Subgroups — #14652802

Order 2816.

570 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	#13077014	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊(C₁₁⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₂×#10622119	C₂	1	4	yes	maximal
4	C₁₁×#1842	C₂	1	1	yes	maximal, characteristic
5	C₂×#66922	C₄	1	4	yes	—
6	C₂×#66922	C₂²	1	2	yes	—
7	C₂×C₁₁×C₄⋊C₈	—	2	2	no	—
8	C₂×C₁₁×C₄⋊C₈	C₂²	1	2	yes	—
9	C₄³×C₁₁	C₂²	1	1	yes	characteristic
10	#10622548	—	2	1	no	—
11	#10622548	—	2	1	no	—
12	#10622548	—	2	2	no	—
13	C₂×C₄⋊(C₁₁⋊C₈)	C₂²	1	2	yes	—
14	C₂×C₄×C₁₁⋊C₈	—	2	1	no	—
15	#10622162	—	2	1	no	—
16	#10622119	—	2	8	no	—
17	C₂²×Dic₂₂	C₂×C₄	1	4	yes	—
18	C₂²×C₈×C₁₁	—	2	2	no	—
19	C₂×C₄²×C₁₁	C₂×C₄	1	2	yes	—
20	C₂×C₄²×C₁₁	D₄	1	1	yes	characteristic
21	C₂×C₄²×C₁₁	D₄	1	1	yes	characteristic
22	C₂×C₄²×C₁₁	D₄	1	2	yes	—
23	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
24	C₂×C₄⋊Dic₁₁	—	2	4	no	—
25	C₂×C₄⋊Dic₁₁	—	2	4	no	—
26	C₂×C₄⋊Dic₁₁	—	2	2	no	—
27	C₂×C₄⋊Dic₁₁	—	2	2	no	—
28	C₂²×C₁₁⋊C₈	—	2	2	no	—
29	#66922	—	2	4	no	—
30	#66922	—	2	4	no	—
31	#66922	—	4	8	no	—
32	C₁₁×C₄⋊C₈	—	2	4	no	—
33	C₁₁×C₄⋊C₈	—	4	4	no	—
34	C₄⋊(C₁₁⋊C₈)	—	2	2	no	—
35	C₄⋊(C₁₁⋊C₈)	D₄	1	4	yes	—
36	C₄×C₁₁⋊C₈	—	4	2	no	—
37	#9691	—	11	1	no	maximal, Sylow
38	C₂²×C₄×C₁₁	C₂²⋊C₄	1	2	yes	—
39	C₂²×C₄×C₁₁	C₂²⋊C₄	1	2	yes	—
40	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
43	C₂²×Dic₁₁	—	2	4	no	—
44	C₂×Dic₂₂	—	2	8	no	—
45	C₂×Dic₂₂	—	4	8	no	—
46	C₂×C₈×C₁₁	—	4	2	no	—
47	C₂×C₈×C₁₁	—	4	4	no	—
48	C₄²×C₁₁	—	2	4	no	—
49	C₄²×C₁₁	—	2	2	no	—
50	C₄²×C₁₁	SD₁₆	1	2	yes	—
51	C₄²×C₁₁	SD₁₆	1	4	yes	—
52	C₄²×C₁₁	Q₁₆	1	2	yes	—
53	C₄²×C₁₁	Q₁₆	1	4	yes	—
54	C₄²×C₁₁	C₂×D₄	1	2	yes	—
55	C₄²×C₁₁	#41	1	2	yes	—
56	C₄⋊Dic₁₁	—	4	8	no	—
57	C₄⋊Dic₁₁	—	4	8	no	—
58	C₄⋊Dic₁₁	—	4	4	no	—
59	C₄⋊Dic₁₁	—	4	2	no	—
60	C₄⋊Dic₁₁	—	4	2	no	—
61	C₂×C₁₁⋊C₈	—	2	4	no	—
62	C₂×C₁₁⋊C₈	—	4	2	no	—
63	C₂×C₁₁⋊C₈	—	4	2	no	—
64	#2868	—	11	1	no	—
65	#1842	D₁₁	1	1	yes	characteristic
66	C₄×C₄⋊C₈	—	11	1	no	—
67	C₂×#333	—	11	4	no	—
68	C₂³×C₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
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	#100	1	4	yes	—
73	C₂×C₄×C₁₁	#103	1	4	yes	—
74	C₂×C₄×C₁₁	#103	1	4	yes	—
75	C₂×C₄×C₁₁	#103	1	4	yes	—
76	C₂×C₄×C₁₁	C₄×D₄	1	2	yes	—
77	C₂×C₄×C₁₁	#121	1	2	yes	—
