Subgroups — #15675875

Order 2944.

500 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13092324	C₂	1	2	yes	maximal
2	#13092316	C₂	1	2	yes	maximal
3	#13092303	C₂	1	4	yes	maximal
4	C₄×C₄⋊Dic₂₃	C₂	1	2	yes	maximal
5	C₄×C₄⋊Dic₂₃	C₂	1	1	yes	maximal, characteristic
6	#13092263	C₂	1	2	yes	maximal
7	C₄²×Dic₂₃	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×Q₈×C₂₃	C₂	1	1	yes	maximal, characteristic
9	C₂²×Q₈×C₂₃	C₂²	1	1	yes	characteristic
10	C₄×Q₈×C₂₃	C₄	1	8	yes	complemented
11	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
12	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
13	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₂₃	C₂²	1	2	yes	—
15	C₂×C₄⋊Dic₂₃	C₂²	1	4	yes	—
16	C₂×C₄⋊Dic₂₃	C₂²	1	2	yes	—
17	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
18	C₂×C₄×Dic₂₃	C₂²	1	2	yes	—
19	C₂×C₄×Dic₂₃	C₂²	1	2	yes	—
20	C₂×C₄×Dic₂₃	C₂²	1	2	yes	—
21	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
22	C₂₃⋊#95	C₂²	1	4	yes	—
23	C₂₃⋊#95	C₂²	1	4	yes	—
24	C₂₃⋊#95	C₂²	1	2	yes	—
25	C₂₃⋊#95	C₂²	1	4	yes	—
26	C₂×Q₈×C₂₃	—	2	4	no	—
27	C₂×Q₈×C₂₃	C₂×C₄	1	4	yes	—
28	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
29	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
30	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
31	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
32	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
33	C₂²×Dic₂₃	C₂³	1	4	yes	—
34	C₂²×Dic₂₃	C₂³	1	2	yes	—
35	C₂²×Dic₂₃	C₂³	1	2	yes	—
36	C₄⋊C₄×C₂₃	C₂×C₄	1	8	yes	—
37	C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
38	C₄²×C₂₃	C₂×C₄	1	8	yes	—
39	C₄²×C₂₃	C₂×C₄	1	4	yes	—
40	C₄⋊Dic₂₃	—	2	4	no	—
41	C₄⋊Dic₂₃	—	2	8	no	—
42	C₄×Dic₂₃	—	2	4	no	—
43	C₄×Dic₂₃	—	2	2	no	—
44	C₄×Dic₂₃	—	2	4	no	—
45	C₄×Dic₂₃	—	2	4	no	—
46	C₄×Dic₂₃	Q₈	1	4	yes	complemented
47	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
48	Q₈×C₂₃	—	2	8	no	—
49	C₂×C₄×C₂₃	—	2	2	no	—
50	C₂×C₄×C₂₃	—	2	4	no	—
51	C₂×C₄×C₂₃	—	2	4	no	—
52	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
53	C₂×C₄×C₂₃	C₂²×C₄	1	4	yes	—
54	C₂×C₄×C₂₃	C₂²×C₄	1	4	yes	—
55	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
56	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
57	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
58	C₂×C₄×C₂₃	#41	1	4	yes	—
59	C₂×C₄×C₂₃	#41	1	2	yes	—
60	C₂×Dic₂₃	—	2	2	no	—
61	C₂×Dic₂₃	—	2	2	no	—
62	C₂×Dic₂₃	—	2	2	no	—
63	C₂×Dic₂₃	—	2	2	no	—
64	C₂×Dic₂₃	—	2	4	no	—
65	C₂×Dic₂₃	—	2	4	no	—
66	C₂×Dic₂₃	—	2	4	no	—
67	C₂×Dic₂₃	C₂×Q₈	1	4	yes	—
68	C₂×Dic₂₃	#41	1	4	yes	—
69	#2295	—	23	1	no	maximal, Sylow
70	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
71	C₂²×C₂₃	C₂²×Q₈	1	1	yes	characteristic
72	C₂²×C₂₃	C₂×#41	1	2	yes	—
73	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
74	C₂²×C₂₃	C₂×#41	1	2	yes	—
75	C₄×C₂₃	—	2	4	no	—
76	C₄×C₂₃	—	2	2	no	—
77	C₄×C₂₃	—	2	4	no	—
78	C₄×C₂₃	#117	1	4	yes	—
79	C₄×C₂₃	C₄×Q₈	1	4	yes	—
80	Dic₂₃	—	2	4	no	—
81	Dic₂₃	—	2	4	no	—
82	Dic₂₃	—	4	4	no	—
83	C₂×C₄×Q₈	D₂₃	1	1	yes	characteristic
84	#391	—	23	2	no	—
85	#389	—	23	2	no	—
86	#387	—	23	4	no	—
87	C₄×C₄⋊C₄	—	23	1	no	—
