Subgroups — C₂×#13093124 #15686928

Order 2944.

1200 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13093124	C₂	1	16	yes	maximal, direct product factor
2	C₂×#10625040	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#10625061	C₂	1	2	yes	maximal, complemented
4	C₂×#10625095	C₂	1	2	yes	maximal, complemented
5	C₂×#10625049	C₂	1	4	yes	maximal, complemented
6	C₂×C₄×Dic₄₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×D₄×Dic₂₃	C₂	1	2	yes	maximal
8	C₂²×C₄⋊Dic₂₃	C₂	1	2	yes	maximal, complemented
9	C₂×C₄×D₄×C₂₃	C₂	1	1	yes	maximal, characteristic
10	C₂²×D₄×C₂₃	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₂₃	C₂²	1	2	yes	—
12	C₂³×Dic₂₃	C₂²	1	4	yes	—
13	C₂²×Dic₄₆	C₂²	1	1	yes	characteristic
14	C₂²×Dic₄₆	C₂²	1	2	yes	—
15	C₄×D₄×C₂₃	C₂²	1	8	yes	—
16	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
17	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
18	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
19	C₂×C₂²⋊Dic₂₃	C₂²	1	4	yes	—
20	D₄×Dic₂₃	C₂²	1	16	yes	—
21	C₂×C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊Dic₂₃	C₂²	1	4	yes	—
23	C₂×C₄⋊Dic₂₃	C₂²	1	2	yes	—
24	C₂×C₄⋊Dic₂₃	C₂²	1	2	yes	—
25	C₂×C₄⋊Dic₂₃	C₂²	1	16	yes	complemented
26	#10625095	C₂²	1	16	yes	complemented
27	C₂×Dic₂₃⋊C₄	C₂²	1	2	yes	—
28	C₂×Dic₂₃⋊C₄	C₂²	1	4	yes	—
29	C₂×C₄×Dic₂₃	C₂²	1	2	yes	—
30	#10625061	C₂²	1	16	yes	complemented
31	#10625049	C₂²	1	32	yes	complemented
32	#10625040	C₂²	1	8	yes	complemented
33	C₄×Dic₄₆	C₂²	1	8	yes	complemented
34	C₂⁴×C₂₃	C₂³	1	2	yes	—
35	C₂×D₄×C₂₃	Q₈	1	8	yes	complemented
36	C₂×D₄×C₂₃	C₂³	1	4	yes	—
37	C₂²×C₄×C₂₃	—	2	8	no	—
38	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
39	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₂₃	C₂³	1	8	yes	—
41	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
43	C₂²×Dic₂₃	—	4	8	no	—
44	C₂²×Dic₂₃	C₂³	1	2	yes	—
45	C₂²×Dic₂₃	C₂³	1	2	yes	—
46	C₂²×Dic₂₃	C₂³	1	4	yes	—
47	C₂²×Dic₂₃	C₂³	1	16	yes	—
48	C₂×Dic₄₆	—	2	4	no	—
49	C₂×Dic₄₆	—	4	4	no	—
50	C₂×Dic₄₆	C₂³	1	4	yes	—
51	C₂×Dic₄₆	C₂³	1	8	yes	—
52	C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
53	C₂²⋊C₄×C₂₃	C₂³	1	8	yes	—
54	C₄²×C₂₃	C₂³	1	4	yes	—
55	C₂²⋊Dic₂₃	C₂³	1	16	yes	—
56	C₄⋊Dic₂₃	C₂³	1	4	yes	—
57	C₄⋊Dic₂₃	C₂³	1	8	yes	—
58	C₄⋊Dic₂₃	C₂³	1	8	yes	—
59	C₄⋊Dic₂₃	C₂³	1	16	yes	—
60	Dic₂₃⋊C₄	C₂³	1	8	yes	—
61	Dic₂₃⋊C₄	C₂³	1	16	yes	—
62	C₄×Dic₂₃	C₂³	1	8	yes	—
63	C₂³×C₂₃	—	2	4	no	—
64	C₂³×C₂₃	—	2	4	no	—
65	C₂³×C₂₃	C₂×Q₈	1	8	yes	—
66	C₂³×C₂₃	C₂⁴	1	4	yes	—
67	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
68	D₄×C₂₃	C₂×Q₈	1	16	yes	complemented
69	C₂×C₄×C₂₃	—	2	4	no	—
70	C₂×C₄×C₂₃	—	2	2	no	—
71	C₂×C₄×C₂₃	—	2	16	no	—
72	C₂×C₄×C₂₃	C₂×Q₈	1	4	yes	—
