Subgroups — #15682227

Order 2944.

506 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	C₂×#10625079	C₂	1	1	yes	maximal, characteristic
2	#13092898	C₂	1	1	yes	maximal, characteristic, complemented
3	#13092891	C₂	1	1	yes	maximal, characteristic, complemented
4	#13092528	C₂	1	2	yes	maximal, complemented
5	#13092516	C₂	1	2	yes	maximal, complemented
6	#13092504	C₂	1	2	yes	maximal, complemented
7	#13092499	C₂	1	2	yes	maximal, complemented
8	#13092411	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#10624991	C₂	1	1	yes	maximal, characteristic, complemented
10	#13092375	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#428	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#67288	C₂²	1	1	yes	characteristic
13	C₂²×D₉₂	C₂²	1	1	yes	characteristic
14	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
15	C₂₃×#117	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
17	C₄×C₂²⋊D₂₃	—	2	2	no	—
18	#10625097	C₂²	1	1	yes	characteristic
19	C₂²×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
20	#10625079	C₄	1	8	yes	complemented
21	C₂×D₁₈₄	C₂²	1	2	yes	—
22	C₂×#67263	C₂²	1	2	yes	—
23	#10625064	C₂²	1	2	yes	—
24	#10625064	C₂²	1	2	yes	—
25	#10625064	C₂²	1	2	yes	complemented
26	#10625063	—	2	1	no	—
27	C₄⋊C₄×D₂₃	—	2	1	no	—
28	#10625058	C₂²	1	2	yes	complemented
29	C₂²⋊C₄×D₂₃	—	2	2	no	—
30	C₄×D₉₂	—	2	1	no	—
31	C₄×D₉₂	—	2	1	no	—
32	C₂₃×#107	C₂²	1	2	yes	complemented
33	C₂₃×#106	C₂²	1	2	yes	complemented
34	#10624997	C₂²	1	2	yes	complemented
35	C₈×Dic₂₃	C₂²	1	2	yes	complemented
36	#10624992	C₂²	1	1	yes	characteristic, complemented
37	#10624992	C₂²	1	1	yes	characteristic, complemented
38	#10624991	C₂²	1	2	yes	—
39	#10624991	C₂²	1	2	yes	—
40	#10624991	C₂²	1	1	yes	characteristic, complemented
41	#10624991	C₂²	1	1	yes	characteristic, complemented
42	C₂³×D₂₃	—	2	1	no	—
43	C₂²×C₄×C₂₃	—	2	1	no	—
44	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
45	C₂×C₂²⋊D₂₃	—	2	1	no	—
46	#67288	—	2	2	no	—
47	#67288	C₂×C₄	1	4	yes	complemented
48	C₂×D₉₂	—	2	1	no	—
49	C₂×D₉₂	—	2	2	no	—
50	C₂×D₉₂	C₂×C₄	1	4	yes	complemented
51	C₂×D₉₂	C₂³	1	2	yes	—
52	C₂×D₉₂	C₂³	1	1	yes	characteristic
53	C₂×C₄×D₂₃	—	2	1	no	—
54	C₂×C₄×D₂₃	—	2	2	no	—
55	C₂×C₄×D₂₃	—	2	2	no	—
56	C₂×C₄×D₂₃	—	2	2	no	—
57	C₂×Dic₄₆	C₂³	1	1	yes	characteristic
58	M₁₆×C₂₃	C₂×C₄	1	4	yes	complemented
59	C₂×C₈×C₂₃	C₂³	1	2	yes	—
60	C₄⋊C₄×C₂₃	—	2	1	no	—
