Subgroups — C₂×#13092277 #15675770

Order 2944.

1232 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊Dic₂₃	C₂	1	2	yes	maximal, complemented
2	C₂²×C₂²⋊Dic₂₃	C₂	1	2	yes	maximal
3	#13092277	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₂₃⋊#95	C₂	1	2	yes	maximal, complemented
5	C₂⁴×C₄×C₂₃	C₂	1	1	yes	maximal, characteristic
6	C₂⁵×C₂₃	C₂²	1	1	yes	characteristic
7	C₂³×C₄×C₂₃	—	2	8	no	—
8	C₂³×C₄×C₂₃	C₄	1	8	yes	complemented
9	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
10	C₂³×C₄×C₂₃	C₂²	1	4	yes	—
11	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
12	C₂³×Dic₂₃	C₂²	1	4	yes	—
13	C₂×C₂²⋊Dic₂₃	—	2	8	no	—
14	C₂×C₂²⋊Dic₂₃	C₂²	1	8	yes	—
15	C₂×C₄⋊Dic₂₃	—	2	8	no	—
16	C₂×C₄⋊Dic₂₃	C₂²	1	8	yes	complemented
17	C₂₃⋊#95	C₂²	1	8	yes	complemented
18	C₂⁴×C₂₃	—	2	4	no	—
19	C₂⁴×C₂₃	—	2	4	no	—
20	C₂⁴×C₂₃	C₂×C₄	1	4	yes	—
21	C₂⁴×C₂₃	D₄	1	4	yes	—
22	C₂⁴×C₂₃	Q₈	1	4	yes	—
23	C₂⁴×C₂₃	C₂³	1	2	yes	—
24	C₂⁴×C₂₃	C₂³	1	1	yes	characteristic
25	C₂²×C₄×C₂₃	—	2	4	no	—
26	C₂²×C₄×C₂₃	—	2	32	no	—
27	C₂²×C₄×C₂₃	—	2	16	no	—
28	C₂²×C₄×C₂₃	C₂×C₄	1	4	yes	—
29	C₂²×C₄×C₂₃	C₂×C₄	1	4	yes	—
30	C₂²×C₄×C₂₃	C₂×C₄	1	16	yes	complemented
31	C₂²×C₄×C₂₃	D₄	1	8	yes	—
32	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
33	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
34	C₂²×Dic₂₃	—	2	8	no	—
35	C₂²×Dic₂₃	—	2	8	no	—
36	C₂²×Dic₂₃	—	2	8	no	—
37	C₂²×Dic₂₃	C₂³	1	8	yes	—
38	C₂²⋊Dic₂₃	—	2	16	no	—
39	C₄⋊Dic₂₃	—	2	16	no	—
40	C₂³×C₂₃	—	2	8	no	—
41	C₂³×C₂₃	—	2	8	no	—
42	C₂³×C₂₃	—	2	16	no	—
43	C₂³×C₂₃	—	2	16	no	—
44	C₂³×C₂₃	—	2	4	no	—
45	C₂³×C₂₃	—	2	4	no	—
46	C₂³×C₂₃	C₄⋊C₄	1	16	yes	complemented
47	C₂³×C₂₃	C₂²×C₄	1	4	yes	—
48	C₂³×C₂₃	C₂²×C₄	1	2	yes	—
49	C₂³×C₂₃	C₂×D₄	1	4	yes	—
50	C₂³×C₂₃	C₂×D₄	1	2	yes	—
51	C₂³×C₂₃	C₂×D₄	1	4	yes	—
52	C₂³×C₂₃	C₂×Q₈	1	4	yes	—
53	C₂³×C₂₃	C₂×Q₈	1	2	yes	—
54	C₂³×C₂₃	#41	1	4	yes	—
55	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
56	C₂×C₄×C₂₃	—	2	8	no	—
57	C₂×C₄×C₂₃	—	2	4	no	—
58	C₂×C₄×C₂₃	—	2	32	no	—
59	C₂×C₄×C₂₃	C₂²⋊C₄	1	16	yes	complemented
60	C₂×C₄×C₂₃	C₂²×C₄	1	4	yes	—
61	C₂×C₄×C₂₃	C₂²×C₄	1	4	yes	—
62	C₂×C₄×C₂₃	C₂×D₄	1	8	yes	—
63	C₂×Dic₂₃	—	2	8	no	—
64	C₂×Dic₂₃	—	2	8	no	—
65	C₂×Dic₂₃	—	2	8	no	—
66	C₂×Dic₂₃	—	4	16	no	—
67	C₂×#380	—	23	1	no	maximal, Sylow
68	C₂²×C₂₃	—	2	4	no	—
69	C₂²×C₂₃	—	2	4	no	—
70	C₂²×C₂₃	—	2	8	no	—
71	C₂²×C₂₃	—	2	8	no	—
72	C₂²×C₂₃	—	2	32	no	—
73	C₂²×C₂₃	C₂×C₂²⋊C₄	1	4	yes	—
74	C₂²×C₂₃	C₂×C₄⋊C₄	1	4	yes	—
75	C₂²×C₂₃	C₂×C₄⋊C₄	1	8	yes	complemented
76	C₂²×C₂₃	#117	1	4	yes	—
77	C₂²×C₂₃	#121	1	8	yes	—
78	C₂²×C₂₃	C₂²⋊Q₈	1	8	yes	—
79	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
80	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
81	C₂²×C₂₃	C₂²×D₄	1	2	yes	—
82	C₂²×C₂₃	C₂²×Q₈	1	1	yes	characteristic
83	C₂²×C₂₃	C₂×#41	1	2	yes	—
84	C₄×C₂₃	—	2	4	no	—
85	C₄×C₂₃	C₂×C₂²⋊C₄	1	8	yes	complemented
86	Dic₂₃	—	4	8	no	—
87	C₂⁴×C₄	D₂₃	1	1	yes	characteristic
88	C₂²×C₄⋊C₄	—	23	2	no	—
89	C₂²×C₂²⋊C₄	—	23	2	no	—
90	#380	—	23	8	no	—
91	C₂×#95	—	23	2	no	—
92	C₂×C₂₃	—	2	8	no	—
93	C₂×C₂₃	#380	1	8	yes	complemented
