Subgroups — #15681991

Order 2944.

476 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10625078	C₂	1	1	yes	maximal, characteristic
2	#13092898	C₂	1	1	yes	maximal, characteristic
3	#13092897	C₂	1	1	yes	maximal, characteristic
4	#13092459	C₂	1	1	yes	maximal, characteristic, complemented
5	#13092458	C₂	1	2	yes	maximal, complemented
6	#13092456	C₂	1	2	yes	maximal, complemented
7	#13092454	C₂	1	2	yes	maximal, complemented
8	#13092453	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#10624990	C₂	1	1	yes	maximal, characteristic, complemented
10	#106×D₂₃	C₂	1	2	yes	maximal, complemented
11	C₂₃×#428	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#67288	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	C₄×C₂²⋊D₂₃	—	2	2	no	—
17	#10625097	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
19	C₂²×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
20	#10625078	C₄	1	8	yes	complemented
21	C₂×#67262	C₂²	1	2	yes	—
22	C₂×C₈×D₂₃	C₂²	1	2	yes	—
23	#10625064	—	2	1	no	—
24	#10625063	C₂²	1	2	yes	—
25	#10625063	C₂²	1	2	yes	—
26	C₄⋊C₄×D₂₃	C₂²	1	2	yes	—
27	C₄⋊C₄×D₂₃	C₂²	1	2	yes	—
28	#10625058	—	2	1	no	—
29	(C₂²⋊D₂₃)⋊C₄	—	2	2	no	—
30	C₄×D₉₂	—	2	1	no	—
31	C₄×Dic₄₆	—	2	1	no	—
32	C₂₃×#107	C₂²	1	2	yes	—
33	C₂₃×#106	C₂²	1	2	yes	—
34	#10625000	C₂²	1	2	yes	—
35	#10624999	C₂²	1	2	yes	—
36	#10624990	C₂²	1	1	yes	characteristic
37	#10624990	C₂²	1	2	yes	—
38	#10624990	C₂²	1	2	yes	—
39	#10624990	C₂²	1	1	yes	characteristic
40	#10624989	C₂²	1	1	yes	characteristic
41	#10624989	C₂²	1	1	yes	characteristic
42	C₂²×C₄×C₂₃	—	2	1	no	—
43	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
44	C₂×C₂²⋊D₂₃	D₄	1	2	yes	—
45	C₂²×Dic₂₃	—	2	1	no	—
46	#67288	C₂×C₄	1	4	yes	—
47	#67288	Q₈	1	4	yes	—
48	C₂×D₉₂	D₄	1	1	yes	characteristic
49	C₂×C₄×D₂₃	—	2	2	no	—
50	C₂×C₄×D₂₃	—	2	2	no	—
51	C₂×C₄×D₂₃	C₂³	1	2	yes	—
52	C₂×Dic₄₆	D₄	1	1	yes	characteristic
53	M₁₆×C₂₃	C₂×C₄	1	4	yes	complemented
54	C₂×C₈×C₂₃	C₂³	1	2	yes	—
55	C₄⋊C₄×C₂₃	—	2	1	no	—
56	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
57	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
58	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
59	C₂²⋊C₄×C₂₃	—	2	1	no	—
60	C₄²×C₂₃	—	2	1	no	—
61	C₂²⋊Dic₂₃	—	2	1	no	—
62	(C₂×D₂₃)⋊C₄	—	2	2	no	—
63	(C₂×D₂₃)⋊C₄	—	2	2	no	—
64	C₄⋊Dic₂₃	—	2	1	no	—
65	C₄⋊Dic₂₃	C₂³	1	1	yes	characteristic
66	C₄⋊Dic₂₃	C₂³	1	2	yes	—
67	C₄⋊Dic₂₃	C₂³	1	1	yes	characteristic
68	Dic₂₃⋊C₄	—	2	2	no	—
69	Dic₂₃⋊C₄	—	2	2	no	—
70	C₄×Dic₂₃	—	2	1	no	—
71	C₄×Dic₂₃	—	2	2	no	—
72	C₄×Dic₂₃	—	2	2	no	—
73	C₂×C₂₃⋊C₈	C₂×C₄	1	4	yes	complemented
74	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
75	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
76	#67262	C₂×C₄	1	8	yes	complemented
77	C₈×D₂₃	C₂×C₄	1	8	yes	complemented
78	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
