Subgroups — #15681984

Order 2944.

492 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₁₆×D₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂×#10625063	C₂	1	1	yes	maximal, characteristic
3	#117×D₂₃	C₂	1	1	yes	maximal, characteristic
4	#13092460	C₂	1	1	yes	maximal, characteristic, complemented
5	#13092459	C₂	1	1	yes	maximal, characteristic, complemented
6	#13092458	C₂	1	2	yes	maximal, complemented
7	#13092457	C₂	1	1	yes	maximal, characteristic, complemented
8	#13092456	C₂	1	2	yes	maximal, complemented
9	#13092454	C₂	1	2	yes	maximal, complemented
10	#13092452	C₂	1	2	yes	maximal, complemented
11	C₂₃×#428	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₄×D₂₃	C₂²	1	1	yes	characteristic
13	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
14	C₂₃×#117	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
16	C₂×(C₂×D₂₃)⋊C₄	—	2	1	no	—
17	#10625097	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
19	C₂×C₄×Dic₂₃	—	2	1	no	—
20	C₂×#67267	C₂²	1	1	yes	characteristic
21	M₁₆×D₂₃	C₄	1	8	yes	complemented
22	C₂×#67262	C₂²	1	2	yes	—
23	C₂×C₈×D₂₃	C₂²	1	2	yes	—
24	#10625063	—	2	1	no	—
25	#10625063	—	2	1	no	—
26	#10625063	C₂²	1	1	yes	characteristic
27	#10625063	C₂²	1	1	yes	characteristic
28	#10625063	C₂²	1	2	yes	—
29	#10625063	C₂²	1	2	yes	—
30	C₄⋊C₄×D₂₃	C₂²	1	1	yes	characteristic
31	C₄⋊C₄×D₂₃	C₂²	1	1	yes	characteristic
32	C₂²⋊C₄×D₂₃	—	2	1	no	—
33	#10625048	—	2	1	no	—
34	#10625043	—	2	1	no	—
35	C₄²×D₂₃	—	2	1	no	—
36	C₂₃×#107	C₂²	1	2	yes	complemented
37	C₂₃×#106	C₂²	1	2	yes	complemented
38	#10625000	C₂²	1	2	yes	complemented
39	#10624999	C₂²	1	2	yes	complemented
40	#10624990	C₂²	1	2	yes	complemented
41	#10624990	C₂²	1	2	yes	complemented
42	#10624989	C₂²	1	2	yes	complemented
43	#10624989	C₂²	1	2	yes	complemented
44	C₂³×D₂₃	D₄	1	1	yes	characteristic
45	C₂²×C₄×C₂₃	—	2	1	no	—
46	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
47	C₂²×Dic₂₃	—	2	1	no	—
48	C₂²×Dic₂₃	D₄	1	1	yes	characteristic
49	C₂×C₄×D₂₃	—	2	1	no	—
50	C₂×C₄×D₂₃	—	2	1	no	—
51	C₂×C₄×D₂₃	C₂×C₄	1	4	yes	—
52	C₂×C₄×D₂₃	D₄	1	1	yes	characteristic
53	C₂×C₄×D₂₃	D₄	1	1	yes	characteristic
54	C₂×C₄×D₂₃	Q₈	1	4	yes	—
55	C₂×C₄×D₂₃	C₂³	1	1	yes	characteristic
56	C₂×C₄×D₂₃	C₂³	1	1	yes	characteristic
57	M₁₆×C₂₃	C₂×C₄	1	4	yes	complemented
58	C₂×C₈×C₂₃	C₂³	1	2	yes	—
59	C₄⋊C₄×C₂₃	—	2	1	no	—
60	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
61	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
62	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
