Subgroups — #15682139

Order 2944.

492 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10625078	C₂	1	1	yes	maximal, characteristic, complemented
2	#13092963	C₂	1	1	yes	maximal, characteristic
3	#13092493	C₂	1	2	yes	maximal, complemented
4	#13092488	C₂	1	2	yes	maximal, complemented
5	#13092486	C₂	1	2	yes	maximal, complemented
6	C₈×C₂²⋊D₂₃	C₂	1	2	yes	maximal, complemented
7	C₂×Dic₂₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	#13092384	C₂	1	1	yes	maximal, characteristic, complemented
9	#13092383	C₂	1	1	yes	maximal, characteristic, complemented
10	M₁₆×Dic₂₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₂₃×#131	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₂₃	C₂²	1	1	yes	characteristic
13	C₂×#67288	C₂²	1	1	yes	characteristic
14	C₂₃×#131	—	2	4	no	—
15	C₂×M₁₆×C₂₃	C₂²	1	2	yes	—
16	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
17	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
18	C₂²×C₈×C₂₃	C₂²	1	2	yes	—
19	#10625118	C₄	1	1	yes	characteristic
20	#10625115	C₄	1	1	yes	characteristic
21	#10625111	C₄	1	1	yes	characteristic
22	#10625110	C₄	1	2	yes	—
23	#10625110	C₄	1	2	yes	—
24	#10625107	C₄	1	1	yes	characteristic
25	C₄×C₂²⋊D₂₃	C₂²	1	2	yes	—
26	C₄×C₂²⋊D₂₃	C₂²	1	2	yes	—
27	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
28	C₂×#67267	C₂²	1	1	yes	characteristic
29	C₂²×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
30	#10625078	—	2	4	no	—
31	C₂×#67262	C₂²	1	2	yes	—
32	C₂×C₈×D₂₃	C₂²	1	2	yes	—
33	C₂²⋊(C₂₃⋊C₈)	C₂²	1	2	yes	—
34	C₂²⋊(C₂₃⋊C₈)	C₂²	1	2	yes	—
35	(C₂×D₂₃)⋊C₈	C₂²	1	2	yes	—
36	(C₂×D₂₃)⋊C₈	C₂²	1	2	yes	—
37	#10624997	C₂²	1	2	yes	—
38	Dic₂₃⋊C₈	C₂²	1	1	yes	characteristic
39	Dic₂₃⋊C₈	C₂²	1	1	yes	characteristic
40	Dic₂₃⋊C₈	C₂²	1	1	yes	characteristic
41	Dic₂₃⋊C₈	C₂²	1	1	yes	characteristic
42	C₈×Dic₂₃	C₂²	1	2	yes	—
43	#41×C₂₃	—	2	2	no	—
44	#41×C₂₃	—	2	2	no	—
45	C₂×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
46	C₂×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
47	C₂×D₄×C₂₃	C₂×C₄	1	2	yes	—
48	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
49	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
50	C₂×C₂²⋊D₂₃	C₂×C₄	1	2	yes	—
51	C₂×C₂²⋊D₂₃	C₂×C₄	1	2	yes	—
52	C₂×C₂²⋊D₂₃	C₂×C₄	1	2	yes	—
53	C₂²×Dic₂₃	C₂×C₄	1	1	yes	characteristic
54	C₂²×Dic₂₃	C₂×C₄	1	1	yes	characteristic
55	#67288	—	2	2	no	—
56	#67288	—	2	2	no	—
57	C₂×D₉₂	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄×D₂₃	C₂³	1	2	yes	—
