Subgroups — #15682312

Order 2944.

452 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10625121	C₂	1	1	yes	maximal, characteristic
2	#13093183	C₂	1	1	yes	maximal, characteristic, complemented
3	#13092537	C₂	1	4	yes	maximal, complemented
4	#13092532	C₂	1	4	yes	maximal, complemented
5	C₂×#10625004	C₂	1	1	yes	maximal, characteristic, complemented
6	#13092434	C₂	1	1	yes	maximal, characteristic, complemented
7	#13092433	C₂	1	1	yes	maximal, characteristic, complemented
8	#13092375	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#568	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#41×C₂₃	C₂²	1	1	yes	characteristic
11	C₂×#67288	C₂²	1	1	yes	characteristic
12	C₂₃×#131	—	2	2	no	—
13	C₂₃×#131	C₂²	1	4	yes	complemented
14	C₂×M₁₆×C₂₃	—	2	2	no	—
15	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
16	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
17	#10625121	C₄	1	2	yes	complemented
18	#10625121	C₄	1	2	yes	complemented
19	#10625121	C₂²	1	2	yes	—
20	#10625121	C₂²	1	2	yes	—
21	C₂×#67275	C₄	1	1	yes	characteristic, complemented
22	C₂×#67274	C₄	1	1	yes	characteristic, complemented
23	C₂×#67273	C₄	1	1	yes	characteristic, complemented
24	C₂×#67272	C₄	1	1	yes	characteristic, complemented
25	#10625097	C₂²	1	1	yes	characteristic
26	C₂×#67267	C₂²	1	1	yes	characteristic
27	C₂²×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
28	#10625078	—	2	2	no	—
29	M₁₆×D₂₃	—	2	2	no	—
30	#10625072	C₂²	1	4	yes	complemented
31	#10625019	C₂²	1	4	yes	complemented
32	#10625007	C₂²	1	4	yes	complemented
33	#10625004	C₂²	1	2	yes	complemented
34	#10625004	C₂²	1	2	yes	complemented
35	#10624997	C₂²	1	2	yes	complemented
36	C₈×Dic₂₃	C₂²	1	2	yes	complemented
37	#41×C₂₃	—	2	1	no	—
38	#41×C₂₃	—	2	1	no	—
39	#41×C₂₃	C₂×C₄	1	2	yes	—
40	#41×C₂₃	C₂³	1	2	yes	—
41	C₂×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
42	C₂×D₄×C₂₃	—	2	1	no	—
43	C₂×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₂₃	—	2	1	no	—
45	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
46	C₂×C₂²⋊D₂₃	—	2	1	no	—
47	#67288	—	2	1	no	—
48	#67288	—	2	1	no	—
49	#67288	C₂×C₄	1	2	yes	—
50	#67288	C₂³	1	2	yes	—
51	C₂×D₉₂	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄×D₂₃	—	2	1	no	—
53	C₂×Dic₄₆	C₂×C₄	1	1	yes	characteristic
54	M₁₆×C₂₃	—	2	1	no	—
55	M₁₆×C₂₃	—	2	4	no	—
56	M₁₆×C₂₃	D₄	1	2	yes	—
57	M₁₆×C₂₃	D₄	1	2	yes	—
58	M₁₆×C₂₃	C₂³	1	2	yes	—
59	C₂×C₈×C₂₃	—	2	1	no	—
60	C₂×C₈×C₂₃	—	2	4	no	—
61	C₂×C₈×C₂₃	C₂³	1	2	yes	—
