Subgroups — #14646705

Order 2816.

398 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13068612	C₂	1	2	yes	maximal
2	#13067375	C₂	1	2	yes	maximal
3	#13066491	C₂	1	2	yes	maximal
4	C₂×C₈×Q₈×C₁₁	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×Q₈×C₁₁	C₂²	1	1	yes	characteristic
6	C₈×Q₈×C₁₁	—	2	2	no	—
7	C₈×Q₈×C₁₁	—	2	2	no	—
8	C₂×C₁₁×C₄⋊C₈	C₄	1	2	yes	—
9	C₂×C₁₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈×C₁₁	C₄	1	2	yes	—
11	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
12	C₂×#66900	—	2	2	no	—
13	#10622537	C₂²	1	4	yes	—
14	#10622168	—	2	2	no	—
15	#10622096	—	2	2	no	—
16	#10622096	—	2	2	no	—
17	C₂²×Q₈×C₁₁	D₄	1	1	yes	characteristic
18	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₁	C₂×C₄	1	2	yes	—
20	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
21	C₄×Q₈×C₁₁	—	2	2	no	—
22	C₄×Q₈×C₁₁	—	2	2	no	—
23	C₂×C₁₁×C₄⋊C₄	D₄	1	1	yes	characteristic
24	C₂×C₁₁×C₄⋊C₄	Q₈	1	2	yes	—
25	C₂×C₄²×C₁₁	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
27	C₂×#4318	—	2	4	no	—
28	C₁₁×C₄⋊C₈	—	2	4	no	—
29	C₁₁×C₄⋊C₈	—	2	2	no	—
30	C₄×C₈×C₁₁	—	2	2	no	—
31	C₄×C₈×C₁₁	—	2	2	no	—
32	C₄×C₈×C₁₁	D₄	1	2	yes	—
33	C₄×C₈×C₁₁	D₄	1	2	yes	—
34	#66900	—	4	4	no	—
35	C₄⋊(C₁₁⋊C₈)	—	2	4	no	—
36	C₄⋊(C₁₁⋊C₈)	—	2	4	no	—
37	#8617	—	11	1	no	maximal, Sylow
38	C₂×Q₈×C₁₁	—	2	2	no	—
39	C₂×Q₈×C₁₁	—	2	2	no	—
40	C₂×Q₈×C₁₁	D₈	1	2	yes	—
41	C₂×Q₈×C₁₁	Q₁₆	1	2	yes	—
42	C₂²×C₄×C₁₁	C₄⋊C₄	1	2	yes	—
43	C₂²×C₄×C₁₁	C₄⋊C₄	1	2	yes	—
44	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₁₁	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₈×C₁₁	—	2	2	no	—
48	C₂×C₈×C₁₁	—	2	2	no	—
49	C₂×C₈×C₁₁	—	4	1	no	—
50	C₂×C₈×C₁₁	—	4	2	no	—
51	C₂×C₈×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
52	C₂×C₈×C₁₁	C₂²⋊C₄	1	2	yes	—
53	C₂×C₈×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
54	C₁₁×C₄⋊C₄	—	2	2	no	—
55	C₁₁×C₄⋊C₄	—	2	4	no	—
56	C₄²×C₁₁	—	2	2	no	—
57	C₄²×C₁₁	—	2	2	no	—
58	C₄²×C₁₁	C₂×D₄	1	2	yes	—
59	C₄²×C₁₁	#41	1	2	yes	—
60	#4318	—	4	4	no	—
61	#4318	—	4	4	no	—
62	C₂×C₁₁⋊C₈	—	4	4	no	—
63	C₂×C₈×Q₈	D₁₁	1	1	yes	characteristic
64	#1801	—	11	2	no	—
65	#1567	—	11	2	no	—
66	#1292	—	11	2	no	—
67	C₂³×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
68	Q₈×C₁₁	—	2	4	no	—
69	Q₈×C₁₁	—	2	4	no	—
70	C₂×C₄×C₁₁	—	2	2	no	—
71	C₂×C₄×C₁₁	—	2	2	no	—