59	C₂×Dic₄₆	C₂×C₄	1	1	yes	characteristic
60	M₁₆×C₂₃	—	2	4	no	—
61	M₁₆×C₂₃	D₄	1	4	yes	complemented
62	C₂×C₈×C₂₃	—	2	2	no	—
63	C₂×C₈×C₂₃	—	2	4	no	—
64	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
65	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
66	C₂×C₈×C₂₃	C₂³	1	2	yes	—
67	C₂²⋊Dic₂₃	C₂×C₄	1	2	yes	—
68	C₂²⋊Dic₂₃	C₂×C₄	1	2	yes	—
69	(C₂×D₂₃)⋊C₄	C₂×C₄	1	2	yes	—
70	(C₂×D₂₃)⋊C₄	C₂×C₄	1	2	yes	—
71	Dic₂₃⋊C₄	C₂×C₄	1	2	yes	—
72	Dic₂₃⋊C₄	C₂×C₄	1	2	yes	—
73	C₄×Dic₂₃	C₂×C₄	1	1	yes	characteristic
74	C₄×Dic₂₃	C₂×C₄	1	1	yes	characteristic
75	C₄×Dic₂₃	C₂³	1	1	yes	characteristic
76	C₄×Dic₂₃	C₂³	1	1	yes	characteristic
77	#67267	—	2	2	no	—
78	C₂×C₂₃⋊C₈	—	2	2	no	—
79	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
80	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
81	C₂×C₂₃⋊C₈	C₂³	1	2	yes	—
82	#67262	—	2	4	no	—
83	C₈×D₂₃	—	2	4	no	—
84	C₂³×C₂₃	C₂²×C₄	1	2	yes	—
85	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
86	C₂²×D₂₃	C₂²×C₄	1	2	yes	—
87	Q₈×C₂₃	—	2	2	no	—
88	D₄×C₂₃	—	2	4	no	—
89	D₄×C₂₃	—	2	2	no	—
90	C₂×C₄×C₂₃	—	2	2	no	—
91	C₂×C₄×C₂₃	—	2	2	no	—
92	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
93	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
95	C₂×C₄×C₂₃	#41	1	2	yes	—
96	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
97	C₂²⋊D₂₃	—	2	4	no	—
98	C₂²⋊D₂₃	—	2	4	no	—
99	C₂²⋊D₂₃	—	4	2	no	—
100	C₂×Dic₂₃	—	2	2	no	—
101	C₂×Dic₂₃	—	4	1	no	—
102	C₂×Dic₂₃	C₂²×C₄	1	1	yes	characteristic
103	C₂×Dic₂₃	C₂²×C₄	1	1	yes	characteristic
104	C₂×Dic₂₃	C₂²×C₄	1	1	yes	characteristic
105	C₂×Dic₂₃	C₂²×C₄	1	2	yes	—
106	C₂×Dic₂₃	C₂²×C₄	1	1	yes	characteristic
107	D₉₂	—	2	2	no	—
108	C₄×D₂₃	—	2	2	no	—
109	C₄×D₂₃	—	2	2	no	—
110	Dic₄₆	—	2	2	no	—
111	C₈×C₂₃	—	2	1	no	—
112	C₈×C₂₃	—	2	1	no	—
113	C₈×C₂₃	C₂×D₄	1	4	yes	—
114	C₂₃⋊C₈	—	2	1	no	—
115	C₂₃⋊C₈	—	2	1	no	—
116	C₂₃⋊C₈	—	2	2	no	—
117	#2935	—	23	1	no	maximal, Sylow
118	C₂²×C₂₃	—	2	1	no	—
119	C₂²×C₂₃	—	2	1	no	—
120	C₂²×C₂₃	—	2	2	no	—
121	C₂²×C₂₃	—	2	2	no	—
122	C₂²×C₂₃	—	2	2	no	—
123	C₂²×C₂₃	C₄×D₄	1	2	yes	—
124	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
125	C₂×D₂₃	—	2	2	no	—
126	C₂×D₂₃	—	2	2	no	—
127	C₂×D₂₃	—	2	2	no	—
128	C₄×C₂₃	—	2	2	no	—
129	C₄×C₂₃	C₄×D₄	1	2	yes	—
130	C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
131	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
132	Dic₂₃	—	2	2	no	—
133	Dic₂₃	—	2	1	no	—
134	Dic₂₃	—	2	1	no	—
135	Dic₂₃	—	2	1	no	—
