Subgroups — C₂×#13070890 #14909249

Order 2816.

558 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×#10622487	C₂	1	2	yes	maximal
2	#13070890	C₂	1	8	yes	maximal, direct product factor
3	C₂×#10622142	C₂	1	1	yes	maximal, characteristic
4	C₂×#10622127	C₂	1	2	yes	maximal
5	C₂×C₁₆×Dic₁₁	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₁×#366	C₂	1	1	yes	maximal, characteristic
7	C₂×#66941	—	2	2	no	—
8	C₂²×C₁₁×C₁₆	C₂²	1	1	yes	characteristic
9	C₂×C₁₁×#107	C₂²	1	2	yes	—
10	C₂×#66882	C₂²	1	2	yes	—
11	C₂×C₈×Dic₁₁	C₂²	1	1	yes	characteristic
12	C₂²×C₁₁⋊C₁₆	C₂²	1	1	yes	characteristic
13	C₂×#66871	—	2	2	no	—
14	#10622487	—	2	4	no	—
15	#10622487	C₂²	1	8	yes	—
16	C₁₁×#366	C₂²	1	4	yes	—
17	#10622142	C₂²	1	4	yes	—
18	C₁₆×Dic₁₁	C₂²	1	4	yes	—
19	#10622127	C₂²	1	8	yes	—
20	C₂²×Dic₂₂	—	4	2	no	—
21	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
22	C₂×C₁₁×C₄⋊C₄	—	2	2	no	—
23	C₂×C₄⋊Dic₁₁	—	2	2	no	—
24	C₂×Dic₁₁⋊C₄	—	4	2	no	—
25	C₂×C₄×Dic₁₁	D₄	1	1	yes	characteristic
26	C₂²×C₁₁⋊C₈	D₄	1	1	yes	characteristic
27	#66941	—	2	8	no	—
28	#66941	—	4	4	no	—
29	C₂×C₁₁×C₁₆	Q₈	1	4	yes	complemented
30	C₂×C₁₁×C₁₆	C₂³	1	2	yes	—
31	C₁₁×#107	—	2	2	no	—
32	C₁₁×#107	C₂³	1	4	yes	—
33	#66882	—	2	2	no	—
34	#66882	C₂³	1	4	yes	—
35	C₈×Dic₁₁	D₄	1	2	yes	—
36	C₈×Dic₁₁	C₂³	1	2	yes	—
37	C₂×C₁₁⋊C₁₆	Q₈	1	4	yes	complemented
38	C₂×C₁₁⋊C₁₆	C₂³	1	2	yes	—
39	#66871	—	2	8	no	—
40	C₂×#2252	—	11	1	no	maximal, Sylow
41	C₂²×C₄×C₁₁	—	4	2	no	—
42	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
43	C₂²×Dic₁₁	—	4	2	no	—
44	C₂²×Dic₁₁	D₈	1	1	yes	characteristic
45	C₂²×Dic₁₁	D₈	1	1	yes	characteristic
46	C₂×Dic₂₂	—	4	8	no	—
47	C₂×Dic₂₂	—	8	8	no	—
48	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
49	C₂×C₈×C₁₁	C₂×Q₈	1	2	yes	—
50	C₂×C₈×C₁₁	C₂×Q₈	1	2	yes	—
51	C₂×C₈×C₁₁	C₂⁴	1	1	yes	characteristic
52	C₁₁×C₄⋊C₄	—	2	4	no	—
53	C₁₁×C₄⋊C₄	—	4	2	no	—
54	C₄⋊Dic₁₁	—	2	4	no	—
55	C₄⋊Dic₁₁	—	4	2	no	—
56	Dic₁₁⋊C₄	—	4	8	no	—
57	C₄×Dic₁₁	D₈	1	2	yes	—
58	C₄×Dic₁₁	C₂×D₄	1	2	yes	—
59	C₂×C₁₁⋊C₈	Q₁₆	1	4	yes	—
60	C₂×C₁₁⋊C₈	C₂×D₄	1	2	yes	—
61	C₁₁×C₁₆	C₂×Q₈	1	4	yes	complemented
62	C₁₁⋊C₁₆	C₂×Q₈	1	4	yes	complemented
63	C₂×#500	—	11	2	no	—
64	#2252	—	11	8	no	—
65	C₂×#366	—	11	1	no	—
66	C₂×#366	—	11	2	no	—
67	C₂×#366	D₁₁	1	1	yes	characteristic
