Subgroups — #15683923

Order 2944.

340 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13093045	C₂	1	1	yes	maximal, characteristic
2	#13092827	C₂	1	2	yes	maximal, complemented
3	#13092823	C₂	1	2	yes	maximal, complemented
4	#13092816	C₂	1	2	yes	maximal, complemented
5	#13092813	C₂	1	2	yes	maximal, complemented
6	#13092460	C₂	1	1	yes	maximal, characteristic
7	#13092457	C₂	1	1	yes	maximal, characteristic
8	#13092453	C₂	1	1	yes	maximal, characteristic
9	C₂×#10624990	C₂	1	1	yes	maximal, characteristic
10	#13092373	C₂	1	1	yes	maximal, characteristic
11	C₂₃×#544	C₂	1	1	yes	maximal, characteristic
12	C₂₃×C₄⋊Q₈	—	2	1	no	—
13	C₂₃×C₄⋊Q₈	C₂²	1	2	yes	—
14	C₂₃×#125	—	2	1	no	—
15	C₂₃×#125	C₂²	1	2	yes	—
16	C₂₃×C₂²⋊Q₈	—	2	2	no	—
17	C₂₃×#117	C₂²	1	1	yes	characteristic
18	C₂₃×#117	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
20	#10625097	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
22	C₂×#67267	C₂²	1	1	yes	characteristic
23	C₂²×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
24	#10625061	—	2	1	no	—
25	#10625059	—	2	1	no	—
26	#10625049	—	2	2	no	—
27	#10625041	C₂²	1	2	yes	—
28	#10625040	C₂²	1	2	yes	—
29	#10624990	C₂²	1	2	yes	—
30	#10624990	C₂²	1	2	yes	—
31	#10624990	C₂²	1	1	yes	characteristic
32	#10624990	C₂²	1	1	yes	characteristic
33	#10624990	C₂²	1	2	yes	—
34	#10624990	C₂²	1	2	yes	—
35	#10624989	C₂²	1	1	yes	characteristic
36	#10624989	C₂²	1	1	yes	characteristic
37	#10624989	C₂²	1	2	yes	—
38	#10624989	C₂²	1	2	yes	—
39	#10624985	C₂²	1	2	yes	—
40	C₄×C₂₃⋊C₈	C₂²	1	2	yes	—
41	C₂×Q₈×C₂₃	—	2	2	no	—
42	C₂²×C₄×C₂₃	—	2	1	no	—
43	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
44	C₂²×Dic₂₃	—	2	1	no	—
45	C₂×Dic₄₆	—	2	2	no	—
46	C₄⋊C₄×C₂₃	—	2	1	no	—
47	C₄⋊C₄×C₂₃	—	2	2	no	—
48	C₄⋊C₄×C₂₃	—	2	2	no	—
49	C₄⋊C₄×C₂₃	—	2	2	no	—
50	C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
51	C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
52	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
53	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
54	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
55	C₂²⋊C₄×C₂₃	—	2	1	no	—
56	C₂²⋊C₄×C₂₃	D₄	1	2	yes	—
57	C₄²×C₂₃	—	2	1	no	—
58	C₄²×C₂₃	C₂³	1	2	yes	—
59	C₂²⋊Dic₂₃	—	2	1	no	—
60	C₄⋊Dic₂₃	—	2	2	no	—
61	C₄⋊Dic₂₃	—	2	1	no	—
62	C₄⋊Dic₂₃	C₂³	1	1	yes	characteristic
63	C₄⋊Dic₂₃	C₂³	1	2	yes	—
64	C₄⋊Dic₂₃	C₂³	1	1	yes	characteristic
65	Dic₂₃⋊C₄	—	2	2	no	—
66	Dic₂₃⋊C₄	—	2	2	no	—
67	C₄×Dic₂₃	—	2	1	no	—
68	#67267	Q₈	1	4	yes	complemented
69	C₂×C₂₃⋊C₈	Q₈	1	4	yes	complemented
70	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