136	Dic₂₃	#131	1	2	yes	complemented
137	C₂×#131	—	23	1	no	—
138	C₂×#131	D₂₃	1	1	yes	characteristic, complemented
139	#532	—	23	1	no	—
140	#436	—	23	2	no	—
141	#435	—	23	2	no	—
142	#435	—	23	2	no	—
143	C₈×D₄	—	23	2	no	—
144	C₂×C₄⋊C₈	—	23	1	no	—
145	#408	—	23	1	no	—
146	#408	—	23	1	no	—
147	C₄×M₁₆	—	23	1	no	—
148	C₂×C₂₃	—	2	2	no	—
149	C₂×C₂₃	—	4	2	no	—
150	C₂×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
151	C₂×C₂₃	C₂×#131	1	1	yes	characteristic
152	C₂×C₂₃	#568	1	1	yes	characteristic
153	D₂₃	—	4	2	no	—
154	C₂×#41	—	23	1	no	—
155	C₂×#41	C₂×D₂₃	1	1	yes	characteristic
156	#131	—	2	4	no	—
157	#131	—	46	4	no	—
158	C₂×M₁₆	—	23	1	no	—
159	C₂×M₁₆	—	23	2	no	—
160	C₂×M₁₆	C₂×D₂₃	1	2	yes	—
161	C₂×M₁₆	C₂×D₂₃	1	1	yes	characteristic
162	C₂²×C₈	—	23	1	no	—
163	C₂²×C₈	—	23	2	no	—
164	C₂²×C₈	C₂×D₂₃	1	1	yes	characteristic
165	C₂²×C₈	C₂×D₂₃	1	2	yes	—
166	C₄⋊Q₈	—	23	1	no	—
167	#127	—	23	1	no	—
168	#124	—	23	1	no	—
169	#124	—	23	1	no	—
170	#121	—	23	2	no	—
171	#121	—	23	2	no	—
172	C₄×D₄	—	23	2	no	—
173	C₄×D₄	—	23	2	no	—
174	C₂×C₄²	—	23	1	no	—
175	C₄⋊C₈	—	23	1	no	—
176	C₄⋊C₈	—	23	1	no	—
177	C₄⋊C₈	—	23	1	no	—
178	C₄⋊C₈	—	23	1	no	—
179	C₂²⋊C₈	—	23	2	no	—
180	C₂²⋊C₈	—	23	2	no	—
181	C₂²⋊C₈	—	23	2	no	—
182	C₂²⋊C₈	—	23	2	no	—
183	#97	—	23	2	no	—
184	C₄×C₈	—	23	2	no	—
185	C₂₃	#2935	1	1	yes	characteristic, complemented, Sylow
186	#41	—	2	2	no	—
187	#41	—	2	2	no	—
188	#41	—	46	2	no	—
189	#41	—	46	2	no	—
190	C₂×Q₈	—	23	1	no	—
191	C₂×Q₈	C₄×D₂₃	1	1	yes	characteristic
192	C₂×D₄	—	23	2	no	—
193	C₂×D₄	—	23	2	no	—
194	C₂×D₄	—	23	2	no	—
195	C₂×D₄	—	23	1	no	—
196	C₂×D₄	C₄×D₂₃	1	2	yes	—
197	C₂×D₄	C₄×D₂₃	1	1	yes	characteristic
198	C₂²×C₄	—	23	1	no	—
199	C₂²×C₄	—	23	1	no	—
200	C₂²×C₄	—	23	2	no	—
201	C₂²×C₄	C₂²×D₂₃	1	2	yes	—
202	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
203	M₁₆	—	2	4	no	—
204	M₁₆	—	46	2	no	—
205	M₁₆	—	46	4	no	—
206	M₁₆	C₂²⋊D₂₃	1	4	yes	complemented
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	—	2	4	no	—
209	C₂×C₈	—	23	2	no	—
210	C₂×C₈	—	23	1	no	—
211	C₂×C₈	—	23	1	no	—
212	C₂×C₈	—	46	2	no	—
213	C₂×C₈	—	46	4	no	—
214	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
215	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
216	C₂×C₈	C₂²×D₂₃	1	2	yes	—
217	C₄⋊C₄	—	23	2	no	—
218	C₄⋊C₄	—	23	2	no	—
219	C₂²⋊C₄	—	23	2	no	—
220	C₂²⋊C₄	—	23	2	no	—
221	C₂²⋊C₄	—	23	2	no	—
222	C₂²⋊C₄	—	23	2	no	—
223	C₄²	—	23	1	no	—
224	C₄²	—	23	1	no	—
225	C₄²	—	23	1	no	—
226	C₄²	—	23	1	no	—
