Subgroups — #15687554

Order 2944.

1336 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13093156	C₂	1	3	yes	maximal, complemented
2	#13093097	C₂	1	3	yes	maximal, complemented
3	#13093060	C₂	1	6	yes	maximal, complemented
4	#13093049	C₂	1	3	yes	maximal, complemented
5	#13093001	C₂	1	6	yes	maximal, complemented
6	#13092985	C₂	1	3	yes	maximal, complemented
7	#13092977	C₂	1	3	yes	maximal, complemented
8	#120×D₂₃	C₂	1	3	yes	maximal, complemented
9	C₂₃×H(3,𝔽₄)	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂⁴×D₂₃	C₂²	1	1	yes	characteristic, complemented
11	C₂²×C₂²⋊D₂₃	C₂²	1	3	yes	complemented
12	C₂×#67292	C₂²	1	3	yes	complemented
13	C₂×D₄×D₂₃	C₂²	1	3	yes	complemented
14	C₂×D₄×D₂₃	C₂²	1	6	yes	complemented
15	C₂₃×#124	C₂²	1	3	yes	complemented
16	C₂₃×#124	C₂²	1	6	yes	complemented
17	C₂₃×#120	C₂²	1	3	yes	complemented
18	C₂₃×#120	C₂²	1	3	yes	complemented
19	#10625124	C₂²	1	3	yes	complemented
20	#10625124	C₂²	1	1	yes	characteristic, complemented
21	#10625118	C₂²	1	3	yes	complemented
22	#10625117	C₂²	1	6	yes	complemented
23	#10625111	C₂²	1	3	yes	complemented
24	#10625110	C₂²	1	6	yes	complemented
25	#10625109	C₂²	1	6	yes	complemented
26	#10625108	C₂²	1	3	yes	—
27	#10625108	C₂²	1	6	yes	complemented
28	#10625106	C₂²	1	3	yes	complemented
29	#10625057	C₂²	1	3	yes	complemented
30	#10625055	C₂²	1	6	yes	complemented
31	#10625055	C₂²	1	6	yes	complemented
32	#10625055	C₂²	1	6	yes	complemented
33	#10625054	C₂²	1	6	yes	complemented
34	#10625054	C₂²	1	6	yes	complemented
35	#10625053	C₂²	1	6	yes	complemented
36	#10625053	C₂²	1	3	yes	complemented
37	#10625053	C₂²	1	6	yes	complemented
38	(C₂²⋊D₂₃)⋊C₄	C₂²	1	6	yes	complemented
39	C₂²⋊C₄×D₂₃	C₂²	1	3	yes	complemented
40	C₂²⋊C₄×D₂₃	C₂²	1	6	yes	complemented
41	C₂²⋊C₄×D₂₃	C₂²	1	3	yes	complemented
42	#10625050	C₂²	1	6	yes	complemented
43	#10625047	C₂²	1	2	yes	complemented
44	#10625045	C₂²	1	1	yes	characteristic, complemented
45	C₄×D₉₂	C₂²	1	6	yes	complemented
46	#10625043	C₂²	1	3	yes	complemented
47	C₂⁴×C₂₃	C₂³	1	1	yes	characteristic
48	C₂⁴×C₂₃	C₂³	1	1	yes	characteristic, complemented
49	C₂³×D₂₃	—	4	6	no	—
50	C₂³×D₂₃	C₂³	1	3	yes	—
51	C₂³×D₂₃	C₂³	1	3	yes	—
52	C₂³×D₂₃	C₂³	1	3	yes	—
