Subgroups — #15682304

Order 2944.

458 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13093215	C₂	1	1	yes	maximal, characteristic
2	#13093183	C₂	1	1	yes	maximal, characteristic, complemented
3	#13092537	C₂	1	4	yes	maximal, complemented
4	#13092532	C₂	1	4	yes	maximal, complemented
5	#13092465	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#10625019	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#10625007	C₂	1	1	yes	maximal, characteristic, complemented
8	M₁₆×Dic₂₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#568	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#41×C₂₃	C₂²	1	1	yes	characteristic
11	C₂×#67288	C₂²	1	1	yes	characteristic
12	C₂₃×#131	—	2	2	no	—
13	C₂₃×#131	C₂²	1	4	yes	complemented
14	C₂×M₁₆×C₂₃	—	2	2	no	—
15	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
16	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
17	#10625122	C₄	1	2	yes	complemented
18	#10625121	C₂²	1	4	yes	—
19	#10625120	C₄	1	2	yes	complemented
20	#10625113	C₄	1	2	yes	complemented
21	#10625103	C₄	1	2	yes	complemented
22	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
23	C₂×#67267	C₂²	1	1	yes	characteristic
24	C₂×#67267	C₂²	1	1	yes	characteristic
25	#10625078	—	2	2	no	—
26	M₁₆×D₂₃	—	2	2	no	—
27	#10625072	C₂²	1	4	yes	complemented
28	#10625019	C₂²	1	2	yes	complemented
29	#10625019	C₂²	1	2	yes	complemented
30	#10625007	C₂²	1	2	yes	complemented
31	#10625007	C₂²	1	2	yes	complemented
32	#10625004	C₂²	1	4	yes	complemented
33	#10624997	C₂²	1	2	yes	complemented
34	C₈×Dic₂₃	C₂²	1	2	yes	complemented
35	#41×C₂₃	—	2	1	no	—
36	#41×C₂₃	—	2	1	no	—
37	#41×C₂₃	C₂×C₄	1	2	yes	—
38	#41×C₂₃	C₂³	1	2	yes	—
39	C₂×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
40	C₂×D₄×C₂₃	—	2	1	no	—
41	C₂×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₂₃	—	2	1	no	—
43	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
44	C₂×C₂²⋊D₂₃	—	2	1	no	—
45	C₂²×Dic₂₃	—	2	1	no	—
46	#67288	—	2	1	no	—
47	#67288	—	2	1	no	—
48	#67288	C₂×C₄	1	2	yes	—
49	#67288	C₂³	1	2	yes	—
50	C₂×D₉₂	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄×D₂₃	—	2	1	no	—
52	C₂×Dic₄₆	C₂×C₄	1	1	yes	characteristic
53	M₁₆×C₂₃	—	2	1	no	—
54	M₁₆×C₂₃	—	2	4	no	—
55	M₁₆×C₂₃	D₄	1	2	yes	—
56	M₁₆×C₂₃	D₄	1	2	yes	—
57	M₁₆×C₂₃	C₂³	1	2	yes	—
58	C₂×C₈×C₂₃	—	2	1	no	—
59	C₂×C₈×C₂₃	—	2	4	no	—
60	C₂×C₈×C₂₃	C₂³	1	2	yes	—
61	#67275	C₂×C₄	1	4	yes	complemented
62	#67274	C₂×C₄	1	4	yes	complemented
63	#67273	C₂×C₄	1	4	yes	complemented
64	#67272	C₂×C₄	1	4	yes	complemented
65	C₄×Dic₂₃	—	2	1	no	—
66	C₄×Dic₂₃	C₂³	1	1	yes	characteristic
67	C₄×Dic₂₃	C₂³	1	1	yes	characteristic
68	#67267	—	2	1	no	—
69	#67267	C₂×C₄	1	2	yes	—
70	#67267	C₂×C₄	1	2	yes	—
71	#67267	C₂³	1	2	yes	—
72	C₂×C₂₃⋊C₈	—	2	1	no	—
73	C₂×C₂₃⋊C₈	C₂³	1	2	yes	—
74	#67262	—	2	4	no	—
75	C₈×D₂₃	—	2	4	no	—
76	C₂³×C₂₃	—	2	1	no	—
77	C₂³×C₂₃	#41	1	1	yes	characteristic
78	C₂²×D₂₃	—	2	1	no	—
79	Q₈×C₂₃	—	2	1	no	—
80	Q₈×C₂₃	C₂²×C₄	1	2	yes	—
81	D₄×C₂₃	—	2	2	no	—
82	D₄×C₂₃	—	2	2	no	—
83	D₄×C₂₃	—	2	1	no	—
84	D₄×C₂₃	C₂²×C₄	1	2	yes	—
85	C₂×C₄×C₂₃	—	2	1	no	—
86	C₂×C₄×C₂₃	—	2	2	no	—
87	C₂×C₄×C₂₃	—	2	2	no	—
88	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
91	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₄×C₂₃	#41	1	1	yes	characteristic
93	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
94	C₂²⋊D₂₃	—	2	2	no	—
95	C₂²⋊D₂₃	—	2	2	no	—
96	C₂×Dic₂₃	—	2	1	no	—
97	C₂×Dic₂₃	—	2	1	no	—
98	C₂×Dic₂₃	—	2	1	no	—
99	C₂×Dic₂₃	—	4	2	no	—
100	D₉₂	—	2	1	no	—
101	D₉₂	C₂²×C₄	1	2	yes	—
102	C₄×D₂₃	—	2	2	no	—
103	C₄×D₂₃	—	2	2	no	—
104	Dic₄₆	—	2	1	no	—
105	Dic₄₆	C₂²×C₄	1	2	yes	—
106	C₈×C₂₃	—	2	2	no	—
107	C₈×C₂₃	C₂×D₄	1	4	yes	—
108	C₂₃⋊C₈	—	2	2	no	—
109	C₂₃⋊C₈	C₂²×C₄	1	4	yes	—
110	#2956	—	23	1	no	maximal, Sylow
111	C₂²×C₂₃	—	2	2	no	—
112	C₂²×C₂₃	—	4	1	no	—
113	C₂²×C₂₃	—	8	1	no	—
114	C₂²×C₂₃	C₄×D₄	1	1	yes	characteristic
115	C₂²×C₂₃	C₄×D₄	1	1	yes	characteristic
116	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
117	C₂×D₂₃	—	2	2	no	—
118	C₂×D₂₃	—	8	1	no	—
119	C₄×C₂₃	—	2	2	no	—
120	C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
121	C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
