Subgroups — C₂×#13070423 #14906848

Order 2816.

610 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13070423	C₂	1	8	yes	maximal, direct product factor
2	C₂×C₄⋊(C₁₁⋊C₁₆)	C₂	1	2	yes	maximal, complemented
3	C₂×C₂²⋊(C₁₁⋊C₁₆)	C₂	1	2	yes	maximal
4	C₂×#10622099	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄×C₁₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₁₁×#405	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₁₁×M₁₆	C₄	1	1	yes	characteristic
8	C₂³×C₈×C₁₁	C₂²	1	1	yes	characteristic
9	C₂×C₁₁×#117	C₄	1	1	yes	characteristic
10	C₁₁×#405	C₄	1	4	yes	—
11	C₁₁×#405	C₂²	1	4	yes	—
12	C₂×C₁₁×#97	C₄	1	1	yes	characteristic
13	C₂×C₁₁×#97	C₄	1	1	yes	characteristic
14	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
16	C₂²×C₁₁⋊C₁₆	C₂²	1	2	yes	—
17	C₂²×C₁₁⋊C₁₆	C₂²	1	2	yes	—
18	C₂²⋊(C₁₁⋊C₁₆)	C₂²	1	8	yes	—
19	C₄⋊(C₁₁⋊C₁₆)	C₂²	1	8	yes	complemented
20	#10622099	C₂²	1	4	yes	complemented
21	C₄×C₁₁⋊C₁₆	C₂²	1	4	yes	complemented
22	C₂³×C₄×C₁₁	C₂×C₄	1	1	yes	characteristic
23	C₂×C₁₁×M₁₆	—	2	4	no	—
24	C₂×C₁₁×M₁₆	C₂×C₄	1	4	yes	—
25	C₂²×C₈×C₁₁	—	2	4	no	—
26	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₁	C₂×C₄	1	2	yes	—
29	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₁₁	C₂³	1	2	yes	—
31	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
32	C₁₁×#117	C₈	1	4	yes	—
33	C₁₁×#117	C₂×C₄	1	4	yes	—
34	C₂×C₁₁×C₄⋊C₄	C₈	1	2	yes	—
35	C₂×C₁₁×C₂²⋊C₄	C₈	1	2	yes	—
36	C₂×C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
38	C₁₁×#97	C₂×C₄	1	4	yes	—
39	C₁₁×#97	C₂×C₄	1	4	yes	—
40	C₄×C₈×C₁₁	C₂×C₄	1	2	yes	—
41	C₄×C₈×C₁₁	C₂×C₄	1	2	yes	—
42	C₄×C₈×C₁₁	C₂³	1	2	yes	—
43	C₄×C₈×C₁₁	C₂³	1	2	yes	—
44	C₂×C₁₁⋊C₁₆	—	2	4	no	—
45	C₂×C₁₁⋊C₁₆	—	2	4	no	—
46	C₂×C₁₁⋊C₁₆	C₂³	1	4	yes	—
47	C₂×C₁₁⋊C₁₆	C₂³	1	4	yes	—
48	C₂×#2164	—	11	1	no	maximal, Sylow
49	C₂⁴×C₁₁	C₂×C₈	1	1	yes	characteristic
50	C₂²×C₄×C₁₁	—	2	2	no	—
51	C₂²×C₄×C₁₁	—	2	2	no	—
52	C₂²×C₄×C₁₁	C₂×C₈	1	2	yes	—
53	C₂²×C₄×C₁₁	C₂×C₈	1	2	yes	—
54	C₂²×C₄×C₁₁	C₂×C₈	1	1	yes	characteristic
55	C₂²×C₄×C₁₁	C₂×C₈	1	2	yes	—
56	C₂²×C₄×C₁₁	C₂²×C₄	1	2	yes	—
57	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