62	C₂²⋊Dic₂₃	—	2	1	no	—
63	#67275	C₂×C₄	1	2	yes	—
64	#67275	C₂×C₄	1	2	yes	—
65	#67274	C₂×C₄	1	2	yes	—
66	#67274	C₂×C₄	1	2	yes	—
67	#67273	C₂×C₄	1	2	yes	—
68	#67273	C₂×C₄	1	2	yes	—
69	#67272	C₂×C₄	1	2	yes	—
70	#67272	C₂×C₄	1	2	yes	—
71	C₄⋊Dic₂₃	—	2	1	no	—
72	C₄×Dic₂₃	C₂³	1	2	yes	—
73	#67267	—	2	1	no	—
74	#67267	C₂³	1	2	yes	—
75	C₂×C₂₃⋊C₈	—	2	1	no	—
76	C₂×C₂₃⋊C₈	C₂×C₄	1	1	yes	characteristic
77	C₂×C₂₃⋊C₈	C₂×C₄	1	1	yes	characteristic
78	C₂×C₂₃⋊C₈	C₂×C₄	1	1	yes	characteristic
79	C₂×C₂₃⋊C₈	C₂×C₄	1	1	yes	characteristic
80	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
81	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
82	#67262	—	2	4	no	—
83	C₈×D₂₃	—	2	4	no	—
84	C₂³×C₂₃	—	2	1	no	—
85	C₂³×C₂₃	#41	1	1	yes	characteristic
86	C₂²×D₂₃	—	2	1	no	—
87	Q₈×C₂₃	—	2	1	no	—
88	Q₈×C₂₃	C₂²×C₄	1	2	yes	—
89	D₄×C₂₃	—	2	2	no	—
90	D₄×C₂₃	—	2	2	no	—
91	D₄×C₂₃	—	2	1	no	—
92	D₄×C₂₃	C₂²×C₄	1	2	yes	—
93	C₂×C₄×C₂₃	—	2	1	no	—
94	C₂×C₄×C₂₃	—	2	2	no	—
95	C₂×C₄×C₂₃	—	2	2	no	—
96	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
99	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
100	C₂×C₄×C₂₃	#41	1	1	yes	characteristic
101	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
102	C₂²⋊D₂₃	—	2	2	no	—
103	C₂²⋊D₂₃	—	2	2	no	—
104	C₂×Dic₂₃	—	2	1	no	—
105	C₂×Dic₂₃	—	2	2	no	—
106	D₉₂	—	2	1	no	—
107	D₉₂	C₂²×C₄	1	2	yes	—
108	C₄×D₂₃	—	2	2	no	—
109	C₄×D₂₃	—	2	2	no	—
110	Dic₄₆	—	2	1	no	—
111	Dic₄₆	C₂²×C₄	1	2	yes	—
112	C₈×C₂₃	—	2	2	no	—
113	C₈×C₂₃	C₂×D₄	1	4	yes	—
114	C₂₃⋊C₈	—	2	2	no	—
115	C₂₃⋊C₈	C₂²×C₄	1	1	yes	characteristic
116	C₂₃⋊C₈	C₂²×C₄	1	1	yes	characteristic
117	C₂₃⋊C₈	C₂²×C₄	1	1	yes	characteristic
118	C₂₃⋊C₈	C₂²×C₄	1	1	yes	characteristic
119	#2957	—	23	1	no	maximal, Sylow
120	C₂²×C₂₃	—	2	2	no	—
121	C₂²×C₂₃	—	4	1	no	—
122	C₂²×C₂₃	—	8	1	no	—
123	C₂²×C₂₃	C₄×D₄	1	1	yes	characteristic
124	C₂²×C₂₃	C₄×D₄	1	1	yes	characteristic
125	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
126	C₂×D₂₃	—	2	2	no	—
127	C₂×D₂₃	—	8	1	no	—
128	C₄×C₂₃	—	2	2	no	—
129	C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
130	C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
131	C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
132	C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
133	Dic₂₃	—	2	2	no	—
134	Dic₂₃	—	4	2	no	—
135	C₂×#135	—	23	1	no	—
136	#568	—	23	1	no	—
137	#568	D₂₃	1	1	yes	characteristic, complemented
138	#444	—	23	4	no	—
139	#443	—	23	4	no	—
140	C₂×#108	—	23	1	no	—
