Subgroups — #15682212

Order 2944.

484 conjugacy classes of subgroups.

The subgroup lattice has 288 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	#13092885	C₂	1	1	yes	maximal, characteristic
3	C₂×Q₈×Dic₂₃	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	#13092460	C₂	1	1	yes	maximal, characteristic, complemented
9	#13092419	C₂	1	1	yes	maximal, characteristic
10	M₁₆×Dic₂₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#419	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	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
15	C₂₃×#117	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	C₂×C₄⋊Dic₂₃	—	2	1	no	—
24	C₂×C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
25	C₂×C₄×Dic₂₃	—	2	1	no	—
26	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
27	C₂×#67267	C₂²	1	1	yes	characteristic
28	#10625064	C₂²	1	1	yes	characteristic
29	#10625064	C₂²	1	1	yes	characteristic
30	#10625058	C₂²	1	1	yes	characteristic
31	#10625058	C₂²	1	1	yes	characteristic
32	(C₂²⋊D₂₃)⋊C₄	—	2	1	no	—
33	(C₂²⋊D₂₃)⋊C₄	—	2	1	no	—
34	#10625043	—	2	1	no	—
35	C₄²×D₂₃	—	2	1	no	—
36	C₂₃×#103	C₂²	1	2	yes	—
37	C₂₃×#103	C₂²	1	2	yes	—
38	#10625003	C₂²	1	2	yes	—
39	#10624998	C₂²	1	2	yes	—
40	#10624997	C₂²	1	2	yes	—
41	C₈×Dic₂₃	C₂²	1	2	yes	—
42	#10624990	C₂²	1	2	yes	—
43	#10624989	C₂²	1	2	yes	—
44	C₂×Q₈×C₂₃	—	2	1	no	—
45	C₂×Q₈×C₂₃	—	2	2	no	—
46	C₂×Q₈×C₂₃	C₂×C₄	1	4	yes	complemented
47	C₂×Q₈×C₂₃	C₂³	1	2	yes	—
48	C₂²×C₄×C₂₃	—	2	1	no	—
49	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
50	C₂×C₂²⋊D₂₃	—	2	1	no	—
51	C₂²×Dic₂₃	—	2	1	no	—
52	C₂²×Dic₂₃	D₄	1	1	yes	characteristic
53	C₂²×Dic₂₃	D₄	1	1	yes	characteristic
54	#67288	—	2	2	no	—
55	#67288	C₂×C₄	1	4	yes	—
56	C₂×D₉₂	C₂³	1	1	yes	characteristic
57	C₂×C₄×D₂₃	—	2	1	no	—
58	C₂×C₄×D₂₃	—	2	1	no	—
59	C₂×C₄×D₂₃	—	2	1	no	—
60	C₂×Dic₄₆	C₂³	1	1	yes	characteristic
61	M₁₆×C₂₃	—	2	2	no	—
62	C₂×C₈×C₂₃	C₂³	1	2	yes	—
63	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
64	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
65	C₂²⋊C₄×C₂₃	—	2	1	no	—
66	C₄²×C₂₃	—	2	1	no	—
67	#67275	C₂×C₄	1	8	yes	complemented
68	#67274	C₂×C₄	1	8	yes	complemented
69	(C₂×D₂₃)⋊C₄	—	2	1	no	—
70	(C₂×D₂₃)⋊C₄	—	2	1	no	—
71	C₄⋊Dic₂₃	—	2	1	no	—
72	C₄⋊Dic₂₃	—	2	2	no	—
73	C₄⋊Dic₂₃	—	2	2	no	—