53	C₂×Q₈×C₂₃	C₂³	1	3	yes	complemented
54	C₂×D₄×C₂₃	C₂³	1	3	yes	—
55	C₂×D₄×C₂₃	C₂³	1	6	yes	—
56	C₂×C₂²⋊D₂₃	—	4	6	no	—
57	C₂×C₂²⋊D₂₃	C₂³	1	6	yes	—
58	C₂×C₂²⋊D₂₃	C₂³	1	6	yes	—
59	C₂×C₂²⋊D₂₃	C₂³	1	3	yes	—
60	C₂×C₂²⋊D₂₃	C₂³	1	3	yes	—
61	C₂×C₂²⋊D₂₃	C₂³	1	6	yes	complemented
62	C₂²×Dic₂₃	C₂³	1	3	yes	complemented
63	#67292	—	4	6	no	—
64	D₄×D₂₃	—	4	6	no	—
65	D₄×D₂₃	—	4	12	no	—
66	C₂×D₉₂	C₂³	1	6	yes	—
67	C₂×D₉₂	C₂³	1	3	yes	—
68	C₂×D₉₂	C₂³	1	3	yes	complemented
69	C₂×C₄×D₂₃	C₂³	1	3	yes	—
70	C₂×C₄×D₂₃	C₂³	1	6	yes	—
71	C₂×C₄×D₂₃	C₂³	1	6	yes	complemented
72	C₂²⋊C₄×C₂₃	C₂³	1	6	yes	—
73	C₂²⋊C₄×C₂₃	C₂³	1	3	yes	—
74	C₂²⋊C₄×C₂₃	C₂³	1	3	yes	complemented
75	C₂²⋊C₄×C₂₃	C₂³	1	6	yes	complemented
76	C₄²×C₂₃	C₂³	1	1	yes	characteristic, complemented
77	C₄²×C₂₃	C₂³	1	2	yes	complemented
78	C₂²⋊Dic₂₃	C₂³	1	3	yes	—
79	C₂²⋊Dic₂₃	C₂³	1	6	yes	—
80	C₂²⋊Dic₂₃	C₂³	1	3	yes	complemented
81	(C₂×D₂₃)⋊C₄	C₂³	1	6	yes	—
82	(C₂×D₂₃)⋊C₄	C₂³	1	6	yes	—
83	(C₂×D₂₃)⋊C₄	C₂³	1	6	yes	—
84	(C₂×D₂₃)⋊C₄	C₂³	1	6	yes	complemented
85	C₄⋊Dic₂₃	C₂³	1	6	yes	complemented
86	Dic₂₃⋊C₄	C₂³	1	6	yes	complemented
87	Dic₂₃⋊C₄	C₂³	1	6	yes	complemented
88	C₄×Dic₂₃	C₂³	1	3	yes	complemented
89	C₂³×C₂₃	—	4	3	no	—
90	C₂³×C₂₃	—	4	3	no	—
91	C₂³×C₂₃	C₂⁴	1	3	yes	—
92	C₂³×C₂₃	C₂⁴	1	3	yes	—
93	C₂²×D₂₃	—	2	12	no	—
94	C₂²×D₂₃	—	2	6	no	—
95	C₂²×D₂₃	—	2	6	no	—
96	C₂²×D₂₃	—	4	3	no	—
97	C₂²×D₂₃	—	4	6	no	—
98	C₂²×D₂₃	—	4	6	no	—
99	C₂²×D₂₃	—	4	16	no	—
100	C₂²×D₂₃	C₂⁴	1	1	yes	characteristic
101	C₂²×D₂₃	C₂⁴	1	3	yes	—
102	C₂²×D₂₃	C₂⁴	1	3	yes	—
103	Q₈×C₂₃	—	4	3	no	—
104	D₄×C₂₃	—	4	3	no	—
105	D₄×C₂₃	—	4	6	no	—
106	C₂×C₄×C₂₃	C₂⁴	1	3	yes	—
107	C₂×C₄×C₂₃	C₂⁴	1	6	yes	—
108	C₂²⋊D₂₃	—	2	12	no	—
109	C₂²⋊D₂₃	—	4	3	no	—
110	C₂²⋊D₂₃	—	4	3	no	—
111	C₂²⋊D₂₃	—	4	6	no	—
112	C₂²⋊D₂₃	—	4	6	no	—
113	C₂×Dic₂₃	—	4	3	no	—
114	C₂×Dic₂₃	C₂⁴	1	6	yes	—
115	C₂×Dic₂₃	C₂⁴	1	3	yes	—
116	D₉₂	—	2	12	no	—
117	D₉₂	—	4	3	no	—
118	D₉₂	—	4	3	no	—
119	C₄×D₂₃	—	2	6	no	—
120	C₄×D₂₃	—	4	6	no	—
121	C₄×D₂₃	—	4	6	no	—
122	#3532	—	23	1	no	maximal, Sylow
123	C₂²×C₂₃	—	2	3	no	—
124	C₂²×C₂₃	—	2	6	no	—
125	C₂²×C₂₃	—	2	3	no	—
126	C₂²×C₂₃	—	2	6	no	—
127	C₂²×C₂₃	—	4	4	no	—
128	C₂²×C₂₃	—	8	2	no	—
129	C₂²×C₂₃	C₂⁵	1	1	yes	characteristic
130	C₂×D₂₃	—	2	6	no	—