227	C₂³	—	23	2	no	—
228	C₂³	C₂×C₄×D₂₃	1	2	yes	—
229	C₂³	C₂×C₄×D₂₃	1	1	yes	characteristic
230	Q₈	—	2	2	no	—
231	Q₈	—	46	2	no	—
232	D₄	—	2	4	no	—
233	D₄	—	2	2	no	—
234	D₄	—	46	4	no	—
235	D₄	—	46	4	no	—
236	D₄	—	46	2	no	—
237	D₄	—	92	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	23	1	no	—
241	C₂×C₄	—	23	1	no	—
242	C₂×C₄	—	23	1	no	—
243	C₂×C₄	—	23	1	no	—
244	C₂×C₄	—	23	2	no	—
245	C₂×C₄	—	46	2	no	—
246	C₂×C₄	—	46	2	no	—
247	C₂×C₄	—	46	2	no	—
248	C₂×C₄	—	92	1	no	—
249	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
250	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
251	C₂×C₄	#67288	1	2	yes	—
252	C₂×C₄	C₂×C₂²⋊D₂₃	1	2	yes	—
253	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
254	C₈	—	2	1	no	—
255	C₈	—	2	1	no	—
256	C₈	—	46	2	no	—
257	C₈	—	46	1	no	—
258	C₈	—	46	1	no	—
259	C₈	C₂×C₂²⋊D₂₃	1	4	yes	—
260	C₂²	—	2	1	no	—
261	C₂²	—	2	1	no	—
262	C₂²	—	2	2	no	—
263	C₂²	—	2	2	no	—
264	C₂²	—	2	2	no	—
265	C₂²	—	46	2	no	—
266	C₂²	—	46	2	no	—
267	C₂²	—	46	2	no	—
268	C₂²	C₄×C₂²⋊D₂₃	1	2	yes	—
269	C₂²	C₂²×C₄×D₂₃	1	1	yes	characteristic
270	C₄	—	2	2	no	—
271	C₄	—	23	2	no	—
272	C₄	—	46	2	no	—
273	C₄	—	46	1	no	—
274	C₄	—	46	1	no	—
275	C₄	—	46	1	no	—
276	C₄	C₄×C₂²⋊D₂₃	1	2	yes	—
277	C₄	C₂×#67288	1	1	yes	characteristic
278	C₄	C₂²×C₂²⋊D₂₃	1	1	yes	characteristic
279	C₂	—	2	2	no	—
280	C₂	—	4	2	no	—
281	C₂	—	92	2	no	—
282	C₂	C₂×C₄×C₂²⋊D₂₃	1	1	yes	characteristic
283	C₂	#131×D₂₃	1	1	yes	characteristic
284	C₂	#13093182	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₃
commutator, socle	C₂²×C₂₃	C₂³×C₄
Fitting	C₂×C₂₃×#131	C₂
prime radical	#13092963	C₂
cosocle	C₂×C₄×C₂₃	C₂⁴
hypercenter	C₂×#131	D₂₃
hypocenter	C₂₃	#2935
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#131

Complements

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

N	H ≅ G/N	type
1. C₂×#10625078	C₂	semidirect
3. #13092493	C₂	semidirect
4. #13092488	C₂	semidirect
5. #13092486	C₂	semidirect
6. C₈×C₂²⋊D₂₃	C₂	semidirect
7. C₂×Dic₂₃⋊C₈	C₂	semidirect
8. #13092384	C₂	semidirect
9. #13092383	C₂	semidirect
10. M₁₆×Dic₂₃	C₂	semidirect
11. C₂×C₂₃×#131	C₂	semidirect
61. M₁₆×C₂₃	D₄	semidirect
136. Dic₂₃	#131	semidirect
138. C₂×#131	D₂₃	semidirect
185. C₂₃	#2935	semidirect
206. M₁₆	C₂²⋊D₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10625078	2
#13092963	2
#13092493	2
#13092488	2
#13092486	2
C₈×C₂²⋊D₂₃	2
C₂×Dic₂₃⋊C₈	2
#13092384	2
#13092383	2
M₁₆×Dic₂₃	2
C₂×C₂₃×#131	2
#2935	23