74	C₄⋊Dic₂₃	C₂³	1	2	yes	—
75	Dic₂₃⋊C₄	—	2	1	no	—
76	Dic₂₃⋊C₄	—	2	1	no	—
77	C₄×Dic₂₃	—	2	1	no	—
78	C₄×Dic₂₃	—	2	1	no	—
79	C₄×Dic₂₃	—	2	2	no	—
80	C₄×Dic₂₃	—	2	2	no	—
81	C₄×Dic₂₃	D₄	1	1	yes	characteristic
82	C₄×Dic₂₃	D₄	1	1	yes	characteristic
83	C₄×Dic₂₃	C₂³	1	1	yes	characteristic
84	C₄×Dic₂₃	C₂³	1	1	yes	characteristic
85	#67267	C₂×C₄	1	4	yes	complemented
86	C₂×C₂₃⋊C₈	C₂³	1	2	yes	—
87	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
88	C₂²×D₂₃	—	2	1	no	—
89	Q₈×C₂₃	—	2	4	no	—
90	Q₈×C₂₃	—	2	2	no	—
91	Q₈×C₂₃	C₂²×C₄	1	4	yes	—
92	C₂×C₄×C₂₃	—	2	4	no	—
93	C₂×C₄×C₂₃	—	2	2	no	—
94	C₂×C₄×C₂₃	—	2	1	no	—
95	C₂×C₄×C₂₃	—	2	1	no	—
96	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
97	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄×C₂₃	#41	1	2	yes	—
99	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
100	C₂²⋊D₂₃	—	2	4	no	—
101	C₂×Dic₂₃	—	2	1	no	—
102	C₂×Dic₂₃	—	2	2	no	—
103	C₂×Dic₂₃	—	2	2	no	—
104	C₂×Dic₂₃	—	4	1	no	—
105	C₂×Dic₂₃	—	4	2	no	—
106	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
107	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
108	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
109	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
110	D₉₂	—	2	1	no	—
111	D₉₂	C₂²×C₄	1	2	yes	—
112	C₄×D₂₃	—	2	4	no	—
113	C₄×D₂₃	—	4	2	no	—
114	C₄×D₂₃	—	4	2	no	—
115	Dic₄₆	—	2	1	no	—
116	Dic₄₆	C₂²×C₄	1	2	yes	—
117	C₈×C₂₃	—	2	2	no	—
118	C₂₃⋊C₈	C₂²×C₄	1	4	yes	—
119	C₄×#137	—	23	1	no	maximal, Sylow
120	C₂²×C₂₃	—	2	1	no	—
121	C₂²×C₂₃	—	2	1	no	—
122	C₂²×C₂₃	C₄×D₄	1	2	yes	—
123	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
124	C₂×D₂₃	—	2	2	no	—
125	C₂×D₂₃	—	4	1	no	—
126	C₂×D₂₃	—	4	1	no	—
127	C₄×C₂₃	—	2	4	no	—
128	C₄×C₂₃	—	4	1	no	—
129	C₄×C₂₃	—	4	1	no	—
130	C₄×C₂₃	C₄×D₄	1	2	yes	—
131	C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
132	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
133	Dic₂₃	—	2	2	no	—
134	Dic₂₃	—	2	1	no	—
135	Dic₂₃	—	2	1	no	—
136	Dic₂₃	—	2	1	no	—
137	Dic₂₃	—	4	2	no	—
138	Dic₂₃	#137	1	2	yes	complemented
139	C₂×#137	—	23	1	no	—
140	C₄×#41	—	23	1	no	—
141	C₂×C₄×Q₈	—	23	1	no	—
142	#441	—	23	2	no	—
143	#440	—	23	2	no	—
144	C₄×Q₁₆	—	23	2	no	—
145	C₄×SD₁₆	—	23	2	no	—
146	#428	—	23	1	no	—
147	#419	D₂₃	1	1	yes	characteristic, complemented
148	#417	—	23	1	no	—