72	C₂×C₄×C₁₁	—	2	1	no	—
73	C₂×C₄×C₁₁	—	2	1	no	—
74	C₂×C₄×C₁₁	—	2	2	no	—
75	C₂×C₄×C₁₁	—	4	1	no	—
76	C₂×C₄×C₁₁	—	4	2	no	—
77	C₂×C₄×C₁₁	#107	1	4	yes	—
78	C₂×C₄×C₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₁₁	#117	1	1	yes	characteristic
80	C₂×C₄×C₁₁	#121	1	2	yes	—
81	C₂×C₄×C₁₁	C₂²⋊Q₈	1	2	yes	—
82	C₂×C₄×C₁₁	C₂×D₈	1	1	yes	characteristic
83	C₂×C₄×C₁₁	C₂×Q₁₆	1	1	yes	characteristic
84	C₂×C₄×C₁₁	#135	1	2	yes	—
85	C₈×C₁₁	—	2	1	no	—
86	C₈×C₁₁	—	2	2	no	—
87	C₈×C₁₁	—	4	2	no	—
88	C₈×C₁₁	#101	1	2	yes	complemented
89	C₁₁⋊C₈	—	8	4	no	—
90	C₂×C₄×Q₈	C₂×D₁₁	1	1	yes	characteristic
91	C₈×Q₈	—	2	2	no	—
92	C₈×Q₈	—	2	2	no	—
93	#423	—	11	4	no	—
94	C₂×C₄⋊C₈	Dic₁₁	1	2	yes	—
95	C₂×C₄⋊C₈	C₂×D₁₁	1	1	yes	characteristic
96	C₂×#101	—	22	2	no	—
97	C₂×C₄×C₈	Dic₁₁	1	2	yes	—
98	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
99	#343	—	22	2	no	—
100	#335	—	22	2	no	—
101	#335	—	22	2	no	—
102	C₂²×C₁₁	—	2	1	no	—
103	C₂²×C₁₁	—	2	1	no	—
104	C₂²×C₁₁	#380	1	1	yes	characteristic
105	C₂²×C₁₁	C₂×#107	1	1	yes	characteristic
106	C₂²×C₁₁	#427	1	1	yes	characteristic
107	C₄×C₁₁	—	2	1	no	—
108	C₄×C₁₁	—	2	4	no	—
109	C₄×C₁₁	—	2	2	no	—
110	C₄×C₁₁	—	4	2	no	—
111	C₄×C₁₁	C₂×#101	1	1	yes	characteristic
112	C₄×C₁₁	#413	1	1	yes	characteristic
113	C₄×C₁₁	#466	1	2	yes	—
114	C₄×C₁₁	#467	1	2	yes	—
115	C₂²×Q₈	D₄₄	1	1	yes	characteristic
116	C₂×M₁₆	—	22	4	no	—
117	C₂²×C₈	C₂×Dic₁₁	1	1	yes	characteristic
118	C₂²×C₈	C₂×Dic₁₁	1	2	yes	—
119	C₂²×C₈	C₂×Dic₁₁	1	1	yes	characteristic
120	C₄×Q₈	—	2	2	no	—
121	C₄×Q₈	—	2	2	no	—
122	C₂×C₄⋊C₄	Dic₂₂	1	2	yes	—
123	C₂×C₄⋊C₄	D₄₄	1	1	yes	characteristic
124	C₂×C₄²	C₂×Dic₁₁	1	2	yes	—
125	C₂×C₄²	C₂²×D₁₁	1	1	yes	characteristic
126	C₄⋊C₈	—	2	4	no	—
127	C₄⋊C₈	—	2	2	no	—
128	C₄⋊C₈	—	22	4	no	—
129	C₄⋊C₈	—	22	4	no	—
130	#101	—	44	4	no	—
131	C₄×C₈	—	2	2	no	—
132	C₄×C₈	—	2	2	no	—
133	C₄×C₈	C₂²⋊D₁₁	1	2	yes	—
134	C₄×C₈	C₂²⋊D₁₁	1	2	yes	—
135	C₂×C₁₁	—	2	1	no	—
136	C₂×C₁₁	—	2	1	no	—
137	C₂×C₁₁	#1801	1	1	yes	characteristic
138	C₂×C₁₁	#1808	1	1	yes	characteristic
139	C₂×C₁₁	#1813	1	1	yes	characteristic
140	C₂×Q₈	—	2	2	no	—
141	C₂×Q₈	—	2	2	no	—
142	C₂×Q₈	D₈₈	1	2	yes	—
143	C₂×Q₈	Dic₄₄	1	2	yes	—
144	C₂²×C₄	C₄⋊Dic₁₁	1	2	yes	—
145	C₂²×C₄	C₄⋊Dic₁₁	1	2	yes	—
146	C₂²×C₄	C₂×Dic₂₂	1	1	yes	characteristic
147	C₂²×C₄	C₂×D₄₄	1	1	yes	characteristic