94	C₂×C₂₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₂₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
96	C₂×C₂₃	C₂×#117	1	1	yes	characteristic
97	C₂×C₂₃	C₂×#121	1	2	yes	—
98	C₂×C₂₃	C₂×C₂²⋊Q₈	1	2	yes	—
99	C₂⁵	C₂×D₂₃	1	1	yes	characteristic
100	C₂³×C₄	—	2	8	no	—
101	C₂³×C₄	—	23	4	no	—
102	C₂³×C₄	Dic₂₃	1	8	yes	complemented
103	C₂³×C₄	C₂×D₂₃	1	1	yes	characteristic
104	C₂³×C₄	C₂×D₂₃	1	4	yes	—
105	C₂³×C₄	C₂×D₂₃	1	1	yes	characteristic
106	C₂×C₄⋊C₄	—	23	8	no	—
107	C₂×C₄⋊C₄	—	46	8	no	—
108	C₂×C₂²⋊C₄	—	23	8	no	—
109	C₂×C₂²⋊C₄	—	46	8	no	—
110	#95	—	23	8	no	—
111	C₂₃	C₂×#380	1	1	yes	characteristic, complemented, Sylow
112	C₂⁴	—	2	4	no	—
113	C₂⁴	—	2	4	no	—
114	C₂⁴	Dic₄₆	1	4	yes	—
115	C₂⁴	D₉₂	1	4	yes	—
116	C₂⁴	C₂×Dic₂₃	1	4	yes	—
117	C₂⁴	C₂²×D₂₃	1	2	yes	—
118	C₂⁴	C₂²×D₂₃	1	1	yes	characteristic
119	C₂²×C₄	—	2	4	no	—
120	C₂²×C₄	—	2	32	no	—
121	C₂²×C₄	—	2	16	no	—
122	C₂²×C₄	—	23	8	no	—
123	C₂²×C₄	—	46	8	no	—
124	C₂²×C₄	—	46	8	no	—
125	C₂²×C₄	—	46	8	no	—
126	C₂²×C₄	C₂×Dic₂₃	1	4	yes	—
127	C₂²×C₄	C₂×Dic₂₃	1	4	yes	—
128	C₂²×C₄	C₂×Dic₂₃	1	16	yes	complemented
129	C₂²×C₄	C₂²⋊D₂₃	1	8	yes	—
130	C₂²×C₄	C₂²×D₂₃	1	2	yes	—
131	C₂²×C₄	C₂²×D₂₃	1	2	yes	—
132	C₄⋊C₄	—	46	16	no	—
133	C₂²⋊C₄	—	46	16	no	—
134	C₂³	—	2	16	no	—
135	C₂³	—	2	8	no	—
136	C₂³	—	2	16	no	—
137	C₂³	—	2	8	no	—
138	C₂³	—	2	4	no	—
139	C₂³	—	2	4	no	—
140	C₂³	C₄⋊Dic₂₃	1	16	yes	complemented
141	C₂³	C₂×Dic₄₆	1	4	yes	—
142	C₂³	C₂×Dic₄₆	1	2	yes	—
143	C₂³	C₂×D₉₂	1	4	yes	—
144	C₂³	C₂×D₉₂	1	2	yes	—
145	C₂³	#67288	1	4	yes	—
146	C₂³	C₂²×Dic₂₃	1	4	yes	—
147	C₂³	C₂²×Dic₂₃	1	2	yes	—
148	C₂³	C₂×C₂²⋊D₂₃	1	4	yes	—
149	C₂³	C₂³×D₂₃	1	1	yes	characteristic
150	C₂×C₄	—	2	8	no	—
151	C₂×C₄	—	2	4	no	—
152	C₂×C₄	—	2	32	no	—
153	C₂×C₄	—	46	8	no	—
154	C₂×C₄	—	46	8	no	—
155	C₂×C₄	—	46	8	no	—
156	C₂×C₄	—	92	16	no	—
157	C₂×C₄	C₂²⋊Dic₂₃	1	16	yes	complemented
158	C₂×C₄	C₂²×Dic₂₃	1	4	yes	—
159	C₂×C₄	C₂²×Dic₂₃	1	4	yes	—
160	C₂×C₄	C₂×C₂²⋊D₂₃	1	8	yes	—
161	C₂²	—	2	4	no	—
162	C₂²	—	2	4	no	—
163	C₂²	—	2	8	no	—
164	C₂²	—	2	8	no	—
165	C₂²	—	2	32	no	—
166	C₂²	#10625095	1	8	yes	—
167	C₂²	C₂×C₄⋊Dic₂₃	1	4	yes	—
168	C₂²	C₂×C₄⋊Dic₂₃	1	8	yes	complemented
169	C₂²	#10625097	1	4	yes	—
170	C₂²	#10625101	1	8	yes	—
171	C₂²	C₂×C₂²⋊Dic₂₃	1	4	yes	—
172	C₂²	C₂²×Dic₄₆	1	1	yes	characteristic
173	C₂²	C₂²×D₉₂	1	1	yes	characteristic
174	C₂²	C₂×#67288	1	2	yes	—
175	C₂²	C₂³×Dic₂₃	1	1	yes	characteristic
176	C₂²	C₂²×C₂²⋊D₂₃	1	2	yes	—
177	C₄	—	2	4	no	—
178	C₄	—	92	8	no	—
179	C₄	C₂×C₂²⋊Dic₂₃	1	8	yes	complemented
180	C₂	—	2	8	no	—
181	C₂	#13092277	1	8	yes	direct product factor
182	C₂	C₂²×C₂²⋊Dic₂₃	1	1	yes	characteristic
183	C₂	C₂²×C₄⋊Dic₂₃	1	1	yes	characteristic
184	C₂	C₂×#10625097	1	1	yes	characteristic
185	C₂	C₂×#10625101	1	2	yes	—
186	C₂	C₂×#10625095	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂²×D₂₃
commutator	C₂²×C₂₃	C₂³×C₄
Frattini	C₂³	C₂³×D₂₃
Fitting	C₂⁴×C₄×C₂₃	C₂
socle	C₂⁴×C₂₃	C₂³
prime radical	C₂⁵×C₂₃	C₂²
cosocle	C₂³×C₂₃	C₂⁴
hypercenter	C₂⁴×C₄	D₂₃
hypocenter	C₂₃	C₂×#380
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂⁴ ◁ C₂⁴×C₄