58	C₁₁×M₁₆	—	2	8	no	—
59	C₂×C₈×C₁₁	—	2	4	no	—
60	C₂×C₈×C₁₁	—	2	4	no	—
61	C₂×C₈×C₁₁	—	2	2	no	—
62	C₂×C₈×C₁₁	—	2	2	no	—
63	C₂×C₈×C₁₁	M₁₆	1	4	yes	—
64	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
65	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
66	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
68	C₂×C₈×C₁₁	#41	1	4	yes	—
69	C₂×C₈×C₁₁	C₂⁴	1	1	yes	characteristic
70	C₁₁×C₄⋊C₄	C₂×C₈	1	8	yes	—
71	C₁₁×C₂²⋊C₄	C₂×C₈	1	8	yes	—
72	C₄²×C₁₁	C₂×C₈	1	2	yes	—
73	C₄²×C₁₁	C₂×C₈	1	2	yes	—
74	C₄²×C₁₁	C₂²×C₄	1	2	yes	—
75	C₄²×C₁₁	C₂²×C₄	1	2	yes	—
76	C₁₁⋊C₁₆	—	2	4	no	—
77	C₁₁⋊C₁₆	—	2	4	no	—
78	C₂×#405	D₁₁	1	1	yes	characteristic
79	#2164	—	11	8	no	—
80	C₂×C₄⋊C₁₆	—	11	2	no	—
81	C₂×C₂²⋊C₁₆	—	11	2	no	—
82	C₂×#346	—	11	1	no	—
83	C₂×C₄×C₁₆	—	11	1	no	—
84	C₂³×C₁₁	—	2	2	no	—
85	C₂³×C₁₁	—	2	4	no	—
86	C₂³×C₁₁	C₂²×C₈	1	2	yes	—
87	C₂³×C₁₁	C₂²×C₈	1	1	yes	characteristic
88	C₂×C₄×C₁₁	—	2	2	no	—
89	C₂×C₄×C₁₁	—	2	4	no	—
90	C₂×C₄×C₁₁	—	2	4	no	—
91	C₂×C₄×C₁₁	—	2	2	no	—
92	C₂×C₄×C₁₁	—	2	2	no	—
93	C₂×C₄×C₁₁	—	2	4	no	—
94	C₂×C₄×C₁₁	#117	1	2	yes	—
95	C₂×C₄×C₁₁	C₂²×C₈	1	4	yes	—
96	C₂×C₄×C₁₁	C₂²×C₈	1	4	yes	—
97	C₂×C₄×C₁₁	C₂²×C₈	1	2	yes	—
98	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
99	C₂×C₄×C₁₁	C₂×M₁₆	1	2	yes	—
100	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
101	C₈×C₁₁	—	2	2	no	—
102	C₈×C₁₁	—	2	2	no	—
103	C₈×C₁₁	#117	1	2	yes	—
104	C₈×C₁₁	C₂×M₁₆	1	4	yes	—
105	C₈×C₁₁	C₂×#41	1	2	yes	—
106	C₂²×M₁₆	Dic₁₁	1	1	yes	characteristic
107	C₂³×C₈	C₂×D₁₁	1	1	yes	characteristic
108	C₂×#117	Dic₁₁	1	1	yes	characteristic
109	C₂²×C₁₆	—	11	2	no	—
110	C₂²×C₁₆	—	11	2	no	—
111	#405	Dic₁₁	1	4	yes	—
112	#405	C₂×D₁₁	1	4	yes	—
113	C₂×#97	Dic₁₁	1	1	yes	characteristic
114	C₂×#97	Dic₁₁	1	1	yes	characteristic
115	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
116	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
117	C₄⋊C₁₆	—	11	8	no	—
118	C₂²⋊C₁₆	—	11	8	no	—
119	#346	—	11	4	no	—
120	C₄×C₁₆	—	11	4	no	—
121	C₂²×C₁₁	—	2	4	no	—
122	C₂²×C₁₁	—	2	2	no	—
123	C₂²×C₁₁	—	4	4	no	—
124	C₂²×C₁₁	#431	1	2	yes	—
125	C₂²×C₁₁	#504	1	4	yes	—
