Subgroups — #15682043

Order 2944.

384 conjugacy classes of subgroups.

The subgroup lattice has 230 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	#13092477	C₂	1	2	yes	maximal, complemented
3	#13092475	C₂	1	2	yes	maximal, complemented
4	#13092474	C₂	1	1	yes	maximal, characteristic, complemented
5	#13092472	C₂	1	1	yes	maximal, characteristic, complemented
6	#13092472	C₂	1	1	yes	maximal, characteristic, complemented
7	#13092442	C₂	1	2	yes	maximal, complemented
8	#13092382	C₂	1	2	yes	maximal
9	#13092369	C₂	1	1	yes	maximal, characteristic, complemented
10	#97×D₂₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#432	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#67288	C₂²	1	1	yes	characteristic
13	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
14	C₄×C₂²⋊D₂₃	C₄	1	2	yes	—
15	C₄×C₂²⋊D₂₃	C₄	1	2	yes	—
16	#10625097	C₂²	1	1	yes	characteristic
17	C₂×#67262	C₂²	1	2	yes	—
18	C₂×C₈×D₂₃	C₂²	1	2	yes	—
19	C₄×D₉₂	C₄	1	2	yes	—
20	#10625043	C₂²	1	1	yes	characteristic
21	#10625043	C₂²	1	1	yes	characteristic
22	C₄²×D₂₃	C₂²	1	1	yes	characteristic
23	C₄²×D₂₃	C₂²	1	1	yes	characteristic
24	C₄×Dic₄₆	C₄	1	2	yes	—
25	C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
26	C₂₃×C₂²⋊C₈	C₂²	1	2	yes	—
27	C₂₃×#97	C₂²	1	1	yes	characteristic
28	C₂₃×#97	C₂²	1	1	yes	characteristic
29	C₂²⋊(C₂₃⋊C₈)	C₂²	1	2	yes	—
30	(C₂×D₂₃)⋊C₈	C₂²	1	2	yes	—
31	(C₂×D₂₃)⋊C₈	C₂²	1	2	yes	—
32	#10624997	C₂²	1	2	yes	—
33	Dic₂₃⋊C₈	C₂²	1	2	yes	—
34	Dic₂₃⋊C₈	C₂²	1	2	yes	—
35	C₈×Dic₂₃	C₂²	1	2	yes	—
36	C₄⋊(C₂₃⋊C₈)	C₂²	1	2	yes	—
37	#10624985	C₂²	1	1	yes	characteristic
38	#10624985	C₂²	1	1	yes	characteristic
39	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
40	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
41	C₂×C₂²⋊D₂₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₂²⋊D₂₃	C₂×C₄	1	1	yes	characteristic
43	#67288	—	2	4	no	—
44	C₂×D₉₂	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×D₂₃	C₂×C₄	1	2	yes	—
46	C₂×C₄×D₂₃	C₂×C₄	1	2	yes	—
47	C₂×C₄×D₂₃	C₂³	1	1	yes	characteristic
48	C₂×C₄×D₂₃	C₂³	1	1	yes	characteristic
49	C₂×Dic₄₆	C₂×C₄	1	1	yes	characteristic
50	C₂×C₈×C₂₃	C₂³	1	2	yes	—
51	C₂×C₈×C₂₃	C₂³	1	2	yes	—
52	C₄²×C₂₃	C₂×C₄	1	2	yes	—
53	C₄²×C₂₃	C₂³	1	1	yes	characteristic
54	C₄²×C₂₃	C₂³	1	1	yes	characteristic
55	C₂²⋊Dic₂₃	C₂×C₄	1	2	yes	—
56	(C₂×D₂₃)⋊C₄	C₂×C₄	1	2	yes	—
57	(C₂×D₂₃)⋊C₄	C₂×C₄	1	2	yes	—
58	C₄⋊Dic₂₃	C₂×C₄	1	2	yes	—
59	Dic₂₃⋊C₄	C₂×C₄	1	2	yes	—
60	Dic₂₃⋊C₄	C₂×C₄	1	2	yes	—
61	C₄×Dic₂₃	C₂×C₄	1	2	yes	—
62	C₄×Dic₂₃	C₂×C₄	1	2	yes	—
63	C₄×Dic₂₃	C₂³	1	1	yes	characteristic
64	C₄×Dic₂₃	C₂³	1	1	yes	characteristic
