Subgroups — #15682107

Order 2944.

500 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10625072	C₂	1	1	yes	maximal, characteristic, complemented
2	#13092963	C₂	1	1	yes	maximal, characteristic
3	#13092493	C₂	1	4	yes	maximal, complemented
4	C₈×C₂²⋊D₂₃	C₂	1	4	yes	maximal, complemented
5	#13092440	C₂	1	1	yes	maximal, characteristic, complemented
6	#13092385	C₂	1	1	yes	maximal, characteristic, complemented
7	#13092384	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₈×Dic₂₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₂₃×#131	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#41×C₂₃	C₂²	1	1	yes	characteristic
11	C₂×#67288	C₂²	1	1	yes	characteristic
12	C₂₃×#131	—	2	4	no	—
13	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
14	C₂×M₁₆×C₂₃	C₂²	1	2	yes	—
15	C₂²×C₈×C₂₃	C₂²	1	2	yes	—
16	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
17	#10625118	C₄	1	1	yes	characteristic
18	#10625115	C₄	1	1	yes	characteristic
19	#10625111	C₄	1	1	yes	characteristic
20	#10625110	C₄	1	4	yes	—
21	#10625107	C₄	1	1	yes	characteristic
22	C₄×C₂²⋊D₂₃	C₂²	1	4	yes	—
23	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
24	C₂×#67267	C₂²	1	1	yes	characteristic
25	C₂²×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
26	#10625072	—	2	4	no	—
27	C₂×#67262	C₂²	1	2	yes	—
28	C₂×C₈×D₂₃	C₂²	1	2	yes	—
29	C₂²⋊(C₂₃⋊C₈)	C₂²	1	4	yes	—
30	(C₂×D₂₃)⋊C₈	C₂²	1	4	yes	—
31	Dic₂₃⋊C₈	C₂²	1	4	yes	—
32	C₈×Dic₂₃	C₂²	1	2	yes	—
33	C₈×Dic₂₃	C₂²	1	2	yes	—
34	#41×C₂₃	—	2	2	no	—
35	#41×C₂₃	—	2	2	no	—
36	C₂×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
37	C₂×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
38	C₂×D₄×C₂₃	C₂×C₄	1	2	yes	—
39	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
40	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
41	C₂×C₂²⋊D₂₃	C₂×C₄	1	4	yes	—
42	C₂×C₂²⋊D₂₃	C₂×C₄	1	2	yes	—
43	C₂²×Dic₂₃	C₂×C₄	1	2	yes	—
44	#67288	—	2	2	no	—
45	#67288	—	2	2	no	—
46	C₂×D₉₂	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄×D₂₃	C₂³	1	2	yes	—
48	C₂×Dic₄₆	C₂×C₄	1	1	yes	characteristic
49	M₁₆×C₂₃	—	2	2	no	—
50	M₁₆×C₂₃	—	2	4	no	—
51	C₂×C₈×C₂₃	—	2	4	no	—
52	C₂×C₈×C₂₃	D₄	1	4	yes	complemented
53	C₂×C₈×C₂₃	C₂³	1	2	yes	—
54	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
55	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
56	C₂²⋊Dic₂₃	C₂×C₄	1	4	yes	—
57	(C₂×D₂₃)⋊C₄	C₂×C₄	1	4	yes	—
58	Dic₂₃⋊C₄	C₂×C₄	1	4	yes	—
59	C₄×Dic₂₃	C₂×C₄	1	1	yes	characteristic
60	C₄×Dic₂₃	C₂×C₄	1	1	yes	characteristic
61	C₄×Dic₂₃	C₂³	1	2	yes	—
62	#67267	—	2	2	no	—
63	C₂×C₂₃⋊C₈	—	2	2	no	—
64	C₂×C₂₃⋊C₈	C₂³	1	2	yes	—
65	C₂×C₂₃⋊C₈	C₂³	1	2	yes	—
66	#67262	—	2	4	no	—
67	C₈×D₂₃	—	2	4	no	—
68	C₂³×C₂₃	C₂²×C₄	1	2	yes	—