126	C₂²×C₁₁	C₂³×C₈	1	1	yes	characteristic
127	C₄×C₁₁	—	2	4	no	—
128	C₄×C₁₁	—	2	4	no	—
129	C₄×C₁₁	—	2	2	no	—
130	C₄×C₁₁	#431	1	2	yes	—
131	C₄×C₁₁	C₂×#117	1	1	yes	characteristic
132	C₄×C₁₁	C₂²×M₁₆	1	1	yes	characteristic
133	C₂³×C₄	C₂×Dic₁₁	1	1	yes	characteristic
134	C₂×M₁₆	—	2	4	no	—
135	C₂×M₁₆	C₂×Dic₁₁	1	4	yes	—
136	C₂²×C₈	—	2	4	no	—
137	C₂²×C₈	C₂×Dic₁₁	1	1	yes	characteristic
138	C₂²×C₈	C₂×Dic₁₁	1	2	yes	—
139	C₂²×C₈	C₂×Dic₁₁	1	1	yes	characteristic
140	C₂²×C₈	C₂×Dic₁₁	1	1	yes	characteristic
141	C₂²×C₈	C₂²×D₁₁	1	2	yes	—
142	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
143	#117	C₁₁⋊C₈	1	4	yes	—
144	#117	C₂×Dic₁₁	1	4	yes	—
145	C₂×C₄⋊C₄	C₁₁⋊C₈	1	2	yes	—
146	C₂×C₂²⋊C₄	C₁₁⋊C₈	1	2	yes	—
147	C₂×C₄²	C₂×Dic₁₁	1	1	yes	characteristic
148	C₂×C₄²	C₂×Dic₁₁	1	1	yes	characteristic
149	C₂×C₁₆	—	11	4	no	—
150	C₂×C₁₆	—	11	4	no	—
151	C₂×C₁₆	—	22	4	no	—
152	C₂×C₁₆	—	22	4	no	—
153	#97	C₂×Dic₁₁	1	4	yes	—
154	#97	C₂×Dic₁₁	1	4	yes	—
155	C₄×C₈	C₂×Dic₁₁	1	2	yes	—
156	C₄×C₈	C₂×Dic₁₁	1	2	yes	—
157	C₄×C₈	C₂²×D₁₁	1	2	yes	—
158	C₄×C₈	C₂²×D₁₁	1	2	yes	—
159	C₂×C₁₁	—	4	2	no	—
160	C₂×C₁₁	#2164	1	4	yes	complemented
161	C₂×C₁₁	C₂×#431	1	1	yes	characteristic
162	C₂×C₁₁	C₂×#504	1	2	yes	—
163	C₂⁴	C₂×C₁₁⋊C₈	1	1	yes	characteristic
164	C₂²×C₄	—	2	2	no	—
165	C₂²×C₄	—	2	2	no	—
166	C₂²×C₄	C₂×C₁₁⋊C₈	1	2	yes	—
167	C₂²×C₄	C₂×C₁₁⋊C₈	1	2	yes	—
168	C₂²×C₄	C₂×C₁₁⋊C₈	1	1	yes	characteristic
169	C₂²×C₄	C₂×C₁₁⋊C₈	1	2	yes	—
170	C₂²×C₄	C₂²×Dic₁₁	1	2	yes	—
171	C₂²×C₄	C₂²×Dic₁₁	1	1	yes	characteristic
172	M₁₆	—	2	8	no	—
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	—	2	2	no	—
175	C₂×C₈	—	2	4	no	—
176	C₂×C₈	—	2	4	no	—
177	C₂×C₈	#4318	1	4	yes	—
178	C₂×C₈	#4339	1	4	yes	—
179	C₂×C₈	C₂²×Dic₁₁	1	2	yes	—
180	C₂×C₈	C₂²×Dic₁₁	1	2	yes	—
181	C₂×C₈	C₂²×Dic₁₁	1	2	yes	—
182	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
183	C₂×C₈	C₂³×D₁₁	1	1	yes	characteristic
184	C₄⋊C₄	C₂×C₁₁⋊C₈	1	8	yes	—
185	C₂²⋊C₄	C₂×C₁₁⋊C₈	1	8	yes	—
186	C₄²	C₂×C₁₁⋊C₈	1	2	yes	—
187	C₄²	C₂×C₁₁⋊C₈	1	2	yes	—
188	C₄²	C₂²×Dic₁₁	1	2	yes	—
189	C₄²	C₂²×Dic₁₁	1	2	yes	—
190	C₁₆	—	22	4	no	—
191	C₁₆	—	22	4	no	—