65	C₂×C₂₃⋊C₈	C₂³	1	2	yes	—
66	C₂×C₂₃⋊C₈	C₂³	1	2	yes	—
67	#67262	—	2	4	no	—
68	C₈×D₂₃	—	2	4	no	—
69	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
70	C₂²×D₂₃	C₂²×C₄	1	1	yes	characteristic
71	C₂²×D₂₃	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₄×C₂₃	—	2	2	no	—
73	C₂×C₄×C₂₃	—	4	2	no	—
74	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
75	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
76	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
78	C₂²⋊D₂₃	—	2	2	no	—
79	C₂²⋊D₂₃	—	2	2	no	—
80	C₂×Dic₂₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×Dic₂₃	C₂²×C₄	1	2	yes	—
82	C₂×Dic₂₃	C₂²×C₄	1	2	yes	—
83	C₂×Dic₂₃	C₂²×C₄	1	1	yes	characteristic
84	D₉₂	—	2	2	no	—
85	C₄×D₂₃	—	2	2	no	—
86	C₄×D₂₃	—	4	2	no	—
87	C₄×D₂₃	—	4	2	no	—
88	C₄×D₂₃	#41	1	4	yes	—
89	Dic₄₆	—	2	2	no	—
90	C₈×C₂₃	—	2	2	no	—
91	C₈×C₂₃	—	2	2	no	—
92	C₂₃⋊C₈	—	2	2	no	—
93	C₂₃⋊C₈	—	2	2	no	—
94	#2923	—	23	1	no	maximal, Sylow
95	C₂²×C₂₃	—	2	1	no	—
96	C₂²×C₂₃	—	2	2	no	—
97	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
98	C₂×D₂₃	—	2	2	no	—
99	C₂×D₂₃	—	2	1	no	—
100	C₂×D₂₃	—	2	2	no	—
101	C₂×D₂₃	#117	1	2	yes	—
102	C₄×C₂₃	—	2	1	no	—
103	C₄×C₂₃	—	2	2	no	—
104	C₄×C₂₃	—	2	2	no	—
105	C₄×C₂₃	C₂×#41	1	2	yes	—
106	Dic₂₃	—	2	1	no	—
107	Dic₂₃	—	2	2	no	—
108	Dic₂₃	—	2	2	no	—
109	Dic₂₃	#117	1	2	yes	—
110	C₄×#41	—	23	1	no	—
111	#433	—	23	2	no	—
112	#433	—	23	2	no	—
113	#432	—	23	1	no	—
114	#432	—	23	1	no	—
115	#432	—	23	1	no	—
116	#432	D₂₃	1	1	yes	characteristic, complemented
117	#424	—	23	2	no	—
118	#408	—	23	2	no	—
119	C₄×M₁₆	—	23	1	no	—
120	C₂×#97	—	23	1	no	—
121	C₂×C₂₃	—	4	1	no	—
122	C₂×C₂₃	C₂×#117	1	1	yes	characteristic
123	C₂×C₂₃	#568	1	2	yes	—
124	D₂₃	—	2	2	no	—
125	D₂₃	—	4	1	no	—
126	C₂×#41	—	23	1	no	—
127	C₂×M₁₆	—	23	2	no	—
128	C₂²×C₈	—	23	2	no	—
129	C₄×Q₈	—	23	2	no	—
130	C₄×D₄	—	23	2	no	—
131	C₄×D₄	—	23	2	no	—
132	C₄×D₄	—	23	2	no	—
133	#117	—	23	1	no	—
134	#117	—	23	1	no	—
135	#117	—	23	1	no	—
136	C₂×C₄²	—	23	1	no	—
137	C₂×C₄²	—	23	1	no	—
138	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
139	C₄⋊C₈	—	23	2	no	—
140	C₄⋊C₈	—	23	2	no	—
141	C₄⋊C₈	—	23	2	no	—
142	C₄⋊C₈	C₂×D₂₃	1	2	yes	—
143	C₂²⋊C₈	—	23	2	no	—
144	C₂²⋊C₈	—	23	2	no	—
145	C₂²⋊C₈	—	23	2	no	—
146	C₂²⋊C₈	C₂×D₂₃	1	2	yes	—
147	#97	—	23	2	no	—
148	#97	—	23	1	no	—
149	#97	—	23	1	no	—
150	#97	C₂×D₂₃	1	1	yes	characteristic
151	#97	C₂×D₂₃	1	1	yes	characteristic
152	C₄×C₈	—	23	2	no	—
153	C₂₃	#2923	1	1	yes	characteristic, complemented, Sylow
154	#41	—	46	4	no	—
155	C₂×Q₈	—	23	1	no	—
156	C₂×D₄	—	23	1	no	—
157	C₂×D₄	—	23	1	no	—
158	C₂×D₄	—	23	1	no	—
159	C₂²×C₄	—	23	1	no	—