78	C₂×C₄×C₁₁	#124	1	1	yes	characteristic
79	C₂×C₄×C₁₁	#127	1	1	yes	characteristic
80	C₂×C₄×C₁₁	C₂×SD₁₆	1	1	yes	characteristic
81	C₂×C₄×C₁₁	C₂×SD₁₆	1	2	yes	—
82	C₂×C₄×C₁₁	C₂×Q₁₆	1	1	yes	characteristic
83	C₂×C₄×C₁₁	C₂×Q₁₆	1	2	yes	—
84	C₂×C₄×C₁₁	#137	1	2	yes	—
85	C₂×Dic₁₁	—	4	4	no	—
86	C₂×Dic₁₁	—	4	8	no	—
87	Dic₂₂	—	8	8	no	—
88	C₈×C₁₁	—	8	2	no	—
89	C₁₁⋊C₈	—	4	4	no	—
90	C₂×C₄⋊Q₈	—	11	2	no	—
91	C₂×C₄⋊Q₈	—	11	4	no	—
92	C₂×C₄⋊C₈	—	2	2	no	—
93	C₂×C₄⋊C₈	—	11	2	no	—
94	C₂×C₄⋊C₈	C₂×D₁₁	1	2	yes	—
95	C₂×C₄×C₈	—	22	1	no	—
96	#384	—	22	1	no	—
97	#384	—	22	1	no	—
98	#384	—	22	2	no	—
99	C₄³	C₂×D₁₁	1	1	yes	characteristic
100	#336	—	22	1	no	—
101	#333	—	22	8	no	—
102	C₂²×C₁₁	#390	1	1	yes	characteristic
103	C₂²×C₁₁	C₂×#100	1	2	yes	—
104	C₂²×C₁₁	C₂×#103	1	2	yes	—
105	C₂²×C₁₁	C₂×#103	1	1	yes	characteristic
106	C₂²×C₁₁	#419	1	1	yes	characteristic
107	C₄×C₁₁	—	2	4	no	—
108	C₄×C₁₁	—	2	2	no	—
109	C₄×C₁₁	#333	1	8	yes	complemented
110	C₄×C₁₁	#461	1	2	yes	—
111	C₄×C₁₁	#462	1	2	yes	—
112	C₄×C₁₁	#487	1	2	yes	—
113	C₄×C₁₁	#492	1	1	yes	characteristic
114	C₄×C₁₁	#494	1	1	yes	characteristic
115	Dic₁₁	—	8	4	no	—
116	C₂²×Q₈	—	11	4	no	—
117	C₂²×C₈	—	2	2	no	—
118	C₂²×C₈	—	22	2	no	—
119	C₄⋊Q₈	—	22	4	no	—
120	C₄⋊Q₈	—	22	4	no	—
121	C₄⋊Q₈	—	44	8	no	—
122	C₂×C₄⋊C₄	—	22	4	no	—
123	C₂×C₄⋊C₄	—	22	4	no	—
124	C₂×C₄⋊C₄	—	22	2	no	—
125	C₂×C₄⋊C₄	—	22	2	no	—
126	C₂×C₄²	C₄×D₁₁	1	2	yes	—
127	C₂×C₄²	D₄₄	1	1	yes	characteristic
128	C₂×C₄²	D₄₄	1	1	yes	characteristic
129	C₂×C₄²	C₂²⋊D₁₁	1	2	yes	—
130	C₂×C₄²	C₂²×D₁₁	1	1	yes	characteristic
131	C₄⋊C₈	—	2	4	no	—
132	C₄⋊C₈	—	4	4	no	—
133	C₄⋊C₈	—	11	4	no	—
134	C₄⋊C₈	—	22	2	no	—
135	C₄×C₈	—	44	2	no	—
136	C₂×C₁₁	C₂×#333	1	2	yes	—
137	C₂×C₁₁	#1548	1	2	yes	—
138	C₂×C₁₁	#1956	1	1	yes	characteristic
139	C₂×C₁₁	#1965	1	1	yes	characteristic
140	C₂×C₁₁	#1981	1	1	yes	characteristic
141	C₂×Q₈	—	22	8	no	—
142	C₂×Q₈	—	44	8	no	—
143	C₂²×C₄	—	22	4	no	—
144	C₂²×C₄	(C₂×D₁₁)⋊C₄	1	2	yes	—
145	C₂²×C₄	(C₂×D₁₁)⋊C₄	1	2	yes	—
146	C₂²×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
147	C₂²×C₄	C₂×D₄₄	1	1	yes	characteristic
148	C₂²×C₄	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
149	C₂×C₈	—	4	4	no	—
150	C₂×C₈	—	4	2	no	—
151	C₂×C₈	—	22	4	no	—
152	C₂×C₈	—	44	2	no	—
153	C₂×C₈	—	44	2	no	—
154	C₄⋊C₄	—	44	8	no	—