68	C₂×C₄×C₁₆	—	11	1	no	—
69	C₂³×C₁₁	C₂×D₈	1	1	yes	characteristic
70	C₂×C₄×C₁₁	—	4	4	no	—
71	C₂×C₄×C₁₁	—	8	4	no	—
72	C₂×C₄×C₁₁	C₄⋊Q₈	1	2	yes	—
73	C₂×C₄×C₁₁	C₂×D₈	1	1	yes	characteristic
74	C₂×C₄×C₁₁	C₂×Q₁₆	1	2	yes	—
75	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
76	C₂×Dic₁₁	—	4	4	no	—
77	C₂×Dic₁₁	—	8	4	no	—
78	C₂×Dic₁₁	D₁₆	1	4	yes	—
79	C₂×Dic₁₁	Q₃₂	1	4	yes	—
80	C₂×Dic₁₁	C₂×D₈	1	2	yes	—
81	C₂×Dic₁₁	C₂×D₈	1	2	yes	—
82	Dic₂₂	—	8	16	no	—
83	C₈×C₁₁	C₄⋊Q₈	1	2	yes	—
84	C₈×C₁₁	C₂²×Q₈	1	1	yes	characteristic
85	C₈×C₁₁	C₂²×Q₈	1	1	yes	characteristic
86	C₁₁⋊C₈	C₂×Q₁₆	1	4	yes	—
87	C₂×C₄⋊Q₈	—	22	2	no	—
88	C₂²×C₁₆	—	11	1	no	—
89	C₂²×C₁₆	C₂×D₁₁	1	1	yes	characteristic
90	#500	—	11	8	no	—
91	#500	—	22	4	no	—
92	C₂×#107	—	11	2	no	—
93	C₂×#107	—	22	2	no	—
94	C₂×#107	C₂×D₁₁	1	2	yes	—
95	C₂×C₄×C₈	—	11	1	no	—
96	#366	—	11	4	no	—
97	#366	—	11	8	no	—
98	#366	C₂×D₁₁	1	4	yes	—
99	C₄×C₁₆	—	11	4	no	—
100	C₂²×C₁₁	#500	1	2	yes	—
101	C₂²×C₁₁	C₂×D₁₆	1	2	yes	—
102	C₂²×C₁₁	C₂×Q₃₂	1	2	yes	—
103	C₂²×C₁₁	C₂²×D₈	1	1	yes	characteristic
104	C₄×C₁₁	—	8	4	no	—
105	C₄×C₁₁	#500	1	2	yes	—
106	C₄×C₁₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
107	C₄×C₁₁	C₂²×Q₁₆	1	1	yes	characteristic
108	Dic₁₁	—	8	4	no	—
109	Dic₁₁	C₂×D₁₆	1	4	yes	—
110	Dic₁₁	C₂×Q₃₂	1	4	yes	—
111	C₂²×Q₈	—	44	2	no	—
112	C₂²×C₈	—	11	1	no	—
113	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
114	C₄⋊Q₈	—	22	8	no	—
115	C₄⋊Q₈	—	44	4	no	—
116	C₂×C₄⋊C₄	—	2	2	no	—
117	C₂×C₄⋊C₄	—	22	2	no	—
118	C₂×C₄⋊C₄	—	44	2	no	—
119	C₂×C₄²	—	11	1	no	—
120	C₂×C₁₆	—	11	4	no	—
121	C₂×C₁₆	—	11	2	no	—
122	C₂×C₁₆	Dic₂₂	1	4	yes	complemented
123	C₂×C₁₆	C₂²×D₁₁	1	2	yes	—
124	#107	—	2	2	no	—
125	#107	—	11	4	no	—
126	#107	—	22	8	no	—
127	#107	—	22	2	no	—
128	#107	C₂²×D₁₁	1	4	yes	—
129	C₄×C₈	—	11	2	no	—
130	C₄×C₈	—	11	2	no	—
131	C₂×C₁₁	#2252	1	4	yes	complemented
132	C₂×C₁₁	C₂×#500	1	1	yes	characteristic
133	C₂×C₁₁	C₂²×D₁₆	1	1	yes	characteristic
134	C₂×C₁₁	C₂²×Q₃₂	1	1	yes	characteristic
135	C₂×Q₈	—	44	8	no	—
136	C₂×Q₈	—	88	8	no	—
137	C₂²×C₄	—	4	2	no	—
138	C₂²×C₄	—	11	1	no	—
139	C₂²×C₄	—	11	1	no	—
140	C₂²×C₄	—	44	2	no	—
141	C₂²×C₄	D₄×D₁₁	1	1	yes	characteristic
142	C₂×C₈	—	11	4	no	—
143	C₂×C₈	—	11	2	no	—
144	C₂×C₈	C₂×Dic₂₂	1	2	yes	—
145	C₂×C₈	D₄×D₁₁	1	1	yes	characteristic