69	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
70	C₂²×D₂₃	C₂²×C₄	1	2	yes	—
71	Q₈×C₂₃	—	2	2	no	—
72	D₄×C₂₃	—	2	4	no	—
73	D₄×C₂₃	—	2	2	no	—
74	C₂×C₄×C₂₃	—	2	2	no	—
75	C₂×C₄×C₂₃	—	2	2	no	—
76	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
77	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
79	C₂×C₄×C₂₃	#41	1	2	yes	—
80	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
81	C₂²⋊D₂₃	—	2	4	no	—
82	C₂²⋊D₂₃	—	2	8	no	—
83	C₂×Dic₂₃	—	2	4	no	—
84	C₂×Dic₂₃	C₂²×C₄	1	2	yes	—
85	C₂×Dic₂₃	C₂²×C₄	1	2	yes	—
86	C₂×Dic₂₃	C₂²×C₄	1	2	yes	—
87	D₉₂	—	2	2	no	—
88	C₄×D₂₃	—	2	2	no	—
89	C₄×D₂₃	—	2	2	no	—
90	Dic₄₆	—	2	2	no	—
91	C₈×C₂₃	—	2	2	no	—
92	C₈×C₂₃	C₂×D₄	1	2	yes	—
93	C₈×C₂₃	C₂×D₄	1	2	yes	—
94	C₂₃⋊C₈	—	2	2	no	—
95	C₂₃⋊C₈	—	2	2	no	—
96	#2931	—	23	1	no	maximal, Sylow
97	C₂²×C₂₃	—	2	2	no	—
98	C₂²×C₂₃	—	2	2	no	—
99	C₂²×C₂₃	—	2	4	no	—
100	C₂²×C₂₃	C₄×D₄	1	2	yes	—
101	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
102	C₂×D₂₃	—	2	2	no	—
103	C₂×D₂₃	—	2	4	no	—
104	C₄×C₂₃	—	2	2	no	—
105	C₄×C₂₃	C₄×D₄	1	2	yes	—
106	C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
107	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
108	Dic₂₃	—	2	2	no	—
109	Dic₂₃	—	2	2	no	—
110	Dic₂₃	#131	1	4	yes	complemented
111	C₂×#131	—	23	1	no	—
112	C₂×#131	D₂₃	1	1	yes	characteristic, complemented
113	#532	—	23	1	no	—
114	#436	—	23	4	no	—
115	C₈×D₄	—	23	4	no	—
116	#423	—	23	1	no	—
117	#408	—	23	1	no	—
118	#408	—	23	1	no	—
119	C₂×C₄×C₈	—	23	1	no	—
120	C₂×C₂₃	—	2	2	no	—
121	C₂×C₂₃	—	4	2	no	—
122	C₂×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
123	C₂×C₂₃	C₂×#131	1	2	yes	—
124	D₂₃	—	4	2	no	—
125	C₂×#41	—	23	1	no	—
126	C₂×#41	C₂×D₂₃	1	1	yes	characteristic
127	#131	—	2	4	no	—
128	#131	—	46	4	no	—
129	C₂×M₁₆	—	23	1	no	—
130	C₂×M₁₆	—	23	2	no	—
131	C₂×M₁₆	C₂×D₂₃	1	1	yes	characteristic
132	C₂×M₁₆	C₂×D₂₃	1	2	yes	—
133	C₂²×C₈	—	23	1	no	—
134	C₂²×C₈	—	23	2	no	—
135	C₂²×C₈	C₂×D₂₃	1	2	yes	—
136	C₂²×C₈	C₂×D₂₃	1	1	yes	characteristic
137	C₄⋊Q₈	—	23	1	no	—
138	#127	—	23	1	no	—
139	#124	—	23	1	no	—
140	#124	—	23	1	no	—
141	#121	—	23	4	no	—
142	C₄×D₄	—	23	4	no	—
143	C₂×C₄²	—	23	1	no	—
144	C₄⋊C₈	—	23	4	no	—
145	C₂²⋊C₈	—	23	4	no	—
146	C₂²⋊C₈	—	23	4	no	—
147	C₄×C₈	—	23	2	no	—
148	C₄×C₈	—	23	2	no	—
149	C₂₃	#2931	1	1	yes	characteristic, complemented, Sylow
150	#41	—	2	2	no	—
151	#41	—	2	2	no	—
152	#41	—	46	2	no	—
153	#41	—	46	2	no	—
154	C₂×Q₈	—	23	1	no	—
155	C₂×Q₈	C₄×D₂₃	1	1	yes	characteristic
156	C₂×D₄	—	23	4	no	—
157	C₂×D₄	—	23	2	no	—
158	C₂×D₄	—	23	1	no	—
159	C₂×D₄	C₄×D₂₃	1	2	yes	—
160	C₂×D₄	C₄×D₂₃	1	1	yes	characteristic
161	C₂²×C₄	—	23	2	no	—
162	C₂²×C₄	—	23	2	no	—
163	C₂²×C₄	C₂²×D₂₃	1	2	yes	—
164	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
