Subgroups — #15682063

Order 2944.

392 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13092478	C₂	1	4	yes	maximal, complemented
2	#13092474	C₂	1	2	yes	maximal, complemented
3	#13092470	C₂	1	2	yes	maximal, complemented
4	#13092441	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄⋊(C₂₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	#13092384	C₂	1	2	yes	maximal
7	#13092373	C₂	1	2	yes	maximal, complemented
8	C₄×#41×C₂₃	C₂	1	1	yes	maximal, characteristic
9	C₂×#41×C₂₃	C₂²	1	1	yes	characteristic
10	C₄×Q₈×C₂₃	C₄	1	1	yes	characteristic
11	C₄×Q₈×C₂₃	C₄	1	1	yes	characteristic
12	C₄×D₄×C₂₃	C₄	1	2	yes	—
13	C₄×D₄×C₂₃	C₄	1	1	yes	characteristic
14	C₄×D₄×C₂₃	C₄	1	2	yes	—
15	C₄×D₄×C₂₃	C₄	1	1	yes	characteristic
16	C₂₃×#117	C₂²	1	1	yes	characteristic
17	C₂₃×#117	C₂²	1	2	yes	—
18	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
19	C₂×C₄²×C₂₃	C₂²	1	2	yes	—
20	C₂×#67267	C₂²	1	2	yes	—
21	C₂²×C₂₃⋊C₈	C₂²	1	2	yes	—
22	C₂²⋊(C₂₃⋊C₈)	C₂²	1	4	yes	—
23	C₂²⋊(C₂₃⋊C₈)	C₂²	1	4	yes	—
24	C₄⋊(C₂₃⋊C₈)	C₂²	1	1	yes	characteristic
25	C₄⋊(C₂₃⋊C₈)	C₂²	1	1	yes	characteristic
26	C₄⋊(C₂₃⋊C₈)	C₂²	1	2	yes	—
27	C₄⋊(C₂₃⋊C₈)	C₂²	1	2	yes	—
28	C₄⋊(C₂₃⋊C₈)	C₂²	1	2	yes	—
29	#10624985	C₂²	1	4	yes	—
30	C₄×C₂₃⋊C₈	C₂²	1	4	yes	—
31	#41×C₂₃	—	2	2	no	—
32	#41×C₂₃	—	2	2	no	—
33	C₂×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
34	C₂×D₄×C₂₃	C₂×C₄	1	2	yes	—
35	C₂×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
37	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
39	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
41	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
42	C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
43	C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
44	C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
45	C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
46	C₂²⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
47	C₂²⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
48	C₂²⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
49	C₄²×C₂₃	C₂×C₄	1	2	yes	—
50	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
51	C₄²×C₂₃	C₂×C₄	1	2	yes	—
52	C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
53	C₄²×C₂₃	C₂³	1	1	yes	characteristic
54	C₄²×C₂₃	C₂³	1	2	yes	—
55	C₄²×C₂₃	C₂³	1	1	yes	characteristic
56	#67267	—	2	4	no	—
57	C₂×C₂₃⋊C₈	—	2	4	no	—
58	C₂×C₂₃⋊C₈	C₂³	1	2	yes	—
59	C₂×C₂₃⋊C₈	C₂³	1	2	yes	—
60	C₂×C₂₃⋊C₈	C₂³	1	4	yes	—
61	C₂³×C₂₃	C₂²×C₄	1	2	yes	—
62	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
63	Q₈×C₂₃	—	2	2	no	—
