Subgroups — #15682165

Order 2944.

500 conjugacy classes of subgroups.

The subgroup lattice has 270 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	#41×Dic₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#13092885	C₂	1	1	yes	maximal, characteristic, complemented
4	#13092509	C₂	1	2	yes	maximal, complemented
5	#13092508	C₂	1	2	yes	maximal, complemented
6	#13092502	C₂	1	2	yes	maximal, complemented
7	#13092497	C₂	1	2	yes	maximal, complemented
8	#13092453	C₂	1	1	yes	maximal, characteristic, complemented
9	#13092414	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₈×Dic₂₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#416	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₂₃	C₂²	1	1	yes	characteristic
13	C₂×#67288	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
15	C₂₃×#117	C₂²	1	1	yes	characteristic
16	#10625121	C₄	1	8	yes	complemented
17	Q₈×Dic₂₃	C₂²	1	2	yes	—
18	C₂×#67275	C₂²	1	1	yes	characteristic
19	C₂×#67274	C₂²	1	1	yes	characteristic
20	D₄×Dic₂₃	—	2	2	no	—
21	D₄×Dic₂₃	C₂²	1	2	yes	—
22	C₂×#67273	C₂²	1	1	yes	characteristic
23	C₂×#67272	C₂²	1	1	yes	characteristic
24	#10625097	—	2	1	no	—
25	#10625097	C₂²	1	1	yes	characteristic
26	C₂×C₄×Dic₂₃	—	2	1	no	—
27	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
28	C₂²×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
29	#10625064	C₂²	1	2	yes	—
30	#10625058	C₂²	1	2	yes	—
31	(C₂²⋊D₂₃)⋊C₄	—	2	2	no	—
32	#10625043	—	2	1	no	—
33	C₄²×D₂₃	—	2	1	no	—
34	C₂₃×#103	C₂²	1	2	yes	—
35	C₂₃×#102	C₂²	1	2	yes	—
36	#10625003	C₂²	1	2	yes	—
37	#10624998	C₂²	1	2	yes	—
38	C₈×Dic₂₃	C₂²	1	2	yes	—
39	C₈×Dic₂₃	C₂²	1	2	yes	—
40	#10624990	C₂²	1	2	yes	—
41	#10624989	C₂²	1	2	yes	—
42	#41×C₂₃	—	2	2	no	—
43	#41×C₂₃	C₂×C₄	1	4	yes	complemented
44	C₂×Q₈×C₂₃	C₂³	1	1	yes	characteristic
45	C₂×D₄×C₂₃	—	2	1	no	—
46	C₂×D₄×C₂₃	C₂³	1	1	yes	characteristic
47	C₂²×C₄×C₂₃	—	2	1	no	—
48	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
49	C₂×C₂²⋊D₂₃	—	2	1	no	—
50	C₂²×Dic₂₃	—	2	2	no	—
51	C₂²×Dic₂₃	D₄	1	2	yes	—
52	#67288	—	2	2	no	—
53	#67288	C₂×C₄	1	4	yes	complemented
54	C₂×D₉₂	C₂³	1	1	yes	characteristic
55	C₂×C₄×D₂₃	—	2	1	no	—
56	C₂×C₄×D₂₃	—	2	2	no	—
57	C₂×Dic₄₆	C₂³	1	1	yes	characteristic
58	C₂×C₈×C₂₃	—	2	2	no	—
59	C₂×C₈×C₂₃	C₂³	1	2	yes	—
