Subgroups — #14640686

Order 2816.

262 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊(C₁₁⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
2	C₂×#10622099	C₂	1	1	yes	maximal, characteristic
3	C₁₁×#1770	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₁×C₄⋊C₈	C₄	1	1	yes	characteristic
5	C₂×C₁₁×#97	—	2	1	no	—
6	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
7	C₄×C₁₁×C₄⋊C₄	C₄	1	1	yes	characteristic
8	C₂²×C₁₁⋊C₁₆	—	2	1	no	—
9	C₂²×C₁₁⋊C₁₆	—	2	1	no	—
10	C₁₁×#336	—	2	1	no	—
11	C₄⋊(C₁₁⋊C₁₆)	—	2	1	no	—
12	C₄⋊(C₁₁⋊C₁₆)	C₄	1	2	yes	complemented
13	#10622099	—	2	1	no	—
14	#10622099	C₄	1	2	yes	complemented
15	C₂²×C₈×C₁₁	—	2	1	no	—
16	C₂²×C₈×C₁₁	D₄	1	1	yes	characteristic
17	C₂²×C₈×C₁₁	D₄	1	1	yes	characteristic
18	C₂×C₁₁×C₄⋊C₄	C₈	1	2	yes	—
19	C₂×C₄²×C₁₁	—	2	1	no	—
20	C₂×C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
21	C₁₁×C₄⋊C₈	—	2	1	no	—
22	C₁₁×C₄⋊C₈	C₈	1	2	yes	—
23	C₁₁×#97	—	4	1	no	—
24	C₁₁×#97	—	4	1	no	—
25	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₁₁	D₄	1	1	yes	characteristic
28	C₄×C₈×C₁₁	Q₈	1	1	yes	characteristic
29	C₁₁×#95	—	2	1	no	—
30	C₂×C₁₁⋊C₁₆	—	2	2	no	—
31	C₂×C₁₁⋊C₁₆	—	2	2	no	—
32	C₂×C₁₁⋊C₁₆	—	2	2	no	—
33	C₂×C₁₁⋊C₁₆	—	2	2	no	—
34	C₂×C₁₁⋊C₁₆	—	4	1	no	—
35	C₂×C₁₁⋊C₁₆	—	4	1	no	—
36	#7170	—	11	1	no	maximal, Sylow
37	C₂²×C₄×C₁₁	—	2	2	no	—
38	C₂²×C₄×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
39	C₂²×C₄×C₁₁	C₂×C₈	1	1	yes	characteristic
40	C₂²×C₄×C₁₁	M₁₆	1	1	yes	characteristic
41	C₂×C₈×C₁₁	—	2	2	no	—
42	C₂×C₈×C₁₁	—	2	1	no	—
43	C₂×C₈×C₁₁	—	2	1	no	—
44	C₂×C₈×C₁₁	—	4	1	no	—
45	C₂×C₈×C₁₁	—	4	1	no	—
46	C₂×C₈×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₈×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₈×C₁₁	C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₈×C₁₁	C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₈×C₁₁	D₈	1	1	yes	characteristic
51	C₂×C₈×C₁₁	SD₁₆	1	1	yes	characteristic
52	C₂×C₈×C₁₁	SD₁₆	1	1	yes	characteristic
53	C₂×C₈×C₁₁	Q₁₆	1	1	yes	characteristic
54	C₁₁×C₄⋊C₄	—	4	2	no	—
55	C₄²×C₁₁	—	4	1	no	—
56	C₄²×C₁₁	—	4	1	no	—
57	C₄²×C₁₁	C₄²	1	1	yes	characteristic
58	C₄²×C₁₁	C₄⋊C₄	1	1	yes	characteristic
59	C₄²×C₁₁	C₂×C₈	1	1	yes	characteristic
60	C₄²×C₁₁	M₁₆	1	1	yes	characteristic
61	C₁₁⋊C₁₆	—	2	4	no	—
62	C₁₁⋊C₁₆	—	4	2	no	—
63	C₂×C₄⋊C₁₆	—	11	1	no	—
64	C₂×#346	—	11	1	no	—
65	#1770	D₁₁	1	1	yes	characteristic
66	C₂³×C₁₁	C₂²⋊C₈	1	1	yes	characteristic
67	C₂×C₄×C₁₁	—	2	2	no	—
68	C₂×C₄×C₁₁	—	2	2	no	—
69	C₂×C₄×C₁₁	—	4	2	no	—
70	C₂×C₄×C₁₁	—	4	2	no	—
71	C₂×C₄×C₁₁	—	4	2	no	—
72	C₂×C₄×C₁₁	#95	1	1	yes	characteristic
