Subgroups — #15682238

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×#10625080	C₂	1	1	yes	maximal, characteristic
2	#13092898	C₂	1	1	yes	maximal, characteristic
3	#13092894	C₂	1	1	yes	maximal, characteristic, complemented
4	#13092523	C₂	1	2	yes	maximal, complemented
5	#13092516	C₂	1	2	yes	maximal, complemented
6	#13092511	C₂	1	2	yes	maximal, complemented
7	#13092504	C₂	1	2	yes	maximal, complemented
8	#13092411	C₂	1	1	yes	maximal, characteristic
9	C₂×#10624992	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×Dic₄₆	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	#10625080	C₄	1	8	yes	complemented
21	C₂×Dic₉₂	C₂²	1	2	yes	—
22	C₂×#67263	C₂²	1	2	yes	—
23	#10625064	C₂²	1	2	yes	—
24	#10625063	—	2	1	no	—
25	C₄⋊C₄×D₂₃	—	2	1	no	—
26	#10625058	C₂²	1	2	yes	—
27	#10625058	C₂²	1	2	yes	—
28	#10625058	C₂²	1	2	yes	—
29	#10625048	—	2	2	no	—
30	C₄×Dic₄₆	—	2	1	no	—
31	C₄×Dic₄₆	—	2	1	no	—
32	C₂₃×#107	C₂²	1	2	yes	—
33	C₂₃×#106	C₂²	1	2	yes	—
34	#10624997	C₂²	1	2	yes	—
35	C₈×Dic₂₃	C₂²	1	2	yes	—
36	#10624992	C₂²	1	2	yes	—
37	#10624992	C₂²	1	2	yes	—
38	#10624992	C₂²	1	1	yes	characteristic
39	#10624992	C₂²	1	1	yes	characteristic
40	#10624991	C₂²	1	1	yes	characteristic
41	#10624991	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	—	2	1	no	—
45	C₂²×Dic₂₃	—	2	1	no	—
46	#67288	—	2	2	no	—
47	#67288	C₂×C₄	1	4	yes	—
48	C₂×D₉₂	C₂³	1	1	yes	characteristic
49	C₂×C₄×D₂₃	—	2	1	no	—
50	C₂×C₄×D₂₃	—	2	2	no	—
51	C₂×Dic₄₆	—	2	1	no	—
52	C₂×Dic₄₆	—	2	2	no	—
53	C₂×Dic₄₆	C₂×C₄	1	4	yes	complemented
54	C₂×Dic₄₆	C₂³	1	2	yes	—
55	C₂×Dic₄₆	C₂³	1	1	yes	characteristic
56	M₁₆×C₂₃	C₂×C₄	1	4	yes	complemented
57	C₂×C₈×C₂₃	C₂³	1	2	yes	—
58	C₄⋊C₄×C₂₃	—	2	1	no	—
59	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
60	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
61	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
62	C₂²⋊C₄×C₂₃	—	2	1	no	—
63	C₄²×C₂₃	—	2	1	no	—
64	C₂²⋊Dic₂₃	D₄	1	2	yes	—
65	(C₂×D₂₃)⋊C₄	—	2	2	no	—
66	C₄⋊Dic₂₃	D₄	1	1	yes	characteristic
67	C₄⋊Dic₂₃	D₄	1	1	yes	characteristic
68	Dic₂₃⋊C₄	—	2	2	no	—
69	Dic₂₃⋊C₄	—	2	2	no	—
70	Dic₂₃⋊C₄	—	2	2	no	—
71	C₄×Dic₂₃	—	2	2	no	—
72	C₄×Dic₂₃	—	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	Dic₉₂	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	C₂×C₄×C₂₃	—	2	1	no	—
83	C₂×C₄×C₂₃	—	2	1	no	—
84	C₂×C₄×C₂₃	—	2	2	no	—
