Subgroups — C₂×#13066756 #14673974

Order 2816.

306 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10622137	C₂	1	2	yes	maximal, complemented
2	#13066756	C₂	1	4	yes	maximal, direct product factor
3	C₂²×C₁₁×M₃₂	C₂	1	1	yes	maximal, characteristic
4	C₂³×C₈×C₁₁	C₄	1	1	yes	characteristic
5	C₂×C₁₁×M₃₂	—	2	2	no	—
6	C₂×C₁₁×M₃₂	—	2	2	no	—
7	C₂×C₁₁×M₃₂	C₄	1	2	yes	—
8	C₂×C₁₁×M₃₂	C₂²	1	2	yes	—
9	C₂²×C₁₁×C₁₆	C₄	1	1	yes	characteristic
10	C₂²×C₁₁×C₁₆	C₂²	1	1	yes	characteristic
11	#10622137	—	2	2	no	—
12	#10622137	C₂²	1	4	yes	complemented
13	C₂×C₁₁⋊C₃₂	—	2	2	no	—
14	C₂³×C₄×C₁₁	C₈	1	1	yes	characteristic
15	C₂²×C₈×C₁₁	—	2	1	no	—
16	C₂²×C₈×C₁₁	—	2	1	no	—
17	C₂²×C₈×C₁₁	—	2	1	no	—
18	C₂²×C₈×C₁₁	—	2	1	no	—
19	C₂²×C₈×C₁₁	C₈	1	2	yes	—
20	C₂²×C₈×C₁₁	C₈	1	1	yes	characteristic
21	C₂²×C₈×C₁₁	C₂×C₄	1	2	yes	—
22	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
23	C₁₁×M₃₂	—	2	4	no	—
24	C₁₁×M₃₂	—	2	4	no	—
25	C₂×C₁₁×C₁₆	—	2	1	no	—
26	C₂×C₁₁×C₁₆	—	2	1	no	—
27	C₂×C₁₁×C₁₆	C₂×C₄	1	2	yes	—
28	C₂×C₁₁×C₁₆	C₂×C₄	1	1	yes	characteristic
29	C₂×C₁₁×C₁₆	D₄	1	2	yes	—
30	C₂×C₁₁×C₁₆	D₄	1	2	yes	—
31	C₂×C₁₁×C₁₆	C₂³	1	1	yes	characteristic
32	C₁₁⋊C₃₂	—	2	4	no	—
33	C₂×#1400	—	11	1	no	maximal, Sylow
34	C₂⁴×C₁₁	C₁₆	1	1	yes	characteristic
35	C₂²×C₄×C₁₁	—	2	1	no	—
36	C₂²×C₄×C₁₁	—	2	1	no	—
37	C₂²×C₄×C₁₁	—	2	1	no	—
38	C₂²×C₄×C₁₁	—	2	1	no	—
39	C₂²×C₄×C₁₁	C₁₆	1	1	yes	characteristic
40	C₂²×C₄×C₁₁	C₁₆	1	2	yes	—
41	C₂²×C₄×C₁₁	C₂×C₈	1	2	yes	—
42	C₂²×C₄×C₁₁	C₂×C₈	1	1	yes	characteristic
43	C₂×C₈×C₁₁	—	2	2	no	—
44	C₂×C₈×C₁₁	—	2	2	no	—
45	C₂×C₈×C₁₁	—	2	2	no	—
46	C₂×C₈×C₁₁	—	2	2	no	—
47	C₂×C₈×C₁₁	—	2	1	no	—
48	C₂×C₈×C₁₁	—	2	1	no	—
49	C₂×C₈×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₈×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₈×C₁₁	C₂×C₈	1	2	yes	—
52	C₂×C₈×C₁₁	C₂×C₈	1	1	yes	characteristic
53	C₂×C₈×C₁₁	M₁₆	1	1	yes	characteristic
54	C₂×C₈×C₁₁	M₁₆	1	1	yes	characteristic
55	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
56	C₁₁×C₁₆	—	2	2	no	—
57	C₁₁×C₁₆	C₂²⋊C₄	1	2	yes	—
58	C₁₁×C₁₆	C₂×D₄	1	2	yes	—
59	C₂²×M₃₂	D₁₁	1	1	yes	characteristic
60	C₂×M₆₄	—	11	2	no	—
61	#1400	—	11	4	no	—
62	C₂³×C₁₁	—	2	1	no	—
63	C₂³×C₁₁	—	2	1	no	—
64	C₂³×C₁₁	—	4	2	no	—
65	C₂³×C₁₁	C₂×C₁₆	1	1	yes	characteristic
66	C₂³×C₁₁	C₂×C₁₆	1	2	yes	—
67	C₂×C₄×C₁₁	—	2	1	no	—
68	C₂×C₄×C₁₁	—	2	1	no	—
69	C₂×C₄×C₁₁	—	2	2	no	—
70	C₂×C₄×C₁₁	—	2	2	no	—
71	C₂×C₄×C₁₁	—	2	2	no	—
72	C₂×C₄×C₁₁	—	2	2	no	—