141	#421	—	23	1	no	—
142	#421	—	23	1	no	—
143	#405	—	23	1	no	—
144	C₂×C₂₃	—	2	1	no	—
145	C₂×C₂₃	—	2	1	no	—
146	C₂×C₂₃	—	2	1	no	—
147	C₂×C₂₃	—	4	2	no	—
148	C₂×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
149	D₂₃	—	4	2	no	—
150	C₂×#41	—	23	1	no	—
151	C₂×#41	C₂×D₂₃	1	1	yes	characteristic
152	#135	—	23	2	no	—
153	#135	—	23	2	no	—
154	#135	—	23	2	no	—
155	#135	—	23	2	no	—
156	C₂×Q₁₆	—	23	1	no	—
157	C₂×SD₁₆	—	23	1	no	—
158	C₂×SD₁₆	—	23	1	no	—
159	C₂×D₈	—	23	1	no	—
160	#131	—	2	2	no	—
161	#131	—	23	4	no	—
162	#131	—	46	2	no	—
163	#131	C₂×D₂₃	1	4	yes	complemented
164	C₂×M₁₆	—	2	2	no	—
165	C₂×M₁₆	—	23	1	no	—
166	C₂×M₁₆	—	46	2	no	—
167	C₂×M₁₆	C₂×D₂₃	1	1	yes	characteristic
168	C₂×M₁₆	C₂×D₂₃	1	1	yes	characteristic
169	C₂²×C₈	—	23	1	no	—
170	#117	—	23	1	no	—
171	#108	—	23	2	no	—
172	#108	—	23	2	no	—
173	#104	—	23	4	no	—
174	#104	—	23	4	no	—
175	#97	—	23	2	no	—
176	C₄×C₈	—	23	2	no	—
177	C₂₃	#2957	1	1	yes	characteristic, complemented, Sylow
178	#41	—	2	1	no	—
179	#41	—	2	1	no	—
180	#41	—	23	2	no	—
181	#41	—	23	2	no	—
182	#41	—	46	1	no	—
183	#41	—	46	1	no	—
184	#41	C₄×D₂₃	1	2	yes	—
185	#41	C₂²×D₂₃	1	2	yes	—
186	C₂×Q₈	—	23	1	no	—
187	C₂×Q₈	C₄×D₂₃	1	1	yes	characteristic
188	C₂×D₄	—	2	1	no	—
189	C₂×D₄	—	23	1	no	—
190	C₂×D₄	—	46	1	no	—
191	C₂×D₄	C₄×D₂₃	1	1	yes	characteristic
192	C₂²×C₄	—	2	1	no	—
193	C₂²×C₄	—	46	1	no	—
194	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
195	Q₁₆	—	23	2	no	—
196	Q₁₆	—	23	2	no	—
197	SD₁₆	—	23	2	no	—
198	SD₁₆	—	23	2	no	—
199	SD₁₆	—	23	2	no	—
200	SD₁₆	—	23	2	no	—
201	D₈	—	23	2	no	—
202	D₈	—	23	2	no	—
203	M₁₆	—	2	1	no	—
204	M₁₆	—	2	4	no	—
205	M₁₆	—	23	2	no	—
206	M₁₆	—	46	1	no	—
207	M₁₆	—	46	4	no	—
208	M₁₆	C₂²⋊D₂₃	1	2	yes	—
209	M₁₆	C₂²⋊D₂₃	1	2	yes	—
210	M₁₆	C₂²×D₂₃	1	2	yes	—
211	C₂×C₈	—	2	1	no	—
212	C₂×C₈	—	2	4	no	—
213	C₂×C₈	—	23	1	no	—
214	C₂×C₈	—	23	1	no	—
215	C₂×C₈	—	23	1	no	—
216	C₂×C₈	—	23	1	no	—
217	C₂×C₈	—	23	1	no	—
218	C₂×C₈	—	23	1	no	—
219	C₂×C₈	—	46	1	no	—
220	C₂×C₈	—	46	4	no	—
221	C₂×C₈	C₂²×D₂₃	1	2	yes	—
222	C₄⋊C₄	—	46	1	no	—
223	C₂²⋊C₄	—	46	1	no	—
224	C₄²	—	23	2	no	—
225	C₂³	—	2	1	no	—
226	C₂³	—	46	1	no	—
227	C₂³	#67288	1	1	yes	characteristic
228	Q₈	—	2	1	no	—
229	Q₈	—	23	2	no	—
230	Q₈	—	46	1	no	—
231	Q₈	C₂×C₄×D₂₃	1	2	yes	—
232	D₄	—	2	2	no	—
233	D₄	—	2	2	no	—
234	D₄	—	2	1	no	—
235	D₄	—	23	2	no	—
236	D₄	—	46	2	no	—
237	D₄	—	46	2	no	—
238	D₄	—	46	1	no	—
239	D₄	C₂×C₄×D₂₃	1	2	yes	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	46	1	no	—
