Subgroups — #15671624

Order 2944.

568 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃⋊#95	C₂	1	2	yes	maximal
2	C₂×C₂₃⋊#95	C₂	1	2	yes	maximal
3	C₂×C₂₃⋊#95	C₂	1	2	yes	maximal
4	C₂²×C₄²×C₂₃	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
6	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
7	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
8	C₂³×Dic₂₃	C₂²	1	4	yes	—
9	C₂×C₄²×C₂₃	—	2	4	no	—
10	C₂×C₄²×C₂₃	C₄	1	4	yes	complemented
11	C₂₃⋊#95	—	2	4	no	—
12	C₂₃⋊#95	—	2	4	no	—
13	C₂₃⋊#95	C₄	1	8	yes	complemented
14	C₂⁴×C₂₃	C₂³	1	1	yes	characteristic
15	C₂²×C₄×C₂₃	—	2	2	no	—
16	C₂²×C₄×C₂₃	—	2	2	no	—
17	C₂²×C₄×C₂₃	—	2	2	no	—
18	C₂²×C₄×C₂₃	—	2	2	no	—
19	C₂²×C₄×C₂₃	—	2	2	no	—
20	C₂²×C₄×C₂₃	—	2	2	no	—
21	C₂²×C₄×C₂₃	C₂×C₄	1	4	yes	—
22	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₂₃	D₄	1	2	yes	—
26	C₂²×C₄×C₂₃	D₄	1	4	yes	—
27	C₂²×C₄×C₂₃	Q₈	1	2	yes	—
28	C₂²×Dic₂₃	—	2	4	no	—
29	C₂²×Dic₂₃	—	2	4	no	—
30	C₂²×Dic₂₃	—	2	4	no	—
31	C₂²×Dic₂₃	—	2	4	no	—
32	C₂²×Dic₂₃	—	2	4	no	—
33	C₂²×Dic₂₃	—	2	4	no	—
34	C₂²×Dic₂₃	C₂×C₄	1	8	yes	—
35	C₄²×C₂₃	—	2	8	no	—
36	C₂³×C₂₃	C₂²×C₄	1	2	yes	—
37	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
38	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
39	C₂³×C₂₃	C₂×D₄	1	2	yes	—
40	C₂³×C₂₃	C₂×Q₈	1	1	yes	characteristic
41	C₂³×C₂₃	#41	1	2	yes	—
42	C₂³×C₂₃	#41	1	2	yes	—
43	C₂³×C₂₃	#41	1	2	yes	—
44	C₂³×C₂₃	#41	1	2	yes	—
45	C₂×C₄×C₂₃	—	2	2	no	—
46	C₂×C₄×C₂₃	—	2	2	no	—
47	C₂×C₄×C₂₃	—	2	2	no	—
48	C₂×C₄×C₂₃	—	2	2	no	—
49	C₂×C₄×C₂₃	—	2	2	no	—
50	C₂×C₄×C₂₃	—	2	2	no	—
51	C₂×C₄×C₂₃	—	2	8	no	—
52	C₂×C₄×C₂₃	—	2	4	no	—
53	C₂×C₄×C₂₃	—	2	4	no	—
54	C₂×C₄×C₂₃	—	2	8	no	—
55	C₂×C₄×C₂₃	C₄²	1	4	yes	—
56	C₂×C₄×C₂₃	C₂²⋊C₄	1	4	yes	—
57	C₂×C₄×C₂₃	C₄⋊C₄	1	4	yes	—
58	C₂×Dic₂₃	—	2	8	no	—
59	C₂×Dic₂₃	—	2	8	no	—
60	C₂×Dic₂₃	—	2	8	no	—
61	C₂×Dic₂₃	—	4	4	no	—
62	C₂×Dic₂₃	—	4	4	no	—
63	C₂×Dic₂₃	—	4	4	no	—
64	C₂×Dic₂₃	—	4	4	no	—
65	#1434	—	23	1	no	maximal, Sylow
66	C₂²×C₂₃	C₂×C₄²	1	1	yes	characteristic