85	C₂×C₄×C₂₃	—	8	1	no	—
86	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
87	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
88	C₂×C₄×C₂₃	#41	1	2	yes	—
89	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
90	C₂²⋊D₂₃	—	2	4	no	—
91	C₂×Dic₂₃	—	2	4	no	—
92	C₂×Dic₂₃	—	2	2	no	—
93	C₂×Dic₂₃	—	2	1	no	—
94	C₂×Dic₂₃	C₂×D₄	1	2	yes	—
95	C₂×Dic₂₃	C₂×D₄	1	2	yes	—
96	D₉₂	—	2	1	no	—
97	D₉₂	C₂²×C₄	1	2	yes	—
98	C₄×D₂₃	—	2	4	no	—
99	C₄×D₂₃	—	8	2	no	—
100	Dic₄₆	—	2	4	no	—
101	Dic₄₆	—	2	2	no	—
102	Dic₄₆	—	2	1	no	—
103	Dic₄₆	C₂²×C₄	1	4	yes	—
104	Dic₄₆	C₂²×C₄	1	2	yes	—
105	C₈×C₂₃	C₂²×C₄	1	4	yes	—
106	C₂₃⋊C₈	—	2	1	no	—
107	C₂₃⋊C₈	—	2	1	no	—
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	—	4	2	no	—
117	C₄×C₂₃	—	4	1	no	—
118	C₄×C₂₃	—	4	1	no	—
119	C₄×C₂₃	C₄×D₄	1	2	yes	—
120	C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
121	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
122	Dic₂₃	—	2	4	no	—
123	Dic₂₃	—	2	2	no	—
124	Dic₂₃	—	2	2	no	—
125	Dic₂₃	—	2	2	no	—
126	C₂×#137	—	23	1	no	—
127	#520	—	23	1	no	—
128	#519	—	23	1	no	—
129	#441	—	23	2	no	—
130	#440	—	23	2	no	—
131	C₄×Q₁₆	—	23	2	no	—
132	C₄×SD₁₆	—	23	2	no	—
133	#428	D₂₃	1	1	yes	characteristic, complemented
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	—	23	1	no	—
144	#137	—	23	8	no	—
145	C₂×Q₁₆	—	23	2	no	—
146	C₂×SD₁₆	—	23	2	no	—
147	C₂×M₁₆	C₂×D₂₃	1	1	yes	characteristic
148	C₂²×C₈	—	23	1	no	—
149	C₄×Q₈	—	23	2	no	—
150	C₄×Q₈	—	23	2	no	—
151	C₄×Q₈	—	23	2	no	—
152	C₄×Q₈	—	46	1	no	—
153	C₄×Q₈	—	46	1	no	—
154	C₄×D₄	—	23	2	no	—
155	C₄×D₄	—	46	2	no	—
156	#117	—	23	1	no	—
157	#117	—	46	2	no	—
158	#117	—	46	1	no	—
159	#117	C₂×D₂₃	1	1	yes	characteristic
160	C₂×C₄⋊C₄	—	46	1	no	—
161	C₂×C₄⋊C₄	C₂×D₂₃	1	1	yes	characteristic
162	#107	C₂×D₂₃	1	2	yes	—
163	#106	C₂×D₂₃	1	2	yes	—
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	—	23	4	no	—
176	C₂×Q₈	—	23	2	no	—
177	C₂×Q₈	—	23	1	no	—
178	C₂×Q₈	—	46	1	no	—
179	C₂×Q₈	—	46	2	no	—
180	C₂×D₄	—	23	1	no	—
181	C₂×D₄	—	46	1	no	—
182	C₂²×C₄	—	2	1	no	—
183	C₂²×C₄	—	46	2	no	—
184	C₂²×C₄	—	46	1	no	—
185	C₂²×C₄	—	46	1	no	—
186	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
187	Q₁₆	—	23	8	no	—
188	SD₁₆	—	23	8	no	—
189	M₁₆	C₄×D₂₃	1	4	yes	complemented
190	C₂×C₈	—	23	1	no	—
191	C₂×C₈	—	23	1	no	—
192	C₂×C₈	—	46	2	no	—
193	C₂×C₈	C₂²×D₂₃	1	2	yes	—
194	C₄⋊C₄	—	2	1	no	—
