Subgroups — #14669090

Order 2816.

398 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10622149	C₂	1	2	yes	maximal
2	M₃₂×Dic₁₁	C₂	1	2	yes	maximal
3	C₄×C₈×Dic₁₁	C₂	1	1	yes	maximal, characteristic
4	#13066698	C₂	1	1	yes	maximal, characteristic
5	C₁₁×#1376	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₁×M₃₂	C₂²	1	2	yes	—
7	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
8	C₂×C₈×Dic₁₁	C₄	1	1	yes	characteristic
9	C₂×C₈×Dic₁₁	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₁₁	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₁₁	C₂²	1	1	yes	characteristic
12	C₂×#66876	C₂²	1	2	yes	—
13	C₄²×Dic₁₁	C₄	1	1	yes	characteristic
14	C₂×C₄×C₁₁⋊C₈	C₄	1	1	yes	characteristic
15	#10622149	C₄	1	8	yes	complemented
16	#10622140	—	2	2	no	—
17	C₁₆×Dic₁₁	—	2	2	no	—
18	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
20	C₂×C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×Dic₁₁	—	2	2	no	—
22	C₂×C₄×Dic₁₁	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×Dic₁₁	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₁₁⋊C₈	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₁₁⋊C₈	C₂×C₄	1	1	yes	characteristic
26	C₁₁×M₃₂	—	2	2	no	—
27	C₁₁×M₃₂	C₂×C₄	1	4	yes	—
28	C₂×C₁₁×C₁₆	—	2	2	no	—
29	C₄×C₈×C₁₁	—	2	2	no	—
30	C₈×Dic₁₁	—	2	2	no	—
31	C₈×Dic₁₁	C₈	1	4	yes	—
32	C₈×Dic₁₁	C₂×C₄	1	4	yes	—
33	C₈×Dic₁₁	D₄	1	1	yes	characteristic
34	C₈×Dic₁₁	D₄	1	1	yes	characteristic
35	C₈×Dic₁₁	Q₈	1	1	yes	characteristic
36	C₈×Dic₁₁	Q₈	1	1	yes	characteristic
37	#66876	—	2	2	no	—
38	#66876	C₂×C₄	1	4	yes	—
39	C₂×C₁₁⋊C₁₆	—	2	2	no	—
40	C₄×C₁₁⋊C₈	C₈	1	4	yes	—
41	C₄×#364	—	11	1	no	maximal, Sylow
42	C₂²×C₄×C₁₁	—	2	1	no	—
43	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
44	C₂²×Dic₁₁	—	2	1	no	—
45	C₂²×Dic₁₁	M₁₆	1	2	yes	—
46	C₂×C₈×C₁₁	—	2	1	no	—
47	C₂×C₈×C₁₁	—	2	1	no	—
48	C₂×C₈×C₁₁	C₂×C₈	1	2	yes	—
49	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
50	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈×C₁₁	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₈×C₁₁	#41	1	2	yes	—
53	C₄²×C₁₁	—	2	1	no	—
54	C₄²×C₁₁	C₂×C₈	1	2	yes	—
55	C₄×Dic₁₁	—	2	8	no	—
56	C₄×Dic₁₁	—	2	1	no	—
57	C₄×Dic₁₁	C₄²	1	2	yes	—
58	C₄×Dic₁₁	C₄⋊C₄	1	1	yes	characteristic
59	C₄×Dic₁₁	C₄⋊C₄	1	1	yes	characteristic
60	C₄×Dic₁₁	C₂×C₈	1	2	yes	—
61	C₂×C₁₁⋊C₈	C₄²	1	2	yes	—
62	C₂×C₁₁⋊C₈	C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₁₁⋊C₈	C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₁₁⋊C₈	C₂×C₈	1	2	yes	—