63	C₂²⋊C₄×C₂₃	—	2	1	no	—
64	C₄²×C₂₃	—	2	1	no	—
65	C₂²⋊Dic₂₃	—	2	1	no	—
66	(C₂×D₂₃)⋊C₄	—	2	1	no	—
67	(C₂×D₂₃)⋊C₄	—	2	1	no	—
68	(C₂×D₂₃)⋊C₄	—	2	2	no	—
69	(C₂×D₂₃)⋊C₄	—	2	2	no	—
70	C₄⋊Dic₂₃	—	2	1	no	—
71	C₄⋊Dic₂₃	C₂³	1	1	yes	characteristic
72	C₄⋊Dic₂₃	C₂³	1	1	yes	characteristic
73	C₄⋊Dic₂₃	C₂³	1	2	yes	—
74	Dic₂₃⋊C₄	—	2	1	no	—
75	Dic₂₃⋊C₄	—	2	1	no	—
76	C₄×Dic₂₃	—	2	1	no	—
77	C₄×Dic₂₃	—	2	1	no	—
78	C₄×Dic₂₃	—	2	1	no	—
79	C₄×Dic₂₃	—	2	2	no	—
80	C₄×Dic₂₃	—	2	2	no	—
81	#67267	C₂×C₄	1	4	yes	complemented
82	C₂×C₂₃⋊C₈	C₂³	1	2	yes	—
83	#67262	C₂×C₄	1	8	yes	complemented
84	C₈×D₂₃	C₂×C₄	1	8	yes	complemented
85	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
86	C₂²×D₂₃	—	4	1	no	—
87	C₂²×D₂₃	—	4	1	no	—
88	C₂²×D₂₃	C₄⋊C₄	1	4	yes	—
89	C₂²×D₂₃	C₂×D₄	1	1	yes	characteristic
90	C₂²×D₂₃	C₂×D₄	1	1	yes	characteristic
91	C₂×C₄×C₂₃	—	2	2	no	—
92	C₂×C₄×C₂₃	—	2	1	no	—
93	C₂×C₄×C₂₃	—	2	1	no	—
94	C₂×C₄×C₂₃	—	4	2	no	—
95	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
96	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
98	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
99	C₂×Dic₂₃	—	2	1	no	—
100	C₂×Dic₂₃	—	2	1	no	—
101	C₂×Dic₂₃	—	2	2	no	—
102	C₂×Dic₂₃	—	4	2	no	—
103	C₂×Dic₂₃	C₄⋊C₄	1	4	yes	—
104	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
105	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
106	C₄×D₂₃	—	4	2	no	—
107	C₄×D₂₃	—	4	2	no	—
108	C₄×D₂₃	C₄⋊C₄	1	4	yes	—
109	C₄×D₂₃	C₄⋊C₄	1	4	yes	—
110	C₄×D₂₃	C₂²×C₄	1	2	yes	—
111	C₄×D₂₃	C₂²×C₄	1	2	yes	—
112	C₄×D₂₃	C₂×Q₈	1	2	yes	—
113	C₄×D₂₃	C₂×Q₈	1	2	yes	—
114	C₈×C₂₃	C₂²×C₄	1	4	yes	—
115	C₂₃⋊C₈	C₂²×C₄	1	4	yes	—
116	#2911	—	23	1	no	maximal, Sylow
117	C₂²×C₂₃	—	2	1	no	—
118	C₂²×C₂₃	—	2	1	no	—
119	C₂²×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
120	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
121	C₂×D₂₃	—	2	1	no	—
122	C₂×D₂₃	—	2	1	no	—
123	C₂×D₂₃	—	2	1	no	—
124	C₂×D₂₃	—	2	1	no	—
125	C₂×D₂₃	—	4	4	no	—
126	C₂×D₂₃	C₂×C₄⋊C₄	1	2	yes	—
127	C₂×D₂₃	C₂×C₄⋊C₄	1	2	yes	—
128	C₄×C₂₃	—	4	1	no	—
129	C₄×C₂₃	—	4	1	no	—
130	C₄×C₂₃	—	4	2	no	—
131	C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
132	C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
133	C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
134	Dic₂₃	—	4	1	no	—
135	Dic₂₃	—	4	1	no	—
136	Dic₂₃	—	4	2	no	—
137	Dic₂₃	C₂×C₄⋊C₄	1	2	yes	—
138	Dic₂₃	C₂×C₄⋊C₄	1	2	yes	—
139	C₂²×M₁₆	—	23	1	no	—
140	C₂×#117	—	23	1	no	—
141	C₂×#117	—	23	1	no	—
142	#428	—	23	1	no	—
143	#428	—	23	1	no	—
144	#428	—	23	1	no	—
145	#428	—	23	2	no	—
