Subgroups — #15686996

Order 2944.

1112 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13093153	C₂	1	3	yes	maximal, complemented
2	#13093146	C₂	1	1	yes	maximal, characteristic, complemented
3	#13093137	C₂	1	3	yes	maximal, complemented
4	#13093114	C₂	1	6	yes	maximal, complemented
5	#13093074	C₂	1	2	yes	maximal, complemented
6	#13092965	C₂	1	3	yes	maximal, complemented
7	#13092948	C₂	1	6	yes	maximal, complemented
8	#41×Dic₂₃	C₂	1	2	yes	maximal, complemented
9	C₄×#67290	C₂	1	3	yes	maximal, complemented
10	C₄×Q₈×D₂₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#550	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₂₃	C₂²	1	2	yes	—
13	C₂×Q₈×D₂₃	C₂²	1	1	yes	characteristic
14	C₂×#67290	C₂²	1	3	yes	—
15	C₂₃×#127	C₂²	1	3	yes	—
16	C₂₃×#121	C₂²	1	6	yes	—
17	C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
18	C₄×D₄×C₂₃	C₂²	1	3	yes	—
19	Q₈×Dic₂₃	C₂²	1	1	yes	characteristic
20	Q₈×Dic₂₃	C₂²	1	2	yes	—
21	#10625109	C₂²	1	6	yes	—
22	#10625109	C₂²	1	6	yes	—
23	#10625107	C₂²	1	6	yes	—
24	#10625107	C₂²	1	6	yes	—
25	#10625106	C₂²	1	12	yes	—
26	D₄×Dic₂₃	C₂²	1	3	yes	—
27	D₄×Dic₂₃	C₂²	1	6	yes	—
28	D₄×Dic₂₃	C₂²	1	6	yes	—
29	D₄×Dic₂₃	C₂²	1	6	yes	—
30	C₄×C₂²⋊D₂₃	C₂²	1	6	yes	—
31	#10625097	C₂²	1	6	yes	—
32	#10625095	C₂²	1	6	yes	—
33	C₂×C₄×Dic₂₃	C₂²	1	6	yes	—
34	#10625063	C₂²	1	1	yes	characteristic
35	C₄⋊C₄×D₂₃	C₂²	1	3	yes	—
36	#10625061	C₂²	1	6	yes	—
37	#10625058	C₂²	1	3	yes	—
38	#10625058	C₂²	1	3	yes	—
39	(C₂²⋊D₂₃)⋊C₄	C₂²	1	6	yes	—
40	#10625050	C₂²	1	12	yes	—
41	#10625048	C₂²	1	6	yes	—
42	C₄²×D₂₃	C₂²	1	3	yes	—
43	#10625040	C₂²	1	3	yes	—
44	C₄×Dic₄₆	C₂²	1	3	yes	—
45	C₄×Dic₄₆	C₂²	1	3	yes	—
46	#41×C₂₃	—	2	8	no	—
47	C₂×Q₈×C₂₃	C₂³	1	1	yes	characteristic
48	C₂×D₄×C₂₃	C₂³	1	6	yes	—
49	C₂×D₄×C₂₃	C₂³	1	3	yes	—
50	C₂×D₄×C₂₃	C₂³	1	6	yes	—
51	C₂²×C₄×C₂₃	C₂³	1	6	yes	—
52	C₂×C₂²⋊D₂₃	C₂³	1	6	yes	—
53	C₂²×Dic₂₃	C₂³	1	6	yes	—
54	C₂²×Dic₂₃	C₂³	1	6	yes	—
55	Q₈×D₂₃	—	2	4	no	—
56	#67290	—	2	12	no	—
57	C₂×C₄×D₂₃	C₂³	1	3	yes	—
58	C₂×C₄×D₂₃	C₂³	1	3	yes	—
59	C₂×C₄×D₂₃	C₂³	1	1	yes	characteristic