67	C₂²×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₂²×C₂₃	#117	1	4	yes	—
70	C₂²×C₂₃	#117	1	1	yes	characteristic
71	C₂²×C₂₃	#117	1	1	yes	characteristic
72	C₂²×C₂₃	#117	1	1	yes	characteristic
73	C₂²×C₂₃	#117	1	1	yes	characteristic
74	C₂²×C₂₃	C₄×D₄	1	2	yes	—
75	C₂²×C₂₃	C₄×D₄	1	4	yes	—
76	C₂²×C₂₃	C₄×Q₈	1	2	yes	—
77	C₂²×C₂₃	#121	1	2	yes	—
78	C₂²×C₂₃	C₂²⋊Q₈	1	2	yes	—
79	C₂²×C₂₃	#123	1	4	yes	—
80	C₂²×C₂₃	#124	1	2	yes	—
81	C₂²×C₂₃	#125	1	2	yes	—
82	C₂²×C₂₃	#126	1	4	yes	—
83	C₄×C₂₃	—	2	4	no	—
84	C₄×C₂₃	—	2	4	no	—
85	C₄×C₂₃	—	2	4	no	—
86	Dic₂₃	—	4	8	no	—
87	C₂²×C₄²	D₂₃	1	1	yes	characteristic
88	C₂×#95	—	23	2	no	—
89	C₂×#95	—	23	2	no	—
90	C₂×#95	—	23	2	no	—
91	C₂×C₂₃	#376	1	1	yes	characteristic
92	C₂×C₂₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₂₃	C₄×C₄⋊C₄	1	1	yes	characteristic
94	C₂×C₂₃	#380	1	1	yes	characteristic
95	C₂×C₂₃	#381	1	1	yes	characteristic
96	C₂×C₂₃	#382	1	1	yes	characteristic
97	C₂×C₂₃	#383	1	1	yes	characteristic
98	C₂×C₂₃	#387	1	4	yes	—
99	C₂×C₂₃	#388	1	4	yes	—
100	C₂³×C₄	—	23	4	no	—
101	C₂³×C₄	C₂×D₂₃	1	1	yes	characteristic
102	C₂³×C₄	C₂×D₂₃	1	1	yes	characteristic
103	C₂³×C₄	C₂×D₂₃	1	1	yes	characteristic
104	C₂×C₄²	—	2	4	no	—
105	C₂×C₄²	Dic₂₃	1	4	yes	complemented
106	#95	—	23	8	no	—
107	#95	—	46	4	no	—
108	#95	—	46	4	no	—
109	C₂₃	#1434	1	1	yes	characteristic, complemented, Sylow
110	C₂⁴	C₂²×D₂₃	1	1	yes	characteristic
111	C₂²×C₄	—	2	2	no	—
112	C₂²×C₄	—	2	2	no	—
113	C₂²×C₄	—	2	2	no	—
114	C₂²×C₄	—	2	2	no	—
115	C₂²×C₄	—	2	2	no	—
116	C₂²×C₄	—	2	2	no	—
117	C₂²×C₄	—	23	8	no	—
118	C₂²×C₄	—	46	4	no	—
119	C₂²×C₄	—	46	4	no	—
120	C₂²×C₄	—	46	4	no	—
121	C₂²×C₄	—	46	4	no	—
122	C₂²×C₄	—	46	4	no	—
123	C₂²×C₄	—	46	4	no	—
124	C₂²×C₄	Dic₄₆	1	2	yes	—
125	C₂²×C₄	C₄×D₂₃	1	4	yes	—
126	C₂²×C₄	D₉₂	1	2	yes	—
127	C₂²×C₄	C₂×Dic₂₃	1	2	yes	—
128	C₂²×C₄	C₂×Dic₂₃	1	2	yes	—
129	C₂²×C₄	C₂×Dic₂₃	1	2	yes	—
130	C₂²×C₄	C₂²⋊D₂₃	1	4	yes	—
131	C₄²	—	2	8	no	—
132	C₂³	C₂×Dic₄₆	1	1	yes	characteristic
133	C₂³	C₂×C₄×D₂₃	1	2	yes	—
134	C₂³	C₂×D₉₂	1	1	yes	characteristic
