Subgroups — C₂×#13092970 #15686684

Order 2944.

1168 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13092970	C₂	1	16	yes	maximal, direct product factor
2	C₂×#10625059	C₂	1	2	yes	maximal, complemented
3	C₂×#10625061	C₂	1	2	yes	maximal, complemented
4	C₂×#10625049	C₂	1	4	yes	maximal
5	C₂×C₄×Dic₄₆	C₂	1	4	yes	maximal, complemented
6	C₂×#10625097	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₄×Dic₂₃	C₂	1	1	yes	maximal, characteristic
8	C₂×C₂₃×#117	C₂	1	1	yes	maximal, characteristic
9	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
10	C₂³×Dic₂₃	C₂²	1	2	yes	—
11	C₂²×Dic₄₆	C₂²	1	4	yes	—
12	C₂₃×#117	C₂²	1	8	yes	—
13	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
14	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
15	C₂×C₄²×C₂₃	C₂²	1	2	yes	—
16	C₂×C₂²⋊Dic₂₃	C₂²	1	2	yes	—
17	#10625097	C₂²	1	8	yes	—
18	C₂×C₄⋊Dic₂₃	C₂²	1	2	yes	—
19	C₂×C₄⋊Dic₂₃	C₂²	1	4	yes	—
20	C₂×Dic₂₃⋊C₄	C₂²	1	4	yes	—
21	C₂×Dic₂₃⋊C₄	C₂²	1	4	yes	—
22	C₂×C₄×Dic₂₃	C₂²	1	2	yes	—
23	C₂×C₄×Dic₂₃	C₂²	1	2	yes	—
24	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
25	C₂×C₄×Dic₂₃	C₂²	1	1	yes	characteristic
26	C₂×C₄×Dic₂₃	C₂²	1	8	yes	—
27	#10625061	C₂²	1	16	yes	complemented
28	#10625059	C₂²	1	16	yes	complemented
29	#10625049	C₂²	1	32	yes	—
30	C₄×Dic₄₆	C₂²	1	32	yes	complemented
31	C₂⁴×C₂₃	C₂³	1	1	yes	characteristic
32	C₂²×C₄×C₂₃	Q₈	1	8	yes	complemented
33	C₂²×C₄×C₂₃	C₂³	1	4	yes	—
34	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
35	C₂²×C₄×C₂₃	C₂³	1	4	yes	—
36	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
37	C₂²×Dic₂₃	—	2	8	no	—
38	C₂²×Dic₂₃	C₂³	1	8	yes	—
39	C₂²×Dic₂₃	C₂³	1	2	yes	—
40	C₂²×Dic₂₃	C₂³	1	4	yes	—
41	C₂²×Dic₂₃	C₂³	1	2	yes	—
42	C₂×Dic₄₆	—	2	16	no	—
43	C₂×Dic₄₆	C₂³	1	16	yes	—
44	C₄⋊C₄×C₂₃	C₂³	1	8	yes	—
45	C₂²⋊C₄×C₂₃	C₂³	1	8	yes	—
46	C₄²×C₂₃	C₂³	1	8	yes	—
47	C₂²⋊Dic₂₃	C₂³	1	8	yes	—
48	C₄⋊Dic₂₃	C₂³	1	16	yes	—
49	C₄⋊Dic₂₃	C₂³	1	8	yes	—
50	Dic₂₃⋊C₄	C₂³	1	16	yes	—
51	Dic₂₃⋊C₄	C₂³	1	16	yes	—
52	C₄×Dic₂₃	C₂³	1	8	yes	—
53	C₄×Dic₂₃	C₂³	1	8	yes	—
54	C₄×Dic₂₃	C₂³	1	4	yes	—
55	C₄×Dic₂₃	C₂³	1	4	yes	—
56	C₂³×C₂₃	—	2	4	no	—
57	C₂³×C₂₃	C₂×Q₈	1	4	yes	—
58	C₂³×C₂₃	C₂⁴	1	2	yes	—
59	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
60	C₂×C₄×C₂₃	—	2	8	no	—