79	C₂²×D₂₃	C₂×D₄	1	2	yes	—
80	C₂×C₄×C₂₃	—	2	1	no	—
81	C₂×C₄×C₂₃	—	2	2	no	—
82	C₂×C₄×C₂₃	—	2	1	no	—
83	C₂×C₄×C₂₃	—	8	1	no	—
84	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
85	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
86	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
87	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
88	C₂²⋊D₂₃	C₄⋊C₄	1	8	yes	—
89	C₂×Dic₂₃	—	2	1	no	—
90	C₂×Dic₂₃	—	2	1	no	—
91	C₂×Dic₂₃	—	2	2	no	—
92	C₂×Dic₂₃	—	8	1	no	—
93	C₂×Dic₂₃	C₂×D₄	1	2	yes	—
94	D₉₂	C₄⋊C₄	1	4	yes	—
95	C₄×D₂₃	—	8	2	no	—
96	C₄×D₂₃	—	8	2	no	—
97	C₄×D₂₃	C₂²×C₄	1	4	yes	—
98	C₄×D₂₃	C₂×Q₈	1	4	yes	—
99	Dic₄₆	C₄⋊C₄	1	4	yes	—
100	C₈×C₂₃	C₂²×C₄	1	4	yes	—
101	C₂₃⋊C₈	C₂²×C₄	1	2	yes	—
102	C₂₃⋊C₈	C₂²×C₄	1	2	yes	—
103	#2912	—	23	1	no	maximal, Sylow
104	C₂²×C₂₃	—	2	1	no	—
105	C₂²×C₂₃	—	2	1	no	—
106	C₂²×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
107	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
108	C₂×D₂₃	—	2	2	no	—
109	C₂×D₂₃	—	2	2	no	—
110	C₂×D₂₃	C₂×C₄⋊C₄	1	4	yes	—
111	C₄×C₂₃	—	4	1	no	—
112	C₄×C₂₃	—	4	2	no	—
113	C₄×C₂₃	—	4	1	no	—
114	C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
115	C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
116	C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
117	Dic₂₃	—	4	1	no	—
118	Dic₂₃	—	4	2	no	—
119	Dic₂₃	—	4	1	no	—
120	Dic₂₃	C₂×C₄⋊C₄	1	4	yes	—
121	C₂×#131	—	23	1	no	—
122	#520	—	23	1	no	—
123	#520	—	23	1	no	—
124	#428	—	23	1	no	—
125	#428	—	23	2	no	—
126	#428	—	23	2	no	—
127	#428	D₂₃	1	1	yes	characteristic, complemented
128	#427	—	23	1	no	—
129	#427	—	23	2	no	—
130	C₂×#106	—	23	1	no	—
131	C₂×#106	—	23	2	no	—
132	C₂×C₂₃	—	2	2	no	—
133	C₂×C₂₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
134	C₂×C₂₃	#577	1	1	yes	characteristic
135	C₂×C₂₃	#577	1	1	yes	characteristic
136	D₂₃	—	2	4	no	—
137	C₂×#41	—	23	1	no	—
138	#131	—	23	8	no	—
139	C₂×M₁₆	—	23	2	no	—
140	C₂×M₁₆	C₂×D₂₃	1	1	yes	characteristic
141	C₂²×C₈	—	23	1	no	—
142	C₂²×C₈	—	23	2	no	—
143	C₄×Q₈	—	46	1	no	—
144	C₄×Q₈	—	46	1	no	—
145	C₄×D₄	—	46	1	no	—
146	C₄×D₄	—	46	2	no	—
147	C₄×D₄	—	46	1	no	—
148	C₄×D₄	—	46	2	no	—
149	#117	—	23	1	no	—
150	#117	—	23	2	no	—
151	#117	—	23	2	no	—
152	#117	C₂×D₂₃	1	1	yes	characteristic
153	C₂×C₄⋊C₄	—	23	1	no	—
154	C₂×C₄⋊C₄	—	23	2	no	—
155	C₂×C₄⋊C₄	—	23	2	no	—
156	C₂×C₄⋊C₄	C₂×D₂₃	1	1	yes	characteristic
157	#107	—	23	2	no	—
158	#107	—	23	1	no	—
159	#107	—	23	1	no	—
160	#107	C₂×D₂₃	1	2	yes	—
161	#106	—	23	2	no	—
162	#106	—	23	2	no	—
163	#106	—	23	1	no	—
164	#106	—	23	2	no	—
165	#106	—	23	1	no	—
166	#106	C₂×D₂₃	1	2	yes	—
167	C₂₃	#2912	1	1	yes	characteristic, complemented, Sylow
168	#41	—	23	4	no	—
169	#41	—	23	4	no	—
170	C₂×Q₈	—	23	1	no	—
171	C₂×D₄	—	23	2	no	—
172	C₂×D₄	—	23	1	no	—
173	C₂²×C₄	—	2	1	no	—
