Subgroups — #15683559

Order 2944.

386 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13092974	C₂	1	1	yes	maximal, characteristic, complemented
2	#13092758	C₂	1	2	yes	maximal, complemented
3	#13092712	C₂	1	1	yes	maximal, characteristic, complemented
4	#13092707	C₂	1	1	yes	maximal, characteristic, complemented
5	#13092627	C₂	1	1	yes	maximal, characteristic, complemented
6	#13092622	C₂	1	1	yes	maximal, characteristic, complemented
7	#13092579	C₂	1	2	yes	maximal
8	#13092522	C₂	1	2	yes	maximal, complemented
9	#13092517	C₂	1	2	yes	maximal, complemented
10	#13092474	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#41×C₂₃	C₂	1	1	yes	maximal, characteristic
12	C₂×#41×C₂₃	C₂²	1	1	yes	characteristic
13	C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
14	C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
15	C₄×D₄×C₂₃	—	2	1	no	—
16	C₄×D₄×C₂₃	—	2	1	no	—
17	C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
18	C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
19	C₂₃×#117	—	2	1	no	—
20	C₂₃×#117	C₂²	1	1	yes	characteristic
21	C₂×C₄²×C₂₃	—	2	1	no	—
22	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
23	C₂×#67275	C₂²	1	2	yes	—
24	C₂×#67273	C₂²	1	2	yes	—
25	#10625097	—	2	1	no	—
26	#10625095	—	2	1	no	—
27	#10625095	C₂²	1	2	yes	—
28	#10625041	—	2	1	no	—
29	#10625040	C₂²	1	1	yes	characteristic
30	#10625040	C₂²	1	1	yes	characteristic
31	C₄×Dic₄₆	—	2	1	no	—
32	C₄×Dic₄₆	C₂²	1	2	yes	—
33	#10625017	C₂²	1	2	yes	—
34	#10625014	C₂²	1	2	yes	—
35	C₂²⋊(C₂₃⋊C₈)	C₂²	1	2	yes	—
36	#10624992	C₂²	1	2	yes	—
37	#10624992	C₂²	1	2	yes	—
38	#10624990	C₂²	1	2	yes	—
39	#10624989	C₂²	1	2	yes	—
40	C₄⋊(C₂₃⋊C₈)	C₂²	1	1	yes	characteristic
41	C₄⋊(C₂₃⋊C₈)	C₂²	1	1	yes	characteristic
42	#10624985	C₂²	1	2	yes	—
43	#41×C₂₃	—	2	2	no	—
44	#41×C₂₃	—	2	2	no	—
45	C₂×Q₈×C₂₃	C₂³	1	1	yes	characteristic
46	C₂×D₄×C₂₃	—	2	1	no	—
47	C₂×D₄×C₂₃	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₂₃	—	2	1	no	—
49	C₂²×C₄×C₂₃	—	2	1	no	—
50	C₂²×C₄×C₂₃	—	2	1	no	—
51	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
52	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
53	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
54	C₂×Dic₄₆	—	2	1	no	—
55	C₂×Dic₄₆	C₂³	1	2	yes	—
56	C₄⋊C₄×C₂₃	—	2	1	no	—
57	C₄⋊C₄×C₂₃	—	2	1	no	—
58	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
59	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
60	C₂²⋊C₄×C₂₃	—	2	1	no	—
61	C₂²⋊C₄×C₂₃	—	2	1	no	—
62	C₂²⋊C₄×C₂₃	—	2	1	no	—
63	C₄²×C₂₃	—	2	1	no	—
64	C₄²×C₂₃	—	2	1	no	—
65	C₄²×C₂₃	—	2	1	no	—
66	C₄²×C₂₃	D₄	1	1	yes	characteristic
67	C₄²×C₂₃	D₄	1	1	yes	characteristic
68	C₄²×C₂₃	C₂³	1	1	yes	characteristic
69	C₄²×C₂₃	C₂³	1	1	yes	characteristic
70	C₂²⋊Dic₂₃	—	2	2	no	—
71	#67275	—	2	4	no	—
72	#67273	—	2	4	no	—
73	C₄⋊Dic₂₃	—	2	1	no	—
74	C₄⋊Dic₂₃	—	2	1	no	—
75	C₄⋊Dic₂₃	—	2	1	no	—
76	C₄⋊Dic₂₃	C₂³	1	2	yes	—
77	Dic₂₃⋊C₄	—	2	2	no	—
78	Dic₂₃⋊C₄	—	2	2	no	—
79	C₄×Dic₂₃	—	2	2	no	—
80	C₂×C₂₃⋊C₈	C₂³	1	2	yes	—
81	C₂×C₂₃⋊C₈	C₂³	1	2	yes	—
82	C₂³×C₂₃	—	2	1	no	—