71	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
72	C₂×C₂₃⋊C₈	C₂³	1	2	yes	—
73	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
74	Q₈×C₂₃	—	8	2	no	—
75	C₂×C₄×C₂₃	—	2	1	no	—
76	C₂×C₄×C₂₃	—	2	1	no	—
77	C₂×C₄×C₂₃	—	2	2	no	—
78	C₂×C₄×C₂₃	—	8	1	no	—
79	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
80	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
81	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
82	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
83	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
84	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
85	C₂×Dic₂₃	—	2	1	no	—
86	C₂×Dic₂₃	—	2	1	no	—
87	C₂×Dic₂₃	—	2	2	no	—
88	C₂×Dic₂₃	—	8	1	no	—
89	Dic₄₆	—	8	2	no	—
90	C₂₃⋊C₈	C₂×Q₈	1	2	yes	—
91	C₂₃⋊C₈	C₂×Q₈	1	2	yes	—
92	C₂₃⋊C₈	C₂×Q₈	1	4	yes	—
93	#3163	—	23	1	no	maximal, Sylow
94	C₂²×C₂₃	—	2	1	no	—
95	C₂²×C₂₃	—	2	1	no	—
96	C₂²×C₂₃	C₄⋊Q₈	1	2	yes	—
97	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
98	C₄×C₂₃	—	2	2	no	—
99	C₄×C₂₃	—	2	2	no	—
100	C₄×C₂₃	—	4	1	no	—
101	C₄×C₂₃	—	4	2	no	—
102	C₄×C₂₃	—	4	1	no	—
103	C₄×C₂₃	C₄⋊Q₈	1	2	yes	—
104	C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
105	C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
106	Dic₂₃	—	4	1	no	—
107	Dic₂₃	—	4	2	no	—
108	Dic₂₃	—	4	1	no	—
109	#544	—	23	1	no	—
110	#544	D₂₃	1	1	yes	characteristic
111	#501	—	23	2	no	—
112	#501	—	23	2	no	—
113	#499	—	23	2	no	—
114	#498	—	23	2	no	—
115	#428	—	23	1	no	—
116	#428	—	23	1	no	—
117	#427	—	23	1	no	—
118	C₂×#106	—	23	1	no	—
119	#405	—	23	1	no	—
120	C₂×C₂₃	—	2	2	no	—
121	C₂×C₂₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
122	C₂×C₂₃	#577	1	1	yes	characteristic
123	C₂×C₂₃	#577	1	1	yes	characteristic
124	C₂×M₁₆	—	23	1	no	—
125	C₂²×C₈	—	23	1	no	—
126	C₄⋊Q₈	—	2	1	no	—
127	C₄⋊Q₈	—	23	2	no	—
128	C₄⋊Q₈	—	46	1	no	—
129	C₄⋊Q₈	C₂×D₂₃	1	2	yes	—
130	#125	—	2	1	no	—
131	#125	—	23	2	no	—
132	#125	—	46	1	no	—
133	#125	C₂×D₂₃	1	2	yes	—
134	C₂²⋊Q₈	—	2	2	no	—
135	C₂²⋊Q₈	—	46	2	no	—
136	#117	—	23	1	no	—
137	#117	C₂×D₂₃	1	1	yes	characteristic
138	#117	C₂×D₂₃	1	1	yes	characteristic
139	C₂×C₄⋊C₄	—	23	1	no	—
140	C₂×C₄⋊C₄	C₂×D₂₃	1	1	yes	characteristic
141	#107	—	23	1	no	—
142	#107	—	23	1	no	—
143	#107	—	23	2	no	—
144	#107	—	23	2	no	—
145	#106	—	23	2	no	—
146	#106	—	23	2	no	—
147	#106	—	23	1	no	—
148	#106	—	23	1	no	—
149	#106	—	23	2	no	—
150	#106	—	23	2	no	—
151	#97	—	23	2	no	—
152	C₄×C₈	—	23	2	no	—
153	C₂₃	#3163	1	1	yes	characteristic, complemented, Sylow
154	C₂×Q₈	—	2	2	no	—
155	C₂×Q₈	—	46	2	no	—
156	C₂²×C₄	—	2	1	no	—
157	C₂²×C₄	—	46	1	no	—
158	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