60	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
61	C₂²⋊C₄×C₂₃	—	2	1	no	—
62	C₄²×C₂₃	—	2	1	no	—
63	C₂²⋊Dic₂₃	—	2	1	no	—
64	C₂²⋊Dic₂₃	—	2	2	no	—
65	#67275	C₂×C₄	1	4	yes	complemented
66	#67274	C₂×C₄	1	4	yes	complemented
67	#67273	C₂×C₄	1	4	yes	complemented
68	#67272	C₂×C₄	1	4	yes	complemented
69	(C₂×D₂₃)⋊C₄	—	2	2	no	—
70	C₄⋊Dic₂₃	—	2	2	no	—
71	C₄⋊Dic₂₃	C₂³	1	2	yes	—
72	Dic₂₃⋊C₄	—	2	2	no	—
73	C₄×Dic₂₃	—	2	1	no	—
74	C₄×Dic₂₃	—	2	2	no	—
75	C₄×Dic₂₃	—	2	2	no	—
76	C₄×Dic₂₃	D₄	1	1	yes	characteristic
77	C₄×Dic₂₃	D₄	1	1	yes	characteristic
78	C₄×Dic₂₃	C₂³	1	2	yes	—
79	C₂×C₂₃⋊C₈	C₂×C₄	1	4	yes	complemented
80	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
81	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
82	C₂³×C₂₃	—	2	1	no	—
83	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
84	C₂²×D₂₃	—	2	1	no	—
85	Q₈×C₂₃	—	2	1	no	—
86	Q₈×C₂₃	C₂²×C₄	1	2	yes	—
87	D₄×C₂₃	—	2	4	no	—
88	D₄×C₂₃	—	2	1	no	—
89	D₄×C₂₃	C₂²×C₄	1	2	yes	—
90	C₂×C₄×C₂₃	—	2	1	no	—
91	C₂×C₄×C₂₃	—	2	4	no	—
92	C₂×C₄×C₂₃	—	2	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	1	no	—
99	C₂×Dic₂₃	—	2	4	no	—
100	C₂×Dic₂₃	—	2	2	no	—
101	C₂×Dic₂₃	—	4	4	no	—
102	C₂×Dic₂₃	C₂×D₄	1	2	yes	—
103	C₂×Dic₂₃	C₂×D₄	1	2	yes	—
104	D₉₂	—	2	1	no	—
105	D₉₂	C₂²×C₄	1	2	yes	—
106	C₄×D₂₃	—	2	4	no	—
107	C₄×D₂₃	—	4	4	no	—
108	Dic₄₆	—	2	1	no	—
109	Dic₄₆	C₂²×C₄	1	2	yes	—
110	C₈×C₂₃	—	2	2	no	—
111	C₂₃⋊C₈	C₂²×C₄	1	2	yes	—
112	C₂₃⋊C₈	C₂²×C₄	1	2	yes	—
113	C₄×#135	—	23	1	no	maximal, Sylow
114	C₂²×C₂₃	—	2	2	no	—
115	C₂²×C₂₃	—	2	2	no	—
116	C₂²×C₂₃	—	4	2	no	—
117	C₂²×C₂₃	C₄×D₄	1	2	yes	—
118	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
119	C₂×D₂₃	—	2	2	no	—
120	C₂×D₂₃	—	4	2	no	—
121	C₄×C₂₃	—	2	2	no	—
122	C₄×C₂₃	—	4	2	no	—
123	C₄×C₂₃	C₄×D₄	1	2	yes	—
124	C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
125	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
126	Dic₂₃	—	2	2	no	—
127	Dic₂₃	—	2	2	no	—
128	Dic₂₃	—	4	2	no	—
129	Dic₂₃	#135	1	4	yes	complemented
130	C₂×#135	—	23	1	no	—
131	C₄×#41	—	23	1	no	—
132	C₄×#41	—	23	1	no	—
133	C₄×Q₁₆	—	23	2	no	—
134	C₄×SD₁₆	—	23	2	no	—
135	C₄×SD₁₆	—	23	2	no	—
136	C₄×D₈	—	23	2	no	—
137	#427	—	23	1	no	—
138	#416	—	23	1	no	—
139	#416	D₂₃	1	1	yes	characteristic, complemented
140	C₂×C₄×C₈	—	23	1	no	—