146	#428	—	23	2	no	—
147	#428	D₂₃	1	1	yes	characteristic, complemented
148	#427	—	23	2	no	—
149	#427	—	23	2	no	—
150	C₂×C₂₃	—	2	2	no	—
151	C₂×C₂₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
152	C₂×C₂₃	#575	1	1	yes	characteristic
153	C₂×C₂₃	#575	1	1	yes	characteristic
154	D₂₃	—	2	2	no	—
155	D₂₃	—	2	2	no	—
156	C₂³×C₄	—	23	1	no	—
157	C₂×M₁₆	—	23	1	no	—
158	C₂×M₁₆	—	23	8	no	—
159	C₂×M₁₆	—	23	2	no	—
160	C₂×M₁₆	C₂×D₂₃	1	1	yes	characteristic
161	C₂²×C₈	—	23	2	no	—
162	#117	—	23	1	no	—
163	#117	—	23	1	no	—
164	#117	—	23	2	no	—
165	#117	—	23	1	no	—
166	#117	—	23	2	no	—
167	#117	—	46	1	no	—
168	#117	—	46	1	no	—
169	#117	—	46	1	no	—
170	#117	—	46	1	no	—
171	#117	C₂×D₂₃	1	1	yes	characteristic
172	C₂×C₄⋊C₄	—	23	1	no	—
173	C₂×C₄⋊C₄	—	23	1	no	—
174	C₂×C₄⋊C₄	—	23	1	no	—
175	C₂×C₄⋊C₄	C₂×D₂₃	1	1	yes	characteristic
176	C₂×C₂²⋊C₄	—	46	1	no	—
177	C₂×C₂²⋊C₄	—	46	1	no	—
178	C₂×C₄²	—	46	1	no	—
179	C₂×C₄²	—	46	1	no	—
180	#107	—	23	2	no	—
181	#107	—	23	2	no	—
182	#107	—	23	2	no	—
183	#107	C₂×D₂₃	1	2	yes	complemented
184	#106	—	23	2	no	—
185	#106	—	23	2	no	—
186	#106	—	23	2	no	—
187	#106	C₂×D₂₃	1	2	yes	complemented
188	C₂₃	#2911	1	1	yes	characteristic, complemented, Sylow
189	C₂⁴	—	23	1	no	—
190	C₂²×C₄	—	2	1	no	—
191	C₂²×C₄	—	23	1	no	—
192	C₂²×C₄	—	23	4	no	—
193	C₂²×C₄	—	23	1	no	—
194	C₂²×C₄	—	23	1	no	—
195	C₂²×C₄	—	23	4	no	—
196	C₂²×C₄	—	23	1	no	—
197	C₂²×C₄	—	23	1	no	—
198	C₂²×C₄	—	46	1	no	—
199	C₂²×C₄	—	46	1	no	—
200	C₂²×C₄	—	46	1	no	—
201	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
202	M₁₆	—	23	4	no	—
203	M₁₆	—	23	8	no	—
204	M₁₆	C₄×D₂₃	1	4	yes	complemented
205	C₂×C₈	—	23	2	no	—
206	C₂×C₈	—	23	8	no	—
207	C₂×C₈	C₂²×D₂₃	1	2	yes	—
208	C₄⋊C₄	—	2	1	no	—
209	C₄⋊C₄	—	23	1	no	—
210	C₄⋊C₄	—	23	2	no	—
211	C₄⋊C₄	—	23	1	no	—
212	C₄⋊C₄	—	46	1	no	—
213	C₄⋊C₄	—	46	1	no	—
214	C₄⋊C₄	—	46	1	no	—
215	C₄⋊C₄	C₂²×D₂₃	1	1	yes	characteristic
216	C₄⋊C₄	C₂²×D₂₃	1	2	yes	—
217	C₄⋊C₄	C₂²×D₂₃	1	1	yes	characteristic
218	C₂²⋊C₄	—	2	1	no	—
219	C₂²⋊C₄	—	46	1	no	—
220	C₂²⋊C₄	—	46	1	no	—
221	C₂²⋊C₄	—	46	2	no	—
222	C₂²⋊C₄	—	46	2	no	—
223	C₂²⋊C₄	—	46	1	no	—
224	C₄²	—	2	1	no	—
225	C₄²	—	46	1	no	—
226	C₄²	—	46	1	no	—
227	C₄²	—	46	2	no	—
228	C₄²	—	46	2	no	—
229	C₄²	—	46	1	no	—
230	C₂³	—	23	1	no	—
231	C₂³	—	23	1	no	—
232	C₂³	—	23	4	no	—
233	C₂³	—	92	1	no	—
234	C₂³	—	92	1	no	—
235	C₂³	D₄×D₂₃	1	1	yes	characteristic
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	—	23	4	no	—
241	C₂×C₄	—	23	1	no	—
242	C₂×C₄	—	23	1	no	—
243	C₂×C₄	—	23	4	no	—
