Subgroups — #15682234

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	#13092898	C₂	1	1	yes	maximal, characteristic
3	#13092895	C₂	1	1	yes	maximal, characteristic, complemented
4	#13092521	C₂	1	2	yes	maximal, complemented
5	#13092520	C₂	1	2	yes	maximal, complemented
6	#13092509	C₂	1	2	yes	maximal, complemented
7	#13092502	C₂	1	2	yes	maximal, complemented
8	#13092457	C₂	1	1	yes	maximal, characteristic, complemented
9	#13092414	C₂	1	1	yes	maximal, characteristic
10	#13092372	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₂₃×#103	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	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
16	Q₈×Dic₂₃	—	2	1	no	—
17	Q₈×Dic₂₃	—	2	1	no	—
18	Q₈×Dic₂₃	C₂²	1	2	yes	—
19	Q₈×Dic₂₃	C₂²	1	2	yes	—
20	C₂×#67275	C₂²	1	2	yes	—
21	#10625113	C₄	1	8	yes	complemented
22	C₂×#67274	C₂²	1	2	yes	—
23	C₄×C₂²⋊D₂₃	—	2	2	no	—
24	#10625097	—	2	2	no	—
25	#10625097	C₂²	1	1	yes	characteristic
26	#10625097	C₂²	1	1	yes	characteristic
27	C₂×#67267	C₂²	1	1	yes	characteristic
28	#10625064	C₂²	1	2	yes	—
29	#10625063	—	2	1	no	—
30	C₄⋊C₄×D₂₃	—	2	1	no	—
31	#10625058	C₂²	1	2	yes	—
32	C₂₃×#103	C₂²	1	2	yes	—
33	C₂₃×#103	C₂²	1	2	yes	—
34	#10625003	C₂²	1	1	yes	characteristic
35	#10625003	C₂²	1	1	yes	characteristic
36	#10624998	C₂²	1	1	yes	characteristic
37	#10624998	C₂²	1	1	yes	characteristic
38	#10624997	C₂²	1	2	yes	—
39	C₈×Dic₂₃	C₂²	1	2	yes	—
40	#10624990	C₂²	1	2	yes	—
41	#10624989	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	—	2	1	no	—
48	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
49	C₂×C₂²⋊D₂₃	—	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	C₂×C₈×C₂₃	—	2	2	no	—
57	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
58	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
59	C₄⋊C₄×C₂₃	—	2	1	no	—
60	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
61	C₂²⋊Dic₂₃	—	2	1	no	—
62	C₂²⋊Dic₂₃	D₄	1	2	yes	—
63	#67275	C₂×C₄	1	8	yes	complemented
64	#67274	C₂×C₄	1	8	yes	complemented
65	(C₂×D₂₃)⋊C₄	—	2	2	no	—
66	C₄⋊Dic₂₃	—	2	2	no	—
67	C₄⋊Dic₂₃	—	2	2	no	—
68	C₄⋊Dic₂₃	D₄	1	1	yes	characteristic
69	C₄⋊Dic₂₃	D₄	1	1	yes	characteristic
70	C₄⋊Dic₂₃	C₂³	1	1	yes	characteristic
71	C₄⋊Dic₂₃	C₂³	1	1	yes	characteristic
72	Dic₂₃⋊C₄	—	2	2	no	—
73	C₄×Dic₂₃	—	2	1	no	—
74	C₄×Dic₂₃	—	2	2	no	—
75	C₄×Dic₂₃	—	2	2	no	—
76	C₄×Dic₂₃	—	2	2	no	—
77	C₄×Dic₂₃	C₂³	1	2	yes	—
78	#67267	C₂×C₄	1	4	yes	complemented
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	2	no	—
88	C₂×C₄×C₂₃	—	8	1	no	—
89	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
90	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
91	C₂×C₄×C₂₃	#41	1	2	yes	—
92	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
93	C₂²⋊D₂₃	—	2	4	no	—
94	C₂×Dic₂₃	—	2	1	no	—
95	C₂×Dic₂₃	—	2	1	no	—
96	C₂×Dic₂₃	—	2	1	no	—
97	C₂×Dic₂₃	C₂×D₄	1	2	yes	—
98	C₂×Dic₂₃	C₂×D₄	1	2	yes	—
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	C₈×C₂₃	—	2	1	no	—
106	C₈×C₂₃	—	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	—	4	2	no	—
118	C₄×C₂₃	C₄×D₄	1	2	yes	—
119	C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
120	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
121	Dic₂₃	—	2	2	no	—
122	Dic₂₃	—	2	2	no	—
123	Dic₂₃	—	2	2	no	—
124	Dic₂₃	—	4	1	no	—
