Subgroups — #15682382

Order 2944.

362 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13092951	C₂	1	1	yes	maximal, characteristic
2	#13092545	C₂	1	2	yes	maximal, complemented
3	#13092493	C₂	1	2	yes	maximal, complemented
4	#13092490	C₂	1	2	yes	maximal, complemented
5	#13092486	C₂	1	2	yes	maximal, complemented
6	#13092476	C₂	1	2	yes	maximal, complemented
7	#13092474	C₂	1	1	yes	maximal, characteristic, complemented
8	#13092371	C₂	1	2	yes	maximal, complemented
9	C₄×M₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×M₁₆×C₂₃	C₂²	1	2	yes	—
11	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
12	#10625101	C₄	1	1	yes	characteristic
13	#10625100	C₄	1	2	yes	—
14	C₄×C₂²⋊D₂₃	C₂²	1	2	yes	—
15	#10625095	C₄	1	1	yes	characteristic
16	C₂×#67262	C₂²	1	2	yes	—
17	C₂×C₈×D₂₃	C₂²	1	2	yes	—
18	#10625047	C₄	1	2	yes	—
19	#10625046	C₄	1	1	yes	characteristic
20	C₄×D₉₂	C₂²	1	1	yes	characteristic
21	#10625043	C₂²	1	2	yes	—
22	#10625041	C₄	1	1	yes	characteristic
23	C₄×Dic₄₆	C₂²	1	1	yes	characteristic
24	C₂₃×#97	C₂²	1	2	yes	—
25	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
26	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
27	C₂²⋊(C₂₃⋊C₈)	C₂²	1	1	yes	characteristic
28	C₂²⋊(C₂₃⋊C₈)	C₂²	1	1	yes	characteristic
29	(C₂×D₂₃)⋊C₈	C₂²	1	2	yes	—
30	(C₂×D₂₃)⋊C₈	C₂²	1	2	yes	—
31	#10624997	C₂²	1	2	yes	—
32	Dic₂₃⋊C₈	C₂²	1	2	yes	—
33	Dic₂₃⋊C₈	C₂²	1	2	yes	—
34	C₈×Dic₂₃	C₂²	1	2	yes	—
35	C₄⋊(C₂₃⋊C₈)	C₂²	1	2	yes	—
36	#10624985	C₂²	1	2	yes	—
37	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
40	C₂×C₂²⋊D₂₃	C₂×C₄	1	2	yes	—
41	C₂×D₉₂	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄×D₂₃	C₂³	1	2	yes	—
43	C₂×Dic₄₆	C₂×C₄	1	1	yes	characteristic
44	M₁₆×C₂₃	—	2	4	no	—
45	C₂×C₈×C₂₃	C₂³	1	2	yes	—
46	C₂×C₈×C₂₃	C₂³	1	2	yes	—
47	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
48	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
49	C₄²×C₂₃	C₂³	1	1	yes	characteristic
50	C₄²×C₂₃	C₂³	1	1	yes	characteristic
51	C₂²⋊Dic₂₃	C₂×C₄	1	2	yes	—
52	(C₂×D₂₃)⋊C₄	C₂×C₄	1	2	yes	—
53	(C₂×D₂₃)⋊C₄	C₂×C₄	1	2	yes	—
54	(C₂×D₂₃)⋊C₄	C₂×C₄	1	2	yes	—
55	(C₂×D₂₃)⋊C₄	C₂×C₄	1	2	yes	—
56	C₄⋊Dic₂₃	C₂×C₄	1	1	yes	characteristic
57	C₄⋊Dic₂₃	C₂×C₄	1	1	yes	characteristic
58	Dic₂₃⋊C₄	C₂×C₄	1	2	yes	—
59	Dic₂₃⋊C₄	C₂×C₄	1	2	yes	—
60	Dic₂₃⋊C₄	C₂×C₄	1	2	yes	—
61	Dic₂₃⋊C₄	C₂×C₄	1	2	yes	—
62	C₄×Dic₂₃	C₂³	1	2	yes	—
63	C₂×C₂₃⋊C₈	C₂³	1	2	yes	—
64	C₂×C₂₃⋊C₈	C₂³	1	2	yes	—
65	#67262	—	2	4	no	—
66	C₈×D₂₃	—	2	4	no	—
