Subgroups — #15682215

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×#10625080	C₂	1	1	yes	maximal, characteristic
2	#13092885	C₂	1	1	yes	maximal, characteristic
3	C₂×#10625058	C₂	1	1	yes	maximal, characteristic, complemented
4	#13092523	C₂	1	2	yes	maximal, complemented
5	#13092516	C₂	1	2	yes	maximal, complemented
6	#13092511	C₂	1	2	yes	maximal, complemented
7	#13092504	C₂	1	2	yes	maximal, complemented
8	#13092425	C₂	1	1	yes	maximal, characteristic, complemented
9	#13092416	C₂	1	1	yes	maximal, characteristic
10	M₁₆×Dic₂₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#428	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#67288	C₂²	1	1	yes	characteristic
13	C₂²×Dic₄₆	C₂²	1	1	yes	characteristic
14	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
15	C₂₃×#117	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
17	C₂×Dic₂₃⋊C₄	—	2	1	no	—
18	C₂×C₄×Dic₂₃	—	2	1	no	—
19	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
20	C₂×#67267	C₂²	1	1	yes	characteristic
21	#10625080	C₄	1	8	yes	complemented
22	C₂×Dic₉₂	C₂²	1	2	yes	—
23	C₂×#67263	C₂²	1	2	yes	—
24	#10625064	C₂²	1	1	yes	characteristic
25	#10625064	C₂²	1	1	yes	characteristic
26	#10625058	—	2	1	no	—
27	#10625058	—	2	1	no	—
28	#10625058	C₂²	1	2	yes	—
29	#10625058	C₂²	1	2	yes	—
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×#107	C₂²	1	2	yes	—
37	C₂₃×#106	C₂²	1	2	yes	—
38	#10624997	C₂²	1	2	yes	—
39	C₈×Dic₂₃	C₂²	1	2	yes	—
40	#10624992	C₂²	1	2	yes	—
41	#10624992	C₂²	1	2	yes	—
42	#10624992	C₂²	1	2	yes	—
43	#10624991	C₂²	1	2	yes	—
44	C₂²×C₄×C₂₃	—	2	1	no	—
45	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
46	C₂×C₂²⋊D₂₃	—	2	1	no	—
47	C₂²×Dic₂₃	—	2	1	no	—
48	C₂²×Dic₂₃	D₄	1	1	yes	characteristic
49	C₂²×Dic₂₃	D₄	1	1	yes	characteristic
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	1	no	—
55	C₂×C₄×D₂₃	—	2	1	no	—
56	C₂×Dic₄₆	—	2	1	no	—
57	C₂×Dic₄₆	—	2	2	no	—
58	C₂×Dic₄₆	C₂×C₄	1	4	yes	complemented
59	C₂×Dic₄₆	C₂³	1	2	yes	—
60	C₂×Dic₄₆	C₂³	1	1	yes	characteristic
61	M₁₆×C₂₃	C₂×C₄	1	4	yes	complemented
62	C₂×C₈×C₂₃	C₂³	1	2	yes	—
63	C₄⋊C₄×C₂₃	—	2	1	no	—
64	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
65	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
66	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
67	C₂²⋊C₄×C₂₃	—	2	1	no	—
68	C₄²×C₂₃	—	2	1	no	—
69	(C₂×D₂₃)⋊C₄	—	2	1	no	—
70	(C₂×D₂₃)⋊C₄	—	2	1	no	—
71	Dic₂₃⋊C₄	—	2	1	no	—
72	Dic₂₃⋊C₄	—	2	1	no	—
73	Dic₂₃⋊C₄	—	2	2	no	—
74	Dic₂₃⋊C₄	—	2	2	no	—
75	C₄×Dic₂₃	—	2	1	no	—
76	C₄×Dic₂₃	—	2	1	no	—
77	C₄×Dic₂₃	—	2	2	no	—
78	C₄×Dic₂₃	—	2	2	no	—
79	C₄×Dic₂₃	D₄	1	1	yes	characteristic
80	C₄×Dic₂₃	D₄	1	1	yes	characteristic
81	C₄×Dic₂₃	C₂³	1	1	yes	characteristic
82	C₄×Dic₂₃	C₂³	1	1	yes	characteristic
83	#67267	—	2	2	no	—
84	C₂×C₂₃⋊C₈	C₂³	1	2	yes	—
85	Dic₉₂	C₂×C₄	1	8	yes	complemented
86	#67263	C₂×C₄	1	8	yes	complemented
87	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
88	C₂²×D₂₃	—	2	1	no	—
89	C₂×C₄×C₂₃	—	2	2	no	—
90	C₂×C₄×C₂₃	—	2	1	no	—
91	C₂×C₄×C₂₃	—	2	1	no	—
92	C₂×C₄×C₂₃	—	4	2	no	—
93	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
94	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
95	C₂×C₄×C₂₃	#41	1	2	yes	—
96	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
97	C₂²⋊D₂₃	—	2	4	no	—
98	C₂×Dic₂₃	—	2	4	no	—
99	C₂×Dic₂₃	—	2	2	no	—
100	C₂×Dic₂₃	—	2	1	no	—
101	C₂×Dic₂₃	—	2	2	no	—
102	C₂×Dic₂₃	—	4	1	no	—
103	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
104	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
105	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
106	C₂×Dic₂₃	C₂×D₄	1	1	yes	characteristic
107	D₉₂	—	2	1	no	—
108	D₉₂	C₂²×C₄	1	2	yes	—
109	C₄×D₂₃	—	2	4	no	—
110	C₄×D₂₃	—	4	2	no	—
