Subgroups — #15682302

Order 2944.

458 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13093214	C₂	1	1	yes	maximal, characteristic
2	#13093182	C₂	1	2	yes	maximal, complemented
3	#13092539	C₂	1	1	yes	maximal, characteristic, complemented
4	#13092538	C₂	1	1	yes	maximal, characteristic, complemented
5	#13092537	C₂	1	2	yes	maximal, complemented
6	#13092535	C₂	1	1	yes	maximal, characteristic, complemented
7	#13092531	C₂	1	2	yes	maximal, complemented
8	#13092464	C₂	1	1	yes	maximal, characteristic, complemented
9	#104×D₂₃	C₂	1	2	yes	maximal, complemented
10	C₄×#67262	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#443	C₂	1	1	yes	maximal, characteristic, complemented
12	#41×D₂₃	C₂²	1	2	yes	—
13	C₂₃×#135	C₂²	1	1	yes	characteristic
14	C₂₃×#131	C₂²	1	2	yes	complemented
15	#10625121	C₂²	1	2	yes	—
16	#10625119	C₂²	1	2	yes	complemented
17	#10625089	C₄	1	2	yes	complemented
18	#10625086	C₄	1	2	yes	complemented
19	#10625085	C₄	1	2	yes	complemented
20	#10625082	C₄	1	2	yes	complemented
21	#10625078	—	2	2	no	—
22	#10625078	C₂²	1	2	yes	complemented
23	M₁₆×D₂₃	—	2	2	no	—
24	M₁₆×D₂₃	C₂²	1	2	yes	—
25	#10625075	C₂²	1	1	yes	characteristic
26	#10625072	—	2	2	no	—
27	#10625072	C₂²	1	2	yes	complemented
28	C₂×#67262	—	2	2	no	—
29	C₂×#67262	C₂²	1	1	yes	characteristic
30	C₂×#67262	C₂²	1	1	yes	characteristic
31	C₄²×D₂₃	C₂²	1	1	yes	characteristic
32	C₂₃×#108	C₂²	1	1	yes	characteristic, complemented
33	C₂₃×#104	C₂²	1	2	yes	complemented
34	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic, complemented
35	#10625019	C₂²	1	2	yes	complemented
36	#10625007	C₂²	1	2	yes	complemented
37	#10625004	C₂²	1	2	yes	complemented
38	#10625001	C₂²	1	1	yes	characteristic, complemented
39	#10624997	C₂²	1	1	yes	characteristic, complemented
40	#10624997	C₂²	1	1	yes	characteristic, complemented
41	#10624987	C₂²	1	2	yes	complemented
42	C₄×C₂₃⋊C₈	C₂²	1	1	yes	characteristic, complemented
43	#41×C₂₃	C₂³	1	2	yes	—
44	#67292	C₂×C₄	1	2	yes	—
45	Q₈×D₂₃	C₂×C₄	1	2	yes	—
46	#67290	—	2	2	no	—
47	#67290	C₂×C₄	1	2	yes	—
48	D₄×D₂₃	—	2	2	no	—
49	D₄×D₂₃	C₂×C₄	1	2	yes	—
50	#67288	—	2	2	no	—
51	#67288	C₂³	1	2	yes	—
52	C₂×C₄×D₂₃	—	2	2	no	—
53	C₂×C₄×D₂₃	—	2	1	no	—
54	C₂×C₄×D₂₃	C₂³	1	1	yes	characteristic
55	Q₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic, complemented
56	SD₁₆×C₂₃	C₂×C₄	1	2	yes	complemented
57	D₈×C₂₃	C₂×C₄	1	1	yes	characteristic, complemented
58	M₁₆×C₂₃	—	2	2	no	—
59	M₁₆×C₂₃	C₂³	1	2	yes	—
60	C₂×C₈×C₂₃	—	2	2	no	—
61	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
62	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
63	C₄²×C₂₃	C₂³	1	1	yes	characteristic
64	#67275	C₂×C₄	1	2	yes	complemented
65	#67274	C₂×C₄	1	2	yes	complemented
66	#67273	C₂×C₄	1	2	yes	complemented
67	#67272	C₂×C₄	1	2	yes	complemented
68	C₄×Dic₂₃	—	2	1	no	—
69	C₄×Dic₂₃	C₂³	1	1	yes	characteristic
70	#67267	—	2	2	no	—
71	#67267	C₂³	1	2	yes	—
72	C₂×C₂₃⋊C₈	—	2	2	no	—