60	C₂×Dic₄₆	C₂³	1	3	yes	—
61	C₂×Dic₄₆	C₂³	1	3	yes	—
62	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
63	C₄⋊C₄×C₂₃	C₂³	1	3	yes	—
64	C₂²⋊C₄×C₂₃	C₂³	1	6	yes	—
65	C₄²×C₂₃	C₂³	1	3	yes	—
66	C₂²⋊Dic₂₃	C₂³	1	6	yes	—
67	C₂²⋊Dic₂₃	C₂³	1	12	yes	—
68	C₂²⋊Dic₂₃	C₂³	1	12	yes	—
69	(C₂×D₂₃)⋊C₄	C₂³	1	2	yes	—
70	C₄⋊Dic₂₃	C₂³	1	1	yes	characteristic
71	C₄⋊Dic₂₃	C₂³	1	3	yes	—
72	C₄⋊Dic₂₃	C₂³	1	6	yes	—
73	C₄⋊Dic₂₃	C₂³	1	6	yes	—
74	Dic₂₃⋊C₄	C₂³	1	6	yes	—
75	Dic₂₃⋊C₄	C₂³	1	6	yes	—
76	Dic₂₃⋊C₄	C₂³	1	6	yes	—
77	C₄×Dic₂₃	C₂³	1	3	yes	—
78	C₄×Dic₂₃	C₂³	1	6	yes	—
79	C₄×Dic₂₃	C₂³	1	3	yes	—
80	C₄×Dic₂₃	C₂³	1	3	yes	—
81	C₄×Dic₂₃	C₂³	1	2	yes	—
82	C₄×Dic₂₃	C₂³	1	6	yes	—
83	C₂³×C₂₃	C₂⁴	1	6	yes	—
84	C₂²×D₂₃	C₂⁴	1	1	yes	characteristic
85	Q₈×C₂₃	#41	1	4	yes	complemented
86	D₄×C₂₃	—	2	6	no	—
87	D₄×C₂₃	—	2	12	no	—
88	D₄×C₂₃	—	4	6	no	—
89	C₂×C₄×C₂₃	—	2	12	no	—
90	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
91	C₂×C₄×C₂₃	C₂⁴	1	3	yes	—
92	C₂×C₄×C₂₃	C₂⁴	1	3	yes	—
93	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
94	C₂²⋊D₂₃	—	2	12	no	—
95	C₂×Dic₂₃	—	2	12	no	—
96	C₂×Dic₂₃	—	4	6	no	—
97	C₂×Dic₂₃	C₂⁴	1	2	yes	—
98	C₂×Dic₂₃	C₂⁴	1	3	yes	—
99	C₂×Dic₂₃	C₂⁴	1	3	yes	—
100	C₂×Dic₂₃	C₂⁴	1	6	yes	—
101	C₂×Dic₂₃	C₂⁴	1	1	yes	characteristic
102	C₄×D₂₃	—	2	6	no	—
103	C₄×D₂₃	—	4	3	no	—
104	C₄×D₂₃	#41	1	4	yes	complemented
105	Dic₄₆	—	2	6	no	—
106	Dic₄₆	—	2	6	no	—
107	#3480	—	23	1	no	maximal, Sylow
108	C₂²×C₂₃	—	2	6	no	—
109	C₂²×C₂₃	—	2	6	no	—
110	C₂²×C₂₃	—	2	6	no	—
111	C₂²×C₂₃	C₂⁵	1	1	yes	characteristic
112	C₂×D₂₃	—	2	1	no	—
113	C₂×D₂₃	—	2	1	no	—
114	C₂×D₂₃	C₂×#41	1	2	yes	—
115	C₄×C₂₃	—	2	2	no	—
116	C₄×C₂₃	—	2	3	no	—
117	C₄×C₂₃	C₂×#41	1	6	yes	—
118	C₄×C₂₃	C₂×#41	1	2	yes	—
119	Dic₂₃	—	2	3	no	—
120	Dic₂₃	—	2	6	no	—
121	Dic₂₃	—	2	3	no	—
122	Dic₂₃	—	2	2	no	—
123	Dic₂₃	C₂×#41	1	2	yes	—
124	#559	—	23	3	no	—
125	#557	—	23	1	no	—
126	#556	—	23	3	no	—
127	#553	—	23	6	no	—
128	#550	D₂₃	1	1	yes	characteristic, complemented
129	#548	—	23	2	no	—
130	#532	—	23	3	no	—
131	#529	—	23	6	no	—