149	C₄×M₁₆	—	23	1	no	—
150	C₂×C₂₃	—	2	2	no	—
151	C₂×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
152	C₂×C₂₃	C₂×#137	1	1	yes	characteristic
153	C₂×C₂₃	#575	1	1	yes	characteristic
154	D₂₃	—	4	2	no	—
155	C₂×#41	—	23	1	no	—
156	C₂²×Q₈	C₂×D₂₃	1	1	yes	characteristic
157	#137	—	23	8	no	—
158	C₂×Q₁₆	—	23	2	no	—
159	C₂×SD₁₆	—	23	2	no	—
160	C₂×M₁₆	—	23	1	no	—
161	C₂×M₁₆	C₂×D₂₃	1	1	yes	characteristic
162	C₄×Q₈	—	23	2	no	—
163	C₄×Q₈	—	23	2	no	—
164	C₄×Q₈	—	23	1	no	—
165	C₄×Q₈	—	23	1	no	—
166	C₄×Q₈	—	46	1	no	—
167	C₄×Q₈	—	46	1	no	—
168	C₄×D₄	—	23	1	no	—
169	C₄×D₄	—	23	1	no	—
170	C₄×D₄	—	46	1	no	—
171	C₄×D₄	—	46	1	no	—
172	#117	—	46	1	no	—
173	#117	C₂×D₂₃	1	1	yes	characteristic
174	C₂×C₄⋊C₄	—	23	1	no	—
175	C₂×C₄⋊C₄	—	46	1	no	—
176	C₂×C₄²	—	23	1	no	—
177	C₂×C₄²	—	46	1	no	—
178	C₂×C₄²	—	46	1	no	—
179	#107	—	23	2	no	—
180	#106	—	23	2	no	—
181	#103	—	23	2	no	—
182	#103	C₂×D₂₃	1	2	yes	—
183	#103	C₂×D₂₃	1	2	yes	—
184	#102	—	23	2	no	—
185	#97	—	23	2	no	—
186	C₄×C₈	—	23	2	no	—
187	C₂₃	C₄×#137	1	1	yes	characteristic, complemented, Sylow
188	#41	—	23	4	no	—
189	#41	—	46	2	no	—
190	C₂×Q₈	—	2	1	no	—
191	C₂×Q₈	—	2	2	no	—
192	C₂×Q₈	—	23	1	no	—
193	C₂×Q₈	C₄×D₂₃	1	4	yes	complemented
194	C₂×Q₈	C₂²×D₂₃	1	2	yes	—
195	C₂×D₄	—	23	1	no	—
196	C₂×D₄	—	46	1	no	—
197	C₂²×C₄	—	2	1	no	—
198	C₂²×C₄	—	23	1	no	—
199	C₂²×C₄	—	23	1	no	—
200	C₂²×C₄	—	46	1	no	—
201	C₂²×C₄	—	46	1	no	—
202	C₂²×C₄	—	46	1	no	—
203	C₂²×C₄	—	46	1	no	—
204	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
205	Q₁₆	—	23	8	no	—
206	SD₁₆	—	23	8	no	—
207	M₁₆	—	2	2	no	—
208	M₁₆	—	23	4	no	—
209	C₂×C₈	—	23	2	no	—
210	C₂×C₈	C₂²×D₂₃	1	2	yes	—
211	C₄⋊C₄	—	23	2	no	—
212	C₄⋊C₄	—	46	2	no	—
213	C₄⋊C₄	—	46	2	no	—
214	C₄⋊C₄	—	46	1	no	—
215	C₄⋊C₄	—	46	1	no	—
216	C₄⋊C₄	—	46	1	no	—
217	C₄⋊C₄	C₂²×D₂₃	1	1	yes	characteristic
218	C₄⋊C₄	C₂²×D₂₃	1	1	yes	characteristic
219	C₂²⋊C₄	—	2	1	no	—
220	C₂²⋊C₄	—	46	1	no	—
221	C₂²⋊C₄	—	46	1	no	—
222	C₄²	—	2	1	no	—
223	C₄²	—	23	1	no	—
224	C₄²	—	23	1	no	—
225	C₄²	—	23	1	no	—
226	C₄²	—	23	1	no	—
227	C₄²	—	46	1	no	—
228	C₄²	—	46	1	no	—
229	C₄²	—	46	2	no	—
230	C₄²	—	46	2	no	—
231	C₂³	—	46	1	no	—
232	C₂³	D₄×D₂₃	1	1	yes	characteristic
233	Q₈	—	2	4	no	—
234	Q₈	—	2	2	no	—
235	Q₈	—	23	2	no	—
236	Q₈	—	46	1	no	—
237	Q₈	C₂×C₄×D₂₃	1	4	yes	—