64	D₄×C₂₃	—	2	4	no	—
65	D₄×C₂₃	—	2	2	no	—
66	C₂×C₄×C₂₃	—	2	2	no	—
67	C₂×C₄×C₂₃	—	2	4	no	—
68	C₂×C₄×C₂₃	—	2	2	no	—
69	C₂×C₄×C₂₃	—	2	2	no	—
70	C₂×C₄×C₂₃	—	4	1	no	—
71	C₂×C₄×C₂₃	—	4	2	no	—
72	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
74	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
75	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
77	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₄×C₂₃	#41	1	2	yes	—
81	C₂×C₄×C₂₃	#41	1	2	yes	—
82	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
83	C₂₃⋊C₈	—	2	2	no	—
84	C₂₃⋊C₈	—	2	2	no	—
85	C₂₃⋊C₈	—	2	4	no	—
86	#2925	—	23	1	no	maximal, Sylow
87	C₂²×C₂₃	—	2	1	no	—
88	C₂²×C₂₃	—	2	1	no	—
89	C₂²×C₂₃	—	2	2	no	—
90	C₂²×C₂₃	—	2	2	no	—
91	C₂²×C₂₃	—	2	2	no	—
92	C₂²×C₂₃	#117	1	2	yes	—
93	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
94	C₄×C₂₃	—	2	2	no	—
95	C₄×C₂₃	—	2	1	no	—
96	C₄×C₂₃	—	2	2	no	—
97	C₄×C₂₃	—	2	1	no	—
98	C₄×C₂₃	—	2	1	no	—
99	C₄×C₂₃	—	2	2	no	—
100	C₄×C₂₃	#117	1	2	yes	—
101	C₄×C₂₃	#131	1	2	yes	—
102	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
103	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
104	C₄×#41	D₂₃	1	1	yes	characteristic
105	#433	—	23	4	no	—
106	#432	—	23	2	no	—
107	#431	—	23	2	no	—
108	#423	—	23	1	no	—
109	C₂×C₄⋊C₈	—	23	1	no	—
110	#408	—	23	2	no	—
111	#405	—	23	2	no	—
112	C₂×C₂₃	—	2	2	no	—
113	C₂×C₂₃	—	4	2	no	—
114	C₂×C₂₃	C₂×#117	1	1	yes	characteristic
115	C₂×C₂₃	C₂×#131	1	1	yes	characteristic
116	C₂×C₂₃	#568	1	1	yes	characteristic
117	C₂×#41	C₂×D₂₃	1	1	yes	characteristic
118	C₂×M₁₆	—	23	2	no	—
119	C₂²×C₈	—	23	2	no	—
120	C₄×Q₈	Dic₂₃	1	1	yes	characteristic
121	C₄×Q₈	Dic₂₃	1	1	yes	characteristic
122	C₄×D₄	Dic₂₃	1	2	yes	—
123	C₄×D₄	Dic₂₃	1	1	yes	characteristic
124	C₄×D₄	Dic₂₃	1	2	yes	—
125	C₄×D₄	Dic₂₃	1	1	yes	characteristic
126	#117	C₂×D₂₃	1	1	yes	characteristic
127	#117	C₂×D₂₃	1	2	yes	—
128	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
129	C₂×C₄²	C₂×D₂₃	1	2	yes	—
130	C₄⋊C₈	—	23	1	no	—
131	C₄⋊C₈	—	23	1	no	—
132	C₄⋊C₈	—	23	2	no	—
133	C₄⋊C₈	—	23	2	no	—
134	C₄⋊C₈	—	23	2	no	—
135	C₂²⋊C₈	—	23	4	no	—
136	C₂²⋊C₈	—	23	4	no	—
137	#97	—	23	4	no	—
138	C₄×C₈	—	23	4	no	—
139	C₂₃	#2925	1	1	yes	characteristic, complemented, Sylow
140	#41	—	2	2	no	—
141	#41	—	2	2	no	—
142	C₂×Q₈	C₂×Dic₂₃	1	1	yes	characteristic
143	C₂×D₄	C₂×Dic₂₃	1	2	yes	—
144	C₂×D₄	C₂×Dic₂₃	1	1	yes	characteristic
145	C₂²×C₄	C₂×Dic₂₃	1	2	yes	—
146	C₂²×C₄	C₂×Dic₂₃	1	1	yes	characteristic
147	C₂²×C₄	C₂×Dic₂₃	1	2	yes	—
148	C₂²×C₄	C₂×Dic₂₃	1	1	yes	characteristic
149	C₂²×C₄	C₂²×D₂₃	1	2	yes	—
