Subgroups — C₂×#13093045 #15686795

Order 2944.

1080 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13093045	C₂	1	16	yes	maximal, direct product factor
2	C₂×#10625040	C₂	1	2	yes	maximal, complemented
3	C₂×#10625059	C₂	1	2	yes	maximal, complemented
4	C₂×#10625041	C₂	1	2	yes	maximal, complemented
5	C₂×#10625061	C₂	1	2	yes	maximal, complemented
6	C₂×#10625049	C₂	1	4	yes	maximal
7	C₂×#10625097	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₄⋊Dic₂₃	C₂	1	1	yes	maximal, characteristic
9	C₂×C₂₃×#117	C₂	1	1	yes	maximal, characteristic
10	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
11	C₂³×Dic₂₃	C₂²	1	2	yes	—
12	C₂²×Dic₄₆	C₂²	1	4	yes	—
13	C₂₃×#117	C₂²	1	8	yes	—
14	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
16	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
17	C₂×C₄²×C₂₃	C₂²	1	2	yes	—
18	C₂×C₂²⋊Dic₂₃	C₂²	1	2	yes	—
19	#10625097	C₂²	1	8	yes	—
20	C₂×C₄⋊Dic₂₃	C₂²	1	2	yes	—
21	C₂×C₄⋊Dic₂₃	C₂²	1	2	yes	—
22	C₂×C₄⋊Dic₂₃	C₂²	1	8	yes	—
23	C₂×C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊Dic₂₃	C₂²	1	4	yes	—
26	C₂×Dic₂₃⋊C₄	C₂²	1	4	yes	—
27	C₂×Dic₂₃⋊C₄	C₂²	1	4	yes	—
28	C₂×C₄×Dic₂₃	C₂²	1	2	yes	—
29	#10625061	C₂²	1	16	yes	complemented
30	#10625059	C₂²	1	16	yes	complemented
31	#10625049	C₂²	1	32	yes	—
32	#10625041	C₂²	1	16	yes	complemented
33	#10625040	C₂²	1	16	yes	complemented
34	C₂⁴×C₂₃	C₂³	1	1	yes	characteristic
35	C₂²×C₄×C₂₃	Q₈	1	8	yes	complemented
36	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
37	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
38	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
39	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₂₃	C₂³	1	4	yes	—
41	C₂²×Dic₂₃	—	4	4	no	—
42	C₂²×Dic₂₃	C₂³	1	2	yes	—
43	C₂²×Dic₂₃	C₂³	1	2	yes	—
44	C₂²×Dic₂₃	C₂³	1	4	yes	—
45	C₂²×Dic₂₃	C₂³	1	8	yes	—
46	C₂×Dic₄₆	—	4	8	no	—
47	C₂×Dic₄₆	C₂³	1	16	yes	—
48	C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
49	C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
50	C₂²⋊C₄×C₂₃	C₂³	1	8	yes	—
51	C₄²×C₂₃	C₂³	1	8	yes	—
52	C₂²⋊Dic₂₃	C₂³	1	8	yes	—
53	C₄⋊Dic₂₃	C₂³	1	4	yes	—
54	C₄⋊Dic₂₃	C₂³	1	8	yes	—
55	C₄⋊Dic₂₃	C₂³	1	16	yes	—
56	C₄⋊Dic₂₃	C₂³	1	4	yes	—
57	C₄⋊Dic₂₃	C₂³	1	8	yes	—
58	Dic₂₃⋊C₄	C₂³	1	16	yes	—
59	Dic₂₃⋊C₄	C₂³	1	16	yes	—
60	C₄×Dic₂₃	C₂³	1	8	yes	—
61	C₂³×C₂₃	—	2	2	no	—
62	C₂³×C₂₃	—	2	2	no	—
63	C₂³×C₂₃	C₂×Q₈	1	4	yes	—
64	C₂³×C₂₃	C₂⁴	1	2	yes	—
65	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
66	C₂×C₄×C₂₃	—	2	4	no	—
67	C₂×C₄×C₂₃	—	2	4	no	—