238	D₄	—	23	2	no	—
239	D₄	—	46	4	no	—
240	D₄	—	46	1	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	23	1	no	—
246	C₂×C₄	—	23	1	no	—
247	C₂×C₄	—	23	1	no	—
248	C₂×C₄	—	23	1	no	—
249	C₂×C₄	—	46	2	no	—
250	C₂×C₄	—	46	1	no	—
251	C₂×C₄	—	46	4	no	—
252	C₂×C₄	—	46	2	no	—
253	C₂×C₄	—	92	2	no	—
254	C₂×C₄	—	92	2	no	—
255	C₂×C₄	—	92	1	no	—
256	C₂×C₄	—	92	2	no	—
257	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
258	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
259	C₂×C₄	#67290	1	2	yes	—
260	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
261	C₈	—	2	2	no	—
262	C₈	—	23	4	no	—
263	C₂²	—	2	1	no	—
264	C₂²	—	2	1	no	—
265	C₂²	—	46	2	no	—
266	C₂²	—	92	1	no	—
267	C₂²	—	92	1	no	—
268	C₂²	(C₂²⋊D₂₃)⋊C₄	1	2	yes	—
269	C₂²	C₂×D₄×D₂₃	1	1	yes	characteristic
270	C₄	—	2	4	no	—
271	C₄	—	4	1	no	—
272	C₄	—	4	1	no	—
273	C₄	—	23	2	no	—
274	C₄	—	46	2	no	—
275	C₄	—	46	1	no	—
276	C₄	—	46	1	no	—
277	C₄	—	46	1	no	—
278	C₄	—	92	2	no	—
279	C₄	(C₂²⋊D₂₃)⋊C₄	1	2	yes	—
280	C₄	C₂²×C₄×D₂₃	1	1	yes	characteristic
281	C₄	C₂×#67290	1	1	yes	characteristic
282	C₂	—	2	2	no	—
283	C₂	—	92	2	no	—
284	C₂	C₂×(C₂²⋊D₂₃)⋊C₄	1	1	yes	characteristic
285	C₂	#137×D₂₃	1	1	yes	characteristic
286	C₂	#13093216	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₂₃
commutator	C₄×C₂₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₂₃
Fitting	C₂₃×#419	C₂
socle	C₂²×C₂₃	C₂²×D₄
prime radical	C₂×#67288	C₂²
cosocle	C₂×C₄×C₂₃	C₂⁴
hypercenter	#419	D₂₃
hypocenter	C₂₃	C₄×#137
LCS	G ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #419

Complements

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

N	H ≅ G/N	type
3. C₂×Q₈×Dic₂₃	C₂	semidirect
4. #13092521	C₂	semidirect
5. #13092520	C₂	semidirect
6. #13092509	C₂	semidirect
7. #13092502	C₂	semidirect
8. #13092460	C₂	semidirect
10. M₁₆×Dic₂₃	C₂	semidirect
11. C₂₃×#419	C₂	semidirect
21. #10625113	C₄	semidirect
46. C₂×Q₈×C₂₃	C₂×C₄	semidirect
67. #67275	C₂×C₄	semidirect
68. #67274	C₂×C₄	semidirect
85. #67267	C₂×C₄	semidirect
138. Dic₂₃	#137	semidirect
147. #419	D₂₃	semidirect
187. C₂₃	C₄×#137	semidirect
193. 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
#13092885	2
C₂×Q₈×Dic₂₃	2
#13092521	2
#13092520	2
#13092509	2
#13092502	2
#13092460	2
#13092419	2
M₁₆×Dic₂₃	2
C₂₃×#419	2
C₄×#137	23