111	C₄×D₂₃	—	4	2	no	—
112	Dic₄₆	—	2	4	no	—
113	Dic₄₆	—	2	2	no	—
114	Dic₄₆	—	2	1	no	—
115	Dic₄₆	C₂²×C₄	1	4	yes	—
116	Dic₄₆	C₂²×C₄	1	2	yes	—
117	C₈×C₂₃	C₂²×C₄	1	4	yes	—
118	C₂₃⋊C₈	—	2	2	no	—
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	—	4	1	no	—
128	C₄×C₂₃	—	4	1	no	—
129	C₄×C₂₃	—	4	2	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	4	no	—
134	Dic₂₃	—	2	2	no	—
135	Dic₂₃	—	2	1	no	—
136	Dic₂₃	—	2	1	no	—
137	Dic₂₃	—	2	1	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	D₂₃	1	1	yes	characteristic, complemented
147	#419	—	23	1	no	—
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	—	23	1	no	—
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	—	46	1	no	—
175	C₂×C₄⋊C₄	C₂×D₂₃	1	1	yes	characteristic
176	C₂×C₄²	—	23	1	no	—
177	C₂×C₄²	—	46	1	no	—
178	C₂×C₄²	—	46	1	no	—
179	#107	C₂×D₂₃	1	2	yes	—
180	#106	C₂×D₂₃	1	2	yes	—
181	#103	—	23	2	no	—
182	#103	—	23	2	no	—
183	#103	—	23	2	no	—
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	—	23	4	no	—
191	C₂×Q₈	—	23	2	no	—
192	C₂×Q₈	—	23	1	no	—
193	C₂×Q₈	—	46	1	no	—
194	C₂×Q₈	—	46	2	no	—
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	—	46	2	no	—
208	M₁₆	C₄×D₂₃	1	4	yes	complemented
209	C₂×C₈	—	23	2	no	—
210	C₂×C₈	C₂²×D₂₃	1	2	yes	—
211	C₄⋊C₄	—	2	1	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	C₂²×D₂₃	1	2	yes	—
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	—	23	4	no	—
234	Q₈	—	23	2	no	—
235	Q₈	—	46	4	no	—
236	Q₈	—	46	2	no	—
237	Q₈	—	46	1	no	—
238	D₄	—	23	2	no	—
239	D₄	—	46	4	no	—
240	D₄	—	46	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	4	2	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	4	no	—
251	C₂×C₄	—	46	1	no	—
252	C₂×C₄	—	46	2	no	—
253	C₂×C₄	—	46	4	no	—
254	C₂×C₄	—	92	2	no	—
255	C₂×C₄	—	92	2	no	—
256	C₂×C₄	—	92	1	no	—
257	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
258	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
259	C₂×C₄	#67292	1	2	yes	—
260	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
261	C₈	—	46	2	no	—
262	C₈	C₂×C₄×D₂₃	1	4	yes	—
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	#10625064	1	2	yes	—
269	C₂²	C₂×D₄×D₂₃	1	1	yes	characteristic
270	C₄	—	4	1	no	—
271	C₄	—	4	2	no	—
272	C₄	—	4	1	no	—
273	C₄	—	23	2	no	—
274	C₄	—	46	2	no	—
275	C₄	—	46	4	no	—
276	C₄	—	46	1	no	—
277	C₄	—	46	1	no	—
278	C₄	—	46	1	no	—
279	C₄	#10625064	1	2	yes	—
280	C₄	C₂²×C₄×D₂₃	1	1	yes	characteristic
281	C₄	C₂×#67292	1	1	yes	characteristic
282	C₂	—	2	2	no	—
283	C₂	—	92	2	no	—
284	C₂	C₂×#10625064	1	1	yes	characteristic
285	C₂	#137×D₂₃	1	1	yes	characteristic
286	C₂	#13093213	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₂₃
commutator	C₄×C₂₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₂₃
Fitting	C₂₃×#428	C₂
socle	C₂²×C₂₃	C₂²×D₄
prime radical	C₂×#67288	C₂²
cosocle	C₂×C₄×C₂₃	C₂⁴
hypercenter	#428	D₂₃
hypocenter	C₂₃	C₄×#137
LCS	G ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #428

Complements

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

N	H ≅ G/N	type
3. C₂×#10625058	C₂	semidirect
4. #13092523	C₂	semidirect
5. #13092516	C₂	semidirect
6. #13092511	C₂	semidirect
7. #13092504	C₂	semidirect
8. #13092425	C₂	semidirect
10. M₁₆×Dic₂₃	C₂	semidirect
11. C₂₃×#428	C₂	semidirect
21. #10625080	C₄	semidirect
58. C₂×Dic₄₆	C₂×C₄	semidirect
61. M₁₆×C₂₃	C₂×C₄	semidirect
85. Dic₉₂	C₂×C₄	semidirect
86. #67263	C₂×C₄	semidirect
138. Dic₂₃	#137	semidirect
146. #428	D₂₃	semidirect
187. C₂₃	C₄×#137	semidirect
208. M₁₆	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×#10625080	2
#13092885	2
C₂×#10625058	2
#13092523	2
#13092516	2
#13092511	2
#13092504	2
#13092425	2
#13092416	2
M₁₆×Dic₂₃	2
C₂₃×#428	2
C₄×#137	23