195	C₄⋊C₄	—	23	1	no	—
196	C₄⋊C₄	—	23	1	no	—
197	C₄⋊C₄	—	46	2	no	—
198	C₄⋊C₄	—	46	2	no	—
199	C₄⋊C₄	—	46	2	no	—
200	C₄⋊C₄	C₂²×D₂₃	1	1	yes	characteristic
201	C₄⋊C₄	C₂²×D₂₃	1	1	yes	characteristic
202	C₄⋊C₄	C₂²×D₂₃	1	2	yes	—
203	C₂²⋊C₄	—	2	1	no	—
204	C₂²⋊C₄	—	23	2	no	—
205	C₂²⋊C₄	—	46	2	no	—
206	C₄²	—	2	1	no	—
207	C₄²	—	23	2	no	—
208	C₄²	—	46	2	no	—
209	C₄²	—	46	2	no	—
210	C₄²	—	46	2	no	—
211	C₂³	—	46	1	no	—
212	C₂³	D₄×D₂₃	1	1	yes	characteristic
213	Q₈	—	23	4	no	—
214	Q₈	—	23	2	no	—
215	Q₈	—	46	4	no	—
216	Q₈	—	46	2	no	—
217	Q₈	—	46	1	no	—
218	D₄	—	23	2	no	—
219	D₄	—	46	4	no	—
220	D₄	—	46	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	8	1	no	—
225	C₂×C₄	—	23	2	no	—
226	C₂×C₄	—	23	2	no	—
227	C₂×C₄	—	46	4	no	—
228	C₂×C₄	—	46	1	no	—
229	C₂×C₄	—	46	2	no	—
230	C₂×C₄	—	46	4	no	—
231	C₂×C₄	—	184	2	no	—
232	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
233	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
234	C₂×C₄	#67292	1	2	yes	—
235	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
236	C₈	—	46	1	no	—
237	C₈	—	46	1	no	—
238	C₈	C₂×C₄×D₂₃	1	4	yes	—
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	#10625064	1	2	yes	—
245	C₂²	C₂×D₄×D₂₃	1	1	yes	characteristic
246	C₄	—	4	2	no	—
247	C₄	—	4	1	no	—
248	C₄	—	4	1	no	—
249	C₄	—	46	2	no	—
250	C₄	—	46	4	no	—
251	C₄	—	46	2	no	—
252	C₄	—	46	2	no	—
253	C₄	#10625064	1	2	yes	—
254	C₄	C₂²×C₄×D₂₃	1	1	yes	characteristic
255	C₄	C₂×#67292	1	1	yes	characteristic
256	C₂	—	2	2	no	—
257	C₂	—	92	2	no	—
258	C₂	C₂×#10625064	1	1	yes	characteristic
259	C₂	#13093227	1	1	yes	characteristic
260	C₂	#13093233	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	#2947
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
3. #13092894	C₂	semidirect
4. #13092523	C₂	semidirect
5. #13092516	C₂	semidirect
6. #13092511	C₂	semidirect
7. #13092504	C₂	semidirect
9. C₂×#10624992	C₂	semidirect
10. #13092375	C₂	semidirect
11. C₂₃×#428	C₂	semidirect
20. #10625080	C₄	semidirect
53. C₂×Dic₄₆	C₂×C₄	semidirect
56. M₁₆×C₂₃	C₂×C₄	semidirect
78. Dic₉₂	C₂×C₄	semidirect
79. #67263	C₂×C₄	semidirect
133. #428	D₂₃	semidirect
172. C₂₃	#2947	semidirect
189. 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×#10625080	2
#13092898	2
#13092894	2
#13092523	2
#13092516	2
#13092511	2
#13092504	2
#13092411	2
C₂×#10624992	2
#13092375	2
C₂₃×#428	2
#2947	23