146	C₂×C₈	Q₈×D₁₁	1	2	yes	—
147	C₂×C₈	C₂³×D₁₁	1	1	yes	characteristic
148	C₄⋊C₄	—	2	4	no	—
149	C₄⋊C₄	—	4	2	no	—
150	C₄⋊C₄	—	22	4	no	—
151	C₄⋊C₄	—	44	8	no	—
152	C₄⋊C₄	—	44	2	no	—
153	C₄²	—	11	2	no	—
154	C₄²	—	11	2	no	—
155	C₁₆	—	11	4	no	—
156	C₁₆	C₂×Dic₂₂	1	4	yes	complemented
157	C₁₁	C₂×#2252	1	1	yes	characteristic, complemented, Sylow
158	C₂³	D₈×D₁₁	1	1	yes	characteristic
159	Q₈	—	88	16	no	—
160	C₂×C₄	—	4	4	no	—
161	C₂×C₄	—	8	4	no	—
162	C₂×C₄	—	11	2	no	—
163	C₂×C₄	—	11	2	no	—
164	C₂×C₄	—	11	4	no	—
165	C₂×C₄	—	11	4	no	—
166	C₂×C₄	—	44	4	no	—
167	C₂×C₄	—	88	4	no	—
168	C₂×C₄	#66941	1	2	yes	—
169	C₂×C₄	D₈×D₁₁	1	1	yes	characteristic
170	C₂×C₄	Q₁₆×D₁₁	1	2	yes	—
171	C₂×C₄	C₂×D₄×D₁₁	1	1	yes	characteristic
172	C₈	—	11	4	no	—
173	C₈	#66941	1	2	yes	—
174	C₈	C₂²×Dic₂₂	1	1	yes	characteristic
175	C₈	C₂×Q₈×D₁₁	1	1	yes	characteristic
176	C₂²	#10622487	1	2	yes	—
177	C₂²	D₁₁×D₁₆	1	2	yes	—
178	C₂²	D₁₁×Q₃₂	1	2	yes	—
179	C₂²	C₂×D₈×D₁₁	1	1	yes	characteristic
180	C₄	—	8	4	no	—
181	C₄	—	11	4	no	—
182	C₄	—	11	4	no	—
183	C₄	—	88	4	no	—
184	C₄	#10622487	1	2	yes	—
185	C₄	C₂×#66941	1	1	yes	characteristic
186	C₄	C₂×Q₁₆×D₁₁	1	1	yes	characteristic
187	C₂	#13070890	1	4	yes	direct product factor
188	C₂	C₂×#10622487	1	1	yes	characteristic
189	C₂	C₂×D₁₁×D₁₆	1	1	yes	characteristic
190	C₂	C₂×D₁₁×Q₃₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₈×D₁₁
commutator, cosocle	C₂×C₈×C₁₁	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₁
Fitting	C₂×C₁₁×#366	C₂
socle, prime radical	C₂³×C₁₁	C₂×D₈
hypercenter	C₂×#366	D₁₁
hypocenter	C₁₁	C₂×#2252
LCS	G ▷ C₂×C₈×C₁₁ ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#366

Complements

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

N	H ≅ G/N	type
2. #13070890	C₂	direct (N × H)
29. C₂×C₁₁×C₁₆	Q₈	semidirect
37. C₂×C₁₁⋊C₁₆	Q₈	semidirect
61. C₁₁×C₁₆	C₂×Q₈	semidirect
62. C₁₁⋊C₁₆	C₂×Q₈	semidirect
122. C₂×C₁₆	Dic₂₂	semidirect
131. C₂×C₁₁	#2252	semidirect
156. C₁₆	C₂×Dic₂₂	semidirect
157. C₁₁	C₂×#2252	semidirect
187. C₂	#13070890	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#10622487	2
#13070890	2
C₂×#10622142	2
C₂×#10622127	2
C₂×C₁₆×Dic₁₁	2
C₂×C₁₁×#366	2
C₂×#2252	11

Subgroups — C2×#13070890 #14909249

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13070890 #14909249