73	C₂×C₄×C₂₃	#41	1	4	yes	—
74	C₂×C₄×C₂₃	C₂⁴	1	4	yes	—
75	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
76	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
77	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
78	C₂×Dic₂₃	—	2	4	no	—
79	C₂×Dic₂₃	—	2	8	no	—
80	C₂×Dic₂₃	—	2	4	no	—
81	C₂×Dic₂₃	—	4	16	no	—
82	C₂×Dic₂₃	C₂⁴	1	4	yes	—
83	C₂×Dic₂₃	C₂⁴	1	4	yes	—
84	C₂×Dic₂₃	C₂⁴	1	8	yes	—
85	Dic₄₆	—	2	8	no	—
86	Dic₄₆	—	4	8	no	—
87	C₂×#554	—	23	1	no	maximal, Sylow
88	C₂²×C₂₃	—	2	16	no	—
89	C₂²×C₂₃	—	2	4	no	—
90	C₂²×C₂₃	—	2	4	no	—
91	C₂²×C₂₃	C₂²×Q₈	1	8	yes	—
92	C₂²×C₂₃	C₂²×Q₈	1	2	yes	—
93	C₂²×C₂₃	C₂×#41	1	2	yes	—
94	C₂²×C₂₃	H(5,𝔽₂)	1	2	yes	—
95	C₂²×C₂₃	C₂⁵	1	1	yes	characteristic
96	C₄×C₂₃	—	2	4	no	—
97	C₄×C₂₃	—	2	2	no	—
98	C₄×C₂₃	C₂²×Q₈	1	4	yes	—
99	C₄×C₂₃	C₂×#41	1	4	yes	—
100	Dic₂₃	—	2	4	no	—
101	Dic₂₃	—	2	8	no	—
102	Dic₂₃	—	2	4	no	—
103	#554	—	23	16	no	—
104	C₂×C₄⋊Q₈	—	23	1	no	—
105	C₂×#125	—	23	2	no	—
106	C₂×C₂²⋊Q₈	—	23	2	no	—
107	C₂×C₂²⋊Q₈	—	23	4	no	—
108	C₂×C₄×Q₈	—	23	1	no	—
109	C₂×C₄×D₄	—	23	2	no	—
110	C₂×C₄×D₄	D₂₃	1	1	yes	characteristic
111	C₂²×C₄⋊C₄	—	23	2	no	—
112	C₂×C₂₃	—	2	8	no	—
113	C₂×C₂₃	#554	1	4	yes	complemented
114	C₂×C₂₃	C₂³×Q₈	1	1	yes	characteristic
115	C₂×C₂₃	C₂²×#41	1	1	yes	characteristic
116	C₂×C₂₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
117	C₂²×Q₈	—	23	1	no	—
118	C₂²×Q₈	—	23	2	no	—
119	C₂²×D₄	C₂×D₂₃	1	1	yes	characteristic
120	C₂³×C₄	—	23	4	no	—
121	C₂³×C₄	C₂×D₂₃	1	2	yes	—
122	C₄⋊Q₈	—	23	8	no	—
123	#125	—	23	16	no	—
124	C₂²⋊Q₈	—	23	16	no	—
125	C₂²⋊Q₈	—	23	32	no	—
126	C₄×Q₈	—	23	8	no	—
127	C₄×D₄	—	23	16	no	—
128	C₄×D₄	C₂×D₂₃	1	8	yes	—
129	C₂×C₄⋊C₄	—	23	1	no	—
130	C₂×C₄⋊C₄	—	23	4	no	—
131	C₂×C₄⋊C₄	—	23	2	no	—
132	C₂×C₄⋊C₄	—	23	16	no	—
133	C₂×C₄⋊C₄	—	23	2	no	—
134	C₂×C₄⋊C₄	—	23	2	no	—
135	C₂×C₄⋊C₄	—	23	4	no	—
136	C₂×C₄⋊C₄	C₂×D₂₃	1	1	yes	characteristic
137	C₂×C₂²⋊C₄	—	23	4	no	—
138	C₂×C₂²⋊C₄	C₂×D₂₃	1	2	yes	—
139	C₂×C₄²	—	23	2	no	—
140	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
141	C₂₃	C₂×#554	1	1	yes	characteristic, complemented, Sylow
142	C₂⁴	C₂²×D₂₃	1	2	yes	—
143	C₂×Q₈	—	23	4	no	—
144	C₂×Q₈	—	23	8	no	—
145	C₂×Q₈	—	46	4	no	—
146	C₂×Q₈	—	92	4	no	—
147	C₂×D₄	Dic₄₆	1	8	yes	complemented
148	C₂×D₄	C₂²×D₂₃	1	4	yes	—
149	C₂²×C₄	—	2	8	no	—
150	C₂²×C₄	—	23	2	no	—
151	C₂²×C₄	—	23	2	no	—
152	C₂²×C₄	—	23	4	no	—
153	C₂²×C₄	—	23	16	no	—
154	C₂²×C₄	—	92	8	no	—
155	C₂²×C₄	C₂²×D₂₃	1	8	yes	—
156	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
157	C₂²×C₄	C₂²×D₂₃	1	2	yes	—
158	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
159	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