155	C₄⋊C₄	—	44	8	no	—
156	C₄⋊C₄	—	44	4	no	—
157	C₄⋊C₄	—	44	2	no	—
158	C₄⋊C₄	—	44	2	no	—
159	C₄²	—	2	4	no	—
160	C₄²	—	2	2	no	—
161	C₄²	#4314	1	2	yes	—
162	C₄²	Dic₄₄	1	2	yes	—
163	C₄²	#4324	1	4	yes	—
164	C₄²	#4326	1	4	yes	—
165	C₄²	D₄×D₁₁	1	2	yes	—
166	C₄²	#4343	1	2	yes	—
167	C₁₁	#9691	1	1	yes	characteristic, complemented, Sylow
168	C₂³	C₂×(C₂×D₁₁)⋊C₄	1	1	yes	characteristic
169	Q₈	—	88	8	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	44	8	no	—
174	C₂×C₄	—	44	4	no	—
175	C₂×C₄	#66874	1	4	yes	—
176	C₂×C₄	#66874	1	4	yes	—
177	C₂×C₄	#66880	1	4	yes	—
178	C₂×C₄	#66887	1	4	yes	—
179	C₂×C₄	#66946	1	2	yes	—
180	C₂×C₄	#66948	1	2	yes	—
181	C₂×C₄	C₂×#4314	1	1	yes	characteristic
182	C₂×C₄	C₂×Dic₄₄	1	1	yes	characteristic
183	C₂×C₄	#66962	1	2	yes	—
184	C₂×C₄	C₂×#4324	1	2	yes	—
185	C₂×C₄	#66993	1	1	yes	characteristic
186	C₂×C₄	C₂×#4326	1	2	yes	—
187	C₂×C₄	#67000	1	1	yes	characteristic
188	C₈	—	8	2	no	—
189	C₈	—	44	4	no	—
190	C₂²	C₂×#66874	1	2	yes	—
191	C₂²	#10622601	1	1	yes	characteristic
192	C₂²	C₂×#66880	1	1	yes	characteristic
193	C₂²	#10622733	1	1	yes	characteristic
194	C₂²	C₂×#66887	1	2	yes	—
195	C₄	—	2	4	no	—
196	C₄	—	2	2	no	—
197	C₄	—	88	4	no	—
198	C₄	#10622119	1	8	yes	complemented
199	C₄	#10622454	1	2	yes	—
200	C₄	#10622462	1	2	yes	—
201	C₄	#10622689	1	1	yes	characteristic
202	C₄	#10622705	1	1	yes	characteristic
203	C₄	#10622706	1	2	yes	—
204	C₂	C₂×#10622119	1	2	yes	—
205	C₂	#13067298	1	2	yes	—
206	C₂	#13069417	1	1	yes	characteristic
207	C₂	#13069455	1	1	yes	characteristic
208	C₂	#13069538	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×(C₂×D₁₁)⋊C₄
commutator	C₂²×C₄×C₁₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₁
Fitting	C₁₁×#1842	C₂
socle, prime radical	C₂³×C₁₁	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₁₁	C₂³
hypercenter	#1842	D₁₁
hypocenter	C₁₁	#9691
LCS	G ▷ C₂²×C₄×C₁₁ ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1842

Complements

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

N	H ≅ G/N	type
109. C₄×C₁₁	#333	semidirect
167. C₁₁	#9691	semidirect
198. C₄	#10622119	semidirect

Maximal subgroups

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

subgroup	index
#13077014	2
C₄×C₄⋊(C₁₁⋊C₈)	2
C₂×#10622119	2
C₁₁×#1842	2
#9691	11