73	C₂×C₄×C₁₁	C₂²⋊C₈	1	1	yes	characteristic
74	C₂×C₄×C₁₁	C₂²⋊C₈	1	1	yes	characteristic
75	C₂×C₄×C₁₁	C₂×C₁₆	1	2	yes	—
76	C₂×C₄×C₁₁	C₂×C₁₆	1	1	yes	characteristic
77	C₂×C₄×C₁₁	M₃₂	1	1	yes	characteristic
78	C₂×C₄×C₁₁	M₃₂	1	1	yes	characteristic
79	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
80	C₈×C₁₁	—	2	2	no	—
81	C₈×C₁₁	C₂²⋊C₈	1	1	yes	characteristic
82	C₈×C₁₁	C₂²⋊C₈	1	1	yes	characteristic
83	C₈×C₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₈×C₁₁	C₂×M₁₆	1	1	yes	characteristic
85	C₂³×C₈	Dic₁₁	1	1	yes	characteristic
86	C₂×M₃₂	—	2	2	no	—
87	C₂×M₃₂	—	2	2	no	—
88	C₂×M₃₂	Dic₁₁	1	2	yes	—
89	C₂×M₃₂	C₂×D₁₁	1	2	yes	—
90	C₂²×C₁₆	Dic₁₁	1	1	yes	characteristic
91	C₂²×C₁₆	C₂×D₁₁	1	1	yes	characteristic
92	M₆₄	—	11	4	no	—
93	M₆₄	—	22	2	no	—
94	C₂×C₃₂	—	22	2	no	—
95	C₂²×C₁₁	—	2	2	no	—
96	C₂²×C₁₁	—	2	2	no	—
97	C₂²×C₁₁	—	4	2	no	—
98	C₂²×C₁₁	—	4	2	no	—
99	C₂²×C₁₁	—	4	2	no	—
100	C₂²×C₁₁	C₂²⋊C₁₆	1	1	yes	characteristic
101	C₂²×C₁₁	C₂²⋊C₁₆	1	1	yes	characteristic
102	C₂²×C₁₁	C₂²×C₁₆	1	1	yes	characteristic
103	C₄×C₁₁	—	2	2	no	—
104	C₄×C₁₁	C₂²⋊C₁₆	1	1	yes	characteristic
105	C₄×C₁₁	C₂²⋊C₁₆	1	1	yes	characteristic
106	C₄×C₁₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
107	C₄×C₁₁	C₂×M₃₂	1	1	yes	characteristic
108	C₂³×C₄	C₁₁⋊C₈	1	1	yes	characteristic
109	C₂²×C₈	—	2	1	no	—
110	C₂²×C₈	—	2	1	no	—
111	C₂²×C₈	—	2	1	no	—
112	C₂²×C₈	—	2	1	no	—
113	C₂²×C₈	C₁₁⋊C₈	1	1	yes	characteristic
114	C₂²×C₈	C₁₁⋊C₈	1	2	yes	—
115	C₂²×C₈	C₂×Dic₁₁	1	2	yes	—
116	C₂²×C₈	C₂×Dic₁₁	1	1	yes	characteristic
117	M₃₂	—	2	4	no	—
118	M₃₂	—	2	4	no	—
119	C₂×C₁₆	—	2	1	no	—
120	C₂×C₁₆	—	2	1	no	—
121	C₂×C₁₆	C₂×Dic₁₁	1	2	yes	—
122	C₂×C₁₆	C₂×Dic₁₁	1	1	yes	characteristic
123	C₂×C₁₆	C₂²⋊D₁₁	1	2	yes	—
124	C₂×C₁₆	C₂²⋊D₁₁	1	2	yes	—
125	C₂×C₁₆	C₂²×D₁₁	1	1	yes	characteristic
126	C₃₂	—	22	4	no	—
127	C₂×C₁₁	—	2	1	no	—
128	C₂×C₁₁	—	2	1	no	—
129	C₂×C₁₁	—	4	2	no	—
130	C₂×C₁₁	#1400	1	2	yes	complemented
131	C₂×C₁₁	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
132	C₂⁴	C₁₁⋊C₁₆	1	1	yes	characteristic
133	C₂²×C₄	—	2	1	no	—
134	C₂²×C₄	—	2	1	no	—
135	C₂²×C₄	—	2	1	no	—
136	C₂²×C₄	—	2	1	no	—
137	C₂²×C₄	C₁₁⋊C₁₆	1	1	yes	characteristic
138	C₂²×C₄	C₁₁⋊C₁₆	1	2	yes	—
139	C₂²×C₄	C₂×C₁₁⋊C₈	1	2	yes	—
140	C₂²×C₄	C₂×C₁₁⋊C₈	1	1	yes	characteristic
141	C₂×C₈	—	2	1	no	—
142	C₂×C₈	—	2	1	no	—
143	C₂×C₈	—	2	2	no	—
144	C₂×C₈	—	2	2	no	—
145	C₂×C₈	—	2	2	no	—
146	C₂×C₈	—	2	2	no	—
147	C₂×C₈	C₂×C₁₁⋊C₈	1	2	yes	—