165	M₁₆	—	2	2	no	—
166	M₁₆	—	2	4	no	—
167	M₁₆	—	46	2	no	—
168	M₁₆	—	46	4	no	—
169	C₂×C₈	—	2	4	no	—
170	C₂×C₈	—	23	2	no	—
171	C₂×C₈	—	23	2	no	—
172	C₂×C₈	—	46	2	no	—
173	C₂×C₈	—	46	4	no	—
174	C₂×C₈	C₂²⋊D₂₃	1	4	yes	complemented
175	C₂×C₈	C₂²×D₂₃	1	2	yes	—
176	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
177	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
178	C₄⋊C₄	—	23	4	no	—
179	C₂²⋊C₄	—	23	4	no	—
180	C₂²⋊C₄	—	23	4	no	—
181	C₄²	—	23	1	no	—
182	C₄²	—	23	1	no	—
183	C₄²	—	23	2	no	—
184	C₂³	—	23	2	no	—
185	C₂³	C₂×C₄×D₂₃	1	2	yes	—
186	C₂³	C₂×C₄×D₂₃	1	1	yes	characteristic
187	Q₈	—	2	2	no	—
188	Q₈	—	46	2	no	—
189	D₄	—	2	4	no	—
190	D₄	—	2	2	no	—
191	D₄	—	46	8	no	—
192	D₄	—	46	4	no	—
193	D₄	—	46	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	23	2	no	—
197	C₂×C₄	—	23	2	no	—
198	C₂×C₄	—	23	2	no	—
199	C₂×C₄	—	46	4	no	—
200	C₂×C₄	—	46	2	no	—
201	C₂×C₄	—	46	2	no	—
202	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
203	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
204	C₂×C₄	#67288	1	2	yes	—
205	C₂×C₄	C₂×C₂²⋊D₂₃	1	2	yes	—
206	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
207	C₈	—	2	2	no	—
208	C₈	—	46	2	no	—
209	C₈	—	46	2	no	—
210	C₈	C₂×C₂²⋊D₂₃	1	2	yes	—
211	C₈	C₂×C₂²⋊D₂₃	1	2	yes	—
212	C₂²	—	2	2	no	—
213	C₂²	—	2	2	no	—
214	C₂²	—	2	4	no	—
215	C₂²	—	46	2	no	—
216	C₂²	—	46	4	no	—
217	C₂²	C₄×C₂²⋊D₂₃	1	2	yes	—
218	C₂²	C₂²×C₄×D₂₃	1	1	yes	characteristic
219	C₄	—	2	2	no	—
220	C₄	—	23	4	no	—
221	C₄	—	46	2	no	—
222	C₄	—	46	2	no	—
223	C₄	C₄×C₂²⋊D₂₃	1	2	yes	—
224	C₄	C₂×#67288	1	1	yes	characteristic
225	C₄	C₂²×C₂²⋊D₂₃	1	1	yes	characteristic
226	C₂	—	2	2	no	—
227	C₂	—	4	2	no	—
228	C₂	—	92	2	no	—
229	C₂	C₂×C₄×C₂²⋊D₂₃	1	1	yes	characteristic
230	C₂	#131×D₂₃	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂²×D₂₃
commutator, socle	C₂²×C₂₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₂₃
Fitting	C₂×C₂₃×#131	C₂
prime radical	#13092963	C₂
cosocle	C₂×C₄×C₂₃	C₂⁴
hypercenter	C₂×#131	D₂₃
hypocenter	C₂₃	#2931
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#131

Complements

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

N	H ≅ G/N	type
1. C₂×#10625072	C₂	semidirect
3. #13092493	C₂	semidirect
4. C₈×C₂²⋊D₂₃	C₂	semidirect
5. #13092440	C₂	semidirect
6. #13092385	C₂	semidirect
7. #13092384	C₂	semidirect
8. C₂×C₈×Dic₂₃	C₂	semidirect
9. C₂×C₂₃×#131	C₂	semidirect
52. C₂×C₈×C₂₃	D₄	semidirect
110. Dic₂₃	#131	semidirect
112. C₂×#131	D₂₃	semidirect
149. C₂₃	#2931	semidirect
174. C₂×C₈	C₂²⋊D₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10625072	2
#13092963	2
#13092493	2
C₈×C₂²⋊D₂₃	2
#13092440	2
#13092385	2
#13092384	2
C₂×C₈×Dic₂₃	2
C₂×C₂₃×#131	2
#2931	23