67	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
68	C₂²×D₂₃	C₂²×C₄	1	2	yes	—
69	C₂×C₄×C₂₃	—	2	2	no	—
70	C₂×C₄×C₂₃	—	4	1	no	—
71	C₂×C₄×C₂₃	—	4	1	no	—
72	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
78	C₂²⋊D₂₃	—	4	2	no	—
79	C₂×Dic₂₃	C₂²×C₄	1	2	yes	—
80	C₂×Dic₂₃	C₂²×C₄	1	2	yes	—
81	C₂×Dic₂₃	C₂²×C₄	1	2	yes	—
82	D₉₂	—	4	1	no	—
83	C₄×D₂₃	—	2	2	no	—
84	C₄×D₂₃	—	2	2	no	—
85	Dic₄₆	—	4	1	no	—
86	C₈×C₂₃	—	2	2	no	—
87	C₈×C₂₃	#41	1	4	yes	—
88	C₂₃⋊C₈	—	2	2	no	—
89	C₂₃⋊C₈	—	2	2	no	—
90	#2967	—	23	1	no	maximal, Sylow
91	C₂²×C₂₃	—	2	1	no	—
92	C₂²×C₂₃	—	2	1	no	—
93	C₂²×C₂₃	—	2	1	no	—
94	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
95	C₂×D₂₃	—	2	2	no	—
96	C₂×D₂₃	—	2	2	no	—
97	C₂×D₂₃	—	2	2	no	—
98	C₄×C₂₃	—	2	1	no	—
99	C₄×C₂₃	—	2	1	no	—
100	C₄×C₂₃	—	2	1	no	—
101	C₄×C₂₃	—	2	1	no	—
102	C₄×C₂₃	—	2	1	no	—
103	C₄×C₂₃	C₂×#41	1	2	yes	—
104	Dic₂₃	—	2	2	no	—
105	Dic₂₃	—	2	2	no	—
106	Dic₂₃	—	2	2	no	—
107	#529	—	23	1	no	—
108	#446	—	23	2	no	—
109	#436	—	23	2	no	—
110	#435	—	23	2	no	—
111	#435	—	23	2	no	—
112	#433	—	23	2	no	—
113	#432	—	23	1	no	—
114	#405	—	23	2	no	—
115	C₄×M₁₆	D₂₃	1	1	yes	characteristic, complemented
116	C₂×C₂₃	—	4	1	no	—
117	C₂×C₂₃	C₄×#41	1	1	yes	characteristic
118	C₂×C₂₃	#568	1	1	yes	characteristic
119	C₂×C₂₃	#568	1	1	yes	characteristic
120	D₂₃	—	4	2	no	—
121	C₂×M₁₆	—	23	2	no	—
122	C₂×M₁₆	C₂×D₂₃	1	2	yes	—
123	C₂²×C₈	—	23	2	no	—
124	#126	—	23	2	no	—
125	#125	—	23	1	no	—
126	#124	—	23	1	no	—
127	#123	—	23	2	no	—
128	C₂²⋊Q₈	—	23	1	no	—
129	#121	—	23	1	no	—
130	C₄×Q₈	—	23	1	no	—
131	C₄×D₄	—	23	2	no	—
132	C₄×D₄	—	23	1	no	—
133	#117	—	23	2	no	—
134	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
135	C₄⋊C₈	—	23	2	no	—
136	C₄⋊C₈	—	23	2	no	—
137	C₄⋊C₈	—	23	2	no	—
138	C₂²⋊C₈	—	23	1	no	—
139	C₂²⋊C₈	—	23	1	no	—
140	C₂²⋊C₈	—	23	2	no	—
141	C₂²⋊C₈	—	23	2	no	—
142	#97	—	23	2	no	—
143	#97	—	23	2	no	—
144	#97	C₂×D₂₃	1	2	yes	—
145	C₄×C₈	—	23	2	no	—
146	C₄×C₈	C₂×D₂₃	1	1	yes	characteristic
147	C₄×C₈	C₂×D₂₃	1	1	yes	characteristic
148	C₂₃	#2967	1	1	yes	characteristic, complemented, Sylow
149	C₂×Q₈	—	23	1	no	—
150	C₂×D₄	—	23	2	no	—
151	C₂×D₄	—	23	1	no	—
152	C₂²×C₄	—	23	2	no	—
153	C₂²×C₄	C₄×D₂₃	1	1	yes	characteristic
154	C₂²×C₄	C₄×D₂₃	1	1	yes	characteristic
155	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
156	M₁₆	—	2	4	no	—
157	M₁₆	—	46	4	no	—
158	C₂×C₈	—	23	2	no	—
159	C₂×C₈	—	23	2	no	—
160	C₂×C₈	—	46	4	no	—
161	C₂×C₈	C₂²×D₂₃	1	2	yes	—
162	C₂×C₈	C₂²×D₂₃	1	2	yes	—
163	C₄⋊C₄	—	23	2	no	—