148	C₂×C₈	C₂×C₁₁⋊C₈	1	1	yes	characteristic
149	C₂×C₈	#4318	1	1	yes	characteristic
150	C₂×C₈	#4318	1	1	yes	characteristic
151	C₂×C₈	C₂²⋊Dic₁₁	1	1	yes	characteristic
152	C₂×C₈	C₂²⋊Dic₁₁	1	1	yes	characteristic
153	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
154	C₁₆	—	2	2	no	—
155	C₁₆	C₂²⋊Dic₁₁	1	2	yes	—
156	C₁₆	C₂×C₂²⋊D₁₁	1	2	yes	—
157	C₁₁	C₂×#1400	1	1	yes	characteristic, complemented, Sylow
158	C₂³	—	2	1	no	—
159	C₂³	—	2	1	no	—
160	C₂³	—	4	2	no	—
161	C₂³	C₂×C₁₁⋊C₁₆	1	1	yes	characteristic
162	C₂³	C₂×C₁₁⋊C₁₆	1	2	yes	—
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	C₂×C₁₁⋊C₁₆	1	2	yes	—
170	C₂×C₄	C₂×C₁₁⋊C₁₆	1	1	yes	characteristic
171	C₂×C₄	#66876	1	1	yes	characteristic
172	C₂×C₄	#66876	1	1	yes	characteristic
173	C₂×C₄	C₂²⋊(C₁₁⋊C₈)	1	1	yes	characteristic
174	C₂×C₄	C₂²⋊(C₁₁⋊C₈)	1	1	yes	characteristic
175	C₂×C₄	C₂²×C₁₁⋊C₈	1	1	yes	characteristic
176	C₈	—	2	2	no	—
177	C₈	C₂²⋊(C₁₁⋊C₈)	1	1	yes	characteristic
178	C₈	C₂²⋊(C₁₁⋊C₈)	1	1	yes	characteristic
179	C₈	C₂×#4318	1	1	yes	characteristic
180	C₈	C₂×C₂²⋊Dic₁₁	1	1	yes	characteristic
181	C₂²	—	2	2	no	—
182	C₂²	—	2	2	no	—
183	C₂²	—	4	2	no	—
184	C₂²	—	4	2	no	—
185	C₂²	—	4	2	no	—
186	C₂²	C₂²⋊(C₁₁⋊C₁₆)	1	1	yes	characteristic
187	C₂²	C₂²⋊(C₁₁⋊C₁₆)	1	1	yes	characteristic
188	C₂²	C₂²×C₁₁⋊C₁₆	1	1	yes	characteristic
189	C₄	—	2	2	no	—
190	C₄	C₂²⋊(C₁₁⋊C₁₆)	1	1	yes	characteristic
191	C₄	C₂²⋊(C₁₁⋊C₁₆)	1	1	yes	characteristic
192	C₄	C₂×#66876	1	1	yes	characteristic
193	C₄	C₂×C₂²⋊(C₁₁⋊C₈)	1	1	yes	characteristic
194	C₂	—	2	1	no	—
195	C₂	—	2	1	no	—
196	C₂	—	4	2	no	—
197	C₂	#13066756	1	2	yes	direct product factor
198	C₂	C₂×C₂²⋊(C₁₁⋊C₁₆)	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²⋊Dic₁₁
commutator	C₂²×C₁₁	C₂²×C₁₆
Frattini	C₂×C₁₆	C₂²×D₁₁
Fitting	C₂²×C₁₁×M₃₂	C₂
socle	C₂²×C₁₁	C₂²⋊C₁₆
prime radical	C₂⁴×C₁₁	C₁₆
cosocle	C₂×C₁₁×C₁₆	C₂³
hypercenter	C₂²×M₃₂	D₁₁
hypocenter	C₁₁	C₂×#1400
LCS	G ▷ C₂²×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₁₆ ◁ C₂²×M₃₂

Complements

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

N	H ≅ G/N	type
1. C₂×#10622137	C₂	semidirect
2. #13066756	C₂	direct (N × H)
12. #10622137	C₂²	semidirect
130. C₂×C₁₁	#1400	semidirect
157. C₁₁	C₂×#1400	semidirect
197. C₂	#13066756	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×#10622137	2
#13066756	2
C₂²×C₁₁×M₃₂	2
C₂×#1400	11

Subgroups — C2×#13066756 #14673974

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13066756 #14673974