61	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
62	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
63	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
64	C₂²⋊C₄×C₂₃	—	2	1	no	—
65	C₄²×C₂₃	—	2	1	no	—
66	C₂²⋊Dic₂₃	D₄	1	2	yes	—
67	(C₂×D₂₃)⋊C₄	—	2	2	no	—
68	(C₂×D₂₃)⋊C₄	—	2	2	no	—
69	(C₂×D₂₃)⋊C₄	—	2	2	no	—
70	C₄⋊Dic₂₃	D₄	1	1	yes	characteristic
71	C₄⋊Dic₂₃	D₄	1	1	yes	characteristic
72	Dic₂₃⋊C₄	—	2	2	no	—
73	C₄×Dic₂₃	—	2	2	no	—
74	C₄×Dic₂₃	C₂³	1	2	yes	—
75	C₂×C₂₃⋊C₈	—	2	2	no	—
76	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
77	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
78	D₁₈₄	C₂×C₄	1	8	yes	complemented
79	#67263	C₂×C₄	1	8	yes	complemented
80	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
81	C₂²×D₂₃	—	2	1	no	—
82	C₂²×D₂₃	—	2	2	no	—
83	C₂²×D₂₃	—	2	4	no	—
84	C₂²×D₂₃	—	8	1	no	—
85	C₂²×D₂₃	—	8	1	no	—
86	C₂×C₄×C₂₃	—	2	2	no	—
87	C₂×C₄×C₂₃	—	2	1	no	—
88	C₂×C₄×C₂₃	—	2	1	no	—
89	C₂×C₄×C₂₃	—	8	1	no	—
90	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
91	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₄×C₂₃	#41	1	2	yes	—
93	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
94	C₂²⋊D₂₃	—	2	4	no	—
95	C₂×Dic₂₃	—	2	1	no	—
96	C₂×Dic₂₃	C₂×D₄	1	2	yes	—
97	C₂×Dic₂₃	C₂×D₄	1	2	yes	—
98	D₉₂	—	2	4	no	—
99	D₉₂	—	2	2	no	—
100	D₉₂	—	2	1	no	—
101	D₉₂	C₂²×C₄	1	4	yes	—
102	D₉₂	C₂²×C₄	1	2	yes	—
103	C₄×D₂₃	—	2	4	no	—
104	C₄×D₂₃	—	8	2	no	—
105	C₄×D₂₃	—	8	2	no	—
106	C₄×D₂₃	—	8	2	no	—
107	Dic₄₆	—	2	1	no	—
108	Dic₄₆	C₂²×C₄	1	2	yes	—
109	C₈×C₂₃	C₂²×C₄	1	4	yes	—
110	C₂₃⋊C₈	—	2	1	no	—
111	C₂₃⋊C₈	—	2	1	no	—
112	#2946	—	23	1	no	maximal, Sylow
113	C₂²×C₂₃	—	2	1	no	—
114	C₂²×C₂₃	—	2	1	no	—
115	C₂²×C₂₃	C₄×D₄	1	2	yes	—
116	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
117	C₂×D₂₃	—	2	2	no	—
118	C₂×D₂₃	—	2	4	no	—
119	C₂×D₂₃	—	4	1	no	—
120	C₂×D₂₃	—	4	2	no	—
121	C₂×D₂₃	—	4	2	no	—
122	C₂×D₂₃	—	4	1	no	—
123	C₂×D₂₃	—	8	4	no	—
124	C₄×C₂₃	—	4	1	no	—
125	C₄×C₂₃	—	4	2	no	—
126	C₄×C₂₃	—	4	1	no	—
127	C₄×C₂₃	C₄×D₄	1	2	yes	—
128	C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
129	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
130	Dic₂₃	—	2	2	no	—
131	Dic₂₃	—	2	2	no	—
132	Dic₂₃	—	2	2	no	—
133	C₂×AGL(1,ℤ/8)	—	23	1	no	—
134	#520	—	23	1	no	—
135	#518	—	23	1	no	—
136	#442	—	23	2	no	—
137	#440	—	23	2	no	—
138	C₄×SD₁₆	—	23	2	no	—
139	C₄×D₈	—	23	2	no	—
140	#428	D₂₃	1	1	yes	characteristic, complemented