132	C₄×#41	—	23	2	no	—
133	C₄×#41	—	23	3	no	—
134	C₂×C₄×Q₈	—	23	1	no	—
135	C₂×C₂₃	—	4	6	no	—
136	C₂×C₂₃	C₂²×#41	1	1	yes	characteristic
137	C₂×C₂₃	C₂²×#41	1	1	yes	characteristic
138	C₂×C₂₃	#585	1	1	yes	characteristic
139	D₂₃	—	2	2	no	—
140	C₂×#41	—	23	3	no	—
141	C₂×#41	C₂×D₂₃	1	2	yes	—
142	C₂²×Q₈	—	23	1	no	—
143	C₄⋊Q₈	—	23	3	no	—
144	#127	C₂×D₂₃	1	3	yes	—
145	#126	—	23	12	no	—
146	#125	—	23	6	no	—
147	#124	—	23	6	no	—
148	#124	—	23	6	no	—
149	#123	—	23	12	no	—
150	C₂²⋊Q₈	—	23	6	no	—
151	#121	—	23	6	no	—
152	#121	—	23	6	no	—
153	#121	C₂×D₂₃	1	6	yes	—
154	C₄×Q₈	—	23	1	no	—
155	C₄×Q₈	—	23	2	no	—
156	C₄×Q₈	—	23	3	no	—
157	C₄×Q₈	—	23	3	no	—
158	C₄×Q₈	—	23	3	no	—
159	C₄×Q₈	—	23	3	no	—
160	C₄×Q₈	C₂×D₂₃	1	1	yes	characteristic
161	C₄×D₄	—	23	6	no	—
162	C₄×D₄	—	23	6	no	—
163	C₄×D₄	—	23	3	no	—
164	C₄×D₄	—	23	6	no	—
165	C₄×D₄	—	23	6	no	—
166	C₄×D₄	—	23	6	no	—
167	C₄×D₄	C₂×D₂₃	1	3	yes	—
168	#117	—	23	6	no	—
169	#117	—	23	1	no	—
170	#117	—	23	6	no	—
171	C₂×C₄⋊C₄	—	23	3	no	—
172	C₂×C₄²	—	23	6	no	—
173	C₂×C₄²	—	23	3	no	—
174	C₂₃	#3480	1	1	yes	characteristic, complemented, Sylow
175	#41	—	2	8	no	—
176	#41	—	46	12	no	—
177	C₂×Q₈	—	23	3	no	—
178	C₂×Q₈	—	23	3	no	—
179	C₂×Q₈	—	46	4	no	—
180	C₂×Q₈	C₂²×D₂₃	1	1	yes	characteristic
181	C₂×D₄	—	23	6	no	—
182	C₂×D₄	C₂²×D₂₃	1	6	yes	—
183	C₂×D₄	C₂²×D₂₃	1	3	yes	—
184	C₂×D₄	C₂²×D₂₃	1	6	yes	—
185	C₂²×C₄	—	23	1	no	—
186	C₂²×C₄	—	23	6	no	—
187	C₂²×C₄	—	23	6	no	—
188	C₂²×C₄	—	23	3	no	—
189	C₂²×C₄	—	23	3	no	—
190	C₂²×C₄	C₂²×D₂₃	1	6	yes	—
191	C₄⋊C₄	—	23	3	no	—
192	C₄⋊C₄	—	23	6	no	—
193	C₄⋊C₄	—	23	1	no	—
194	C₄⋊C₄	—	23	6	no	—
195	C₄⋊C₄	—	23	6	no	—
196	C₄⋊C₄	—	23	6	no	—
197	C₄⋊C₄	—	23	6	no	—
198	C₄⋊C₄	C₂²×D₂₃	1	3	yes	—
199	C₄⋊C₄	C₂²×D₂₃	1	1	yes	characteristic
200	C₂²⋊C₄	—	23	12	no	—
201	C₂²⋊C₄	—	23	6	no	—
202	C₂²⋊C₄	—	23	12	no	—
203	C₂²⋊C₄	—	23	2	no	—
204	C₂²⋊C₄	C₂²×D₂₃	1	6	yes	—
205	C₄²	—	23	2	no	—
206	C₄²	—	23	3	no	—
207	C₄²	—	23	3	no	—
208	C₄²	—	23	6	no	—
209	C₄²	—	23	6	no	—
210	C₄²	—	23	3	no	—
211	C₄²	C₂²×D₂₃	1	3	yes	—
212	C₂³	—	23	1	no	—