150	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
151	M₁₆	—	46	4	no	—
152	C₂×C₈	—	23	2	no	—
153	C₂×C₈	—	23	4	no	—
154	C₂×C₈	—	23	2	no	—
155	C₂×C₈	—	46	4	no	—
156	C₄⋊C₄	C₂×Dic₂₃	1	2	yes	—
157	C₄⋊C₄	C₂×Dic₂₃	1	2	yes	—
158	C₄⋊C₄	C₂×Dic₂₃	1	1	yes	characteristic
159	C₄⋊C₄	C₂×Dic₂₃	1	1	yes	characteristic
160	C₂²⋊C₄	C₂×Dic₂₃	1	2	yes	—
161	C₂²⋊C₄	C₂×Dic₂₃	1	2	yes	—
162	C₂²⋊C₄	C₂×Dic₂₃	1	2	yes	—
163	C₄²	C₂×Dic₂₃	1	2	yes	—
164	C₄²	C₂×Dic₂₃	1	1	yes	characteristic
165	C₄²	C₂×Dic₂₃	1	2	yes	—
166	C₄²	C₂×Dic₂₃	1	1	yes	characteristic
167	C₄²	C₂²×D₂₃	1	2	yes	—
168	C₄²	C₂²×D₂₃	1	1	yes	characteristic
169	C₄²	C₂²×D₂₃	1	1	yes	characteristic
170	C₂³	C₂²×Dic₂₃	1	2	yes	—
171	C₂³	C₂²×Dic₂₃	1	1	yes	characteristic
172	Q₈	—	2	2	no	—
173	D₄	—	2	4	no	—
174	D₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	4	1	no	—
180	C₂×C₄	—	4	2	no	—
181	C₂×C₄	#67288	1	2	yes	—
182	C₂×C₄	#67288	1	2	yes	—
183	C₂×C₄	C₂²×Dic₂₃	1	1	yes	characteristic
184	C₂×C₄	C₂²×Dic₂₃	1	2	yes	—
185	C₂×C₄	C₂²×Dic₂₃	1	2	yes	—
186	C₂×C₄	C₂²×Dic₂₃	1	1	yes	characteristic
187	C₂×C₄	C₂²×Dic₂₃	1	2	yes	—
188	C₂×C₄	C₂²×Dic₂₃	1	1	yes	characteristic
189	C₂×C₄	C₂²×Dic₂₃	1	1	yes	characteristic
190	C₂×C₄	C₂²×Dic₂₃	1	1	yes	characteristic
191	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
192	C₈	—	46	2	no	—
193	C₈	—	46	2	no	—
194	C₈	—	46	4	no	—
195	C₂²	—	2	1	no	—
196	C₂²	—	2	1	no	—
197	C₂²	—	2	2	no	—
198	C₂²	—	2	2	no	—
199	C₂²	—	2	2	no	—
200	C₂²	#10625097	1	2	yes	—
201	C₂²	C₂³×Dic₂₃	1	1	yes	characteristic
202	C₄	—	2	2	no	—
203	C₄	—	2	1	no	—
204	C₄	—	2	2	no	—
205	C₄	—	2	1	no	—
206	C₄	—	2	1	no	—
207	C₄	—	2	2	no	—
208	C₄	#10625097	1	2	yes	—
209	C₄	#10625119	1	2	yes	—
210	C₄	C₂×#67288	1	1	yes	characteristic
211	C₄	C₂×#67288	1	1	yes	characteristic
212	C₂	—	2	2	no	—
213	C₂	—	4	2	no	—
214	C₂	C₂×#10625097	1	1	yes	characteristic
215	C₂	C₂×#10625119	1	1	yes	characteristic
216	C₂	#13093184	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₃
commutator, socle	C₂²×C₂₃	C₂³×C₄
Fitting	C₄×#41×C₂₃	C₂
prime radical	C₂×#41×C₂₃	C₂²
cosocle	C₂×C₄×C₂₃	C₂⁴
hypercenter	C₄×#41	D₂₃
hypocenter	C₂₃	#2925
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
1. #13092478	C₂	semidirect
2. #13092474	C₂	semidirect
3. #13092470	C₂	semidirect
4. #13092441	C₂	semidirect
5. C₂×C₄⋊(C₂₃⋊C₈)	C₂	semidirect
7. #13092373	C₂	semidirect
139. C₂₃	#2925	semidirect

Maximal subgroups

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

subgroup	index
#13092478	2
#13092474	2
#13092470	2
#13092441	2
C₂×C₄⋊(C₂₃⋊C₈)	2
#13092384	2
#13092373	2
C₄×#41×C₂₃	2
#2925	23