122	C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
123	C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
124	Dic₂₃	—	2	2	no	—
125	Dic₂₃	—	2	2	no	—
126	Dic₂₃	—	4	1	no	—
127	#575	—	23	1	no	—
128	#568	—	23	1	no	—
129	#568	D₂₃	1	1	yes	characteristic, complemented
130	#444	—	23	4	no	—
131	#443	—	23	4	no	—
132	#430	—	23	1	no	—
133	C₂×#104	—	23	1	no	—
134	C₂×#104	—	23	1	no	—
135	C₄×M₁₆	—	23	1	no	—
136	C₂×C₂₃	—	2	1	no	—
137	C₂×C₂₃	—	2	1	no	—
138	C₂×C₂₃	—	2	1	no	—
139	C₂×C₂₃	—	4	2	no	—
140	C₂×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
141	D₂₃	—	4	2	no	—
142	C₂×#41	—	23	1	no	—
143	C₂×#41	C₂×D₂₃	1	1	yes	characteristic
144	#137	—	23	2	no	—
145	#137	—	23	2	no	—
146	AGL(1,ℤ/8)	—	23	2	no	—
147	AGL(1,ℤ/8)	—	23	2	no	—
148	#135	—	23	4	no	—
149	#131	—	2	2	no	—
150	#131	—	23	4	no	—
151	#131	—	46	2	no	—
152	#131	C₂×D₂₃	1	4	yes	complemented
153	C₂×M₁₆	—	2	2	no	—
154	C₂×M₁₆	—	23	1	no	—
155	C₂×M₁₆	—	23	1	no	—
156	C₂×M₁₆	—	46	2	no	—
157	C₂×M₁₆	C₂×D₂₃	1	1	yes	characteristic
158	C₂×M₁₆	C₂×D₂₃	1	1	yes	characteristic
159	C₂×C₄²	—	23	1	no	—
160	#108	—	23	4	no	—
161	#104	—	23	2	no	—
162	#104	—	23	2	no	—
163	#104	—	23	2	no	—
164	#104	—	23	2	no	—
165	#97	—	23	2	no	—
166	C₄×C₈	—	23	2	no	—
167	C₂₃	#2956	1	1	yes	characteristic, complemented, Sylow
168	#41	—	2	1	no	—
169	#41	—	2	1	no	—
170	#41	—	23	2	no	—
171	#41	—	23	2	no	—
172	#41	—	46	1	no	—
173	#41	—	46	1	no	—
174	#41	C₄×D₂₃	1	2	yes	—
175	#41	C₂²×D₂₃	1	2	yes	—
176	C₂×Q₈	—	23	1	no	—
177	C₂×Q₈	C₄×D₂₃	1	1	yes	characteristic
178	C₂×D₄	—	2	1	no	—
179	C₂×D₄	—	23	1	no	—
180	C₂×D₄	—	46	1	no	—
181	C₂×D₄	C₄×D₂₃	1	1	yes	characteristic
182	C₂²×C₄	—	2	1	no	—
183	C₂²×C₄	—	46	1	no	—
184	C₂²×C₄	—	46	1	no	—
185	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
186	Q₁₆	—	23	4	no	—
187	SD₁₆	—	23	4	no	—
188	SD₁₆	—	23	4	no	—
189	D₈	—	23	4	no	—
190	M₁₆	—	2	1	no	—
191	M₁₆	—	2	4	no	—
192	M₁₆	—	23	2	no	—
193	M₁₆	—	23	2	no	—
194	M₁₆	—	23	2	no	—
195	M₁₆	—	46	1	no	—
196	M₁₆	—	46	4	no	—
197	M₁₆	C₂²⋊D₂₃	1	2	yes	—
198	M₁₆	C₂²⋊D₂₃	1	2	yes	—
199	M₁₆	C₂²×D₂₃	1	2	yes	—
200	C₂×C₈	—	2	1	no	—
201	C₂×C₈	—	2	4	no	—
202	C₂×C₈	—	23	2	no	—
203	C₂×C₈	—	46	1	no	—
204	C₂×C₈	—	46	4	no	—
205	C₂×C₈	C₂²×D₂₃	1	2	yes	—
206	C₄²	—	23	1	no	—
207	C₄²	—	23	1	no	—
208	C₄²	—	46	1	no	—
209	C₂³	—	2	1	no	—
210	C₂³	—	46	1	no	—
211	C₂³	#67288	1	1	yes	characteristic
212	Q₈	—	2	1	no	—
213	Q₈	—	23	2	no	—
214	Q₈	—	46	1	no	—
215	Q₈	C₂×C₄×D₂₃	1	2	yes	—
216	D₄	—	2	2	no	—
217	D₄	—	2	2	no	—
218	D₄	—	2	1	no	—
219	D₄	—	23	2	no	—
220	D₄	—	46	2	no	—
221	D₄	—	46	2	no	—
222	D₄	—	46	1	no	—
223	D₄	C₂×C₄×D₂₃	1	2	yes	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	46	1	no	—
228	C₂×C₄	—	46	1	no	—
229	C₂×C₄	—	46	2	no	—
230	C₂×C₄	—	46	2	no	—
231	C₂×C₄	—	46	1	no	—
232	C₂×C₄	—	92	2	no	—
233	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
234	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
235	C₂×C₄	#67288	1	1	yes	characteristic
236	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
237	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
238	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
239	C₈	—	2	2	no	—
240	C₈	—	23	4	no	—
241	C₈	—	46	2	no	—
242	C₈	C₂×C₂²⋊D₂₃	1	4	yes	—
243	C₂²	—	2	2	no	—
244	C₂²	—	4	1	no	—
245	C₂²	—	8	1	no	—
246	C₂²	—	46	2	no	—
247	C₂²	—	184	1	no	—
248	C₂²	C₄×C₂²⋊D₂₃	1	1	yes	characteristic
249	C₂²	C₄×C₂²⋊D₂₃	1	1	yes	characteristic
250	C₂²	C₂×#67288	1	1	yes	characteristic
251	C₄	—	2	2	no	—
252	C₄	—	46	2	no	—
253	C₄	—	46	2	no	—
254	C₄	—	92	1	no	—
255	C₄	C₄×C₂²⋊D₂₃	1	1	yes	characteristic
256	C₄	C₄×C₂²⋊D₂₃	1	1	yes	characteristic
257	C₄	C₂²×C₄×D₂₃	1	1	yes	characteristic
258	C₄	C₂²×C₂²⋊D₂₃	1	1	yes	characteristic
259	C₂	—	2	1	no	—
260	C₂	—	2	1	no	—
261	C₂	—	2	1	no	—
262	C₂	—	4	2	no	—
263	C₂	—	92	2	no	—
264	C₂	C₂×C₄×C₂²⋊D₂₃	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²×C₂²⋊D₂₃
commutator	C₄×C₂₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₂₃
Fitting	C₂₃×#568	C₂
socle	C₂×C₂₃	C₂×C₄×D₄
prime radical	#13093215	C₂
cosocle	C₂×C₄×C₂₃	C₂⁴
hypercenter	#568	D₂₃
hypocenter	C₂₃	#2956
LCS	G ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #568

