Subgroups — #15682240

Order 2944.

470 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10625113	C₂	1	1	yes	maximal, characteristic
2	#13092897	C₂	1	1	yes	maximal, characteristic
3	#13092522	C₂	1	2	yes	maximal, complemented
4	#13092515	C₂	1	2	yes	maximal, complemented
5	C₄×#67275	C₂	1	2	yes	maximal, complemented
6	C₄×#67274	C₂	1	2	yes	maximal, complemented
7	#13092460	C₂	1	1	yes	maximal, characteristic, complemented
8	#13092411	C₂	1	1	yes	maximal, characteristic
9	C₂×#10625017	C₂	1	1	yes	maximal, characteristic, complemented
10	#13092373	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#519	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×Q₈×C₂₃	C₂²	1	1	yes	characteristic
13	C₂×#67288	C₂²	1	1	yes	characteristic
14	C₄×Q₈×C₂₃	—	2	1	no	—
15	C₄×Q₈×C₂₃	—	2	1	no	—
16	C₄×Q₈×C₂₃	C₂²	1	2	yes	—
17	C₄×Q₈×C₂₃	C₂²	1	2	yes	—
18	C₂₃×#117	—	2	2	no	—
19	C₂₃×#117	C₂²	1	1	yes	characteristic
20	C₂₃×#117	C₂²	1	1	yes	characteristic
21	C₂×#67275	C₂²	1	2	yes	—
22	#10625113	C₄	1	8	yes	complemented
23	C₂×#67274	C₂²	1	2	yes	—
24	C₂×C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
25	C₂×#67267	C₂²	1	1	yes	characteristic
26	C₂²×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
27	#10625063	—	2	1	no	—
28	C₄⋊C₄×D₂₃	—	2	1	no	—
29	(C₂²⋊D₂₃)⋊C₄	—	2	2	no	—
30	C₄×D₉₂	C₂²	1	2	yes	—
31	C₄×Dic₄₆	C₂²	1	2	yes	—
32	#10625017	C₂²	1	2	yes	—
33	#10625017	C₂²	1	2	yes	—
34	#10624992	C₂²	1	1	yes	characteristic
35	#10624992	C₂²	1	1	yes	characteristic
36	#10624991	C₂²	1	1	yes	characteristic
37	#10624991	C₂²	1	1	yes	characteristic
38	#10624990	C₂²	1	2	yes	—
39	#10624989	C₂²	1	2	yes	—
40	#10624985	C₂²	1	2	yes	—
41	C₄×C₂₃⋊C₈	C₂²	1	2	yes	—
42	C₂×Q₈×C₂₃	—	2	1	no	—
43	C₂×Q₈×C₂₃	—	2	2	no	—
44	C₂×Q₈×C₂₃	C₂×C₄	1	4	yes	complemented
45	C₂×Q₈×C₂₃	C₂³	1	2	yes	—
46	C₂²×C₄×C₂₃	—	2	1	no	—
47	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
48	C₂×C₂²⋊D₂₃	—	2	1	no	—
49	C₂²×Dic₂₃	—	2	1	no	—
50	#67288	—	2	2	no	—
51	#67288	C₂×C₄	1	4	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×Dic₄₆	C₂³	1	1	yes	characteristic
56	C₄⋊C₄×C₂₃	—	2	2	no	—
57	C₄⋊C₄×C₂₃	—	2	2	no	—
58	C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
59	C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
60	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
61	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
62	C₂²⋊C₄×C₂₃	—	2	1	no	—
63	C₂²⋊C₄×C₂₃	D₄	1	2	yes	—
64	C₄²×C₂₃	—	2	2	no	—
65	C₄²×C₂₃	—	2	2	no	—
66	C₄²×C₂₃	—	2	1	no	—
67	C₄²×C₂₃	C₂³	1	2	yes	—
68	#67275	C₂×C₄	1	8	yes	complemented
69	#67274	C₂×C₄	1	8	yes	complemented
70	(C₂×D₂₃)⋊C₄	—	2	2	no	—
71	C₄⋊Dic₂₃	—	2	1	no	—
72	C₄⋊Dic₂₃	C₂³	1	2	yes	—
73	Dic₂₃⋊C₄	—	2	2	no	—
74	C₄×Dic₂₃	—	2	2	no	—
75	#67267	C₂×C₄	1	4	yes	complemented
76	C₂×C₂₃⋊C₈	—	2	2	no	—
77	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
78	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
79	C₂×C₂₃⋊C₈	C₂³	1	2	yes	—
80	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