131	C₂×D₂₃	—	2	6	no	—
132	C₂×D₂₃	—	2	3	no	—
133	C₂×D₂₃	—	2	3	no	—
134	C₂×D₂₃	—	4	12	no	—
135	C₂×D₂₃	—	4	12	no	—
136	C₂×D₂₃	—	8	6	no	—
137	C₂×D₂₃	H(5,𝔽₂)	1	6	yes	—
138	C₄×C₂₃	—	2	3	no	—
139	C₄×C₂₃	—	2	6	no	—
140	Dic₂₃	—	2	3	no	—
141	Dic₂₃	—	2	6	no	—
142	H(3,𝔽₄)	D₂₃	1	1	yes	characteristic, complemented
143	#560	—	23	3	no	—
144	#551	—	23	3	no	—
145	#546	—	23	6	no	—
146	D₄²	—	23	3	no	—
147	#538	—	23	6	no	—
148	#535	—	23	3	no	—
149	#534	—	23	3	no	—
150	C₂×#120	—	23	3	no	—
151	C₂×C₂₃	—	4	3	no	—
152	C₂×C₂₃	—	4	3	no	—
153	C₂×C₂₃	C₂×H(5,𝔽₂)	1	3	yes	—
154	D₂₃	—	2	6	no	—
155	D₂₃	—	4	1	no	—
156	D₂₃	—	4	3	no	—
157	C₂⁵	—	23	1	no	—
158	C₂×#41	—	23	3	no	—
159	C₂²×D₄	—	23	3	no	—
160	C₂²×D₄	—	23	3	no	—
161	C₂²×D₄	—	23	6	no	—
162	#127	—	23	3	no	—
163	#127	—	23	1	no	—
164	#126	—	23	6	no	—
165	#126	—	23	2	no	—
166	#124	—	23	3	no	—
167	#124	C₂×D₂₃	1	3	yes	complemented
168	#124	C₂×D₂₃	1	6	yes	complemented
169	#123	—	23	3	no	—
170	#123	—	23	6	no	—
171	#123	—	23	3	no	—
172	#123	—	23	6	no	—
173	C₂²⋊Q₈	—	23	6	no	—
174	#121	—	23	6	no	—
175	#121	—	23	6	no	—
176	#121	—	23	6	no	—
177	#121	—	23	6	no	—
178	#121	—	23	6	no	—
179	#120	—	23	3	no	—
180	#120	—	23	1	no	—
181	#120	—	23	6	no	—
182	#120	—	23	3	no	—
183	#120	—	23	6	no	—
184	#120	—	23	3	no	—
185	#120	—	23	6	no	—
186	#120	C₂×D₂₃	1	3	yes	complemented
187	#120	C₂×D₂₃	1	3	yes	complemented
188	C₄×D₄	—	23	6	no	—
189	C₄×D₄	—	23	6	no	—
190	#117	—	23	3	no	—
191	C₂×C₂²⋊C₄	—	23	3	no	—
192	C₂×C₂²⋊C₄	—	23	6	no	—
193	C₂×C₂²⋊C₄	—	23	3	no	—
194	C₂₃	#3532	1	1	yes	characteristic, complemented, Sylow
195	C₂⁴	—	23	3	no	—
196	C₂⁴	—	23	3	no	—
197	C₂⁴	—	23	3	no	—
198	C₂⁴	—	92	6	no	—
199	C₂⁴	C₂²×D₂₃	1	1	yes	characteristic
200	C₂⁴	C₂²×D₂₃	1	1	yes	characteristic, complemented
201	#41	—	92	6	no	—
202	C₂×Q₈	C₂²×D₂₃	1	3	yes	complemented
203	C₂×D₄	—	23	6	no	—
204	C₂×D₄	—	23	3	no	—
205	C₂×D₄	—	23	3	no	—
206	C₂×D₄	—	23	6	no	—
207	C₂×D₄	—	23	6	no	—
208	C₂×D₄	—	23	6	no	—
209	C₂×D₄	—	23	3	no	—
210	C₂×D₄	—	23	3	no	—
211	C₂×D₄	—	92	6	no	—
212	C₂×D₄	—	92	12	no	—
213	C₂×D₄	—	92	6	no	—
214	C₂×D₄	C₂²×D₂₃	1	6	yes	—
215	C₂×D₄	C₂²×D₂₃	1	3	yes	—
216	C₂²×C₄	—	23	3	no	—
217	C₂²×C₄	—	23	3	no	—
218	C₂²×C₄	—	23	6	no	—
