Subgroups — #15683693

Order 2944.

358 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13093027	C₂	1	1	yes	maximal, characteristic
2	#13092894	C₂	1	1	yes	maximal, characteristic
3	#13092758	C₂	1	1	yes	maximal, characteristic
4	#13092727	C₂	1	1	yes	maximal, characteristic, complemented
5	#13092723	C₂	1	1	yes	maximal, characteristic, complemented
6	#13092626	C₂	1	1	yes	maximal, characteristic, complemented
7	#13092626	C₂	1	1	yes	maximal, characteristic, complemented
8	#13092575	C₂	1	1	yes	maximal, characteristic
9	#13092572	C₂	1	1	yes	maximal, characteristic
10	#13092523	C₂	1	1	yes	maximal, characteristic, complemented
11	#13092521	C₂	1	1	yes	maximal, characteristic, complemented
12	#13092511	C₂	1	1	yes	maximal, characteristic, complemented
13	#13092509	C₂	1	1	yes	maximal, characteristic, complemented
14	#13092475	C₂	1	1	yes	maximal, characteristic
15	C₂₃×#485	C₂	1	1	yes	maximal, characteristic
16	C₂²×Dic₄₆	C₂²	1	1	yes	characteristic
17	C₂₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
18	C₂₃×#117	C₂²	1	1	yes	characteristic
19	Q₈×Dic₂₃	C₂²	1	1	yes	characteristic
20	#10625115	C₂²	1	1	yes	characteristic
21	C₂×#67275	C₂²	1	1	yes	characteristic
22	C₂×#67275	C₂²	1	1	yes	characteristic
23	#10625097	C₂²	1	1	yes	characteristic
24	#10625095	C₂²	1	1	yes	characteristic
25	C₂×Dic₉₂	C₂²	1	1	yes	characteristic
26	C₂×Dic₉₂	C₂²	1	1	yes	characteristic
27	#10625061	—	2	1	no	—
28	#10625061	C₂²	1	1	yes	characteristic
29	#10625060	—	2	1	no	—
30	#10625058	C₂²	1	1	yes	characteristic
31	#10625058	C₂²	1	1	yes	characteristic
32	#10625058	C₂²	1	1	yes	characteristic
33	#10625058	C₂²	1	1	yes	characteristic
34	#10625058	C₂²	1	1	yes	characteristic
35	#10625050	—	2	1	no	—
36	#10625050	—	2	1	no	—
37	#10625048	—	2	1	no	—
38	#10625048	—	2	1	no	—
39	C₄×Dic₄₆	—	2	1	no	—
40	C₄×Dic₄₆	—	2	1	no	—
41	C₂₃×#107	C₂²	1	1	yes	characteristic
42	C₂₃×#106	C₂²	1	1	yes	characteristic
43	C₂₃×#103	C₂²	1	1	yes	characteristic
44	C₂₃×#103	C₂²	1	1	yes	characteristic
45	C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
46	C₂²⋊(C₂₃⋊C₈)	C₂²	1	1	yes	characteristic
47	#10624998	C₂²	1	1	yes	characteristic
48	#10624998	C₂²	1	1	yes	characteristic
49	#10624997	C₂²	1	1	yes	characteristic
50	Dic₂₃⋊C₈	C₂²	1	1	yes	characteristic
51	Dic₂₃⋊C₈	C₂²	1	1	yes	characteristic
52	C₈×Dic₂₃	C₂²	1	1	yes	characteristic
53	#10624992	C₂²	1	1	yes	characteristic
54	#10624992	C₂²	1	1	yes	characteristic
55	#10624992	C₂²	1	1	yes	characteristic
56	#10624992	C₂²	1	1	yes	characteristic
57	#10624990	C₂²	1	1	yes	characteristic
58	#10624989	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₂₃	C₂³	1	1	yes	characteristic
60	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