141	#416	—	23	1	no	—
142	C₂×#102	—	23	1	no	—
143	#405	—	23	1	no	—
144	C₂×C₂₃	—	2	2	no	—
145	C₂×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
146	C₂×C₂₃	#576	1	1	yes	characteristic
147	C₂×C₂₃	#577	1	1	yes	characteristic
148	D₂₃	—	4	2	no	—
149	D₂₃	—	4	4	no	—
150	C₂×#41	—	23	1	no	—
151	C₂²×D₄	—	23	1	no	—
152	AGL(1,ℤ/8)	—	23	8	no	—
153	C₂×SD₁₆	—	23	2	no	—
154	C₂×D₈	—	23	2	no	—
155	C₂×M₁₆	C₂×D₂₃	1	1	yes	characteristic
156	C₂²×C₈	—	23	1	no	—
157	C₄×Q₈	—	23	2	no	—
158	C₄×D₄	—	23	2	no	—
159	C₄×D₄	—	23	2	no	—
160	C₄×D₄	—	23	2	no	—
161	C₄×D₄	—	46	1	no	—
162	C₄×D₄	—	46	1	no	—
163	C₄×D₄	—	46	2	no	—
164	#117	—	23	1	no	—
165	#117	—	46	1	no	—
166	#117	C₂×D₂₃	1	1	yes	characteristic
167	C₂×C₄⋊C₄	—	46	1	no	—
168	C₂×C₄⋊C₄	C₂×D₂₃	1	1	yes	characteristic
169	C₂×C₂²⋊C₄	—	46	2	no	—
170	#107	C₂×D₂₃	1	2	yes	complemented
171	#106	C₂×D₂₃	1	2	yes	complemented
172	#103	—	23	1	no	—
173	#103	—	23	1	no	—
174	#102	—	23	2	no	—
175	#102	—	23	2	no	—
176	#102	—	23	1	no	—
177	#102	—	23	1	no	—
178	#97	—	23	2	no	—
179	C₄×C₈	—	23	2	no	—
180	C₂₃	#2946	1	1	yes	characteristic, complemented, Sylow
181	C₂⁴	—	46	1	no	—
182	#41	—	23	4	no	—
183	#41	—	46	2	no	—
184	C₂×Q₈	—	23	1	no	—
185	C₂×D₄	—	23	4	no	—
186	C₂×D₄	—	23	1	no	—
187	C₂×D₄	—	23	2	no	—
188	C₂×D₄	—	46	1	no	—
189	C₂×D₄	—	46	1	no	—
190	C₂×D₄	—	46	2	no	—
191	C₂²×C₄	—	2	1	no	—
192	C₂²×C₄	—	46	2	no	—
193	C₂²×C₄	—	46	2	no	—
194	C₂²×C₄	—	46	2	no	—
195	C₂²×C₄	—	46	1	no	—
196	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
197	SD₁₆	—	23	8	no	—
198	D₈	—	23	8	no	—
199	M₁₆	C₄×D₂₃	1	4	yes	complemented
200	C₂×C₈	—	23	1	no	—
201	C₂×C₈	—	23	1	no	—
202	C₂×C₈	—	46	2	no	—
203	C₂×C₈	C₂²×D₂₃	1	2	yes	—
204	C₄⋊C₄	—	2	1	no	—
205	C₄⋊C₄	—	23	1	no	—
206	C₄⋊C₄	—	23	1	no	—
207	C₄⋊C₄	—	46	2	no	—
208	C₄⋊C₄	C₂²×D₂₃	1	2	yes	—
209	C₄⋊C₄	C₂²×D₂₃	1	1	yes	characteristic
210	C₄⋊C₄	C₂²×D₂₃	1	1	yes	characteristic
211	C₂²⋊C₄	—	2	1	no	—
212	C₂²⋊C₄	—	23	2	no	—
213	C₂²⋊C₄	—	46	2	no	—
214	C₂²⋊C₄	—	46	2	no	—
215	C₂²⋊C₄	—	46	2	no	—
216	C₄²	—	2	1	no	—
217	C₄²	—	23	2	no	—
218	C₄²	—	46	2	no	—
219	C₂³	—	46	1	no	—
220	C₂³	—	46	2	no	—
221	C₂³	—	46	4	no	—
222	C₂³	—	184	1	no	—
223	C₂³	—	184	1	no	—
224	C₂³	D₄×D₂₃	1	1	yes	characteristic
225	Q₈	—	23	2	no	—
226	Q₈	—	46	1	no	—
227	D₄	—	23	2	no	—
228	D₄	—	23	4	no	—
229	D₄	—	46	4	no	—
230	D₄	—	46	4	no	—
231	D₄	—	46	2	no	—
232	D₄	—	46	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	8	1	no	—