160	C₂²×C₄	—	23	1	no	—
161	C₂²×C₄	—	23	2	no	—
162	C₂²×C₄	—	23	2	no	—
163	C₂²×C₄	C₄×D₂₃	1	2	yes	—
164	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
165	M₁₆	—	46	4	no	—
166	C₂×C₈	—	23	2	no	—
167	C₂×C₈	—	23	2	no	—
168	C₂×C₈	—	46	4	no	—
169	C₂×C₈	C₂²×D₂₃	1	2	yes	—
170	C₂×C₈	C₂²×D₂₃	1	2	yes	—
171	C₄⋊C₄	—	23	2	no	—
172	C₄⋊C₄	—	23	2	no	—
173	C₄⋊C₄	—	23	2	no	—
174	C₂²⋊C₄	—	23	2	no	—
175	C₂²⋊C₄	—	23	2	no	—
176	C₂²⋊C₄	—	23	2	no	—
177	C₄²	—	23	2	no	—
178	C₄²	—	23	2	no	—
179	C₄²	—	23	1	no	—
180	C₄²	—	23	1	no	—
181	C₄²	C₄×D₂₃	1	2	yes	—
182	C₄²	C₂²×D₂₃	1	1	yes	characteristic
183	C₄²	C₂²×D₂₃	1	1	yes	characteristic
184	C₂³	—	23	1	no	—
185	C₂³	—	23	1	no	—
186	C₂³	C₂×C₄×D₂₃	1	1	yes	characteristic
187	Q₈	—	46	2	no	—
188	D₄	—	46	2	no	—
189	D₄	—	46	2	no	—
190	D₄	—	46	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	4	2	no	—
193	C₂×C₄	—	23	1	no	—
194	C₂×C₄	—	23	4	no	—
195	C₂×C₄	—	23	2	no	—
196	C₂×C₄	—	23	2	no	—
197	C₂×C₄	—	23	1	no	—
198	C₂×C₄	—	46	2	no	—
199	C₂×C₄	—	92	2	no	—
200	C₂×C₄	—	92	2	no	—
201	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
202	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
203	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
204	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
205	C₈	—	2	2	no	—
206	C₈	—	2	2	no	—
207	C₈	—	46	2	no	—
208	C₈	—	46	2	no	—
209	C₂²	—	2	1	no	—
210	C₂²	—	2	2	no	—
211	C₂²	—	23	2	no	—
212	C₂²	—	46	2	no	—
213	C₂²	—	46	1	no	—
214	C₂²	—	46	2	no	—
215	C₂²	C₂²×C₄×D₂₃	1	1	yes	characteristic
216	C₄	—	2	1	no	—
217	C₄	—	2	2	no	—
218	C₄	—	2	2	no	—
219	C₄	—	23	2	no	—
220	C₄	—	46	1	no	—
221	C₄	—	46	2	no	—
222	C₄	—	46	2	no	—
223	C₄	#41×D₂₃	1	2	yes	—
224	C₂	—	4	1	no	—
225	C₂	—	46	2	no	—
226	C₂	—	92	1	no	—
227	C₂	#117×D₂₃	1	1	yes	characteristic
228	C₂	#13093183	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₃
commutator, socle	C₂²×C₂₃	C₂³×C₄
Fitting	C₂₃×#432	C₂
prime radical	C₂×#67288	C₂²
cosocle	C₂×C₄×C₂₃	C₂⁴
hypercenter	#432	D₂₃
hypocenter	C₂₃	#2923
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ #432

Complements

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

N	H ≅ G/N	type
2. #13092477	C₂	semidirect
3. #13092475	C₂	semidirect
4. #13092474	C₂	semidirect
5. #13092472	C₂	semidirect
6. #13092472	C₂	semidirect
7. #13092442	C₂	semidirect
9. #13092369	C₂	semidirect
10. #97×D₂₃	C₂	semidirect
11. C₂₃×#432	C₂	semidirect
116. #432	D₂₃	semidirect
153. C₂₃	#2923	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
#13092477	2
#13092475	2
#13092474	2
#13092472	2
#13092442	2
#13092382	2
#13092369	2
#97×D₂₃	2
C₂₃×#432	2
#2923	23