81	C₂²×D₂₃	—	2	1	no	—
82	Q₈×C₂₃	—	2	4	no	—
83	Q₈×C₂₃	—	2	2	no	—
84	Q₈×C₂₃	C₂²×C₄	1	4	yes	—
85	C₂×C₄×C₂₃	—	2	4	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	C₂²×C₄	1	2	yes	—
90	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
91	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
92	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
93	C₂×C₄×C₂₃	#41	1	2	yes	—
94	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
95	C₂²⋊D₂₃	—	2	4	no	—
96	C₂×Dic₂₃	—	2	1	no	—
97	C₂×Dic₂₃	—	2	2	no	—
98	C₂×Dic₂₃	—	8	1	no	—
99	D₉₂	—	2	1	no	—
100	D₉₂	C₂²×C₄	1	2	yes	—
101	C₄×D₂₃	—	2	4	no	—
102	C₄×D₂₃	—	8	2	no	—
103	Dic₄₆	—	2	1	no	—
104	Dic₄₆	C₂²×C₄	1	2	yes	—
105	C₂₃⋊C₈	—	2	1	no	—
106	C₂₃⋊C₈	—	2	1	no	—
107	C₂₃⋊C₈	C₂²×C₄	1	4	yes	—
108	#2947	—	23	1	no	maximal, Sylow
109	C₂²×C₂₃	—	2	1	no	—
110	C₂²×C₂₃	—	2	1	no	—
111	C₂²×C₂₃	C₄×D₄	1	2	yes	—
112	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
113	C₂×D₂₃	—	2	2	no	—
114	C₂×D₂₃	—	4	1	no	—
115	C₂×D₂₃	—	4	1	no	—
116	C₄×C₂₃	—	2	4	no	—
117	C₄×C₂₃	—	2	2	no	—
118	C₄×C₂₃	—	2	2	no	—
119	C₄×C₂₃	—	4	1	no	—
120	C₄×C₂₃	—	4	1	no	—
121	C₄×C₂₃	C₄×D₄	1	2	yes	—
122	C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
123	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
124	Dic₂₃	—	2	2	no	—
125	Dic₂₃	—	4	2	no	—
126	C₂×#137	—	23	1	no	—
127	#520	—	23	1	no	—
128	#519	D₂₃	1	1	yes	characteristic, complemented
129	#441	—	23	2	no	—
130	#440	—	23	2	no	—
131	C₄×Q₁₆	—	23	2	no	—
132	C₄×SD₁₆	—	23	2	no	—
133	#428	—	23	1	no	—
134	#416	—	23	1	no	—
135	C₂×#103	—	23	1	no	—
136	#405	—	23	1	no	—
137	C₂×C₂₃	—	2	2	no	—
138	C₂×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
139	C₂×C₂₃	#577	1	1	yes	characteristic
140	C₂×C₂₃	#578	1	1	yes	characteristic
141	D₂₃	—	4	2	no	—
142	C₂×#41	—	23	1	no	—
143	C₂²×Q₈	C₂×D₂₃	1	1	yes	characteristic
144	#137	—	23	8	no	—
145	C₂×Q₁₆	—	23	2	no	—
146	C₂×SD₁₆	—	23	2	no	—
147	C₂×M₁₆	—	23	1	no	—
148	C₂²×C₈	—	23	1	no	—
149	C₄×Q₈	—	2	1	no	—
150	C₄×Q₈	—	2	1	no	—
151	C₄×Q₈	—	23	2	no	—
152	C₄×Q₈	C₂×D₂₃	1	2	yes	—
153	C₄×Q₈	C₂×D₂₃	1	2	yes	—
154	C₄×D₄	—	23	2	no	—
155	C₄×D₄	—	46	2	no	—
156	#117	—	2	2	no	—
157	#117	—	46	1	no	—
158	#117	C₂×D₂₃	1	1	yes	characteristic
159	#117	C₂×D₂₃	1	1	yes	characteristic
160	C₂×C₄⋊C₄	—	23	1	no	—
161	C₂×C₄⋊C₄	—	46	1	no	—
162	#107	—	23	2	no	—
163	#106	—	23	2	no	—
164	#103	—	23	2	no	—
165	#103	—	23	2	no	—
166	#103	—	23	1	no	—
167	#103	—	23	1	no	—
168	#102	—	23	1	no	—
169	#102	—	23	1	no	—
170	#97	—	23	2	no	—
171	C₄×C₈	—	23	2	no	—
172	C₂₃	#2947	1	1	yes	characteristic, complemented, Sylow
173	#41	—	23	4	no	—
174	#41	—	46	2	no	—
175	C₂×Q₈	—	2	2	no	—
176	C₂×Q₈	—	2	1	no	—
177	C₂×Q₈	—	23	1	no	—
178	C₂×Q₈	C₄×D₂₃	1	4	yes	complemented
179	C₂×Q₈	C₂²×D₂₃	1	2	yes	—
180	C₂×D₄	—	23	1	no	—
181	C₂×D₄	—	46	1	no	—
182	C₂²×C₄	—	2	1	no	—
183	C₂²×C₄	—	46	1	no	—
184	C₂²×C₄	—	46	1	no	—
185	C₂²×C₄	—	46	2	no	—
186	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