237	C₂×C₄	—	23	2	no	—
238	C₂×C₄	—	23	2	no	—
239	C₂×C₄	—	46	1	no	—
240	C₂×C₄	—	46	4	no	—
241	C₂×C₄	—	184	2	no	—
242	C₂×C₄	—	184	2	no	—
243	C₂×C₄	—	184	2	no	—
244	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
245	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
246	C₂×C₄	#67292	1	2	yes	—
247	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
248	C₈	—	46	1	no	—
249	C₈	—	46	1	no	—
250	C₈	C₂×C₄×D₂₃	1	4	yes	—
251	C₂²	—	2	1	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	46	2	no	—
254	C₂²	—	46	4	no	—
255	C₂²	—	92	1	no	—
256	C₂²	—	92	2	no	—
257	C₂²	—	92	2	no	—
258	C₂²	—	92	1	no	—
259	C₂²	—	184	4	no	—
260	C₂²	#10625064	1	2	yes	—
261	C₂²	C₂×D₄×D₂₃	1	1	yes	characteristic
262	C₄	—	4	2	no	—
263	C₄	—	4	1	no	—
264	C₄	—	4	1	no	—
265	C₄	—	46	2	no	—
266	C₄	—	46	2	no	—
267	C₄	—	46	2	no	—
268	C₄	#10625064	1	2	yes	—
269	C₄	C₂²×C₄×D₂₃	1	1	yes	characteristic
270	C₄	C₂×#67292	1	1	yes	characteristic
271	C₂	—	2	2	no	—
272	C₂	—	92	4	no	—
273	C₂	—	92	2	no	—
274	C₂	C₂×#10625064	1	1	yes	characteristic
275	C₂	#13093221	1	1	yes	characteristic
276	C₂	#13093226	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	C₂×#10625079	C₂
cosocle	C₂×C₄×C₂₃	C₂⁴
hypercenter	#428	D₂₃
hypocenter	C₂₃	#2946
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
2. #13092898	C₂	semidirect
3. #13092891	C₂	semidirect
4. #13092528	C₂	semidirect
5. #13092516	C₂	semidirect
6. #13092504	C₂	semidirect
7. #13092499	C₂	semidirect
8. #13092411	C₂	semidirect
9. C₂×#10624991	C₂	semidirect
10. #13092375	C₂	semidirect
11. C₂₃×#428	C₂	semidirect
20. #10625079	C₄	semidirect
25. #10625064	C₂²	semidirect
28. #10625058	C₂²	semidirect
32. C₂₃×#107	C₂²	semidirect
33. C₂₃×#106	C₂²	semidirect
34. #10624997	C₂²	semidirect
35. C₈×Dic₂₃	C₂²	semidirect
36. #10624992	C₂²	semidirect
37. #10624992	C₂²	semidirect
40. #10624991	C₂²	semidirect
41. #10624991	C₂²	semidirect
47. #67288	C₂×C₄	semidirect
50. C₂×D₉₂	C₂×C₄	semidirect
58. M₁₆×C₂₃	C₂×C₄	semidirect
78. D₁₈₄	C₂×C₄	semidirect
79. #67263	C₂×C₄	semidirect
140. #428	D₂₃	semidirect
170. #107	C₂×D₂₃	semidirect
171. #106	C₂×D₂₃	semidirect
180. C₂₃	#2946	semidirect
199. 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×#10625079	2
#13092898	2
#13092891	2
#13092528	2
#13092516	2
#13092504	2
#13092499	2
#13092411	2
C₂×#10624991	2
#13092375	2
C₂₃×#428	2
#2946	23