83	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
84	Q₈×C₂₃	—	2	1	no	—
85	Q₈×C₂₃	#41	1	2	yes	—
86	D₄×C₂₃	—	2	4	no	—
87	D₄×C₂₃	—	2	1	no	—
88	D₄×C₂₃	#41	1	2	yes	—
89	C₂×C₄×C₂₃	—	2	4	no	—
90	C₂×C₄×C₂₃	—	2	1	no	—
91	C₂×C₄×C₂₃	—	2	2	no	—
92	C₂×C₄×C₂₃	—	2	1	no	—
93	C₂×C₄×C₂₃	—	2	1	no	—
94	C₂×C₄×C₂₃	—	2	1	no	—
95	C₂×C₄×C₂₃	—	2	1	no	—
96	C₂×C₄×C₂₃	—	4	1	no	—
97	C₂×C₄×C₂₃	—	4	2	no	—
98	C₂×C₄×C₂₃	—	4	2	no	—
99	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
100	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
101	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
102	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
104	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
105	C₂×Dic₂₃	—	2	2	no	—
106	C₂×Dic₂₃	—	2	2	no	—
107	Dic₄₆	—	2	2	no	—
108	Dic₄₆	—	2	2	no	—
109	Dic₄₆	—	4	2	no	—
110	C₂₃⋊C₈	—	2	2	no	—
111	C₂₃⋊C₈	—	2	2	no	—
112	#3116	—	23	1	no	maximal, Sylow
113	C₂²×C₂₃	—	2	1	no	—
114	C₂²×C₂₃	—	2	2	no	—
115	C₂²×C₂₃	—	2	1	no	—
116	C₂²×C₂₃	—	2	1	no	—
117	C₂²×C₂₃	—	4	1	no	—
118	C₂²×C₂₃	—	4	1	no	—
119	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
120	C₄×C₂₃	—	2	1	no	—
121	C₄×C₂₃	—	2	2	no	—
122	C₄×C₂₃	—	2	1	no	—
123	C₄×C₂₃	—	2	1	no	—
124	C₄×C₂₃	—	2	1	no	—
125	C₄×C₂₃	—	4	1	no	—
126	C₄×C₂₃	—	4	1	no	—
127	C₄×C₂₃	#137	1	2	yes	—
128	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
129	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
130	Dic₂₃	—	4	2	no	—
131	Dic₂₃	—	4	2	no	—
132	#533	—	23	1	no	—
133	C₄×#41	D₂₃	1	1	yes	characteristic
134	#485	—	23	2	no	—
135	#477	—	23	1	no	—
136	#476	—	23	1	no	—
137	#462	—	23	1	no	—
138	#461	—	23	1	no	—
139	#452	—	23	2	no	—
140	#441	—	23	2	no	—
141	#440	—	23	2	no	—
142	#432	—	23	1	no	—
143	C₂×C₂₃	—	4	1	no	—
144	C₂×C₂₃	—	4	2	no	—
145	C₂×C₂₃	#525	1	1	yes	characteristic
146	C₂×C₂₃	C₂×#137	1	1	yes	characteristic
147	C₂×C₂₃	#575	1	1	yes	characteristic
148	C₂×#41	C₂×D₂₃	1	1	yes	characteristic
149	C₂×Q₁₆	—	23	2	no	—
150	C₂×SD₁₆	—	23	2	no	—
151	C₄⋊Q₈	—	23	1	no	—
152	C₄⋊Q₈	—	23	1	no	—
153	#125	—	46	1	no	—
154	C₂²⋊Q₈	—	23	2	no	—
155	C₂²⋊Q₈	—	46	1	no	—
156	C₄×Q₈	—	23	2	no	—
157	C₄×Q₈	—	46	1	no	—
158	C₄×Q₈	C₂×D₂₃	1	1	yes	characteristic
159	C₄×Q₈	C₂×D₂₃	1	1	yes	characteristic
160	C₄×D₄	—	2	1	no	—
161	C₄×D₄	—	2	1	no	—
162	C₄×D₄	C₂×D₂₃	1	1	yes	characteristic
163	C₄×D₄	C₂×D₂₃	1	1	yes	characteristic
164	#117	—	2	1	no	—
165	#117	—	46	1	no	—
166	#117	C₂×D₂₃	1	1	yes	characteristic
167	C₂×C₄²	—	2	1	no	—
168	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
169	#107	—	23	2	no	—
170	#106	—	23	2	no	—
171	C₄⋊C₈	—	23	1	no	—
172	C₄⋊C₈	—	23	1	no	—
173	#103	—	23	2	no	—
174	#103	—	23	2	no	—
175	#103	—	23	2	no	—
176	#102	—	23	2	no	—
177	C₂²⋊C₈	—	23	2	no	—
178	#97	—	23	2	no	—
179	C₂₃	#3116	1	1	yes	characteristic, complemented, Sylow
180	#41	—	2	2	no	—
181	#41	—	2	2	no	—
182	C₂×Q₈	—	23	2	no	—
183	C₂×Q₈	—	46	1	no	—
184	C₂×Q₈	C₂²×D₂₃	1	1	yes	characteristic
185	C₂×D₄	—	2	1	no	—
186	C₂×D₄	C₂²×D₂₃	1	1	yes	characteristic
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	—	2	1	no	—
189	C₂²×C₄	—	2	1	no	—