219	C₂²×C₄	—	23	6	no	—
220	C₄⋊C₄	—	23	6	no	—
221	C₄⋊C₄	—	23	6	no	—
222	C₄⋊C₄	—	23	6	no	—
223	C₂²⋊C₄	—	23	3	no	—
224	C₂²⋊C₄	—	23	6	no	—
225	C₂²⋊C₄	—	23	3	no	—
226	C₂²⋊C₄	—	23	6	no	—
227	C₂²⋊C₄	—	23	6	no	—
228	C₂²⋊C₄	—	23	6	no	—
229	C₂²⋊C₄	—	23	6	no	—
230	C₂²⋊C₄	C₂²×D₂₃	1	6	yes	—
231	C₂²⋊C₄	C₂²×D₂₃	1	3	yes	—
232	C₂²⋊C₄	C₂²×D₂₃	1	3	yes	complemented
233	C₂²⋊C₄	C₂²×D₂₃	1	6	yes	complemented
234	C₄²	—	23	3	no	—
235	C₄²	C₂²×D₂₃	1	1	yes	characteristic, complemented
236	C₄²	C₂²×D₂₃	1	2	yes	complemented
237	C₂³	—	4	3	no	—
238	C₂³	—	4	3	no	—
239	C₂³	—	23	3	no	—
240	C₂³	—	23	1	no	—
241	C₂³	—	23	3	no	—
242	C₂³	—	46	12	no	—
243	C₂³	—	46	6	no	—
244	C₂³	—	46	6	no	—
245	C₂³	—	92	3	no	—
246	C₂³	—	92	6	no	—
247	C₂³	—	92	6	no	—
248	C₂³	—	92	16	no	—
249	C₂³	C₂³×D₂₃	1	3	yes	—
250	C₂³	C₂³×D₂₃	1	3	yes	—
251	Q₈	—	4	3	no	—
252	D₄	—	4	3	no	—
253	D₄	—	4	6	no	—
254	D₄	—	46	12	no	—
255	D₄	—	46	12	no	—
256	D₄	—	92	3	no	—
257	D₄	—	92	3	no	—
258	D₄	—	92	3	no	—
259	D₄	—	92	3	no	—
260	D₄	—	92	6	no	—
261	D₄	—	92	6	no	—
262	C₂×C₄	—	23	6	no	—
263	C₂×C₄	—	23	3	no	—
264	C₂×C₄	—	46	6	no	—
265	C₂×C₄	—	92	6	no	—
266	C₂×C₄	—	92	6	no	—
267	C₂×C₄	—	92	3	no	—
268	C₂×C₄	C₂³×D₂₃	1	3	yes	—
269	C₂×C₄	C₂³×D₂₃	1	6	yes	—
270	C₂²	—	2	3	no	—
271	C₂²	—	2	6	no	—
272	C₂²	—	2	3	no	—
273	C₂²	—	2	6	no	—
274	C₂²	—	4	4	no	—
275	C₂²	—	8	2	no	—
276	C₂²	—	23	6	no	—
277	C₂²	—	46	6	no	—
278	C₂²	—	46	6	no	—
279	C₂²	—	46	3	no	—
280	C₂²	—	46	3	no	—
281	C₂²	—	92	12	no	—
282	C₂²	—	92	12	no	—
283	C₂²	—	184	6	no	—
284	C₂²	C₂⁴×D₂₃	1	1	yes	characteristic
285	C₄	—	2	3	no	—
286	C₄	—	2	6	no	—
287	C₄	—	46	3	no	—
288	C₄	—	46	6	no	—
289	C₂	—	4	3	no	—
290	C₂	—	4	3	no	—
291	C₂	—	46	6	no	—
292	C₂	—	92	1	no	—
293	C₂	—	92	3	no	—
294	C₂	H(5,𝔽₂)×D₂₃	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₂₃
commutator, socle, cosocle	C₂²×C₂₃	C₂⁵
Fitting	C₂₃×H(3,𝔽₄)	C₂
hypercenter	H(3,𝔽₄)	D₂₃
hypocenter	C₂₃	#3532
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ H(3,𝔽₄)

Complements

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

N	H ≅ G/N	type
1. #13093156	C₂	semidirect
2. #13093097	C₂	semidirect
3. #13093060	C₂	semidirect
4. #13093049	C₂	semidirect
5. #13093001	C₂	semidirect
6. #13092985	C₂	semidirect