73	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
74	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
75	Dic₉₂	C₂×C₄	1	1	yes	characteristic, complemented
76	D₁₈₄	C₂×C₄	1	1	yes	characteristic, complemented
77	#67263	C₂×C₄	1	2	yes	complemented
78	#67262	—	2	2	no	—
79	#67262	—	2	2	no	—
80	#67262	—	2	2	no	—
81	#67262	C₂×C₄	1	2	yes	—
82	#67262	C₂×C₄	1	2	yes	—
83	#67262	D₄	1	2	yes	—
84	#67262	D₄	1	2	yes	—
85	C₈×D₂₃	—	2	2	no	—
86	C₈×D₂₃	—	2	2	no	—
87	C₈×D₂₃	—	2	2	no	—
88	C₂²×D₂₃	—	2	2	no	—
89	C₂²×D₂₃	#41	1	1	yes	characteristic
90	Q₈×C₂₃	C₂²×C₄	1	2	yes	—
91	D₄×C₂₃	—	2	2	no	—
92	D₄×C₂₃	C₂²×C₄	1	2	yes	—
93	C₂×C₄×C₂₃	—	2	1	no	—
94	C₂×C₄×C₂₃	—	2	2	no	—
95	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
96	C₂²⋊D₂₃	—	2	2	no	—
97	C₂²⋊D₂₃	—	2	2	no	—
98	C₂²⋊D₂₃	—	2	2	no	—
99	C₂×Dic₂₃	—	2	2	no	—
100	C₂×Dic₂₃	—	2	1	no	—
101	C₂×Dic₂₃	#41	1	1	yes	characteristic
102	D₉₂	—	2	2	no	—
103	D₉₂	C₂²×C₄	1	2	yes	—
104	C₄×D₂₃	—	2	2	no	—
105	C₄×D₂₃	—	2	2	no	—
106	C₄×D₂₃	—	2	2	no	—
107	C₄×D₂₃	—	4	2	no	—
108	C₄×D₂₃	C₂²×C₄	1	2	yes	—
109	C₄×D₂₃	C₂×D₄	1	2	yes	—
110	Dic₄₆	—	2	2	no	—
111	Dic₄₆	C₂²×C₄	1	2	yes	—
112	C₈×C₂₃	—	2	2	no	—
113	C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
114	C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
115	C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
116	C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
117	C₂₃⋊C₈	—	2	2	no	—
118	C₂₃⋊C₈	C₂²×C₄	1	2	yes	—
119	C₂₃⋊C₈	C₂×D₄	1	2	yes	—
120	#2956	—	23	1	no	maximal, Sylow
121	C₂²×C₂₃	—	2	2	no	—
122	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
123	C₂×D₂₃	—	2	2	no	—
124	C₂×D₂₃	—	4	1	no	—
125	C₂×D₂₃	—	8	2	no	—
126	C₂×D₂₃	C₄×D₄	1	2	yes	—
127	C₄×C₂₃	—	2	2	no	—
128	C₄×C₂₃	—	2	2	no	—
129	C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
130	C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
131	Dic₂₃	—	2	2	no	—
132	Dic₂₃	—	4	1	no	—
133	Dic₂₃	C₄×D₄	1	2	yes	—
134	#575	—	23	1	no	—
135	#568	—	23	2	no	—
136	#444	—	23	1	no	—
137	#444	—	23	2	no	—
138	#444	—	23	1	no	—
139	#443	—	23	2	no	—
140	#443	—	23	1	no	—
141	#443	D₂₃	1	1	yes	characteristic, complemented
142	#430	—	23	1	no	—
143	C₂×#104	—	23	2	no	—
144	C₄×M₁₆	—	23	1	no	—
145	C₂×C₂₃	—	2	1	no	—
146	C₂×C₂₃	—	4	2	no	—
147	C₂×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
148	D₂₃	—	2	2	no	—
149	D₂₃	—	4	2	no	—
150	C₂×#41	—	23	2	no	—
151	#137	—	23	2	no	—
152	#137	—	23	2	no	—
153	AGL(1,ℤ/8)	—	23	2	no	—
154	AGL(1,ℤ/8)	—	23	2	no	—
155	#135	—	23	2	no	—
156	#135	—	23	1	no	—
157	#135	C₂×D₂₃	1	1	yes	characteristic
158	#131	—	23	2	no	—
159	#131	—	23	2	no	—
160	#131	—	23	2	no	—
161	#131	—	46	2	no	—
162	#131	—	46	2	no	—
163	#131	C₂×D₂₃	1	2	yes	complemented
164	C₂×M₁₆	—	23	1	no	—
165	C₂×M₁₆	—	23	1	no	—
166	C₂×M₁₆	—	23	2	no	—
167	C₂×M₁₆	—	46	2	no	—
168	C₂×M₁₆	—	46	2	no	—
169	C₂×C₄²	—	23	1	no	—
170	#108	—	23	2	no	—
171	#108	—	23	1	no	—
172	#108	C₂×D₂₃	1	1	yes	characteristic, complemented
173	#104	—	23	2	no	—
174	#104	—	23	2	no	—