190	C₂²×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
191	C₂²×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
192	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
193	Q₁₆	—	46	4	no	—
194	SD₁₆	—	46	4	no	—
195	C₂×C₈	—	23	2	no	—
196	C₂×C₈	—	23	2	no	—
197	C₄⋊C₄	—	2	1	no	—
198	C₄⋊C₄	—	2	1	no	—
199	C₄⋊C₄	—	23	2	no	—
200	C₄⋊C₄	—	46	1	no	—
201	C₄⋊C₄	—	46	1	no	—
202	C₄⋊C₄	—	46	2	no	—
203	C₄⋊C₄	—	46	2	no	—
204	C₄⋊C₄	—	46	1	no	—
205	C₄⋊C₄	C₂²×D₂₃	1	1	yes	characteristic
206	C₄⋊C₄	C₂²×D₂₃	1	1	yes	characteristic
207	C₂²⋊C₄	—	2	1	no	—
208	C₂²⋊C₄	—	2	1	no	—
209	C₂²⋊C₄	—	2	1	no	—
210	C₂²⋊C₄	—	46	2	no	—
211	C₄²	—	2	1	no	—
212	C₄²	—	2	1	no	—
213	C₄²	—	2	1	no	—
214	C₄²	—	46	2	no	—
215	C₄²	C₂²⋊D₂₃	1	1	yes	characteristic
216	C₄²	C₂²⋊D₂₃	1	1	yes	characteristic
217	C₄²	C₂²×D₂₃	1	1	yes	characteristic
218	C₄²	C₂²×D₂₃	1	1	yes	characteristic
219	C₂³	—	2	1	no	—
220	C₂³	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
221	Q₈	—	2	1	no	—
222	Q₈	—	46	2	no	—
223	Q₈	—	46	2	no	—
224	Q₈	—	92	2	no	—
225	Q₈	#67288	1	2	yes	—
226	D₄	—	2	4	no	—
227	D₄	—	2	1	no	—
228	D₄	#67288	1	2	yes	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	4	1	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	46	2	no	—
240	C₂×C₄	—	46	2	no	—
241	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
242	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
243	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
244	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
245	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
246	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
247	C₈	—	46	2	no	—
248	C₈	—	46	2	no	—
249	C₂²	—	2	1	no	—
250	C₂²	—	2	2	no	—
251	C₂²	—	2	1	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	4	1	no	—
254	C₂²	—	4	1	no	—
255	C₂²	C₂²×C₂²⋊D₂₃	1	1	yes	characteristic
256	C₄	—	2	1	no	—
257	C₄	—	2	2	no	—
258	C₄	—	2	1	no	—
259	C₄	—	2	1	no	—
260	C₄	—	2	1	no	—
261	C₄	—	4	1	no	—
262	C₄	—	4	1	no	—
263	C₄	—	92	2	no	—
264	C₄	—	92	2	no	—
265	C₄	#10625122	1	2	yes	—
266	C₄	C₂×#67288	1	1	yes	characteristic
267	C₄	C₂×#67288	1	1	yes	characteristic
268	C₂	—	4	1	no	—
269	C₂	—	4	2	no	—
270	C₂	#13092928	1	1	yes	characteristic
271	C₂	C₂×#10625122	1	1	yes	characteristic
272	C₂	#13093215	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₂₃
commutator, cosocle	C₂×C₄×C₂₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₃
Fitting	C₄×#41×C₂₃	C₂
socle	C₂²×C₂₃	C₂²×D₄
prime radical	C₂×#41×C₂₃	C₂²
hypercenter	C₄×#41	D₂₃
hypocenter	C₂₃	#3116
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
1. #13092974	C₂	semidirect
2. #13092758	C₂	semidirect
3. #13092712	C₂	semidirect
4. #13092707	C₂	semidirect
5. #13092627	C₂	semidirect
6. #13092622	C₂	semidirect
8. #13092522	C₂	semidirect
9. #13092517	C₂	semidirect
10. #13092474	C₂	semidirect
179. C₂₃	#3116	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#13092974	2
#13092758	2
#13092712	2
#13092707	2
#13092627	2
#13092622	2
#13092579	2
#13092522	2
#13092517	2
#13092474	2
C₄×#41×C₂₃	2
#3116	23