7. #13092977	C₂	semidirect
8. #120×D₂₃	C₂	semidirect
9. C₂₃×H(3,𝔽₄)	C₂	semidirect
10. C₂⁴×D₂₃	C₂²	semidirect
11. C₂²×C₂²⋊D₂₃	C₂²	semidirect
12. C₂×#67292	C₂²	semidirect
13. C₂×D₄×D₂₃	C₂²	semidirect
14. C₂×D₄×D₂₃	C₂²	semidirect
15. C₂₃×#124	C₂²	semidirect
16. C₂₃×#124	C₂²	semidirect
17. C₂₃×#120	C₂²	semidirect
18. C₂₃×#120	C₂²	semidirect
19. #10625124	C₂²	semidirect
20. #10625124	C₂²	semidirect
21. #10625118	C₂²	semidirect
22. #10625117	C₂²	semidirect
23. #10625111	C₂²	semidirect
24. #10625110	C₂²	semidirect
25. #10625109	C₂²	semidirect
27. #10625108	C₂²	semidirect
28. #10625106	C₂²	semidirect
29. #10625057	C₂²	semidirect
30. #10625055	C₂²	semidirect
31. #10625055	C₂²	semidirect
32. #10625055	C₂²	semidirect
33. #10625054	C₂²	semidirect
34. #10625054	C₂²	semidirect
35. #10625053	C₂²	semidirect
36. #10625053	C₂²	semidirect
37. #10625053	C₂²	semidirect
38. (C₂²⋊D₂₃)⋊C₄	C₂²	semidirect
39. C₂²⋊C₄×D₂₃	C₂²	semidirect
40. C₂²⋊C₄×D₂₃	C₂²	semidirect
41. C₂²⋊C₄×D₂₃	C₂²	semidirect
42. #10625050	C₂²	semidirect
43. #10625047	C₂²	semidirect
44. #10625045	C₂²	semidirect
45. C₄×D₉₂	C₂²	semidirect
46. #10625043	C₂²	semidirect
48. C₂⁴×C₂₃	C₂³	semidirect
53. C₂×Q₈×C₂₃	C₂³	semidirect
61. C₂×C₂²⋊D₂₃	C₂³	semidirect
62. C₂²×Dic₂₃	C₂³	semidirect
68. C₂×D₉₂	C₂³	semidirect
71. C₂×C₄×D₂₃	C₂³	semidirect
74. C₂²⋊C₄×C₂₃	C₂³	semidirect
75. C₂²⋊C₄×C₂₃	C₂³	semidirect
76. C₄²×C₂₃	C₂³	semidirect
77. C₄²×C₂₃	C₂³	semidirect
80. C₂²⋊Dic₂₃	C₂³	semidirect
84. (C₂×D₂₃)⋊C₄	C₂³	semidirect
85. C₄⋊Dic₂₃	C₂³	semidirect
86. Dic₂₃⋊C₄	C₂³	semidirect
87. Dic₂₃⋊C₄	C₂³	semidirect
88. C₄×Dic₂₃	C₂³	semidirect
142. H(3,𝔽₄)	D₂₃	semidirect
167. #124	C₂×D₂₃	semidirect
168. #124	C₂×D₂₃	semidirect
186. #120	C₂×D₂₃	semidirect
187. #120	C₂×D₂₃	semidirect
194. C₂₃	#3532	semidirect
200. C₂⁴	C₂²×D₂₃	semidirect
202. C₂×Q₈	C₂²×D₂₃	semidirect
232. C₂²⋊C₄	C₂²×D₂₃	semidirect
233. C₂²⋊C₄	C₂²×D₂₃	semidirect
235. C₄²	C₂²×D₂₃	semidirect
236. C₄²	C₂²×D₂₃	semidirect

Maximal subgroups

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

subgroup	index
#13093156	2
#13093097	2
#13093060	2
#13093049	2
#13093001	2
#13092985	2
#13092977	2
#120×D₂₃	2
C₂₃×H(3,𝔽₄)	2
#3532	23