160	C₄⋊C₄	—	23	4	no	—
161	C₄⋊C₄	—	23	16	no	—
162	C₄⋊C₄	—	23	8	no	—
163	C₄⋊C₄	—	23	8	no	—
164	C₄⋊C₄	—	23	8	no	—
165	C₄⋊C₄	—	23	16	no	—
166	C₄⋊C₄	C₂²×D₂₃	1	4	yes	—
167	C₂²⋊C₄	—	23	16	no	—
168	C₂²⋊C₄	C₂²×D₂₃	1	8	yes	—
169	C₄²	—	23	8	no	—
170	C₄²	C₂²×D₂₃	1	4	yes	—
171	C₂³	—	2	4	no	—
172	C₂³	—	2	4	no	—
173	C₂³	C₂×Dic₄₆	1	8	yes	—
174	C₂³	C₂³×D₂₃	1	4	yes	—
175	C₂³	C₂³×D₂₃	1	1	yes	characteristic
176	Q₈	—	46	8	no	—
177	Q₈	—	92	8	no	—
178	D₄	C₂×Dic₄₆	1	16	yes	complemented
179	C₂×C₄	—	2	4	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	16	no	—
182	C₂×C₄	—	23	4	no	—
183	C₂×C₄	—	23	4	no	—
184	C₂×C₄	—	23	8	no	—
185	C₂×C₄	—	46	4	no	—
186	C₂×C₄	—	46	4	no	—
187	C₂×C₄	—	46	8	no	—
188	C₂×C₄	—	92	16	no	—
189	C₂×C₄	C₂×Dic₄₆	1	4	yes	—
190	C₂×C₄	#67290	1	4	yes	—
191	C₂×C₄	C₂³×D₂₃	1	2	yes	—
192	C₂×C₄	C₂³×D₂₃	1	4	yes	—
193	C₂×C₄	C₂³×D₂₃	1	2	yes	—
194	C₂×C₄	C₂³×D₂₃	1	2	yes	—
195	C₂²	—	2	16	no	—
196	C₂²	—	2	4	no	—
197	C₂²	—	2	4	no	—
198	C₂²	C₂²×Dic₄₆	1	8	yes	—
199	C₂²	C₂²×Dic₄₆	1	2	yes	—
200	C₂²	C₂×#67290	1	2	yes	—
201	C₂²	#10625160	1	2	yes	—
202	C₂²	C₂⁴×D₂₃	1	1	yes	characteristic
203	C₄	—	2	4	no	—
204	C₄	—	2	2	no	—
205	C₄	—	46	4	no	—
206	C₄	—	46	4	no	—
207	C₄	—	46	8	no	—
208	C₄	C₂²×Dic₄₆	1	4	yes	—
209	C₄	C₂×#67290	1	4	yes	—
210	C₂	—	2	8	no	—
211	C₂	#13093124	1	4	yes	direct product factor
212	C₂	C₂³×Dic₄₆	1	1	yes	characteristic
213	C₂	C₂²×#67290	1	1	yes	characteristic
214	C₂	C₂×#10625160	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₂₃
commutator, cosocle	C₂²×C₂₃	C₂⁵
Frattini	C₂²	C₂⁴×D₂₃
Fitting	C₂×C₄×D₄×C₂₃	C₂
socle	C₂³×C₂₃	C₂⁴
prime radical	C₂²×D₄×C₂₃	C₂²
hypercenter	C₂×C₄×D₄	D₂₃
hypocenter	C₂₃	C₂×#554
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
1. #13093124	C₂	direct (N × H)
2. C₂×#10625040	C₂	semidirect
3. C₂×#10625061	C₂	semidirect
4. C₂×#10625095	C₂	semidirect
5. C₂×#10625049	C₂	semidirect
6. C₂×C₄×Dic₄₆	C₂	semidirect
8. C₂²×C₄⋊Dic₂₃	C₂	semidirect
25. C₂×C₄⋊Dic₂₃	C₂²	semidirect
26. #10625095	C₂²	semidirect
30. #10625061	C₂²	semidirect
31. #10625049	C₂²	semidirect
32. #10625040	C₂²	semidirect
33. C₄×Dic₄₆	C₂²	semidirect
35. C₂×D₄×C₂₃	Q₈	semidirect
68. D₄×C₂₃	C₂×Q₈	semidirect
113. C₂×C₂₃	#554	semidirect
141. C₂₃	C₂×#554	semidirect
147. C₂×D₄	Dic₄₆	semidirect
178. D₄	C₂×Dic₄₆	semidirect
211. C₂	#13093124	direct (N × H)

Maximal subgroups

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

subgroup	index
#13093124	2
C₂×#10625040	2
C₂×#10625061	2
C₂×#10625095	2
C₂×#10625049	2
C₂×C₄×Dic₄₆	2
C₂×D₄×Dic₂₃	2
C₂²×C₄⋊Dic₂₃	2
C₂×C₄×D₄×C₂₃	2
C₂×#554	23

Subgroups — C2×#13093124 #15686928

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13093124 #15686928