Subgroups — #15682358

Order 2944.

412 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#67288	C₂	1	1	yes	maximal, characteristic
2	#13092544	C₂	1	2	yes	maximal, complemented
3	#13092494	C₂	1	2	yes	maximal, complemented
4	#13092485	C₂	1	4	yes	maximal, complemented
5	#13092468	C₂	1	2	yes	maximal, complemented
6	C₄×#67262	C₂	1	2	yes	maximal, complemented
7	C₂×C₄×C₂₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#431	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#67288	C₄	1	1	yes	characteristic
10	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
11	C₄×C₂²⋊D₂₃	C₄	1	2	yes	—
12	C₄×C₂²⋊D₂₃	C₂²	1	2	yes	—
13	#10625097	C₄	1	1	yes	characteristic
14	C₂²×C₂₃⋊C₈	C₂²	1	2	yes	—
15	C₂×#67262	C₂²	1	4	yes	—
16	C₄×D₉₂	C₄	1	1	yes	characteristic
17	C₄×D₉₂	C₂²	1	1	yes	characteristic
18	#10625043	C₄	1	2	yes	—
19	C₄²×D₂₃	C₂²	1	2	yes	—
20	C₄×Dic₄₆	C₄	1	1	yes	characteristic
21	C₄×Dic₄₆	C₂²	1	1	yes	characteristic
22	C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
23	C₂₃×C₂²⋊C₈	C₂²	1	2	yes	—
24	C₄×C₈×C₂₃	C₂²	1	2	yes	—
25	(C₂×D₂₃)⋊C₈	C₂²	1	4	yes	—
26	#10624997	C₂²	1	4	yes	—
27	Dic₂₃⋊C₈	C₂²	1	4	yes	—
28	C₄×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
29	C₄×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
30	C₄×C₂₃⋊C₈	C₂²	1	2	yes	—
31	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
34	C₂×C₂²⋊D₂₃	C₂×C₄	1	2	yes	—
35	#67288	—	2	4	no	—
36	C₂×D₉₂	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄×D₂₃	C₂×C₄	1	2	yes	—
38	C₂×C₄×D₂₃	C₂×C₄	1	2	yes	—
39	C₂×C₄×D₂₃	C₂³	1	2	yes	—
40	C₂×Dic₄₆	C₂×C₄	1	1	yes	characteristic
41	C₂×C₈×C₂₃	C₂³	1	4	yes	—
42	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
43	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
44	C₄²×C₂₃	C₂³	1	1	yes	characteristic
45	C₄²×C₂₃	C₂³	1	1	yes	characteristic
46	C₂²⋊Dic₂₃	C₂×C₄	1	2	yes	—
47	(C₂×D₂₃)⋊C₄	C₂×C₄	1	2	yes	—
48	(C₂×D₂₃)⋊C₄	C₂×C₄	1	2	yes	—
49	C₄⋊Dic₂₃	C₂×C₄	1	1	yes	characteristic
50	C₄⋊Dic₂₃	C₂×C₄	1	1	yes	characteristic
51	Dic₂₃⋊C₄	C₂×C₄	1	2	yes	—
52	Dic₂₃⋊C₄	C₂×C₄	1	2	yes	—
53	C₄×Dic₂₃	C₂×C₄	1	2	yes	—
54	C₄×Dic₂₃	C₂×C₄	1	2	yes	—
55	C₄×Dic₂₃	C₂³	1	2	yes	—
56	C₂×C₂₃⋊C₈	—	2	4	no	—
57	C₂×C₂₃⋊C₈	C₂³	1	2	yes	—
58	C₂×C₂₃⋊C₈	C₂³	1	2	yes	—
59	#67262	—	2	8	no	—
60	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
61	C₂²×D₂₃	C₂²×C₄	1	2	yes	—
62	C₂×C₄×C₂₃	—	2	2	no	—
63	C₂×C₄×C₂₃	—	2	2	no	—
64	C₂×C₄×C₂₃	M₁₆	1	4	yes	complemented
