋊C₈	—	23	4	no	—
142	C₂²⋊C₈	—	23	4	no	—
143	C₂²⋊C₈	—	23	4	no	—
144	#97	—	23	2	no	—
145	C₄×C₈	—	23	2	no	—
146	C₂₃	#2929	1	1	yes	characteristic, complemented, Sylow
147	C₂⁴	C₂×Dic₂₃	1	1	yes	characteristic
148	#41	—	2	2	no	—
149	#41	—	2	2	no	—
150	C₂×Q₈	C₂×Dic₂₃	1	1	yes	characteristic
151	C₂×D₄	C₂×Dic₂₃	1	4	yes	—
152	C₂×D₄	C₂×Dic₂₃	1	2	yes	—
153	C₂×D₄	C₂×Dic₂₃	1	1	yes	characteristic
154	C₂²×C₄	—	2	2	no	—
155	C₂²×C₄	—	2	2	no	—
156	C₂²×C₄	C₂×Dic₂₃	1	2	yes	—
157	C₂²×C₄	C₂×Dic₂₃	1	1	yes	characteristic
158	C₂²×C₄	C₂²×D₂₃	1	2	yes	—
159	C₂²×C₄	C₂²×D₂₃	1	2	yes	—
160	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
161	M₁₆	—	46	4	no	—
162	M₁₆	—	46	2	no	—
163	M₁₆	—	46	2	no	—
164	C₂×C₈	—	23	4	no	—
165	C₂×C₈	—	23	1	no	—
166	C₂×C₈	—	23	1	no	—
167	C₂×C₈	—	23	2	no	—
168	C₂×C₈	—	23	2	no	—
169	C₂×C₈	—	23	2	no	—
170	C₂×C₈	—	46	4	no	—
171	C₂×C₈	—	46	4	no	—
172	C₂×C₈	—	46	4	no	—
173	C₂×C₈	—	46	2	no	—
174	C₄⋊C₄	C₂×Dic₂₃	1	2	yes	—
175	C₄⋊C₄	C₂×Dic₂₃	1	4	yes	—
176	C₂²⋊C₄	C₂×Dic₂₃	1	2	yes	—
177	C₂²⋊C₄	C₂×Dic₂₃	1	4	yes	—
178	C₂²⋊C₄	C₂×Dic₂₃	1	4	yes	—
179	C₄²	C₂²×D₂₃	1	2	yes	—
180	C₂³	—	2	2	no	—
181	C₂³	—	2	4	no	—
182	C₂³	C₂²×Dic₂₃	1	2	yes	—
183	C₂³	C₂²×Dic₂₃	1	1	yes	characteristic
184	C₂³	C₂²×Dic₂₃	1	2	yes	—
185	Q₈	—	2	2	no	—
186	D₄	—	2	4	no	—
187	D₄	—	2	8	no	—
188	D₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	4	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	D₄×D₂₃	1	2	yes	—
196	C₂×C₄	#67290	1	2	yes	—
197	C₂×C₄	C₂²×Dic₂₃	1	4	yes	—
198	C₂×C₄	C₂²×Dic₂₃	1	2	yes	—
199	C₂×C₄	C₂²×Dic₂₃	1	2	yes	—
200	C₂×C₄	C₂²×Dic₂₃	1	1	yes	characteristic
201	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
202	C₈	—	23	2	no	—
203	C₈	—	23	2	no	—
204	C₈	—	46	2	no	—
205	C₈	—	46	2	no	—
206	C₈	—	46	2	no	—
207	C₂²	—	2	8	no	—
208	C₂²	—	2	2	no	—
209	C₂²	—	2	2	no	—
210	C₂²	—	2	2	no	—
211	C₂²	—	2	2	no	—
212	C₂²	—	2	4	no	—
213	C₂²	D₄×Dic₂₃	1	2	yes	—
214	C₂²	#10625119	1	4	yes	complemented
215	C₂²	C₂³×Dic₂₃	1	1	yes	characteristic
216	C₄	—	2	2	no	—
217	C₄	—	2	2	no	—
218	C₄	—	2	4	no	—
219	C₄	D₄×Dic₂₃	1	2	yes	—
220	C₄	C₂×D₄×D₂₃	1	1	yes	characteristic
221	C₄	C₂×#67290	1	1	yes	characteristic
222	C₂	—	2	2	no	—
223	C₂	—	2	4	no	—
224	C₂	—	4	2	no	—
225	C₂	C₂×D₄×Dic₂₃	1	1	yes	characteristic
226	C₂	C₂×#10625119	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₃
commutator, socle	C₂²×C₂₃	C₂³×C₄
Fitting, prime radical	C₂₃×#525	C₂
cosocle	C₂×C₄×C₂₃	C₂⁴
hypercenter	#525	D₂₃
hypocenter	C₂₃	#2929
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ #525

Complements

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

N	H ≅ G/N	type
1. C₂×#10625119	C₂	semidirect
2. C₂³×C₂₃⋊C₈	C₂	semidirect
3. #13092489	C₂	semidirect
4. D₄×C₂₃⋊C₈	C₂	semidirect
5. #13092444	C₂	semidirect
6. #13092384	C₂	semidirect
7. #13092381	C₂	semidirect
8. #13092373	C₂	semidirect
21. C₂×#67267	C₂²	semidirect
27. C₂²×C₂₃⋊C₈	C₂²	semidirect
29. C₂²⋊(C₂₃⋊C₈)	C₂²	semidirect
30. C₄⋊(C₂₃⋊C₈)	C₂²	semidirect
31. #10624985	C₂²	semidirect
32. C₄×C₂₃⋊C₈	C₂²	semidirect
60. C₂×C₂₃⋊C₈	D₄	semidirect
101. C₂²×C₂₃	#131	semidirect
146. C₂₃	#2929	semidirect
214. C₂²	#10625119	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10625119	2
C₂³×C₂₃⋊C₈	2
#13092489	2
D₄×C₂₃⋊C₈	2
#13092444	2
#13092384	2
#13092381	2
#13092373	2
C₂₃×#525	2
#2929	23