Subgroups — #15686730

Order 2944.

1556 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#67288	C₂	1	2	yes	maximal, complemented
2	#13092998	C₂	1	4	yes	maximal, complemented
3	#13092993	C₂	1	4	yes	maximal, complemented
4	#13092986	C₂	1	4	yes	maximal, complemented
5	#13092979	C₂	1	4	yes	maximal, complemented
6	#13092967	C₂	1	4	yes	maximal, complemented
7	#13092929	C₂	1	8	yes	maximal, complemented
8	C₂×C₂₃×#117	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
10	C₂²×C₂²⋊D₂₃	C₂²	1	4	yes	—
11	C₂×#67288	—	2	16	no	—
12	C₂×#67288	C₂²	1	4	yes	—
13	C₂×#67288	C₂²	1	4	yes	—
14	C₂×#67288	C₂²	1	4	yes	complemented
15	C₂×#67288	C₂²	1	4	yes	complemented
16	C₂²×D₉₂	C₂²	1	2	yes	complemented
17	C₂²×C₄×D₂₃	C₂²	1	4	yes	complemented
18	C₂²×Dic₄₆	C₂²	1	2	yes	complemented
19	C₂₃×#117	C₂²	1	4	yes	complemented
20	C₂₃×#117	C₂²	1	4	yes	complemented
21	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	complemented
22	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
23	C₂×C₄²×C₂₃	C₂²	1	2	yes	complemented
24	#10625101	C₂²	1	16	yes	complemented
25	#10625068	C₂²	1	16	yes	complemented
26	#10625066	C₂²	1	16	yes	complemented
27	#10625057	C₂²	1	16	yes	complemented
28	#10625053	C₂²	1	16	yes	complemented
29	#10625046	C₂²	1	8	yes	complemented
30	#10625045	C₂²	1	4	yes	complemented
31	C₄×D₉₂	C₂²	1	16	yes	complemented
32	#10625040	C₂²	1	4	yes	complemented
33	C₂⁴×C₂₃	C₂³	1	1	yes	characteristic
34	C₂³×D₂₃	C₂³	1	4	yes	—
35	C₂²×C₄×C₂₃	D₄	1	8	yes	complemented
36	C₂²×C₄×C₂₃	C₂³	1	4	yes	—
37	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
38	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
39	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
40	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
41	C₂×C₂²⋊D₂₃	—	2	16	no	—
42	C₂×C₂²⋊D₂₃	C₂³	1	8	yes	—
43	C₂×C₂²⋊D₂₃	C₂³	1	8	yes	—
44	C₂²×Dic₂₃	C₂³	1	4	yes	—
45	#67288	—	2	16	no	—
46	#67288	—	2	16	no	—
47	#67288	—	2	16	no	—
48	#67288	—	2	16	no	—
49	C₂×D₉₂	—	2	8	no	—
50	C₂×D₉₂	C₂³	1	4	yes	—
51	C₂×D₉₂	C₂³	1	4	yes	—
52	C₂×D₉₂	C₂³	1	16	yes	—
53	C₂×C₄×D₂₃	—	2	16	no	—
54	C₂×C₄×D₂₃	C₂³	1	8	yes	—
55	C₂×C₄×D₂₃	C₂³	1	8	yes	—
56	C₂×Dic₄₆	—	2	8	no	—
57	C₂×Dic₄₆	C₂³	1	4	yes	—
58	C₂×Dic₄₆	C₂³	1	4	yes	—
59	C₄⋊C₄×C₂₃	C₂³	1	8	yes	—
60	C₂²⋊C₄×C₂₃	C₂³	1	8	yes	—
61	C₄²×C₂₃	C₂³	1	4	yes	—
62	C₄²×C₂₃	C₂³	1	4	yes	—
63	(C₂×D₂₃)⋊C₄	C₂³	1	32	yes	—
64	C₄⋊Dic₂₃	C₂³	1	16	yes	—
65	C₂³×C₂₃	—	4	2	no	—
66	C₂³×C₂₃	C₂×D₄	1	4	yes	—
67	C₂³×C₂₃	C₂⁴	1	2	yes	—
68	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
69	C₂²×D₂₃	—	2	8	no	—
70	C₂²×D₂₃	—	4	8	no	—
71	C₂²×D₂₃	C₂⁴	1	8	yes	—
72	C₂×C₄×C₂₃	—	4	4	no	—
73	C₂×C₄×C₂₃	C₂×D₄	1	8	yes	—
74	C₂×C₄×C₂₃	C₂×D₄	1	8	yes	—