244	C₂×C₄	—	46	2	no	—
245	C₂×C₄	—	46	2	no	—
246	C₂×C₄	—	46	2	no	—
247	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
248	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
249	C₂×C₄	#67288	1	1	yes	characteristic
250	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
251	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
252	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
253	C₈	—	2	2	no	—
254	C₈	—	23	1	no	—
255	C₈	—	23	1	no	—
256	C₈	—	23	1	no	—
257	C₈	—	23	1	no	—
258	C₈	—	46	2	no	—
259	C₈	C₂×C₂²⋊D₂₃	1	4	yes	—
260	C₂²	—	2	2	no	—
261	C₂²	—	4	1	no	—
262	C₂²	—	8	1	no	—
263	C₂²	—	46	2	no	—
264	C₂²	—	184	1	no	—
265	C₂²	C₄×C₂²⋊D₂₃	1	1	yes	characteristic
266	C₂²	C₄×C₂²⋊D₂₃	1	1	yes	characteristic
267	C₂²	C₂×#67288	1	1	yes	characteristic
268	C₄	—	2	2	no	—
269	C₄	—	46	2	no	—
270	C₄	—	92	2	no	—
271	C₄	C₄×C₂²⋊D₂₃	1	1	yes	characteristic
272	C₄	C₄×C₂²⋊D₂₃	1	1	yes	characteristic
273	C₄	C₂²×C₄×D₂₃	1	1	yes	characteristic
274	C₄	C₂²×C₂²⋊D₂₃	1	1	yes	characteristic
275	C₂	—	2	1	no	—
276	C₂	—	2	1	no	—
277	C₂	—	2	1	no	—
278	C₂	—	4	2	no	—
279	C₂	—	92	2	no	—
280	C₂	C₂×C₄×C₂²⋊D₂₃	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²×C₂²⋊D₂₃
commutator	C₄×C₂₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₂₃
Fitting	C₂₃×#568	C₂
socle	C₂×C₂₃	C₂×C₄×D₄
prime radical	C₂×#10625121	C₂
cosocle	C₂×C₄×C₂₃	C₂⁴
hypercenter	#568	D₂₃
hypocenter	C₂₃	#2957
LCS	G ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #568

Complements

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

N	H ≅ G/N	type
2. #13093183	C₂	semidirect
3. #13092537	C₂	semidirect
4. #13092532	C₂	semidirect
5. C₂×#10625004	C₂	semidirect
6. #13092434	C₂	semidirect
7. #13092433	C₂	semidirect
8. #13092375	C₂	semidirect
9. C₂₃×#568	C₂	semidirect
13. C₂₃×#131	C₂²	semidirect
17. #10625121	C₄	semidirect
18. #10625121	C₄	semidirect
21. C₂×#67275	C₄	semidirect
22. C₂×#67274	C₄	semidirect
23. C₂×#67273	C₄	semidirect
24. C₂×#67272	C₄	semidirect
30. #10625072	C₂²	semidirect
31. #10625019	C₂²	semidirect
32. #10625007	C₂²	semidirect
33. #10625004	C₂²	semidirect
34. #10625004	C₂²	semidirect
35. #10624997	C₂²	semidirect
36. C₈×Dic₂₃	C₂²	semidirect
137. #568	D₂₃	semidirect
163. #131	C₂×D₂₃	semidirect
177. C₂₃	#2957	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10625121	2
#13093183	2
#13092537	2
#13092532	2
C₂×#10625004	2
#13092434	2
#13092433	2
#13092375	2
C₂₃×#568	2
#2957	23