174	C₂²×C₄	—	23	2	no	—
175	C₂²×C₄	—	46	2	no	—
176	C₂²×C₄	—	46	2	no	—
177	C₂²×C₄	—	46	1	no	—
178	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
179	M₁₆	—	23	8	no	—
180	M₁₆	C₄×D₂₃	1	4	yes	complemented
181	C₂×C₈	—	23	4	no	—
182	C₂×C₈	—	23	1	no	—
183	C₂×C₈	—	23	1	no	—
184	C₂×C₈	—	23	8	no	—
185	C₂×C₈	C₂²×D₂₃	1	2	yes	—
186	C₄⋊C₄	—	2	1	no	—
187	C₄⋊C₄	—	23	1	no	—
188	C₄⋊C₄	—	23	2	no	—
189	C₄⋊C₄	—	23	1	no	—
190	C₄⋊C₄	—	46	2	no	—
191	C₄⋊C₄	—	46	2	no	—
192	C₄⋊C₄	—	46	1	no	—
193	C₄⋊C₄	C₂²×D₂₃	1	2	yes	—
194	C₄⋊C₄	C₂²×D₂₃	1	1	yes	characteristic
195	C₄⋊C₄	C₂²×D₂₃	1	1	yes	characteristic
196	C₂²⋊C₄	—	2	1	no	—
197	C₂²⋊C₄	—	46	2	no	—
198	C₂²⋊C₄	—	46	2	no	—
199	C₂²⋊C₄	—	46	1	no	—
200	C₄²	—	2	1	no	—
201	C₄²	—	46	2	no	—
202	C₄²	—	46	2	no	—
203	C₄²	—	46	1	no	—
204	C₂³	—	23	2	no	—
205	C₂³	D₄×D₂₃	1	1	yes	characteristic
206	Q₈	—	23	4	no	—
207	D₄	—	23	8	no	—
208	D₄	—	23	4	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	8	1	no	—
213	C₂×C₄	—	23	2	no	—
214	C₂×C₄	—	23	4	no	—
215	C₂×C₄	—	23	4	no	—
216	C₂×C₄	—	46	1	no	—
217	C₂×C₄	—	46	2	no	—
218	C₂×C₄	—	46	1	no	—
219	C₂×C₄	—	184	2	no	—
220	C₂×C₄	—	184	2	no	—
221	C₂×C₄	—	184	1	no	—
222	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
223	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
224	C₂×C₄	Q₈×D₂₃	1	2	yes	—
225	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
226	C₈	—	23	2	no	—
227	C₈	—	23	2	no	—
228	C₈	C₂×C₄×D₂₃	1	4	yes	—
229	C₂²	—	2	1	no	—
230	C₂²	—	2	1	no	—
231	C₂²	—	23	4	no	—
232	C₂²	—	46	2	no	—
233	C₂²	—	46	2	no	—
234	C₂²	C₄⋊C₄×D₂₃	1	2	yes	—
235	C₂²	C₂×D₄×D₂₃	1	1	yes	characteristic
236	C₄	—	4	1	no	—
237	C₄	—	4	2	no	—
238	C₄	—	4	1	no	—
239	C₄	—	23	4	no	—
240	C₄	—	92	2	no	—
241	C₄	—	92	1	no	—
242	C₄	—	92	1	no	—
243	C₄	C₄⋊C₄×D₂₃	1	2	yes	—
244	C₄	C₂²×C₄×D₂₃	1	1	yes	characteristic
245	C₄	C₂×Q₈×D₂₃	1	1	yes	characteristic
246	C₂	—	2	2	no	—
247	C₂	—	46	4	no	—
248	C₂	C₂×C₄⋊C₄×D₂₃	1	1	yes	characteristic
249	C₂	#13093226	1	1	yes	characteristic
250	C₂	#13093227	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×#67288	C₂²
cosocle	C₂×C₄×C₂₃	C₂⁴
hypercenter	#428	D₂₃
hypocenter	C₂₃	#2912
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. #13092459	C₂	semidirect
5. #13092458	C₂	semidirect
6. #13092456	C₂	semidirect
7. #13092454	C₂	semidirect
8. #13092453	C₂	semidirect
9. C₂×#10624990	C₂	semidirect
10. #106×D₂₃	C₂	semidirect
11. C₂₃×#428	C₂	semidirect
20. #10625078	C₄	semidirect
53. M₁₆×C₂₃	C₂×C₄	semidirect
73. C₂×C₂₃⋊C₈	C₂×C₄	semidirect
76. #67262	C₂×C₄	semidirect
77. C₈×D₂₃	C₂×C₄	semidirect
127. #428	D₂₃	semidirect
167. C₂₃	#2912	semidirect
180. 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×#10625078	2
#13092898	2
#13092897	2
#13092459	2
#13092458	2
#13092456	2
#13092454	2
#13092453	2
C₂×#10624990	2
#106×D₂₃	2
C₂₃×#428	2
#2912	23