187	Q₁₆	—	23	8	no	—
188	SD₁₆	—	23	8	no	—
189	M₁₆	—	23	4	no	—
190	C₂×C₈	—	23	2	no	—
191	C₂×C₈	—	23	1	no	—
192	C₂×C₈	—	23	1	no	—
193	C₂×C₈	—	46	2	no	—
194	C₄⋊C₄	—	2	2	no	—
195	C₄⋊C₄	—	2	2	no	—
196	C₄⋊C₄	—	23	2	no	—
197	C₄⋊C₄	—	46	1	no	—
198	C₄⋊C₄	—	46	2	no	—
199	C₄⋊C₄	C₂²⋊D₂₃	1	1	yes	characteristic
200	C₄⋊C₄	C₂²⋊D₂₃	1	1	yes	characteristic
201	C₄⋊C₄	C₂²×D₂₃	1	1	yes	characteristic
202	C₄⋊C₄	C₂²×D₂₃	1	1	yes	characteristic
203	C₂²⋊C₄	—	2	1	no	—
204	C₂²⋊C₄	—	46	2	no	—
205	C₂²⋊C₄	C₂²⋊D₂₃	1	2	yes	—
206	C₄²	—	2	2	no	—
207	C₄²	—	2	2	no	—
208	C₄²	—	2	1	no	—
209	C₄²	—	46	2	no	—
210	C₄²	C₂²×D₂₃	1	2	yes	—
211	C₂³	—	46	1	no	—
212	C₂³	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
213	Q₈	—	2	4	no	—
214	Q₈	—	2	2	no	—
215	Q₈	—	23	2	no	—
216	Q₈	—	46	1	no	—
217	Q₈	C₂×C₄×D₂₃	1	4	yes	—
218	D₄	—	23	2	no	—
219	D₄	—	46	4	no	—
220	D₄	—	46	1	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	46	1	no	—
226	C₂×C₄	—	46	4	no	—
227	C₂×C₄	—	46	2	no	—
228	C₂×C₄	—	184	2	no	—
229	C₂×C₄	—	184	1	no	—
230	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
231	C₂×C₄	#67288	1	2	yes	—
232	C₂×C₄	C₂×C₂²⋊D₂₃	1	2	yes	—
233	C₂×C₄	C₂×C₂²⋊D₂₃	1	2	yes	—
234	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
235	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
236	C₈	—	23	4	no	—
237	C₈	—	46	1	no	—
238	C₈	—	46	1	no	—
239	C₂²	—	2	1	no	—
240	C₂²	—	2	1	no	—
241	C₂²	—	46	2	no	—
242	C₂²	—	92	1	no	—
243	C₂²	—	92	1	no	—
244	C₂²	C₄×C₂²⋊D₂₃	1	2	yes	—
245	C₂²	C₂²×C₂²⋊D₂₃	1	1	yes	characteristic
246	C₄	—	2	4	no	—
247	C₄	—	2	2	no	—
248	C₄	—	2	2	no	—
249	C₄	—	4	1	no	—
250	C₄	—	4	1	no	—
251	C₄	—	46	2	no	—
252	C₄	—	92	2	no	—
253	C₄	C₄×C₂²⋊D₂₃	1	2	yes	—
254	C₄	C₂²×C₄×D₂₃	1	1	yes	characteristic
255	C₄	C₂×#67288	1	1	yes	characteristic
256	C₂	—	2	2	no	—
257	C₂	—	92	2	no	—
258	C₂	C₂×C₄×C₂²⋊D₂₃	1	1	yes	characteristic
259	C₂	#13093225	1	1	yes	characteristic
260	C₂	#13093232	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₂₃
commutator	C₄×C₂₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₂₃
Fitting	C₂₃×#519	C₂
socle	C₂²×C₂₃	C₂²×D₄
prime radical	C₂×#67288	C₂²
cosocle	C₂×C₄×C₂₃	C₂⁴
hypercenter	#519	D₂₃
hypocenter	C₂₃	#2947
LCS	G ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #519

Complements

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

N	H ≅ G/N	type
3. #13092522	C₂	semidirect
4. #13092515	C₂	semidirect
5. C₄×#67275	C₂	semidirect
6. C₄×#67274	C₂	semidirect
7. #13092460	C₂	semidirect
9. C₂×#10625017	C₂	semidirect
10. #13092373	C₂	semidirect
11. C₂₃×#519	C₂	semidirect
22. #10625113	C₄	semidirect
44. C₂×Q₈×C₂₃	C₂×C₄	semidirect
68. #67275	C₂×C₄	semidirect
69. #67274	C₂×C₄	semidirect
75. #67267	C₂×C₄	semidirect
128. #519	D₂₃	semidirect
172. C₂₃	#2947	semidirect
178. C₂×Q₈	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×#10625113	2
#13092897	2
#13092522	2
#13092515	2
C₄×#67275	2
C₄×#67274	2
#13092460	2
#13092411	2
C₂×#10625017	2
#13092373	2
C₂₃×#519	2
#2947	23