65	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
71	C₂²⋊D₂₃	—	2	4	no	—
72	C₂×Dic₂₃	C₂²×C₄	1	2	yes	—
73	C₂×Dic₂₃	C₂²×C₄	1	2	yes	—
74	C₂×Dic₂₃	C₂²×C₄	1	2	yes	—
75	D₉₂	—	2	2	no	—
76	C₄×D₂₃	—	2	4	no	—
77	C₄×D₂₃	—	2	4	no	—
78	C₄×D₂₃	—	2	4	no	—
79	Dic₄₆	—	2	2	no	—
80	C₈×C₂₃	—	2	4	no	—
81	C₂₃⋊C₈	—	2	2	no	—
82	C₂₃⋊C₈	#41	1	4	yes	complemented
83	#2965	—	23	1	no	maximal, Sylow
84	C₂²×C₂₃	—	2	1	no	—
85	C₂²×C₂₃	—	2	1	no	—
86	C₂²×C₂₃	C₂×M₁₆	1	2	yes	—
87	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
88	C₂×D₂₃	—	2	2	no	—
89	C₂×D₂₃	—	2	2	no	—
90	C₂×D₂₃	—	2	2	no	—
91	C₄×C₂₃	—	2	1	no	—
92	C₄×C₂₃	—	2	1	no	—
93	C₄×C₂₃	C₂×M₁₆	1	2	yes	—
94	C₄×C₂₃	C₂×M₁₆	1	2	yes	—
95	C₄×C₂₃	#131	1	2	yes	—
96	C₄×C₂₃	C₂×#41	1	2	yes	—
97	Dic₂₃	—	2	2	no	—
98	Dic₂₃	—	2	2	no	—
99	Dic₂₃	—	2	2	no	—
100	C₄×#41	—	23	1	no	—
101	#446	—	23	2	no	—
102	#436	—	23	2	no	—
103	#435	—	23	4	no	—
104	#431	—	23	2	no	—
105	#431	D₂₃	1	1	yes	characteristic, complemented
106	C₄×M₁₆	—	23	2	no	—
107	C₂×C₄×C₈	—	23	1	no	—
108	C₂×C₂₃	—	2	2	no	—
109	C₂×C₂₃	C₄×#41	1	1	yes	characteristic
110	C₂×C₂₃	C₂²×M₁₆	1	1	yes	characteristic
111	C₂×C₂₃	C₂×#131	1	1	yes	characteristic
112	D₂₃	—	4	2	no	—
113	C₂×#41	—	23	1	no	—
114	C₂×M₁₆	—	23	4	no	—
115	C₂²×C₈	—	23	2	no	—
116	C₄×Q₈	—	23	1	no	—
117	C₄×Q₈	—	23	1	no	—
118	C₄×D₄	—	23	2	no	—
119	C₄×D₄	—	23	2	no	—
120	C₄×D₄	—	23	1	no	—
121	C₄×D₄	—	23	1	no	—
122	#117	—	23	1	no	—
123	#117	—	23	2	no	—
124	C₂×C₄²	—	23	2	no	—
125	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
126	C₄⋊C₈	—	23	4	no	—
127	C₄⋊C₈	C₂×D₂₃	1	2	yes	—
128	C₂²⋊C₈	—	23	4	no	—
129	C₂²⋊C₈	C₂×D₂₃	1	2	yes	—
130	#97	—	23	4	no	—
131	C₄×C₈	—	23	2	no	—
132	C₄×C₈	—	23	1	no	—
133	C₄×C₈	—	23	1	no	—
134	C₄×C₈	C₂×D₂₃	1	2	yes	—
135	C₂₃	#2965	1	1	yes	characteristic, complemented, Sylow
136	#41	—	46	4	no	—
137	C₂×Q₈	—	23	1	no	—
138	C₂×D₄	—	23	2	no	—
139	C₂×D₄	—	23	1	no	—
140	C₂²×C₄	—	23	2	no	—
141	C₂²×C₄	—	23	2	no	—
142	C₂²×C₄	—	23	2	no	—
143	C₂²×C₄	C₄×D₂₃	1	1	yes	characteristic
144	C₂²×C₄	C₄×D₂₃	1	1	yes	characteristic
145	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
146	M₁₆	—	46	8	no	—
147	C₂×C₈	—	23	2	no	—
148	C₂×C₈	—	23	2	no	—
149	C₂×C₈	—	46	4	no	—
150	C₂×C₈	C₂²×D₂₃	1	4	yes	—
151	C₄⋊C₄	—	23	2	no	—
152	C₄⋊C₄	—	23	2	no	—
153	C₄⋊C₄	—	23	1	no	—
154	C₄⋊C₄	—	23	1	no	—
155	C₂²⋊C₄	—	23	2	no	—
156	C₂²⋊C₄	—	23	2	no	—
157	C₂²⋊C₄	—	23	2	no	—
158	C₄²	—	23	2	no	—