61	C₂²×Dic₂₃	—	2	1	no	—
62	C₂×Dic₄₆	—	2	1	no	—
63	C₂×Dic₄₆	—	2	2	no	—
64	C₂×Dic₄₆	—	2	2	no	—
65	C₂×Dic₄₆	C₂³	1	1	yes	characteristic
66	C₂×Dic₄₆	C₂³	1	1	yes	characteristic
67	C₂×Dic₄₆	C₂³	1	1	yes	characteristic
68	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
69	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
70	C₄⋊C₄×C₂₃	—	2	1	no	—
71	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
72	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
73	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
74	C₂²⋊C₄×C₂₃	—	2	1	no	—
75	C₂²⋊C₄×C₂₃	—	2	1	no	—
76	C₄²×C₂₃	—	2	1	no	—
77	C₂²⋊Dic₂₃	—	2	1	no	—
78	C₂²⋊Dic₂₃	D₄	1	1	yes	characteristic
79	C₂²⋊Dic₂₃	D₄	1	1	yes	characteristic
80	#67275	—	2	2	no	—
81	#67275	—	2	2	no	—
82	C₄⋊Dic₂₃	—	2	1	no	—
83	C₄⋊Dic₂₃	—	2	1	no	—
84	C₄⋊Dic₂₃	D₄	1	1	yes	characteristic
85	C₄⋊Dic₂₃	D₄	1	1	yes	characteristic
86	C₄⋊Dic₂₃	C₂³	1	1	yes	characteristic
87	Dic₂₃⋊C₄	—	2	1	no	—
88	Dic₂₃⋊C₄	—	2	1	no	—
89	Dic₂₃⋊C₄	—	2	1	no	—
90	Dic₂₃⋊C₄	—	2	1	no	—
91	Dic₂₃⋊C₄	—	2	1	no	—
92	Dic₂₃⋊C₄	—	2	1	no	—
93	Dic₂₃⋊C₄	—	2	1	no	—
94	Dic₂₃⋊C₄	—	2	1	no	—
95	Dic₂₃⋊C₄	—	2	1	no	—
96	C₄×Dic₂₃	—	2	1	no	—
97	C₄×Dic₂₃	—	2	1	no	—
98	C₄×Dic₂₃	—	2	1	no	—
99	C₄×Dic₂₃	—	2	1	no	—
100	C₄×Dic₂₃	—	2	1	no	—
101	C₄×Dic₂₃	—	2	1	no	—
102	C₄×Dic₂₃	C₂³	1	1	yes	characteristic
103	C₄×Dic₂₃	C₂³	1	1	yes	characteristic
104	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
105	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
106	Dic₉₂	—	2	2	no	—
107	Dic₉₂	—	2	2	no	—
108	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
109	Q₈×C₂₃	—	2	1	no	—
110	Q₈×C₂₃	—	2	1	no	—
111	C₂×C₄×C₂₃	—	2	1	no	—
112	C₂×C₄×C₂₃	—	2	1	no	—
113	C₂×C₄×C₂₃	—	2	1	no	—
114	C₂×C₄×C₂₃	—	2	1	no	—
115	C₂×C₄×C₂₃	—	2	1	no	—
116	C₂×C₄×C₂₃	—	2	1	no	—
117	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
118	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
119	C₂×Dic₂₃	—	2	4	no	—
120	C₂×Dic₂₃	—	2	1	no	—
121	C₂×Dic₂₃	—	2	1	no	—
122	C₂×Dic₂₃	—	2	1	no	—
123	C₂×Dic₂₃	—	2	1	no	—
124	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
125	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
126	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
127	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
128	Dic₄₆	—	2	4	no	—
129	Dic₄₆	—	2	1	no	—
130	Dic₄₆	—	2	1	no	—
131	Dic₄₆	—	2	1	no	—
132	Dic₄₆	—	2	1	no	—
133	Dic₄₆	#41	1	2	yes	—
134	Dic₄₆	#41	1	2	yes	—
135	C₈×C₂₃	—	2	1	no	—
136	C₈×C₂₃	—	2	1	no	—
137	C₂₃⋊C₈	—	2	1	no	—
138	C₂₃⋊C₈	—	2	1	no	—
139	#3127	—	23	1	no	maximal, Sylow
140	C₂²×C₂₃	—	2	1	no	—
141	C₂²×C₂₃	—	2	1	no	—
142	C₂²×C₂₃	—	2	1	no	—
143	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
144	C₄×C₂₃	—	2	1	no	—