141	C₂×C₂₃	—	2	2	no	—
142	C₂×C₂₃	—	4	2	no	—
143	C₂×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
144	C₂×C₂₃	C₂×#135	1	2	yes	—
145	D₂₃	—	4	2	no	—
146	C₂×#41	—	23	1	no	—
147	C₂×#41	C₂×D₂₃	1	1	yes	characteristic
148	#135	—	23	8	no	—
149	C₂×Q₁₆	—	23	1	no	—
150	C₂×SD₁₆	—	23	1	no	—
151	C₂×SD₁₆	—	23	1	no	—
152	C₂×D₈	—	23	1	no	—
153	C₂²×C₈	—	23	1	no	—
154	C₂²×C₈	C₂×D₂₃	1	1	yes	characteristic
155	C₄×Q₈	—	23	2	no	—
156	C₄×Q₈	—	23	2	no	—
157	C₄×D₄	—	23	2	no	—
158	C₄×D₄	—	23	2	no	—
159	C₄×D₄	—	46	2	no	—
160	C₄×D₄	—	46	2	no	—
161	#117	—	23	1	no	—
162	#117	—	46	1	no	—
163	#117	—	46	1	no	—
164	#117	C₂×D₂₃	1	1	yes	characteristic
165	C₂×C₄²	—	23	1	no	—
166	C₂×C₄²	—	46	1	no	—
167	C₂×C₄²	—	46	1	no	—
168	#107	—	23	2	no	—
169	#106	—	23	2	no	—
170	#103	—	23	2	no	—
171	#103	C₂×D₂₃	1	2	yes	—
172	#102	—	23	2	no	—
173	#102	C₂×D₂₃	1	2	yes	—
174	C₄×C₈	—	23	2	no	—
175	C₄×C₈	—	23	2	no	—
176	C₂₃	C₄×#135	1	1	yes	characteristic, complemented, Sylow
177	#41	—	2	2	no	—
178	#41	—	23	4	no	—
179	#41	—	46	2	no	—
180	#41	C₄×D₂₃	1	4	yes	complemented
181	C₂×Q₈	—	23	1	no	—
182	C₂×Q₈	C₂²×D₂₃	1	1	yes	characteristic
183	C₂×D₄	—	2	1	no	—
184	C₂×D₄	—	23	1	no	—
185	C₂×D₄	—	46	1	no	—
186	C₂×D₄	C₂²×D₂₃	1	1	yes	characteristic
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	—	23	2	no	—
189	C₂²×C₄	—	46	2	no	—
190	C₂²×C₄	—	46	2	no	—
191	C₂²×C₄	—	46	1	no	—
192	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
193	Q₁₆	—	23	4	no	—
194	SD₁₆	—	23	4	no	—
195	SD₁₆	—	23	4	no	—
196	D₈	—	23	4	no	—
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	—	23	4	no	—
199	C₂×C₈	—	23	1	no	—
200	C₂×C₈	—	23	1	no	—
201	C₂×C₈	C₂²×D₂₃	1	2	yes	—
202	C₄⋊C₄	—	23	2	no	—
203	C₄⋊C₄	—	46	2	no	—
204	C₄⋊C₄	—	46	2	no	—
205	C₄⋊C₄	C₂²×D₂₃	1	2	yes	—
206	C₂²⋊C₄	—	2	1	no	—
207	C₂²⋊C₄	—	46	2	no	—
208	C₂²⋊C₄	—	46	2	no	—
209	C₂²⋊C₄	—	46	1	no	—
210	C₄²	—	2	1	no	—
211	C₄²	—	23	1	no	—
212	C₄²	—	23	1	no	—
213	C₄²	—	23	2	no	—
214	C₄²	—	46	2	no	—
215	C₄²	—	46	2	no	—
216	C₄²	—	46	1	no	—
217	C₂³	—	2	1	no	—
218	C₂³	—	46	1	no	—
219	C₂³	D₄×D₂₃	1	1	yes	characteristic
220	Q₈	—	2	1	no	—
221	Q₈	—	23	2	no	—
222	Q₈	—	46	1	no	—
223	Q₈	C₂×C₄×D₂₃	1	2	yes	—
224	D₄	—	2	4	no	—
225	D₄	—	2	1	no	—
226	D₄	—	23	2	no	—
227	D₄	—	46	4	no	—
228	D₄	—	46	1	no	—