148	C₂²×C₄	C₂²×Dic₁₁	1	1	yes	characteristic
149	M₁₆	—	44	4	no	—
150	M₁₆	—	44	4	no	—
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	—	2	2	no	—
153	C₂×C₈	—	4	1	no	—
154	C₂×C₈	—	4	2	no	—
155	C₂×C₈	—	44	4	no	—
156	C₂×C₈	C₂²⋊Dic₁₁	1	1	yes	characteristic
157	C₂×C₈	C₂²⋊Dic₁₁	1	2	yes	—
158	C₂×C₈	C₂²⋊Dic₁₁	1	1	yes	characteristic
159	C₄⋊C₄	—	2	4	no	—
160	C₄⋊C₄	—	2	2	no	—
161	C₄²	—	2	2	no	—
162	C₄²	—	2	2	no	—
163	C₄²	#4339	1	2	yes	—
164	C₄²	C₂×C₂²⋊D₁₁	1	2	yes	—
165	C₁₁	#8617	1	1	yes	characteristic, complemented, Sylow
166	C₂³	C₂×C₄⋊Dic₁₁	1	1	yes	characteristic
167	Q₈	—	2	4	no	—
168	Q₈	—	2	4	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	2	1	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	4	1	no	—
175	C₂×C₄	—	4	2	no	—
176	C₂×C₄	#66882	1	4	yes	—
177	C₂×C₄	C₂×D₈₈	1	1	yes	characteristic
178	C₂×C₄	#66957	1	2	yes	—
179	C₂×C₄	C₂×Dic₄₄	1	1	yes	characteristic
180	C₂×C₄	#66977	1	2	yes	—
181	C₂×C₄	#66979	1	1	yes	characteristic
182	C₂×C₄	#66983	1	2	yes	—
183	C₂×C₄	C₂×C₂²⋊Dic₁₁	1	1	yes	characteristic
184	C₈	—	2	1	no	—
185	C₈	—	2	2	no	—
186	C₈	—	4	2	no	—
187	C₈	—	88	4	no	—
188	C₈	#66900	1	2	yes	complemented
189	C₂²	—	2	1	no	—
190	C₂²	—	2	1	no	—
191	C₂²	C₂×#66882	1	1	yes	characteristic
192	C₂²	#10622719	1	1	yes	characteristic
193	C₂²	#10622825	1	1	yes	characteristic
194	C₄	—	2	1	no	—
195	C₄	—	2	4	no	—
196	C₄	—	2	2	no	—
197	C₄	—	4	2	no	—
198	C₄	#10622728	1	2	yes	—
199	C₄	#10622729	1	2	yes	—
200	C₄	C₂×#66900	1	1	yes	characteristic
201	C₄	#10622850	1	1	yes	characteristic
202	C₂	—	2	1	no	—
203	C₂	—	2	1	no	—
204	C₂	#13068612	1	1	yes	characteristic
205	C₂	#13068645	1	1	yes	characteristic
206	C₂	#13068666	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#10622825
commutator	C₂²×C₄×C₁₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₁
Fitting	C₂×C₈×Q₈×C₁₁	C₂
socle	C₂²×C₁₁	#380
prime radical	C₂³×C₁₁	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₁₁	C₂³
hypercenter	C₂×C₈×Q₈	D₁₁
hypocenter	C₁₁	#8617
LCS	G ▷ C₂²×C₄×C₁₁ ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
88. C₈×C₁₁	#101	semidirect
165. C₁₁	#8617	semidirect
188. C₈	#66900	semidirect

Maximal subgroups

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

subgroup	index
#13068612	2
#13067375	2
#13066491	2
C₂×C₈×Q₈×C₁₁	2
#8617	11