73	C₂×C₄×C₁₁	C₄×C₈	1	1	yes	characteristic
74	C₂×C₄×C₁₁	#97	1	1	yes	characteristic
75	C₂×C₄×C₁₁	C₂²⋊C₈	1	1	yes	characteristic
76	C₂×C₄×C₁₁	#102	1	1	yes	characteristic
77	C₂×C₄×C₁₁	#103	1	1	yes	characteristic
78	C₂×C₄×C₁₁	#104	1	1	yes	characteristic
79	C₂×C₄×C₁₁	#104	1	1	yes	characteristic
80	C₂×C₄×C₁₁	C₄⋊C₈	1	1	yes	characteristic
81	C₂×C₄×C₁₁	C₄⋊C₈	1	1	yes	characteristic
82	C₈×C₁₁	—	2	1	no	—
83	C₈×C₁₁	—	2	1	no	—
84	C₈×C₁₁	—	4	2	no	—
85	C₈×C₁₁	#102	1	1	yes	characteristic
86	C₈×C₁₁	#103	1	1	yes	characteristic
87	C₈×C₁₁	#106	1	1	yes	characteristic
88	C₈×C₁₁	#107	1	1	yes	characteristic
89	C₂²×C₁₆	—	22	1	no	—
90	C₂²×C₁₆	—	22	1	no	—
91	C₂×C₄⋊C₈	Dic₁₁	1	1	yes	characteristic
92	C₂×#97	—	2	1	no	—
93	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
94	C₄×C₄⋊C₄	Dic₁₁	1	1	yes	characteristic
95	C₄⋊C₁₆	—	11	2	no	—
96	C₄⋊C₁₆	—	22	1	no	—
97	#346	—	11	2	no	—
98	#346	—	22	1	no	—
99	#336	—	2	1	no	—
100	C₂²×C₁₁	D₄⋊C₈	1	1	yes	characteristic
101	C₂²×C₁₁	Q₈⋊C₈	1	1	yes	characteristic
102	C₂²×C₁₁	#336	1	1	yes	characteristic
103	C₂²×C₁₁	#349	1	2	yes	—
104	C₂²×C₁₁	#350	1	2	yes	—
105	C₄×C₁₁	—	2	1	no	—
106	C₄×C₁₁	—	2	1	no	—
107	C₄×C₁₁	—	2	1	no	—
108	C₄×C₁₁	—	2	1	no	—
109	C₄×C₁₁	—	8	2	no	—
110	C₄×C₁₁	D₄⋊C₈	1	1	yes	characteristic
111	C₄×C₁₁	Q₈⋊C₈	1	1	yes	characteristic
112	C₄×C₁₁	#339	1	1	yes	characteristic
113	C₄×C₁₁	#340	1	1	yes	characteristic
114	C₂²×C₈	—	2	1	no	—
115	C₂²×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
116	C₂²×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
117	C₂×C₄⋊C₄	C₁₁⋊C₈	1	2	yes	—
118	C₂×C₄²	—	2	1	no	—
119	C₂×C₄²	C₂×Dic₁₁	1	1	yes	characteristic
120	C₂×C₁₆	—	22	2	no	—
121	C₂×C₁₆	—	22	2	no	—
122	C₂×C₁₆	—	22	2	no	—
123	C₂×C₁₆	—	22	2	no	—
124	C₂×C₁₆	—	44	1	no	—
125	C₂×C₁₆	—	44	1	no	—
126	C₄⋊C₈	—	2	1	no	—
127	C₄⋊C₈	C₁₁⋊C₈	1	2	yes	—
128	#97	—	4	1	no	—
129	#97	—	4	1	no	—
130	C₄×C₈	Dic₂₂	1	1	yes	characteristic
131	C₄×C₈	C₄×D₁₁	1	1	yes	characteristic
132	C₄×C₈	C₄×D₁₁	1	1	yes	characteristic
133	C₄×C₈	D₄₄	1	1	yes	characteristic
134	#95	—	2	1	no	—
135	C₂×C₁₁	#1279	1	1	yes	characteristic
136	C₂×C₁₁	#1330	1	2	yes	—
137	C₂×C₁₁	#1338	1	2	yes	—
138	C₂×C₁₁	#1377	1	1	yes	characteristic
139	C₂×C₁₁	#1379	1	1	yes	characteristic
140	C₂²×C₄	—	2	2	no	—
141	C₂²×C₄	C₂×C₁₁⋊C₈	1	1	yes	characteristic
142	C₂²×C₄	#4318	1	1	yes	characteristic
143	C₂²×C₄	C₂²⋊Dic₁₁	1	1	yes	characteristic
144	C₂×C₈	—	2	1	no	—
145	C₂×C₈	—	2	1	no	—
146	C₂×C₈	—	2	2	no	—
147	C₂×C₈	—	4	1	no	—
148	C₂×C₈	—	4	1	no	—
149	C₂×C₈	Dic₁₁⋊C₄	1	1	yes	characteristic
150	C₂×C₈	Dic₁₁⋊C₄	1	1	yes	characteristic