229	D₄	C₂×C₄×D₂₃	1	2	yes	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	23	2	no	—
234	C₂×C₄	—	23	2	no	—
235	C₂×C₄	—	46	4	no	—
236	C₂×C₄	—	46	1	no	—
237	C₂×C₄	—	46	4	no	—
238	C₂×C₄	—	46	2	no	—
239	C₂×C₄	—	92	4	no	—
240	C₂×C₄	—	92	4	no	—
241	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
242	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
243	C₂×C₄	#67290	1	2	yes	—
244	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
245	C₈	—	2	2	no	—
246	C₈	—	23	2	no	—
247	C₈	—	23	2	no	—
248	C₂²	—	2	2	no	—
249	C₂²	—	2	2	no	—
250	C₂²	—	4	2	no	—
251	C₂²	—	46	2	no	—
252	C₂²	—	92	2	no	—
253	C₂²	(C₂²⋊D₂₃)⋊C₄	1	2	yes	—
254	C₂²	C₂×D₄×D₂₃	1	1	yes	characteristic
255	C₄	—	2	2	no	—
256	C₄	—	4	2	no	—
257	C₄	—	23	4	no	—
258	C₄	—	46	2	no	—
259	C₄	—	46	2	no	—
260	C₄	—	92	2	no	—
261	C₄	(C₂²⋊D₂₃)⋊C₄	1	2	yes	—
262	C₄	C₂²×C₄×D₂₃	1	1	yes	characteristic
263	C₄	C₂×#67290	1	1	yes	characteristic
264	C₂	—	2	2	no	—
265	C₂	—	4	2	no	—
266	C₂	—	92	2	no	—
267	C₂	C₂×(C₂²⋊D₂₃)⋊C₄	1	1	yes	characteristic
268	C₂	#135×D₂₃	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₄×D₂₃
commutator	C₄×C₂₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₂₃
Fitting	C₂₃×#416	C₂
socle	C₂²×C₂₃	C₂²×D₄
prime radical	C₂×#10625121	C₂
cosocle	C₂×C₄×C₂₃	C₂⁴
hypercenter	#416	D₂₃
hypocenter	C₂₃	C₄×#135
LCS	G ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #416

Complements

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

N	H ≅ G/N	type
2. #41×Dic₂₃	C₂	semidirect
3. #13092885	C₂	semidirect
4. #13092509	C₂	semidirect
5. #13092508	C₂	semidirect
6. #13092502	C₂	semidirect
7. #13092497	C₂	semidirect
8. #13092453	C₂	semidirect
9. #13092414	C₂	semidirect
10. C₂×C₈×Dic₂₃	C₂	semidirect
11. C₂₃×#416	C₂	semidirect
16. #10625121	C₄	semidirect
43. #41×C₂₃	C₂×C₄	semidirect
53. #67288	C₂×C₄	semidirect
65. #67275	C₂×C₄	semidirect
66. #67274	C₂×C₄	semidirect
67. #67273	C₂×C₄	semidirect
68. #67272	C₂×C₄	semidirect
79. C₂×C₂₃⋊C₈	C₂×C₄	semidirect
129. Dic₂₃	#135	semidirect
139. #416	D₂₃	semidirect
176. C₂₃	C₄×#135	semidirect
180. #41	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×#10625121	2
#41×Dic₂₃	2
#13092885	2
#13092509	2
#13092508	2
#13092502	2
#13092497	2
#13092453	2
#13092414	2
C₂×C₈×Dic₂₃	2
C₂₃×#416	2
C₄×#135	23