145	C₄×C₂₃	—	4	1	no	—
146	C₄×C₂₃	—	4	1	no	—
147	C₄×C₂₃	—	4	1	no	—
148	C₄×C₂₃	—	4	1	no	—
149	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
150	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
151	Dic₂₃	—	2	2	no	—
152	Dic₂₃	—	2	2	no	—
153	Dic₂₃	—	2	1	no	—
154	Dic₂₃	—	2	1	no	—
155	Dic₂₃	—	2	1	no	—
156	Dic₂₃	—	2	1	no	—
157	Dic₂₃	—	4	1	no	—
158	Dic₂₃	—	4	1	no	—
159	#541	—	23	1	no	—
160	#519	—	23	1	no	—
161	#485	—	23	1	no	—
162	#485	D₂₃	1	1	yes	characteristic
163	#479	—	23	1	no	—
164	#479	—	23	1	no	—
165	#462	—	23	1	no	—
166	#462	—	23	1	no	—
167	#451	—	23	1	no	—
168	#451	—	23	1	no	—
169	#441	—	23	1	no	—
170	#441	—	23	1	no	—
171	C₄×Q₁₆	—	23	1	no	—
172	C₄×Q₁₆	—	23	1	no	—
173	#433	—	23	1	no	—
174	C₂×C₂₃	—	4	1	no	—
175	C₂×C₂₃	#525	1	1	yes	characteristic
176	C₂×C₂₃	#578	1	1	yes	characteristic
177	C₂×C₂₃	#578	1	1	yes	characteristic
178	C₂²×Q₈	—	23	1	no	—
179	C₂×Q₁₆	—	23	1	no	—
180	C₂×Q₁₆	—	23	1	no	—
181	C₂×Q₁₆	—	23	1	no	—
182	C₂×Q₁₆	—	23	1	no	—
183	C₄⋊Q₈	—	23	1	no	—
184	#126	—	46	1	no	—
185	#126	—	46	1	no	—
186	#125	—	23	1	no	—
187	#125	—	46	1	no	—
188	#125	—	46	1	no	—
189	C₂²⋊Q₈	—	23	1	no	—
190	C₂²⋊Q₈	C₂×D₂₃	1	1	yes	characteristic
191	C₄×Q₈	—	23	1	no	—
192	C₄×Q₈	—	23	1	no	—
193	C₄×Q₈	—	23	1	no	—
194	C₄×Q₈	—	23	1	no	—
195	C₄×Q₈	—	23	1	no	—
196	C₄×Q₈	—	23	1	no	—
197	C₄×Q₈	—	46	1	no	—
198	C₄×Q₈	—	46	1	no	—
199	#117	—	23	1	no	—
200	#117	—	46	1	no	—
201	#117	—	46	1	no	—
202	#117	C₂×D₂₃	1	1	yes	characteristic
203	#107	—	23	1	no	—
204	#107	C₂×D₂₃	1	1	yes	characteristic
205	#106	—	23	1	no	—
206	#106	C₂×D₂₃	1	1	yes	characteristic
207	C₄⋊C₈	—	23	1	no	—
208	C₄⋊C₈	—	23	1	no	—
209	#103	—	23	1	no	—
210	#103	—	23	1	no	—
211	#103	—	23	1	no	—
212	#103	—	23	1	no	—
213	#103	—	23	1	no	—
214	#103	—	23	1	no	—
215	#103	C₂×D₂₃	1	1	yes	characteristic
216	#103	C₂×D₂₃	1	1	yes	characteristic
217	C₂²⋊C₈	—	23	1	no	—
218	C₂²⋊C₈	C₂×D₂₃	1	1	yes	characteristic
219	#97	—	23	1	no	—
220	C₄×C₈	—	23	1	no	—
221	C₂₃	#3127	1	1	yes	characteristic, complemented, Sylow
222	C₂×Q₈	—	23	1	no	—
223	C₂×Q₈	—	23	1	no	—
224	C₂×Q₈	—	23	1	no	—
225	C₂×Q₈	—	46	2	no	—
226	C₂×Q₈	—	46	2	no	—
227	C₂×Q₈	—	46	1	no	—
228	C₂×Q₈	C₂²×D₂₃	1	1	yes	characteristic
229	C₂²×C₄	—	46	1	no	—
230	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
231	Q₁₆	—	46	2	no	—
232	Q₁₆	—	46	2	no	—
233	Q₁₆	—	46	2	no	—
234	Q₁₆	—	46	2	no	—
235	C₂×C₈	—	23	1	no	—
236	C₂×C₈	—	23	1	no	—
237	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
238	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
239	C₄⋊C₄	—	2	1	no	—
240	C₄⋊C₄	—	23	1	no	—
241	C₄⋊C₄	—	23	1	no	—
242	C₄⋊C₄	—	23	1	no	—
243	C₄⋊C₄	—	46	1	no	—
244	C₄⋊C₄	—	46	1	no	—
245	C₄⋊C₄	—	46	1	no	—
246	C₄⋊C₄	—	46	1	no	—