244	C₂×C₄	—	23	4	no	—
245	C₂×C₄	—	23	2	no	—
246	C₂×C₄	—	23	2	no	—
247	C₂×C₄	—	23	2	no	—
248	C₂×C₄	—	23	2	no	—
249	C₂×C₄	—	46	1	no	—
250	C₂×C₄	—	46	2	no	—
251	C₂×C₄	—	46	1	no	—
252	C₂×C₄	—	92	2	no	—
253	C₂×C₄	—	92	2	no	—
254	C₂×C₄	—	92	2	no	—
255	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
256	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
257	C₂×C₄	Q₈×D₂₃	1	2	yes	—
258	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
259	C₈	—	23	4	no	—
260	C₈	C₂×C₄×D₂₃	1	4	yes	—
261	C₂²	—	2	1	no	—
262	C₂²	—	2	1	no	—
263	C₂²	—	23	2	no	—
264	C₂²	—	23	2	no	—
265	C₂²	—	46	1	no	—
266	C₂²	—	46	1	no	—
267	C₂²	—	46	1	no	—
268	C₂²	—	46	1	no	—
269	C₂²	—	92	4	no	—
270	C₂²	C₄⋊C₄×D₂₃	1	2	yes	—
271	C₂²	C₂×D₄×D₂₃	1	1	yes	characteristic
272	C₄	—	4	2	no	—
273	C₄	—	4	1	no	—
274	C₄	—	4	1	no	—
275	C₄	—	23	2	no	—
276	C₄	—	23	2	no	—
277	C₄	—	92	1	no	—
278	C₄	—	92	1	no	—
279	C₄	—	92	2	no	—
280	C₄	C₄⋊C₄×D₂₃	1	2	yes	—
281	C₄	C₂²×C₄×D₂₃	1	1	yes	characteristic
282	C₄	C₂×Q₈×D₂₃	1	1	yes	characteristic
283	C₂	—	2	2	no	—
284	C₂	—	46	2	no	—
285	C₂	—	46	2	no	—
286	C₂	C₂×C₄⋊C₄×D₂₃	1	1	yes	characteristic
287	C₂	#13093213	1	1	yes	characteristic
288	C₂	#13093216	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₂₃
commutator	C₄×C₂₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₂₃
Fitting	C₂₃×#428	C₂
socle	C₂²×C₂₃	C₂²×D₄
prime radical	C₂³×D₂₃	D₄
cosocle	C₂×C₄×C₂₃	C₂⁴
hypercenter	#428	D₂₃
hypocenter	C₂₃	#2911
LCS	G ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #428

Complements

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

N	H ≅ G/N	type
4. #13092460	C₂	semidirect
5. #13092459	C₂	semidirect
6. #13092458	C₂	semidirect
7. #13092457	C₂	semidirect
8. #13092456	C₂	semidirect
9. #13092454	C₂	semidirect
10. #13092452	C₂	semidirect
11. C₂₃×#428	C₂	semidirect
21. M₁₆×D₂₃	C₄	semidirect
36. C₂₃×#107	C₂²	semidirect
37. C₂₃×#106	C₂²	semidirect
38. #10625000	C₂²	semidirect
39. #10624999	C₂²	semidirect
40. #10624990	C₂²	semidirect
41. #10624990	C₂²	semidirect
42. #10624989	C₂²	semidirect
43. #10624989	C₂²	semidirect
57. M₁₆×C₂₃	C₂×C₄	semidirect
81. #67267	C₂×C₄	semidirect
83. #67262	C₂×C₄	semidirect
84. C₈×D₂₃	C₂×C₄	semidirect
147. #428	D₂₃	semidirect
183. #107	C₂×D₂₃	semidirect
187. #106	C₂×D₂₃	semidirect
188. C₂₃	#2911	semidirect
204. M₁₆	C₄×D₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×M₁₆×D₂₃	2
C₂×#10625063	2
#117×D₂₃	2
#13092460	2
#13092459	2
#13092458	2
#13092457	2
#13092456	2
#13092454	2
#13092452	2
C₂₃×#428	2
#2911	23