65	C₂×C₁₁⋊C₈	C₂×C₈	1	2	yes	—
66	C₂×C₁₁⋊C₈	C₂×C₈	1	2	yes	—
67	C₂×C₁₁⋊C₈	M₁₆	1	2	yes	—
68	C₁₁×C₁₆	—	2	4	no	—
69	C₁₁⋊C₁₆	—	2	4	no	—
70	C₂×#364	—	11	2	no	—
71	C₄×M₃₂	—	11	2	no	—
72	C₄²×C₈	—	11	1	no	—
73	#1376	—	11	1	no	—
74	#1376	D₁₁	1	1	yes	characteristic
75	C₂³×C₁₁	C₂×M₁₆	1	1	yes	characteristic
76	C₂×C₄×C₁₁	—	2	4	no	—
77	C₂×C₄×C₁₁	—	2	2	no	—
78	C₂×C₄×C₁₁	C₂×C₄²	1	1	yes	characteristic
79	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₁₁	#117	1	1	yes	characteristic
81	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
82	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
83	C₂×C₄×C₁₁	C₂×M₁₆	1	1	yes	characteristic
84	C₂×Dic₁₁	—	2	4	no	—
85	C₂×Dic₁₁	—	2	2	no	—
86	C₂×Dic₁₁	—	2	4	no	—
87	C₂×Dic₁₁	C₄⋊C₈	1	2	yes	—
88	C₂×Dic₁₁	C₄⋊C₈	1	2	yes	—
89	C₈×C₁₁	—	2	2	no	—
90	C₈×C₁₁	C₄×D₄	1	2	yes	—
91	C₈×C₁₁	C₄×Q₈	1	2	yes	—
92	C₁₁⋊C₈	C₄×C₈	1	4	yes	—
93	C₁₁⋊C₈	C₄⋊C₈	1	2	yes	—
94	C₁₁⋊C₈	C₄⋊C₈	1	2	yes	—
95	C₂×M₃₂	—	11	2	no	—
96	C₂×M₃₂	C₂×D₁₁	1	2	yes	—
97	C₂×C₄×C₈	—	11	1	no	—
98	C₂×C₄×C₈	—	11	1	no	—
99	C₂×C₄×C₈	—	11	1	no	—
100	C₂×C₄×C₈	—	11	1	no	—
101	C₂×C₄×C₈	—	11	1	no	—
102	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
103	C₄³	—	11	1	no	—
104	#364	—	11	8	no	—
105	#346	—	22	2	no	—
106	C₄×C₁₆	—	22	2	no	—
107	C₂²×C₁₁	—	2	2	no	—
108	C₂²×C₁₁	C₄×M₁₆	1	1	yes	characteristic
109	C₂²×C₁₁	C₂×C₄⋊C₈	1	1	yes	characteristic
110	C₂²×C₁₁	#431	1	1	yes	characteristic
111	C₄×C₁₁	—	2	4	no	—
112	C₄×C₁₁	C₄×C₄⋊C₄	1	1	yes	characteristic
113	C₄×C₁₁	C₂×C₄×C₈	1	1	yes	characteristic
114	C₄×C₁₁	C₂×C₄⋊C₈	1	1	yes	characteristic
115	C₄×C₁₁	#431	1	1	yes	characteristic
116	Dic₁₁	—	2	4	no	—
117	Dic₁₁	—	2	2	no	—
118	Dic₁₁	#364	1	4	yes	complemented
119	C₂²×C₈	—	11	1	no	—
120	C₂²×C₈	—	11	1	no	—
121	C₂²×C₈	C₄×D₁₁	1	1	yes	characteristic
122	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
123	C₂×C₄²	—	11	1	no	—
124	C₂×C₄²	—	11	1	no	—
125	C₂×C₄²	—	22	2	no	—
126	C₂×C₄²	C₄×D₁₁	1	1	yes	characteristic
127	M₃₂	—	2	2	no	—
128	M₃₂	—	11	4	no	—
129	M₃₂	—	22	2	no	—
130	M₃₂	C₄×D₁₁	1	4	yes	—
131	C₂×C₁₆	—	2	2	no	—
132	C₂×C₁₆	—	22	2	no	—
133	C₄×C₈	—	2	2	no	—
134	C₄×C₈	—	11	4	no	—
135	C₄×C₈	—	11	4	no	—
136	C₄×C₈	—	11	1	no	—
137	C₄×C₈	—	11	1	no	—
138	C₄×C₈	—	11	1	no	—
139	C₄×C₈	—	11	1	no	—
140	C₄×C₈	—	11	4	no	—
141	C₄×C₈	—	22	2	no	—
142	C₂×C₁₁	—	2	1	no	—
143	C₂×C₁₁	—	2	1	no	—
144	C₂×C₁₁	C₄×C₄⋊C₈	1	1	yes	characteristic
145	C₂×C₁₁	C₂×#364	1	2	yes	—
146	C₂²×C₄	—	2	1	no	—