68	C₂×C₄×C₂₃	C₂×Q₈	1	4	yes	—
69	C₂×C₄×C₂₃	C₂×Q₈	1	4	yes	—
70	C₂×C₄×C₂₃	C₂×Q₈	1	16	yes	complemented
71	C₂×C₄×C₂₃	C₂⁴	1	4	yes	—
72	C₂×C₄×C₂₃	C₂⁴	1	4	yes	—
73	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
74	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
75	C₂×Dic₂₃	—	2	4	no	—
76	C₂×Dic₂₃	—	2	8	no	—
77	C₂×Dic₂₃	—	2	4	no	—
78	C₂×Dic₂₃	—	4	8	no	—
79	C₂×Dic₂₃	C₂⁴	1	4	yes	—
80	C₂×Dic₂₃	C₂⁴	1	4	yes	—
81	C₂×Dic₂₃	C₂⁴	1	8	yes	—
82	Dic₄₆	—	4	16	no	—
83	C₂×#544	—	23	1	no	maximal, Sylow
84	C₂²×C₂₃	—	2	8	no	—
85	C₂²×C₂₃	—	2	2	no	—
86	C₂²×C₂₃	—	2	2	no	—
87	C₂²×C₂₃	C₂²×Q₈	1	4	yes	—
88	C₂²×C₂₃	C₂²×Q₈	1	2	yes	—
89	C₂²×C₂₃	H(5,𝔽₂)	1	2	yes	—
90	C₂²×C₂₃	2₋¹⁺⁴	1	2	yes	—
91	C₂²×C₂₃	C₂⁵	1	1	yes	characteristic
92	C₄×C₂₃	—	2	4	no	—
93	C₄×C₂₃	—	2	4	no	—
94	C₄×C₂₃	C₂²×Q₈	1	4	yes	—
95	C₄×C₂₃	C₂²×Q₈	1	4	yes	—
96	Dic₂₃	—	2	4	no	—
97	Dic₂₃	—	2	8	no	—
98	Dic₂₃	—	2	4	no	—
99	#544	—	23	16	no	—
100	C₂×C₄⋊Q₈	—	23	2	no	—
101	C₂×C₄⋊Q₈	—	23	2	no	—
102	C₂×#125	—	23	2	no	—
103	C₂×#125	—	23	2	no	—
104	C₂×C₂²⋊Q₈	—	23	4	no	—
105	C₂×#117	—	23	1	no	—
106	C₂×#117	D₂₃	1	1	yes	characteristic
107	C₂²×C₄⋊C₄	—	23	1	no	—
108	C₂×C₂₃	—	2	4	no	—
109	C₂×C₂₃	#544	1	4	yes	complemented
110	C₂×C₂₃	C₂³×Q₈	1	1	yes	characteristic
111	C₂×C₂₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
112	C₂×C₂₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
113	C₂²×Q₈	—	23	4	no	—
114	C₂³×C₄	—	23	2	no	—
115	C₂³×C₄	C₂×D₂₃	1	1	yes	characteristic
116	C₄⋊Q₈	—	23	16	no	—
117	C₄⋊Q₈	—	23	16	no	—
118	#125	—	23	16	no	—
119	#125	—	23	16	no	—
120	C₂²⋊Q₈	—	23	32	no	—
121	#117	—	23	8	no	—
122	#117	C₂×D₂₃	1	8	yes	—
123	C₂×C₄⋊C₄	—	23	2	no	—
124	C₂×C₄⋊C₄	—	23	2	no	—
125	C₂×C₄⋊C₄	—	23	8	no	—
126	C₂×C₄⋊C₄	—	23	1	no	—
127	C₂×C₄⋊C₄	—	23	1	no	—
128	C₂×C₄⋊C₄	—	23	4	no	—
129	C₂×C₄⋊C₄	—	23	4	no	—
130	C₂×C₄⋊C₄	—	23	4	no	—
131	C₂×C₄⋊C₄	C₂×D₂₃	1	1	yes	characteristic
132	C₂×C₄⋊C₄	C₂×D₂₃	1	1	yes	characteristic
133	C₂×C₂²⋊C₄	—	23	2	no	—
134	C₂×C₂²⋊C₄	C₂×D₂₃	1	2	yes	—
135	C₂×C₄²	—	23	2	no	—
136	C₂×C₄²	C₂×D₂₃	1	2	yes	—
137	C₂₃	C₂×#544	1	1	yes	characteristic, complemented, Sylow
138	C₂⁴	C₂²×D₂₃	1	1	yes	characteristic
139	C₂×Q₈	—	23	16	no	—
140	C₂×Q₈	—	92	8	no	—
141	C₂²×C₄	—	23	2	no	—
142	C₂²×C₄	—	23	2	no	—
143	C₂²×C₄	—	23	4	no	—
144	C₂²×C₄	—	23	8	no	—
145	C₂²×C₄	—	92	4	no	—
146	C₂²×C₄	Dic₄₆	1	8	yes	complemented
147	C₂²×C₄	C₂²×D₂₃	1	2	yes	—
148	C₂²×C₄	C₂²×D₂₃	1	2	yes	—
149	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
150	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
151	C₂²×C₄	C₂²×D₂₃	1	4	yes	—