159	C₄²	—	23	2	no	—
160	C₄²	—	23	2	no	—
161	C₄²	C₄×D₂₃	1	1	yes	characteristic
162	C₄²	C₄×D₂₃	1	1	yes	characteristic
163	C₄²	C₂²×D₂₃	1	1	yes	characteristic
164	C₄²	C₂²×D₂₃	1	1	yes	characteristic
165	C₂³	—	23	2	no	—
166	C₂³	C₂×C₄×D₂₃	1	1	yes	characteristic
167	Q₈	—	46	2	no	—
168	D₄	—	46	4	no	—
169	D₄	—	46	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	23	2	no	—
173	C₂×C₄	—	23	2	no	—
174	C₂×C₄	—	23	2	no	—
175	C₂×C₄	—	46	4	no	—
176	C₂×C₄	—	46	4	no	—
177	C₂×C₄	—	46	4	no	—
178	C₂×C₄	#67262	1	4	yes	complemented
179	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
180	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
181	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
182	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
183	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
184	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
185	C₈	—	2	4	no	—
186	C₈	—	23	4	no	—
187	C₈	—	46	2	no	—
188	C₂²	—	2	1	no	—
189	C₂²	—	2	1	no	—
190	C₂²	—	46	2	no	—
191	C₂²	—	46	2	no	—
192	C₂²	—	46	2	no	—
193	C₂²	C₂×#67262	1	2	yes	—
194	C₂²	C₂²×C₄×D₂₃	1	1	yes	characteristic
195	C₄	—	2	1	no	—
196	C₄	—	2	1	no	—
197	C₄	—	46	2	no	—
198	C₄	—	46	2	no	—
199	C₄	—	46	2	no	—
200	C₄	C₂×#67262	1	2	yes	—
201	C₄	C₂×#67262	1	2	yes	—
202	C₄	#10625078	1	2	yes	—
203	C₄	#41×D₂₃	1	2	yes	—
204	C₂	—	2	2	no	—
205	C₂	—	92	2	no	—
206	C₂	#13092885	1	1	yes	characteristic
207	C₂	C₂²×#67262	1	1	yes	characteristic
208	C₂	C₂×#10625078	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₂²×D₂₃
commutator, socle	C₂²×C₂₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₂₃
Fitting	C₂₃×#431	C₂
prime radical	C₂×#67288	C₄
cosocle	C₂×C₄×C₂₃	C₂⁴
hypercenter	#431	D₂₃
hypocenter	C₂₃	#2965
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₄² ◁ #431

Complements

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

N	H ≅ G/N	type
2. #13092544	C₂	semidirect
3. #13092494	C₂	semidirect
4. #13092485	C₂	semidirect
5. #13092468	C₂	semidirect
6. C₄×#67262	C₂	semidirect
7. C₂×C₄×C₂₃⋊C₈	C₂	semidirect
8. C₂₃×#431	C₂	semidirect
64. C₂×C₄×C₂₃	M₁₆	semidirect
82. C₂₃⋊C₈	#41	semidirect
105. #431	D₂₃	semidirect
135. C₂₃	#2965	semidirect
178. C₂×C₄	#67262	semidirect

Maximal subgroups

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

subgroup	index
C₄×#67288	2
#13092544	2
#13092494	2
#13092485	2
#13092468	2
C₄×#67262	2
C₂×C₄×C₂₃⋊C₈	2
C₂₃×#431	2
#2965	23