192	C₁₁	C₂×#2164	1	1	yes	characteristic, complemented, Sylow
193	C₂³	—	2	4	no	—
194	C₂³	—	2	2	no	—
195	C₂³	C₂²×C₁₁⋊C₈	1	2	yes	—
196	C₂³	C₂²×C₁₁⋊C₈	1	1	yes	characteristic
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	2	4	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	4	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	C₂²×C₁₁⋊C₈	1	4	yes	—
204	C₂×C₄	C₂²×C₁₁⋊C₈	1	4	yes	—
205	C₂×C₄	C₂²×C₁₁⋊C₈	1	2	yes	—
206	C₂×C₄	C₂²×C₁₁⋊C₈	1	1	yes	characteristic
207	C₂×C₄	C₂×#4318	1	2	yes	—
208	C₂×C₄	#66979	1	2	yes	—
209	C₂×C₄	C₂³×Dic₁₁	1	1	yes	characteristic
210	C₈	—	2	2	no	—
211	C₈	—	2	2	no	—
212	C₈	C₂×#4318	1	4	yes	—
213	C₈	#66979	1	2	yes	—
214	C₈	C₂×#4339	1	2	yes	—
215	C₂²	—	2	4	no	—
216	C₂²	—	2	2	no	—
217	C₂²	—	4	4	no	—
218	C₂²	#10622538	1	2	yes	—
219	C₂²	#10622753	1	4	yes	—
220	C₂²	C₂³×C₁₁⋊C₈	1	1	yes	characteristic
221	C₄	—	2	4	no	—
222	C₄	—	2	4	no	—
223	C₄	—	2	2	no	—
224	C₄	#10622538	1	2	yes	—
225	C₄	C₂²×#4318	1	1	yes	characteristic
226	C₄	C₂×#66979	1	1	yes	characteristic
227	C₂	—	4	2	no	—
228	C₂	#13070423	1	4	yes	direct product factor
229	C₂	C₂×#10622538	1	1	yes	characteristic
230	C₂	C₂×#10622753	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈	C₂²×D₁₁
commutator	C₂²×C₁₁	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₁₁
Fitting	C₂×C₁₁×#405	C₂
socle	C₂³×C₁₁	C₂²×C₈
prime radical	C₂⁴×C₁₁	C₂×C₈
cosocle	C₂×C₈×C₁₁	C₂⁴
hypercenter	C₂×#405	D₁₁
hypocenter	C₁₁	C₂×#2164
LCS	G ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
1. #13070423	C₂	direct (N × H)
2. C₂×C₄⋊(C₁₁⋊C₁₆)	C₂	semidirect
4. C₂×#10622099	C₂	semidirect
5. C₂×C₄×C₁₁⋊C₁₆	C₂	semidirect
19. C₄⋊(C₁₁⋊C₁₆)	C₂²	semidirect
20. #10622099	C₂²	semidirect
21. C₄×C₁₁⋊C₁₆	C₂²	semidirect
160. C₂×C₁₁	#2164	semidirect
192. C₁₁	C₂×#2164	semidirect
228. C₂	#13070423	direct (N × H)

Maximal subgroups

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

subgroup	index
#13070423	2
C₂×C₄⋊(C₁₁⋊C₁₆)	2
C₂×C₂²⋊(C₁₁⋊C₁₆)	2
C₂×#10622099	2
C₂×C₄×C₁₁⋊C₁₆	2
C₂×C₁₁×#405	2
C₂×#2164	11

Subgroups — C2×#13070423 #14906848

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13070423 #14906848