175	#104	—	23	2	no	—
176	#104	C₂×D₂₃	1	2	yes	complemented
177	#97	—	23	1	no	—
178	#97	—	23	1	no	—
179	C₄×C₈	—	23	1	no	—
180	C₄×C₈	C₂×D₂₃	1	1	yes	characteristic, complemented
181	C₂₃	#2956	1	1	yes	characteristic, complemented, Sylow
182	#41	—	23	2	no	—
183	#41	—	23	2	no	—
184	#41	—	23	2	no	—
185	#41	—	46	2	no	—
186	#41	—	46	2	no	—
187	#41	C₂²×D₂₃	1	2	yes	—
188	C₂×Q₈	—	23	2	no	—
189	C₂×D₄	—	23	2	no	—
190	C₂×D₄	—	46	2	no	—
191	C₂²×C₄	—	23	1	no	—
192	C₂²×C₄	—	46	2	no	—
193	C₂²×C₄	—	46	1	no	—
194	Q₁₆	—	23	2	no	—
195	Q₁₆	—	23	1	no	—
196	Q₁₆	C₄×D₂₃	1	1	yes	characteristic, complemented
197	SD₁₆	—	23	2	no	—
198	SD₁₆	—	23	2	no	—
199	SD₁₆	—	23	2	no	—
200	SD₁₆	C₄×D₂₃	1	2	yes	complemented
201	D₈	—	23	2	no	—
202	D₈	—	23	1	no	—
203	D₈	C₄×D₂₃	1	1	yes	characteristic, complemented
204	M₁₆	—	2	2	no	—
205	M₁₆	—	23	2	no	—
206	M₁₆	—	23	2	no	—
207	M₁₆	—	23	2	no	—
208	M₁₆	—	23	2	no	—
209	M₁₆	—	23	2	no	—
210	M₁₆	—	46	2	no	—
211	M₁₆	—	46	2	no	—
212	M₁₆	—	46	2	no	—
213	M₁₆	—	46	2	no	—
214	M₁₆	C₂²×D₂₃	1	2	yes	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	—	23	1	no	—
217	C₂×C₈	—	23	1	no	—
218	C₂×C₈	—	46	2	no	—
219	C₂×C₈	—	46	2	no	—
220	C₂×C₈	—	46	2	no	—
221	C₂×C₈	—	46	2	no	—
222	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
223	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
224	C₄²	—	23	1	no	—
225	C₄²	—	46	1	no	—
226	C₄²	C₂²×D₂₃	1	1	yes	characteristic
227	C₂³	—	23	1	no	—
228	C₂³	—	46	2	no	—
229	Q₈	—	23	2	no	—
230	Q₈	—	46	2	no	—
231	Q₈	C₂×C₄×D₂₃	1	2	yes	—
232	D₄	—	2	2	no	—
233	D₄	—	23	2	no	—
234	D₄	—	46	2	no	—
235	D₄	—	46	2	no	—
236	D₄	—	46	2	no	—
237	D₄	—	46	2	no	—
238	D₄	C₂×C₄×D₂₃	1	2	yes	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	23	1	no	—
242	C₂×C₄	—	23	2	no	—
243	C₂×C₄	—	23	2	no	—
244	C₂×C₄	—	46	2	no	—
245	C₂×C₄	—	46	2	no	—
246	C₂×C₄	—	46	2	no	—
247	C₂×C₄	—	46	2	no	—
248	C₂×C₄	—	46	1	no	—
249	C₂×C₄	—	92	2	no	—
250	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
251	C₈	—	2	2	no	—
252	C₈	—	23	2	no	—
253	C₈	—	23	2	no	—
254	C₈	—	46	2	no	—
255	C₈	C₂×C₄×D₂₃	1	1	yes	characteristic
256	C₈	C₂×C₄×D₂₃	1	1	yes	characteristic
257	C₈	D₄×D₂₃	1	1	yes	characteristic
258	C₈	D₄×D₂₃	1	1	yes	characteristic
259	C₂²	—	2	2	no	—
260	C₂²	—	23	2	no	—
261	C₂²	—	46	2	no	—
262	C₂²	—	92	1	no	—
263	C₂²	—	184	2	no	—
264	C₂²	#41×D₂₃	1	1	yes	characteristic
265	C₄	—	2	2	no	—
266	C₄	—	2	2	no	—
267	C₄	—	23	2	no	—
268	C₄	—	46	2	no	—
269	C₄	—	92	1	no	—
270	C₄	C₂²×C₄×D₂₃	1	1	yes	characteristic
271	C₄	C₂×D₄×D₂₃	1	1	yes	characteristic
272	C₂	—	2	1	no	—
273	C₂	—	4	2	no	—
274	C₂	—	46	2	no	—
275	C₂	—	92	2	no	—
276	C₂	C₄×D₄×D₂₃	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₄×D₂₃
commutator	C₄×C₂₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₂₃
Fitting	C₂₃×#443	C₂
socle	C₂×C₂₃	C₂×C₄×D₄
prime radical	#13093214	C₂
cosocle	C₂×C₄×C₂₃	C₂⁴
hypercenter	#443	D₂₃
hypocenter	C₂₃	#2956
LCS	G ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ #443