61	C₂×C₄×C₂₃	C₂×Q₈	1	4	yes	—
62	C₂×C₄×C₂₃	C₂×Q₈	1	4	yes	—
63	C₂×C₄×C₂₃	C₂×Q₈	1	16	yes	complemented
64	C₂×C₄×C₂₃	C₂⁴	1	8	yes	—
65	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
66	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
67	C₂×Dic₂₃	—	2	8	no	—
68	C₂×Dic₂₃	—	2	4	no	—
69	C₂×Dic₂₃	—	2	16	no	—
70	C₂×Dic₂₃	#41	1	8	yes	—
71	C₂×Dic₂₃	C₂⁴	1	4	yes	—
72	C₂×Dic₂₃	C₂⁴	1	8	yes	—
73	C₂×Dic₂₃	C₂⁴	1	4	yes	—
74	Dic₄₆	—	2	32	no	—
75	C₂×#533	—	23	1	no	maximal, Sylow
76	C₂²×C₂₃	—	2	8	no	—
77	C₂²×C₂₃	—	2	4	no	—
78	C₂²×C₂₃	C₂²×Q₈	1	4	yes	—
79	C₂²×C₂₃	C₂²×Q₈	1	2	yes	—
80	C₂²×C₂₃	C₂×#41	1	4	yes	—
81	C₂²×C₂₃	C₂⁵	1	1	yes	characteristic
82	C₄×C₂₃	—	2	8	no	—
83	C₄×C₂₃	C₂²×Q₈	1	4	yes	—
84	C₄×C₂₃	C₂²×Q₈	1	4	yes	—
85	Dic₂₃	—	2	8	no	—
86	Dic₂₃	—	2	4	no	—
87	Dic₂₃	C₂×#41	1	8	yes	—
88	#533	—	23	16	no	—
89	C₂×C₄⋊Q₈	—	23	2	no	—
90	C₂×#125	—	23	2	no	—
91	C₂×C₂²⋊Q₈	—	23	4	no	—
92	C₂×C₄×Q₈	—	23	4	no	—
93	C₂×#117	—	23	1	no	—
94	C₂×#117	D₂₃	1	1	yes	characteristic
95	C₂²×C₄²	—	23	1	no	—
96	C₂×C₂₃	—	2	4	no	—
97	C₂×C₂₃	#533	1	4	yes	complemented
98	C₂×C₂₃	C₂³×Q₈	1	1	yes	characteristic
99	C₂×C₂₃	C₂²×#41	1	2	yes	—
100	C₂²×Q₈	—	23	4	no	—
101	C₂³×C₄	—	23	2	no	—
102	C₂³×C₄	C₂×D₂₃	1	1	yes	characteristic
103	C₄⋊Q₈	—	23	16	no	—
104	#125	—	23	16	no	—
105	C₂²⋊Q₈	—	23	32	no	—
106	C₄×Q₈	—	23	32	no	—
107	#117	—	23	8	no	—
108	#117	C₂×D₂₃	1	8	yes	—
109	C₂×C₄⋊C₄	—	23	2	no	—
110	C₂×C₄⋊C₄	—	23	4	no	—
111	C₂×C₄⋊C₄	—	23	4	no	—
112	C₂×C₄⋊C₄	—	23	4	no	—
113	C₂×C₄⋊C₄	C₂×D₂₃	1	2	yes	—
114	C₂×C₂²⋊C₄	—	23	2	no	—
115	C₂×C₂²⋊C₄	C₂×D₂₃	1	2	yes	—
116	C₂×C₄²	—	23	1	no	—
117	C₂×C₄²	—	23	1	no	—
118	C₂×C₄²	—	23	8	no	—
119	C₂×C₄²	—	23	2	no	—
120	C₂×C₄²	—	23	2	no	—
121	C₂×C₄²	C₂×D₂₃	1	2	yes	—
122	C₂₃	C₂×#533	1	1	yes	characteristic, complemented, Sylow
123	C₂⁴	C₂²×D₂₃	1	1	yes	characteristic
124	C₂×Q₈	—	23	16	no	—
125	C₂×Q₈	—	46	16	no	—
126	C₂²×C₄	—	23	2	no	—
127	C₂²×C₄	—	23	2	no	—
128	C₂²×C₄	—	23	8	no	—
129	C₂²×C₄	—	23	4	no	—
130	C₂²×C₄	—	46	8	no	—
131	C₂²×C₄	Dic₄₆	1	8	yes	complemented
132	C₂²×C₄	C₂²×D₂₃	1	4	yes	—
133	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
134	C₂²×C₄	C₂²×D₂₃	1	4	yes	—
135	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
136	C₄⋊C₄	—	23	8	no	—