75	C₂×C₄×C₂₃	C₂×D₄	1	4	yes	—
76	C₂×C₄×C₂₃	C₂×D₄	1	4	yes	—
77	C₂×C₄×C₂₃	C₂⁴	1	8	yes	—
78	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
79	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
80	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
81	C₂²⋊D₂₃	—	2	16	no	—
82	C₂²⋊D₂₃	—	2	16	no	—
83	C₂×Dic₂₃	—	2	8	no	—
84	C₂×Dic₂₃	C₂⁴	1	8	yes	—
85	D₉₂	—	2	8	no	—
86	D₉₂	—	2	8	no	—
87	D₉₂	—	4	16	no	—
88	C₄×D₂₃	—	2	16	no	—
89	C₄×D₂₃	—	2	16	no	—
90	Dic₄₆	—	2	8	no	—
91	Dic₄₆	—	2	8	no	—
92	#3449	—	23	1	no	maximal, Sylow
93	C₂²×C₂₃	—	2	2	no	—
94	C₂²×C₂₃	—	2	4	no	—
95	C₂²×C₂₃	—	4	4	no	—
96	C₂²×C₂₃	C₂²×D₄	1	4	yes	—
97	C₂²×C₂₃	C₂²×D₄	1	2	yes	—
98	C₂²×C₂₃	C₂⁵	1	1	yes	characteristic
99	C₂×D₂₃	—	2	8	no	—
100	C₂×D₂₃	—	2	16	no	—
101	C₂×D₂₃	—	4	16	no	—
102	C₄×C₂₃	—	2	8	no	—
103	C₄×C₂₃	C₂²×D₄	1	2	yes	—
104	C₄×C₂₃	C₂²×D₄	1	4	yes	—
105	C₄×C₂₃	C₂²×D₄	1	2	yes	—
106	Dic₂₃	—	2	8	no	—
107	C₂²×#41	—	23	2	no	—
108	#537	—	23	4	no	—
109	#536	—	23	4	no	—
110	#535	—	23	4	no	—
111	#534	—	23	4	no	—
112	#532	—	23	4	no	—
113	#525	—	23	8	no	—
114	C₂×#117	D₂₃	1	1	yes	characteristic, complemented
115	C₂×C₂₃	—	2	4	no	—
116	C₂×C₂₃	—	2	2	no	—
117	C₂×C₂₃	C₂³×D₄	1	1	yes	characteristic
118	C₂×C₂₃	#585	1	2	yes	—
119	D₂₃	—	4	8	no	—
120	C₂×#41	—	23	4	no	—
121	C₂×#41	—	23	4	no	—
122	C₂×#41	—	23	4	no	—
123	C₂×#41	—	23	4	no	—
124	C₂×#41	—	46	16	no	—
125	C₂²×Q₈	—	23	2	no	—
126	C₂²×D₄	—	23	4	no	—
127	C₂²×D₄	—	23	2	no	—
128	C₂³×C₄	—	23	4	no	—
129	C₂³×C₄	C₂×D₂₃	1	1	yes	characteristic
130	C₄⋊Q₈	—	23	4	no	—
131	#127	—	23	4	no	—
132	#124	—	23	8	no	—
133	#123	—	23	16	no	—
134	C₂²⋊Q₈	—	23	16	no	—
135	#121	—	23	16	no	—
136	#121	—	23	16	no	—
137	#120	—	23	16	no	—
138	C₄×D₄	—	23	16	no	—
139	#117	C₂×D₂₃	1	4	yes	complemented
140	#117	C₂×D₂₃	1	4	yes	complemented
141	C₂×C₄⋊C₄	C₂×D₂₃	1	2	yes	complemented
142	C₂×C₂²⋊C₄	C₂×D₂₃	1	2	yes	—
143	C₂×C₄²	C₂×D₂₃	1	2	yes	complemented
144	C₂₃	#3449	1	1	yes	characteristic, complemented, Sylow
145	C₂⁴	—	23	4	no	—
146	C₂⁴	C₂²×D₂₃	1	1	yes	characteristic
147	#41	—	46	16	no	—
148	#41	—	46	16	no	—
149	#41	—	46	16	no	—
150	#41	—	46	16	no	—
151	C₂×Q₈	—	23	4	no	—
152	C₂×Q₈	—	23	4	no	—
153	C₂×Q₈	—	46	8	no	—
154	C₂×D₄	—	23	8	no	—
155	C₂×D₄	—	23	8	no	—
156	C₂×D₄	—	23	4	no	—
157	C₂×D₄	—	23	4	no	—
158	C₂×D₄	—	23	16	no	—
159	C₂×D₄	—	46	16	no	—
160	C₂×D₄	—	46	8	no	—
161	C₂²×C₄	—	23	4	no	—
162	C₂²×C₄	—	23	8	no	—
163	C₂²×C₄	—	23	8	no	—
164	C₂²×C₄	—	46	16	no	—
165	C₂²×C₄	D₉₂	1	8	yes	complemented
166	C₂²×C₄	C₂²×D₂₃	1	4	yes	—
167	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
168	C₂²×C₄	C₂²×D₂₃	1	2	yes	—
169	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
170	C₂²×C₄	C₂²×D₂₃	1	2	yes	—
171	C₄⋊C₄	—	23	16	no	—