Complements

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

N	H ≅ G/N	type
2. #13093182	C₂	semidirect
3. #13092539	C₂	semidirect
4. #13092538	C₂	semidirect
5. #13092537	C₂	semidirect
6. #13092535	C₂	semidirect
7. #13092531	C₂	semidirect
8. #13092464	C₂	semidirect
9. #104×D₂₃	C₂	semidirect
10. C₄×#67262	C₂	semidirect
11. C₂₃×#443	C₂	semidirect
14. C₂₃×#131	C₂²	semidirect
16. #10625119	C₂²	semidirect
17. #10625089	C₄	semidirect
18. #10625086	C₄	semidirect
19. #10625085	C₄	semidirect
20. #10625082	C₄	semidirect
22. #10625078	C₂²	semidirect
27. #10625072	C₂²	semidirect
32. C₂₃×#108	C₂²	semidirect
33. C₂₃×#104	C₂²	semidirect
34. C₄×C₈×C₂₃	C₂²	semidirect
35. #10625019	C₂²	semidirect
36. #10625007	C₂²	semidirect
37. #10625004	C₂²	semidirect
38. #10625001	C₂²	semidirect
39. #10624997	C₂²	semidirect
40. #10624997	C₂²	semidirect
41. #10624987	C₂²	semidirect
42. C₄×C₂₃⋊C₈	C₂²	semidirect
55. Q₁₆×C₂₃	C₂×C₄	semidirect
56. SD₁₆×C₂₃	C₂×C₄	semidirect
57. D₈×C₂₃	C₂×C₄	semidirect
64. #67275	C₂×C₄	semidirect
65. #67274	C₂×C₄	semidirect
66. #67273	C₂×C₄	semidirect
67. #67272	C₂×C₄	semidirect
75. Dic₉₂	C₂×C₄	semidirect
76. D₁₈₄	C₂×C₄	semidirect
77. #67263	C₂×C₄	semidirect
141. #443	D₂₃	semidirect
163. #131	C₂×D₂₃	semidirect
172. #108	C₂×D₂₃	semidirect
176. #104	C₂×D₂₃	semidirect
180. C₄×C₈	C₂×D₂₃	semidirect
181. C₂₃	#2956	semidirect
196. Q₁₆	C₄×D₂₃	semidirect
200. SD₁₆	C₄×D₂₃	semidirect
203. D₈	C₄×D₂₃	semidirect

Maximal subgroups

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

subgroup	index
#13093214	2
#13093182	2
#13092539	2
#13092538	2
#13092537	2
#13092535	2
#13092531	2
#13092464	2
#104×D₂₃	2
C₄×#67262	2
C₂₃×#443	2
#2956	23