247	C₄⋊C₄	—	46	1	no	—
248	C₄⋊C₄	—	46	1	no	—
249	C₄⋊C₄	—	46	1	no	—
250	C₄⋊C₄	—	46	1	no	—
251	C₄⋊C₄	—	46	1	no	—
252	C₄⋊C₄	—	46	1	no	—
253	C₄⋊C₄	—	46	1	no	—
254	C₄⋊C₄	C₂²×D₂₃	1	1	yes	characteristic
255	C₄⋊C₄	C₂²×D₂₃	1	1	yes	characteristic
256	C₄⋊C₄	C₂²×D₂₃	1	1	yes	characteristic
257	C₂²⋊C₄	—	2	1	no	—
258	C₂²⋊C₄	—	2	1	no	—
259	C₂²⋊C₄	—	23	1	no	—
260	C₂²⋊C₄	—	23	1	no	—
261	C₂²⋊C₄	—	46	1	no	—
262	C₄²	—	2	1	no	—
263	C₄²	—	23	1	no	—
264	C₄²	—	23	1	no	—
265	C₄²	—	46	1	no	—
266	C₄²	—	46	1	no	—
267	C₄²	—	46	1	no	—
268	C₄²	—	46	1	no	—
269	C₄²	—	46	1	no	—
270	C₄²	—	46	1	no	—
271	C₂³	D₄×D₂₃	1	1	yes	characteristic
272	Q₈	—	2	1	no	—
273	Q₈	—	2	1	no	—
274	Q₈	—	23	2	no	—
275	Q₈	—	23	2	no	—
276	Q₈	—	46	4	no	—
277	Q₈	—	46	1	no	—
278	Q₈	—	46	1	no	—
279	Q₈	—	46	1	no	—
280	Q₈	—	46	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	23	1	no	—
288	C₂×C₄	—	23	1	no	—
289	C₂×C₄	—	23	1	no	—
290	C₂×C₄	—	23	1	no	—
291	C₂×C₄	—	46	4	no	—
292	C₂×C₄	—	46	1	no	—
293	C₂×C₄	—	46	1	no	—
294	C₂×C₄	—	46	1	no	—
295	C₂×C₄	—	46	1	no	—
296	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
297	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
298	C₈	—	2	1	no	—
299	C₈	—	2	1	no	—
300	C₈	—	46	1	no	—
301	C₈	—	46	1	no	—
302	C₂²	—	2	1	no	—
303	C₂²	—	2	1	no	—
304	C₂²	—	2	1	no	—
305	C₂²	C₂×D₄×D₂₃	1	1	yes	characteristic
306	C₄	—	2	1	no	—
307	C₄	—	4	1	no	—
308	C₄	—	4	1	no	—
309	C₄	—	4	1	no	—
310	C₄	—	4	1	no	—
311	C₄	—	46	2	no	—
312	C₄	—	46	2	no	—
313	C₄	—	46	1	no	—
314	C₄	—	46	1	no	—
315	C₄	—	46	1	no	—
316	C₄	—	46	1	no	—
317	C₄	—	92	1	no	—
318	C₄	—	92	1	no	—
319	C₄	#41×D₂₃	1	1	yes	characteristic
320	C₄	#41×D₂₃	1	1	yes	characteristic
321	C₂	—	4	1	no	—
322	C₂	#13092925	1	1	yes	characteristic
323	C₂	#13093231	1	1	yes	characteristic
324	C₂	#13093233	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₂₃
commutator, cosocle	C₂×C₄×C₂₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₃
Fitting	C₂₃×#485	C₂
socle	C₂²×C₂₃	C₂²×D₄
prime radical	C₂³×C₂₃	C₂×D₄
hypercenter	#485	D₂₃
hypocenter	C₂₃	#3127
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #485

Complements

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

N	H ≅ G/N	type
4. #13092727	C₂	semidirect
5. #13092723	C₂	semidirect
6. #13092626	C₂	semidirect
7. #13092626	C₂	semidirect
10. #13092523	C₂	semidirect
11. #13092521	C₂	semidirect
12. #13092511	C₂	semidirect
13. #13092509	C₂	semidirect
221. C₂₃	#3127	semidirect

Maximal subgroups

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

subgroup	index
#13093027	2
#13092894	2
#13092758	2
#13092727	2
#13092723	2
#13092626	2
#13092575	2
#13092572	2
#13092523	2
#13092521	2
#13092511	2
#13092509	2
#13092475	2
C₂₃×#485	2
#3127	23