152	C₄⋊C₄	—	23	8	no	—
153	C₄⋊C₄	—	23	8	no	—
154	C₄⋊C₄	—	23	4	no	—
155	C₄⋊C₄	—	23	4	no	—
156	C₄⋊C₄	—	23	16	no	—
157	C₄⋊C₄	—	23	16	no	—
158	C₄⋊C₄	—	23	16	no	—
159	C₄⋊C₄	C₂²×D₂₃	1	4	yes	—
160	C₄⋊C₄	C₂²×D₂₃	1	4	yes	—
161	C₂²⋊C₄	—	23	8	no	—
162	C₂²⋊C₄	C₂²×D₂₃	1	8	yes	—
163	C₄²	—	23	8	no	—
164	C₄²	C₂²×D₂₃	1	8	yes	—
165	C₂³	—	2	2	no	—
166	C₂³	—	2	2	no	—
167	C₂³	C₂×Dic₄₆	1	4	yes	—
168	C₂³	C₂³×D₂₃	1	2	yes	—
169	C₂³	C₂³×D₂₃	1	1	yes	characteristic
170	Q₈	—	92	16	no	—
171	C₂×C₄	—	2	4	no	—
172	C₂×C₄	—	2	4	no	—
173	C₂×C₄	—	23	4	no	—
174	C₂×C₄	—	23	4	no	—
175	C₂×C₄	—	23	8	no	—
176	C₂×C₄	—	46	8	no	—
177	C₂×C₄	—	46	4	no	—
178	C₂×C₄	—	46	4	no	—
179	C₂×C₄	—	92	8	no	—
180	C₂×C₄	C₂×Dic₄₆	1	4	yes	—
181	C₂×C₄	C₂×Dic₄₆	1	4	yes	—
182	C₂×C₄	C₂×Dic₄₆	1	16	yes	complemented
183	C₂×C₄	C₂³×D₂₃	1	4	yes	—
184	C₂×C₄	C₂³×D₂₃	1	4	yes	—
185	C₂×C₄	C₂³×D₂₃	1	2	yes	—
186	C₂×C₄	C₂³×D₂₃	1	2	yes	—
187	C₂²	—	2	2	no	—
188	C₂²	—	2	2	no	—
189	C₂²	—	2	8	no	—
190	C₂²	C₂²×Dic₄₆	1	4	yes	—
191	C₂²	C₂²×Dic₄₆	1	2	yes	—
192	C₂²	#10625160	1	2	yes	—
193	C₂²	#10625161	1	2	yes	—
194	C₂²	C₂⁴×D₂₃	1	1	yes	characteristic
195	C₄	—	2	4	no	—
196	C₄	—	2	4	no	—
197	C₄	—	46	8	no	—
198	C₄	—	46	4	no	—
199	C₄	—	46	4	no	—
200	C₄	C₂²×Dic₄₆	1	4	yes	—
201	C₄	C₂²×Dic₄₆	1	4	yes	—
202	C₂	—	2	4	no	—
203	C₂	#13093045	1	4	yes	direct product factor
204	C₂	C₂³×Dic₄₆	1	1	yes	characteristic
205	C₂	C₂×#10625160	1	1	yes	characteristic
206	C₂	C₂×#10625161	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×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×#544
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂³ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
1. #13093045	C₂	direct (N × H)
2. C₂×#10625040	C₂	semidirect
3. C₂×#10625059	C₂	semidirect
4. C₂×#10625041	C₂	semidirect
5. C₂×#10625061	C₂	semidirect
29. #10625061	C₂²	semidirect
30. #10625059	C₂²	semidirect
32. #10625041	C₂²	semidirect
33. #10625040	C₂²	semidirect
35. C₂²×C₄×C₂₃	Q₈	semidirect
70. C₂×C₄×C₂₃	C₂×Q₈	semidirect
109. C₂×C₂₃	#544	semidirect
137. C₂₃	C₂×#544	semidirect
146. C₂²×C₄	Dic₄₆	semidirect
182. C₂×C₄	C₂×Dic₄₆	semidirect
203. C₂	#13093045	direct (N × H)

Maximal subgroups

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

subgroup	index
#13093045	2
C₂×#10625040	2
C₂×#10625059	2
C₂×#10625041	2
C₂×#10625061	2
C₂×#10625049	2
C₂×#10625097	2
C₂²×C₄⋊Dic₂₃	2
C₂×C₂₃×#117	2
C₂×#544	23

Subgroups — C2×#13093045 #15686795

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13093045 #15686795