125	Dic₂₃	—	4	1	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	—	23	1	no	—
134	#416	—	23	1	no	—
135	C₂×#103	D₂₃	1	1	yes	characteristic, complemented
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	C₂×D₂₃	1	1	yes	characteristic
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	—	23	1	no	—
158	#117	—	46	2	no	—
159	#117	—	46	1	no	—
160	C₂×C₄⋊C₄	—	46	1	no	—
161	C₂×C₄⋊C₄	C₂×D₂₃	1	1	yes	characteristic
162	#107	—	23	2	no	—
163	#106	—	23	2	no	—
164	#103	—	23	1	no	—
165	#103	—	23	1	no	—
166	#103	C₂×D₂₃	1	2	yes	—
167	#103	C₂×D₂₃	1	2	yes	—
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	1	no	—
176	C₂×Q₈	—	2	2	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	—	2	1	no	—
184	C₂²×C₄	—	46	2	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	—	23	4	no	—
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	—	23	2	no	—
192	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
193	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
194	C₄⋊C₄	—	2	1	no	—
195	C₄⋊C₄	—	23	1	no	—
196	C₄⋊C₄	—	23	1	no	—
197	C₄⋊C₄	—	23	1	no	—
198	C₄⋊C₄	—	23	1	no	—
199	C₄⋊C₄	—	46	2	no	—
200	C₄⋊C₄	—	46	2	no	—
201	C₄⋊C₄	—	46	2	no	—
202	C₄⋊C₄	C₂²×D₂₃	1	2	yes	—
203	C₂²⋊C₄	—	23	2	no	—
204	C₂²⋊C₄	—	46	2	no	—
205	C₂²⋊C₄	—	46	1	no	—
206	C₄²	—	23	2	no	—
207	C₄²	—	46	2	no	—
208	C₄²	—	46	2	no	—
209	C₄²	—	46	2	no	—
210	C₄²	—	46	1	no	—
211	C₂³	—	46	1	no	—
212	C₂³	D₄×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	2	no	—
224	C₂×C₄	—	8	1	no	—
225	C₂×C₄	—	23	2	no	—
226	C₂×C₄	—	23	2	no	—
227	C₂×C₄	—	46	1	no	—
228	C₂×C₄	—	46	4	no	—
229	C₂×C₄	—	46	1	no	—
230	C₂×C₄	—	46	1	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	#67290	1	2	yes	—
235	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
236	C₈	—	2	1	no	—
237	C₈	—	2	1	no	—
238	C₈	—	23	4	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	(C₂²⋊D₂₃)⋊C₄	1	2	yes	—
245	C₂²	C₂×D₄×D₂₃	1	1	yes	characteristic
246	C₄	—	2	4	no	—
247	C₄	—	4	2	no	—
248	C₄	—	46	2	no	—
249	C₄	—	46	2	no	—
250	C₄	—	46	2	no	—
251	C₄	—	92	1	no	—
252	C₄	—	92	1	no	—
253	C₄	(C₂²⋊D₂₃)⋊C₄	1	2	yes	—
254	C₄	C₂²×C₄×D₂₃	1	1	yes	characteristic
255	C₄	C₂×#67290	1	1	yes	characteristic
256	C₂	—	2	2	no	—
257	C₂	—	92	2	no	—
258	C₂	C₂×(C₂²⋊D₂₃)⋊C₄	1	1	yes	characteristic
259	C₂	#13093223	1	1	yes	characteristic
260	C₂	#13093230	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₂₃
commutator	C₄×C₂₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₂₃
Fitting	C₂×C₂₃×#103	C₂
socle	C₂²×C₂₃	C₂²×D₄
prime radical	C₂×#67288	C₂²
cosocle	C₂×C₄×C₂₃	C₂⁴
hypercenter	C₂×#103	D₂₃
hypocenter	C₂₃	#2947
LCS	G ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#103

Complements

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

N	H ≅ G/N	type
3. #13092895	C₂	semidirect
4. #13092521	C₂	semidirect
5. #13092520	C₂	semidirect
6. #13092509	C₂	semidirect
7. #13092502	C₂	semidirect
8. #13092457	C₂	semidirect
10. #13092372	C₂	semidirect
11. C₂×C₂₃×#103	C₂	semidirect
21. #10625113	C₄	semidirect
44. C₂×Q₈×C₂₃	C₂×C₄	semidirect
63. #67275	C₂×C₄	semidirect
64. #67274	C₂×C₄	semidirect
78. #67267	C₂×C₄	semidirect
135. C₂×#103	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
#13092898	2
#13092895	2
#13092521	2
#13092520	2
#13092509	2
#13092502	2
#13092457	2
#13092414	2
#13092372	2
C₂×C₂₃×#103	2
#2947	23