135	C₂³	#67288	1	2	yes	—
136	C₂³	#67288	1	2	yes	—
137	C₂³	#67288	1	2	yes	—
138	C₂³	#67288	1	2	yes	—
139	C₂³	C₂²×Dic₂₃	1	1	yes	characteristic
140	C₂³	C₂×C₂²⋊D₂₃	1	2	yes	—
141	C₂×C₄	—	2	4	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	8	no	—
147	C₂×C₄	—	2	8	no	—
148	C₂×C₄	—	2	4	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	46	8	no	—
152	C₂×C₄	—	46	8	no	—
153	C₂×C₄	—	46	8	no	—
154	C₂×C₄	—	92	4	no	—
155	C₂×C₄	—	92	4	no	—
156	C₂×C₄	—	92	4	no	—
157	C₂×C₄	—	92	4	no	—
158	C₂×C₄	C₄×Dic₂₃	1	4	yes	—
159	C₂×C₄	C₄⋊Dic₂₃	1	4	yes	—
160	C₂×C₄	C₂²⋊Dic₂₃	1	4	yes	—
161	C₂²	C₄×Dic₄₆	1	2	yes	—
162	C₂²	#10625041	1	2	yes	—
163	C₂²	#10625043	1	4	yes	—
164	C₂²	C₄×D₉₂	1	2	yes	—
165	C₂²	#10625046	1	2	yes	—
166	C₂²	#10625047	1	4	yes	—
167	C₂²	C₂×C₄×Dic₂₃	1	1	yes	characteristic
168	C₂²	#10625095	1	2	yes	—
169	C₂²	C₂×C₄⋊Dic₂₃	1	1	yes	characteristic
170	C₂²	#10625097	1	1	yes	characteristic
171	C₂²	#10625097	1	1	yes	characteristic
172	C₂²	#10625097	1	1	yes	characteristic
173	C₂²	#10625097	1	1	yes	characteristic
174	C₂²	C₄×C₂²⋊D₂₃	1	4	yes	—
175	C₂²	#10625100	1	4	yes	—
176	C₂²	#10625101	1	2	yes	—
177	C₂²	C₂×C₂²⋊Dic₂₃	1	1	yes	characteristic
178	C₄	—	2	4	no	—
179	C₄	—	2	4	no	—
180	C₄	—	2	4	no	—
181	C₄	—	92	8	no	—
182	C₂	#13092263	1	1	yes	characteristic
183	C₂	C₄×C₂²⋊Dic₂₃	1	1	yes	characteristic
184	C₂	C₄×C₄⋊Dic₂₃	1	1	yes	characteristic
185	C₂	#13092277	1	1	yes	characteristic
186	C₂	#13092280	1	1	yes	characteristic
187	C₂	#13092283	1	1	yes	characteristic
188	C₂	#13092285	1	1	yes	characteristic
189	C₂	#13092301	1	4	yes	—
190	C₂	#13092308	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂⁴	C₂²×D₂₃
commutator	C₂²×C₂₃	C₂×C₄²
Fitting	C₂²×C₄²×C₂₃	C₂
socle, prime radical, cosocle	C₂⁴×C₂₃	C₂³
hypercenter	C₂²×C₄²	D₂₃
hypocenter	C₂₃	#1434
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
10. C₂×C₄²×C₂₃	C₄	semidirect
13. C₂₃⋊#95	C₄	semidirect
105. C₂×C₄²	Dic₂₃	semidirect
109. C₂₃	#1434	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₃⋊#95	2
C₂²×C₄²×C₂₃	2
#1434	23