151	C₂×C₈	(C₂×D₁₁)⋊C₄	1	1	yes	characteristic
152	C₂×C₈	(C₂×D₁₁)⋊C₄	1	1	yes	characteristic
153	C₂×C₈	#4323	1	1	yes	characteristic
154	C₂×C₈	#4324	1	1	yes	characteristic
155	C₂×C₈	#4325	1	1	yes	characteristic
156	C₂×C₈	#4326	1	1	yes	characteristic
157	C₄⋊C₄	—	4	2	no	—
158	C₄²	—	4	1	no	—
159	C₄²	—	4	1	no	—
160	C₄²	C₂×C₁₁⋊C₈	1	1	yes	characteristic
161	C₄²	#4318	1	1	yes	characteristic
162	C₄²	C₄×Dic₁₁	1	1	yes	characteristic
163	C₄²	C₄⋊Dic₁₁	1	1	yes	characteristic
164	C₁₆	—	22	4	no	—
165	C₁₆	—	44	2	no	—
166	C₁₁	#7170	1	1	yes	characteristic, complemented, Sylow
167	C₂³	C₂²⋊(C₁₁⋊C₈)	1	1	yes	characteristic
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	4	2	no	—
171	C₂×C₄	—	4	2	no	—
172	C₂×C₄	—	4	2	no	—
173	C₂×C₄	C₄×C₁₁⋊C₈	1	1	yes	characteristic
174	C₂×C₄	#66867	1	1	yes	characteristic
175	C₂×C₄	C₄⋊(C₁₁⋊C₈)	1	1	yes	characteristic
176	C₂×C₄	C₄⋊(C₁₁⋊C₈)	1	1	yes	characteristic
177	C₂×C₄	C₂²⋊(C₁₁⋊C₈)	1	1	yes	characteristic
178	C₂×C₄	C₁₁⋊#95	1	1	yes	characteristic
179	C₂×C₄	#66896	1	1	yes	characteristic
180	C₂×C₄	#66899	1	1	yes	characteristic
181	C₂×C₄	#66901	1	1	yes	characteristic
182	C₂×C₄	#66901	1	1	yes	characteristic
183	C₈	—	2	1	no	—
184	C₈	—	2	1	no	—
185	C₈	—	4	2	no	—
186	C₈	#66871	1	1	yes	characteristic
187	C₈	#66872	1	1	yes	characteristic
188	C₈	#66873	1	1	yes	characteristic
189	C₈	#66874	1	1	yes	characteristic
190	C₂²	#10622162	1	1	yes	characteristic
191	C₂²	D₄⋊(C₁₁⋊C₈)	1	1	yes	characteristic
192	C₂²	Q₈⋊(C₁₁⋊C₈)	1	1	yes	characteristic
193	C₂²	#10622191	1	2	yes	—
194	C₂²	#10622199	1	2	yes	—
195	C₄	—	2	1	no	—
196	C₄	—	2	1	no	—
197	C₄	—	2	1	no	—
198	C₄	—	2	1	no	—
199	C₄	—	8	2	no	—
200	C₄	#10622167	1	1	yes	characteristic
201	C₄	#10622170	1	1	yes	characteristic
202	C₄	D₄⋊(C₁₁⋊C₈)	1	1	yes	characteristic
203	C₄	Q₈⋊(C₁₁⋊C₈)	1	1	yes	characteristic
204	C₂	#13066457	1	1	yes	characteristic
205	C₂	#13066574	1	2	yes	—
206	C₂	#13066598	1	2	yes	—
207	C₂	#13066700	1	1	yes	characteristic
208	C₂	#13066705	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²⋊Dic₁₁
commutator	C₂×C₄×C₁₁	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₁₁
Fitting	C₁₁×#1770	C₂
socle, prime radical	C₂³×C₁₁	C₂²⋊C₈
cosocle	C₂×C₄×C₈×C₁₁	C₂²
hypercenter	#1770	D₁₁
hypocenter	C₁₁	#7170
LCS	G ▷ C₂×C₄×C₁₁ ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1770

Complements

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

N	H ≅ G/N	type
12. C₄⋊(C₁₁⋊C₁₆)	C₄	semidirect
14. #10622099	C₄	semidirect
166. C₁₁	#7170	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊(C₁₁⋊C₁₆)	2
C₂×#10622099	2
C₁₁×#1770	2
#7170	11