159	M₁₆	—	23	4	no	—
160	C₂×C₈	—	23	4	no	—
161	C₂×C₈	—	23	1	no	—
162	C₂×C₈	—	23	2	no	—
163	C₂×C₈	—	23	1	no	—
164	C₄⋊C₄	—	2	1	no	—
165	C₄⋊C₄	—	2	2	no	—
166	C₄⋊C₄	—	2	2	no	—
167	C₄⋊C₄	—	2	2	no	—
168	C₄⋊C₄	—	23	2	no	—
169	C₄⋊C₄	—	23	1	no	—
170	C₄⋊C₄	—	23	1	no	—
171	C₄⋊C₄	—	46	1	no	—
172	C₄⋊C₄	—	46	2	no	—
173	C₄⋊C₄	—	46	2	no	—
174	C₄⋊C₄	—	46	2	no	—
175	C₄⋊C₄	C₂²⋊D₂₃	1	1	yes	characteristic
176	C₄⋊C₄	C₂²⋊D₂₃	1	1	yes	characteristic
177	C₄⋊C₄	C₂²×D₂₃	1	2	yes	—
178	C₄⋊C₄	C₂²×D₂₃	1	1	yes	characteristic
179	C₄⋊C₄	C₂²×D₂₃	1	1	yes	characteristic
180	C₂²⋊C₄	—	2	1	no	—
181	C₂²⋊C₄	—	46	1	no	—
182	C₂²⋊C₄	C₂²⋊D₂₃	1	2	yes	—
183	C₄²	—	2	1	no	—
184	C₄²	—	46	1	no	—
185	C₄²	C₂²×D₂₃	1	2	yes	—
186	C₂³	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
187	Q₈	—	8	2	no	—
188	Q₈	—	184	2	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	8	1	no	—
193	C₂×C₄	—	46	1	no	—
194	C₂×C₄	—	46	1	no	—
195	C₂×C₄	—	46	2	no	—
196	C₂×C₄	—	184	1	no	—
197	C₂×C₄	Q₈×D₂₃	1	2	yes	—
198	C₂×C₄	Q₈×D₂₃	1	2	yes	—
199	C₂×C₄	C₂×C₂²⋊D₂₃	1	2	yes	—
200	C₂×C₄	C₂×C₂²⋊D₂₃	1	2	yes	—
201	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
202	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
203	C₈	—	23	2	no	—
204	C₈	—	23	2	no	—
205	C₈	—	23	4	no	—
206	C₂²	—	2	1	no	—
207	C₂²	—	2	1	no	—
208	C₂²	#10625115	1	2	yes	—
209	C₂²	C₂²×C₂²⋊D₂₃	1	1	yes	characteristic
210	C₄	—	2	2	no	—
211	C₄	—	2	2	no	—
212	C₄	—	4	1	no	—
213	C₄	—	4	2	no	—
214	C₄	—	4	1	no	—
215	C₄	—	92	2	no	—
216	C₄	—	92	1	no	—
217	C₄	—	92	1	no	—
218	C₄	#10625115	1	2	yes	—
219	C₄	C₂×Q₈×D₂₃	1	1	yes	characteristic
220	C₄	C₂×Q₈×D₂₃	1	1	yes	characteristic
221	C₂	—	2	2	no	—
222	C₂	C₂×#10625115	1	1	yes	characteristic
223	C₂	#13093225	1	1	yes	characteristic
224	C₂	#13093228	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₂₃
commutator, cosocle	C₂×C₄×C₂₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₃
Fitting	C₂₃×#544	C₂
socle	C₂²×C₂₃	C₂²×D₄
prime radical	C₂³×C₂₃	C₂×D₄
hypercenter	#544	D₂₃
hypocenter	C₂₃	#3163
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #544

Complements

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

N	H ≅ G/N	type
2. #13092827	C₂	semidirect
3. #13092823	C₂	semidirect
4. #13092816	C₂	semidirect
5. #13092813	C₂	semidirect
68. #67267	Q₈	semidirect
69. C₂×C₂₃⋊C₈	Q₈	semidirect
153. C₂₃	#3163	semidirect

Maximal subgroups

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

subgroup	index
#13093045	2
#13092827	2
#13092823	2
#13092816	2
#13092813	2
#13092460	2
#13092457	2
#13092453	2
C₂×#10624990	2
#13092373	2
C₂₃×#544	2
#3163	23