88	C₄×C₄⋊C₄	—	23	2	no	—
89	#376	—	23	2	no	—
90	C₄³	—	23	1	no	—
91	C₂×C₂₃	C₂×#117	1	1	yes	characteristic
92	C₂×C₂₃	C₂×C₄×Q₈	1	1	yes	characteristic
93	C₂×C₂₃	C₄×#41	1	1	yes	characteristic
94	C₂×C₂₃	#529	1	2	yes	—
95	C₂×C₂₃	#533	1	2	yes	—
96	C₂²×Q₈	C₂×D₂₃	1	1	yes	characteristic
97	C₄×Q₈	Dic₂₃	1	8	yes	complemented
98	C₂×C₄⋊C₄	—	23	4	no	—
99	C₂×C₄⋊C₄	—	23	2	no	—
100	C₂×C₄⋊C₄	C₂×D₂₃	1	2	yes	—
101	C₂×C₄⋊C₄	C₂×D₂₃	1	1	yes	characteristic
102	C₂×C₄²	—	23	1	no	—
103	C₂×C₄²	—	23	2	no	—
104	C₂×C₄²	—	23	2	no	—
105	C₂×C₄²	—	23	1	no	—
106	C₂×C₄²	—	23	2	no	—
107	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
108	C₂×C₄²	C₂×D₂₃	1	2	yes	—
109	#95	—	23	4	no	—
110	#95	—	23	4	no	—
111	#95	—	23	2	no	—
112	#95	—	23	4	no	—
113	C₂₃	#2295	1	1	yes	characteristic, complemented, Sylow
114	C₂×Q₈	—	2	4	no	—
115	C₂×Q₈	C₂×Dic₂₃	1	4	yes	—
116	C₂²×C₄	—	23	2	no	—
117	C₂²×C₄	—	23	2	no	—
118	C₂²×C₄	—	23	4	no	—
119	C₂²×C₄	C₂²×D₂₃	1	2	yes	—
120	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
121	C₂²×C₄	C₂²×D₂₃	1	2	yes	—
122	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
123	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
124	C₄⋊C₄	—	46	8	no	—
125	C₄⋊C₄	—	46	4	no	—
126	C₄⋊C₄	C₂×Dic₂₃	1	8	yes	—
127	C₄⋊C₄	C₂×Dic₂₃	1	4	yes	—
128	C₄²	—	23	4	no	—
129	C₄²	—	46	4	no	—
130	C₄²	—	46	4	no	—
131	C₄²	—	46	2	no	—
132	C₄²	—	46	4	no	—
133	C₄²	C₂×Dic₂₃	1	8	yes	—
134	C₄²	C₂×Dic₂₃	1	4	yes	—
135	C₂³	C₂³×D₂₃	1	1	yes	characteristic
136	Q₈	—	2	8	no	—
137	C₂×C₄	—	2	4	no	—
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	—	2	4	no	—
140	C₂×C₄	—	23	4	no	—
141	C₂×C₄	—	23	4	no	—
142	C₂×C₄	—	46	2	no	—
143	C₂×C₄	—	46	2	no	—
144	C₂×C₄	—	46	2	no	—
145	C₂×C₄	—	46	2	no	—
146	C₂×C₄	—	46	4	no	—
147	C₂×C₄	—	46	4	no	—
148	C₂×C₄	—	46	4	no	—
149	C₂×C₄	#67288	1	4	yes	—
150	C₂×C₄	Q₈×D₂₃	1	2	yes	—
151	C₂×C₄	#67292	1	2	yes	—
152	C₂×C₄	C₂²×Dic₂₃	1	2	yes	—
153	C₂×C₄	C₂²×Dic₂₃	1	4	yes	—
154	C₂×C₄	C₂²×Dic₂₃	1	4	yes	—
155	C₂×C₄	C₂²×Dic₂₃	1	2	yes	—
156	C₂×C₄	C₂²×Dic₂₃	1	2	yes	—
157	C₂²	C₂×#67288	1	2	yes	—
158	C₂²	C₂×Q₈×D₂₃	1	1	yes	characteristic
159	C₂²	C₂×#67292	1	1	yes	characteristic
160	C₂²	#41×D₂₃	1	2	yes	—
161	C₂²	C₂³×Dic₂₃	1	1	yes	characteristic
162	C₄	—	2	4	no	—
163	C₄	—	2	2	no	—
164	C₄	—	2	4	no	—
165	C₄	—	46	4	no	—
166	C₄	—	46	4	no	—
167	C₄	—	92	4	no	—
168	C₄	#10625097	1	4	yes	—
169	C₄	Q₈×Dic₂₃	1	4	yes	—
170	C₂	C₂×#10625097	1	1	yes	characteristic
171	C₂	C₂×Q₈×Dic₂₃	1	1	yes	characteristic
172	C₂	#41×Dic₂₃	1	1	yes	characteristic
173	C₂	#13092954	1	2	yes	—
174	C₂	#13092971	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²×D₂₃
commutator	C₂²×C₂₃	C₂³×C₄
Frattini	C₂³	C₂³×D₂₃
Fitting	C₂×C₄×Q₈×C₂₃	C₂
socle, prime radical, cosocle	C₂³×C₂₃	C₂⁴
hypercenter	C₂×C₄×Q₈	D₂₃
hypocenter	C₂₃	#2295
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
10. C₄×Q₈×C₂₃	C₄	semidirect
46. C₄×Dic₂₃	Q₈	semidirect
97. C₄×Q₈	Dic₂₃	semidirect
113. C₂₃	#2295	semidirect

Maximal subgroups

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

subgroup	index
#13092324	2
#13092316	2
#13092303	2
C₄×C₄⋊Dic₂₃	2
#13092263	2
C₄²×Dic₂₃	2
C₂×C₄×Q₈×C₂₃	2
#2295	23