213	C₂³	C₂³×D₂₃	1	6	yes	—
214	Q₈	—	46	6	no	—
215	Q₈	—	46	6	no	—
216	Q₈	#67290	1	4	yes	complemented
217	D₄	—	2	12	no	—
218	D₄	—	2	6	no	—
219	D₄	—	4	6	no	—
220	D₄	—	46	12	no	—
221	C₂×C₄	—	2	12	no	—
222	C₂×C₄	—	23	1	no	—
223	C₂×C₄	—	23	4	no	—
224	C₂×C₄	—	23	2	no	—
225	C₂×C₄	—	23	3	no	—
226	C₂×C₄	—	23	3	no	—
227	C₂×C₄	—	23	6	no	—
228	C₂×C₄	—	46	12	no	—
229	C₂×C₄	—	46	6	no	—
230	C₂×C₄	—	92	3	no	—
231	C₂×C₄	—	92	6	no	—
232	C₂×C₄	C₂³×D₂₃	1	2	yes	—
233	C₂×C₄	C₂³×D₂₃	1	3	yes	—
234	C₂×C₄	C₂³×D₂₃	1	3	yes	—
235	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
236	C₂²	—	2	6	no	—
237	C₂²	—	2	6	no	—
238	C₂²	—	2	6	no	—
239	C₂²	—	23	2	no	—
240	C₂²	—	46	1	no	—
241	C₂²	—	46	1	no	—
242	C₂²	C₂⁴×D₂₃	1	1	yes	characteristic
243	C₄	—	2	2	no	—
244	C₄	—	2	3	no	—
245	C₄	—	23	2	no	—
246	C₄	—	46	6	no	—
247	C₄	—	46	3	no	—
248	C₄	—	46	3	no	—
249	C₄	—	46	2	no	—
250	C₄	C₂×#67290	1	6	yes	—
251	C₄	#41×D₂₃	1	2	yes	—
252	C₂	—	4	6	no	—
253	C₂	—	46	2	no	—
254	C₂	C₂×#41×D₂₃	1	1	yes	characteristic
255	C₂	C₂²×#67290	1	1	yes	characteristic
256	C₂	#13093252	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₂₃
commutator, socle, cosocle	C₂²×C₂₃	C₂⁵
Fitting	C₂₃×#550	C₂
hypercenter	#550	D₂₃
hypocenter	C₂₃	#3480
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ #550

Complements

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

N	H ≅ G/N	type
1. #13093153	C₂	semidirect
2. #13093146	C₂	semidirect
3. #13093137	C₂	semidirect
4. #13093114	C₂	semidirect
5. #13093074	C₂	semidirect
6. #13092965	C₂	semidirect
7. #13092948	C₂	semidirect
8. #41×Dic₂₃	C₂	semidirect
9. C₄×#67290	C₂	semidirect
10. C₄×Q₈×D₂₃	C₂	semidirect
11. C₂₃×#550	C₂	semidirect
85. Q₈×C₂₃	#41	semidirect
104. C₄×D₂₃	#41	semidirect
128. #550	D₂₃	semidirect
174. C₂₃	#3480	semidirect
216. Q₈	#67290	semidirect

Maximal subgroups

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

subgroup	index
#13093153	2
#13093146	2
#13093137	2
#13093114	2
#13093074	2
#13092965	2
#13092948	2
#41×Dic₂₃	2
C₄×#67290	2
C₄×Q₈×D₂₃	2
C₂₃×#550	2
#3480	23