137	C₄⋊C₄	—	23	16	no	—
138	C₄⋊C₄	—	23	16	no	—
139	C₄⋊C₄	—	23	16	no	—
140	C₄⋊C₄	C₂²×D₂₃	1	8	yes	—
141	C₂²⋊C₄	—	23	8	no	—
142	C₂²⋊C₄	C₂²×D₂₃	1	8	yes	—
143	C₄²	—	23	8	no	—
144	C₄²	—	23	4	no	—
145	C₄²	—	23	4	no	—
146	C₄²	—	23	8	no	—
147	C₄²	C₂²×D₂₃	1	8	yes	—
148	C₂³	—	2	4	no	—
149	C₂³	C₂×Dic₄₆	1	4	yes	—
150	C₂³	C₂³×D₂₃	1	2	yes	—
151	C₂³	C₂³×D₂₃	1	1	yes	characteristic
152	Q₈	—	46	32	no	—
153	C₂×C₄	—	2	8	no	—
154	C₂×C₄	—	23	8	no	—
155	C₂×C₄	—	23	4	no	—
156	C₂×C₄	—	23	4	no	—
157	C₂×C₄	—	23	8	no	—
158	C₂×C₄	—	46	16	no	—
159	C₂×C₄	—	46	4	no	—
160	C₂×C₄	—	46	8	no	—
161	C₂×C₄	C₂×Dic₄₆	1	4	yes	—
162	C₂×C₄	C₂×Dic₄₆	1	4	yes	—
163	C₂×C₄	C₂×Dic₄₆	1	16	yes	complemented
164	C₂×C₄	C₂³×D₂₃	1	8	yes	—
165	C₂×C₄	C₂³×D₂₃	1	2	yes	—
166	C₂×C₄	C₂³×D₂₃	1	2	yes	—
167	C₂²	—	2	8	no	—
168	C₂²	—	2	4	no	—
169	C₂²	C₂²×Dic₄₆	1	4	yes	—
170	C₂²	C₂²×Dic₄₆	1	2	yes	—
171	C₂²	#41×D₂₃	1	4	yes	—
172	C₂²	C₂⁴×D₂₃	1	1	yes	characteristic
173	C₄	—	2	8	no	—
174	C₄	—	23	8	no	—
175	C₄	—	46	4	no	—
176	C₄	—	46	8	no	—
177	C₄	C₂²×Dic₄₆	1	4	yes	—
178	C₄	C₂²×Dic₄₆	1	4	yes	—
179	C₂	—	2	4	no	—
180	C₂	#13092970	1	4	yes	direct product factor
181	C₂	C₂³×Dic₄₆	1	1	yes	characteristic
182	C₂	C₂×#41×D₂₃	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²×D₂₃
commutator, cosocle	C₂²×C₂₃	C₂⁵
Frattini	C₂²	C₂⁴×D₂₃
Fitting	C₂×C₂₃×#117	C₂
socle	C₂³×C₂₃	C₂⁴
prime radical	C₂⁴×C₂₃	C₂³
hypercenter	C₂×#117	D₂₃
hypocenter	C₂₃	C₂×#533
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
1. #13092970	C₂	direct (N × H)
2. C₂×#10625059	C₂	semidirect
3. C₂×#10625061	C₂	semidirect
5. C₂×C₄×Dic₄₆	C₂	semidirect
27. #10625061	C₂²	semidirect
28. #10625059	C₂²	semidirect
30. C₄×Dic₄₆	C₂²	semidirect
32. C₂²×C₄×C₂₃	Q₈	semidirect
63. C₂×C₄×C₂₃	C₂×Q₈	semidirect
97. C₂×C₂₃	#533	semidirect
122. C₂₃	C₂×#533	semidirect
131. C₂²×C₄	Dic₄₆	semidirect
163. C₂×C₄	C₂×Dic₄₆	semidirect
180. C₂	#13092970	direct (N × H)

Maximal subgroups

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

subgroup	index
#13092970	2
C₂×#10625059	2
C₂×#10625061	2
C₂×#10625049	2
C₂×C₄×Dic₄₆	2
C₂×#10625097	2
C₂²×C₄×Dic₂₃	2
C₂×C₂₃×#117	2
C₂×#533	23

Subgroups — C2×#13092970 #15686684

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13092970 #15686684