147	C₂²×C₄	—	11	2	no	—
148	C₂²×C₄	—	22	1	no	—
149	C₂²×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
150	C₂×C₈	—	2	1	no	—
151	C₂×C₈	—	2	1	no	—
152	C₂×C₈	—	11	2	no	—
153	C₂×C₈	—	11	1	no	—
154	C₂×C₈	—	11	1	no	—
155	C₂×C₈	—	11	2	no	—
156	C₂×C₈	—	11	2	no	—
157	C₂×C₈	—	11	2	no	—
158	C₂×C₈	—	11	2	no	—
159	C₂×C₈	C₈×D₁₁	1	2	yes	—
160	C₂×C₈	C₂×C₄×D₁₁	1	2	yes	—
161	C₂×C₈	D₄×D₁₁	1	1	yes	characteristic
162	C₂×C₈	#4341	1	2	yes	—
163	C₂×C₈	Q₈×D₁₁	1	1	yes	characteristic
164	C₄²	—	2	1	no	—
165	C₄²	—	11	1	no	—
166	C₄²	—	11	1	no	—
167	C₄²	—	11	2	no	—
168	C₄²	—	11	2	no	—
169	C₄²	—	22	8	no	—
170	C₄²	—	22	1	no	—
171	C₄²	C₈×D₁₁	1	2	yes	—
172	C₁₆	—	2	4	no	—
173	C₁₆	—	22	4	no	—
174	C₁₁	C₄×#364	1	1	yes	characteristic, complemented, Sylow
175	C₂³	M₁₆×D₁₁	1	1	yes	characteristic
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	11	2	no	—
179	C₂×C₄	—	11	2	no	—
180	C₂×C₄	—	22	4	no	—
181	C₂×C₄	—	22	4	no	—
182	C₂×C₄	—	22	2	no	—
183	C₂×C₄	C₄²×D₁₁	1	1	yes	characteristic
184	C₂×C₄	#66930	1	1	yes	characteristic
185	C₂×C₄	C₄⋊C₄×D₁₁	1	1	yes	characteristic
186	C₂×C₄	C₂×C₈×D₁₁	1	1	yes	characteristic
187	C₂×C₄	C₂×C₈×D₁₁	1	1	yes	characteristic
188	C₂×C₄	M₁₆×D₁₁	1	1	yes	characteristic
189	C₈	—	2	2	no	—
190	C₈	—	11	4	no	—
191	C₈	—	11	2	no	—
192	C₈	—	11	2	no	—
193	C₈	(C₂²⋊D₁₁)⋊C₄	1	2	yes	—
194	C₈	#66940	1	2	yes	—
195	C₂²	—	2	2	no	—
196	C₂²	#10622320	1	1	yes	characteristic
197	C₂²	#10622331	1	1	yes	characteristic
198	C₂²	D₁₁×C₄⋊C₈	1	1	yes	characteristic
199	C₄	—	2	4	no	—
200	C₄	—	11	4	no	—
201	C₄	—	22	4	no	—
202	C₄	—	22	2	no	—
203	C₄	#10622262	1	1	yes	characteristic
204	C₄	C₄×C₈×D₁₁	1	1	yes	characteristic
205	C₄	#10622331	1	1	yes	characteristic
206	C₄	D₁₁×C₄⋊C₈	1	1	yes	characteristic
207	C₂	—	2	1	no	—
208	C₂	—	2	1	no	—
209	C₂	#13068464	1	1	yes	characteristic
210	C₂	D₁₁×#364	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	D₄×D₁₁
commutator	C₄×C₁₁	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₁₁
Fitting	C₁₁×#1376	C₂
socle	C₂²×C₁₁	C₂×C₄⋊C₈
prime radical	C₂³×C₁₁	C₂×M₁₆
cosocle	C₂²×C₈×C₁₁	C₂³
hypercenter	#1376	D₁₁
hypocenter	C₁₁	C₄×#364
LCS	G ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #1376

Complements

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

N	H ≅ G/N	type
15. #10622149	C₄	semidirect
118. Dic₁₁	#364	semidirect
174. C₁₁	C₄×#364	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10622149	2
M₃₂×Dic₁₁	2
C₄×C₈×Dic₁₁	2
#13066698	2
C₁₁×#1376	2
C₄×#364	11