Complements

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

N	H ≅ G/N	type
1. C₂²×C₄⋊Dic₂₃	C₂	semidirect
3. #13092277	C₂	direct (N × H)
4. C₂×C₂₃⋊#95	C₂	semidirect
8. C₂³×C₄×C₂₃	C₄	semidirect
16. C₂×C₄⋊Dic₂₃	C₂²	semidirect
17. C₂₃⋊#95	C₂²	semidirect
30. C₂²×C₄×C₂₃	C₂×C₄	semidirect
46. C₂³×C₂₃	C₄⋊C₄	semidirect
59. C₂×C₄×C₂₃	C₂²⋊C₄	semidirect
75. C₂²×C₂₃	C₂×C₄⋊C₄	semidirect
85. C₄×C₂₃	C₂×C₂²⋊C₄	semidirect
93. C₂×C₂₃	#380	semidirect
102. C₂³×C₄	Dic₂₃	semidirect
111. C₂₃	C₂×#380	semidirect
128. C₂²×C₄	C₂×Dic₂₃	semidirect
140. C₂³	C₄⋊Dic₂₃	semidirect
157. C₂×C₄	C₂²⋊Dic₂₃	semidirect
168. C₂²	C₂×C₄⋊Dic₂₃	semidirect
179. C₄	C₂×C₂²⋊Dic₂₃	semidirect
181. C₂	#13092277	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊Dic₂₃	2
C₂²×C₂²⋊Dic₂₃	2
#13092277	2
C₂×C₂₃⋊#95	2
C₂⁴×C₄×C₂₃	2
C₂×#380	23

Subgroups — C2×#13092277 #15675770

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13092277 #15675770