164	C₄⋊C₄	—	23	2	no	—
165	C₄⋊C₄	—	23	1	no	—
166	C₄⋊C₄	—	23	2	no	—
167	C₄⋊C₄	—	23	1	no	—
168	C₄⋊C₄	—	23	2	no	—
169	C₂²⋊C₄	—	23	2	no	—
170	C₂²⋊C₄	—	23	2	no	—
171	C₂²⋊C₄	—	23	2	no	—
172	C₂²⋊C₄	—	23	2	no	—
173	C₂²⋊C₄	—	23	2	no	—
174	C₄²	—	23	2	no	—
175	C₄²	C₄×D₂₃	1	1	yes	characteristic
176	C₄²	C₄×D₂₃	1	1	yes	characteristic
177	C₄²	C₂²×D₂₃	1	1	yes	characteristic
178	C₄²	C₂²×D₂₃	1	1	yes	characteristic
179	C₂³	—	23	2	no	—
180	C₂³	C₂×C₄×D₂₃	1	1	yes	characteristic
181	Q₈	—	92	1	no	—
182	D₄	—	92	1	no	—
183	D₄	—	92	2	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	4	1	no	—
186	C₂×C₄	—	4	1	no	—
187	C₂×C₄	—	23	2	no	—
188	C₂×C₄	—	23	2	no	—
189	C₂×C₄	—	23	2	no	—
190	C₂×C₄	—	46	2	no	—
191	C₂×C₄	—	46	2	no	—
192	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
193	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
194	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
195	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
196	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
197	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
198	C₈	—	2	2	no	—
199	C₈	—	46	2	no	—
200	C₈	—	46	2	no	—
201	C₈	#67288	1	4	yes	—
202	C₂²	—	2	1	no	—
203	C₂²	—	2	1	no	—
204	C₂²	—	2	1	no	—
205	C₂²	—	46	2	no	—
206	C₂²	—	46	2	no	—
207	C₂²	—	46	2	no	—
208	C₂²	C₂²×C₄×D₂₃	1	1	yes	characteristic
209	C₄	—	2	1	no	—
210	C₄	—	2	1	no	—
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	46	2	no	—
215	C₄	—	46	2	no	—
216	C₄	—	46	2	no	—
217	C₄	C₂×#67288	1	2	yes	—
218	C₂	—	4	1	no	—
219	C₂	—	92	2	no	—
220	C₂	C₄×#67288	1	1	yes	characteristic
221	C₂	#13093183	1	1	yes	characteristic
222	C₂	#13093183	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₃
commutator, socle	C₂²×C₂₃	C₂³×C₄
Fitting	C₄×M₁₆×C₂₃	C₂
prime radical	#10625101	C₄
cosocle	C₂×C₄×C₂₃	C₂⁴
hypercenter	C₄×M₁₆	D₂₃
hypocenter	C₂₃	#2967
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆

Complements

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

N	H ≅ G/N	type
2. #13092545	C₂	semidirect
3. #13092493	C₂	semidirect
4. #13092490	C₂	semidirect
5. #13092486	C₂	semidirect
6. #13092476	C₂	semidirect
7. #13092474	C₂	semidirect
8. #13092371	C₂	semidirect
9. C₄×M₁₆×C₂₃	C₂	semidirect
115. C₄×M₁₆	D₂₃	semidirect
148. C₂₃	#2967	semidirect

Maximal subgroups

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

subgroup	index
#13092951	2
#13092545	2
#13092493	2
#13092490	2
#13092486	2
#13092476	2
#13092474	2
#13092371	2
C₄×M₁₆×C₂₃	2
#2967	23