Complements

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

N	H ≅ G/N	type
2. #13093183	C₂	semidirect
3. #13092537	C₂	semidirect
4. #13092532	C₂	semidirect
5. #13092465	C₂	semidirect
6. C₂×#10625019	C₂	semidirect
7. C₂×#10625007	C₂	semidirect
8. M₁₆×Dic₂₃	C₂	semidirect
9. C₂₃×#568	C₂	semidirect
13. C₂₃×#131	C₂²	semidirect
17. #10625122	C₄	semidirect
19. #10625120	C₄	semidirect
20. #10625113	C₄	semidirect
21. #10625103	C₄	semidirect
27. #10625072	C₂²	semidirect
28. #10625019	C₂²	semidirect
29. #10625019	C₂²	semidirect
30. #10625007	C₂²	semidirect
31. #10625007	C₂²	semidirect
32. #10625004	C₂²	semidirect
33. #10624997	C₂²	semidirect
34. C₈×Dic₂₃	C₂²	semidirect
61. #67275	C₂×C₄	semidirect
62. #67274	C₂×C₄	semidirect
63. #67273	C₂×C₄	semidirect
64. #67272	C₂×C₄	semidirect
129. #568	D₂₃	semidirect
152. #131	C₂×D₂₃	semidirect
167. C₂₃	#2956	semidirect

Maximal subgroups

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

subgroup	index
#13093215	2
#13093183	2
#13092537	2
#13092532	2
#13092465	2
C₂×#10625019	2
C₂×#10625007	2
M₁₆×Dic₂₃	2
C₂₃×#568	2
#2956	23