172	C₄⋊C₄	C₂²×D₂₃	1	8	yes	—
173	C₂²⋊C₄	—	23	32	no	—
174	C₂²⋊C₄	C₂²×D₂₃	1	8	yes	—
175	C₄²	C₂²×D₂₃	1	4	yes	—
176	C₄²	C₂²×D₂₃	1	4	yes	—
177	C₂³	—	4	2	no	—
178	C₂³	—	23	8	no	—
179	C₂³	—	46	8	no	—
180	C₂³	—	92	8	no	—
181	C₂³	C₂×D₉₂	1	4	yes	—
182	C₂³	C₂³×D₂₃	1	2	yes	—
183	C₂³	C₂³×D₂₃	1	1	yes	characteristic
184	Q₈	—	46	8	no	—
185	Q₈	—	46	8	no	—
186	D₄	—	46	16	no	—
187	D₄	—	46	16	no	—
188	D₄	—	46	8	no	—
189	D₄	—	46	8	no	—
190	D₄	—	92	16	no	—
191	C₂×C₄	—	4	4	no	—
192	C₂×C₄	—	23	8	no	—
193	C₂×C₄	—	46	8	no	—
194	C₂×C₄	—	46	16	no	—
195	C₂×C₄	—	46	16	no	—
196	C₂×C₄	C₂×D₉₂	1	8	yes	—
197	C₂×C₄	C₂×D₉₂	1	8	yes	—
198	C₂×C₄	C₂×D₉₂	1	4	yes	—
199	C₂×C₄	C₂×D₉₂	1	4	yes	—
200	C₂×C₄	C₂³×D₂₃	1	8	yes	—
201	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
202	C₂×C₄	C₂³×D₂₃	1	2	yes	—
203	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
204	C₂²	—	2	2	no	—
205	C₂²	—	2	4	no	—
206	C₂²	—	4	4	no	—
207	C₂²	—	46	8	no	—
208	C₂²	—	46	16	no	—
209	C₂²	—	92	16	no	—
210	C₂²	C₂²×D₉₂	1	4	yes	—
211	C₂²	C₂²×D₉₂	1	2	yes	—
212	C₂²	C₂⁴×D₂₃	1	1	yes	characteristic
213	C₄	—	2	8	no	—
214	C₄	—	46	8	no	—
215	C₄	C₂²×D₉₂	1	2	yes	—
216	C₄	C₂²×D₉₂	1	4	yes	—
217	C₄	C₂²×D₉₂	1	2	yes	—
218	C₂	—	2	4	no	—
219	C₂	—	2	2	no	—
220	C₂	—	92	8	no	—
221	C₂	C₂³×D₉₂	1	1	yes	characteristic
222	C₂	#13093253	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂³×D₂₃
commutator, socle, cosocle	C₂²×C₂₃	C₂⁵
Frattini	C₂²	C₂⁴×D₂₃
Fitting	C₂×C₂₃×#117	C₂
hypercenter	C₂×#117	D₂₃
hypocenter	C₂₃	#3449
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
1. C₂²×#67288	C₂	semidirect
2. #13092998	C₂	semidirect
3. #13092993	C₂	semidirect
4. #13092986	C₂	semidirect
5. #13092979	C₂	semidirect
6. #13092967	C₂	semidirect
7. #13092929	C₂	semidirect
8. C₂×C₂₃×#117	C₂	semidirect
14. C₂×#67288	C₂²	semidirect
15. C₂×#67288	C₂²	semidirect
16. C₂²×D₉₂	C₂²	semidirect
17. C₂²×C₄×D₂₃	C₂²	semidirect
18. C₂²×Dic₄₆	C₂²	semidirect
19. C₂₃×#117	C₂²	semidirect
20. C₂₃×#117	C₂²	semidirect
21. C₂×C₄⋊C₄×C₂₃	C₂²	semidirect
23. C₂×C₄²×C₂₃	C₂²	semidirect
24. #10625101	C₂²	semidirect
25. #10625068	C₂²	semidirect
26. #10625066	C₂²	semidirect
27. #10625057	C₂²	semidirect
28. #10625053	C₂²	semidirect
29. #10625046	C₂²	semidirect
30. #10625045	C₂²	semidirect
31. C₄×D₉₂	C₂²	semidirect
32. #10625040	C₂²	semidirect
35. C₂²×C₄×C₂₃	D₄	semidirect
114. C₂×#117	D₂₃	semidirect
139. #117	C₂×D₂₃	semidirect
140. #117	C₂×D₂₃	semidirect
141. C₂×C₄⋊C₄	C₂×D₂₃	semidirect
143. C₂×C₄²	C₂×D₂₃	semidirect
144. C₂₃	#3449	semidirect
165. C₂²×C₄	D₉₂	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#67288	2
#13092998	2
#13092993	2
#13092986	2
#13092979	2
#13092967	2
#13092929	2
C₂×C₂₃×#117	2
#3449	23