Subgroups — #14672378

Order 2816.

390 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10622295	C₂	1	1	yes	maximal, characteristic
2	#13068780	C₂	1	1	yes	maximal, characteristic
3	#13066708	C₂	1	4	yes	maximal
4	C₂×C₁₁×#346	C₂	1	1	yes	maximal, characteristic
5	C₂×#66922	—	2	1	no	—
6	C₂²×C₁₁×C₁₆	C₂²	1	2	yes	—
7	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
8	C₂×#66882	—	2	1	no	—
9	C₂×#66882	C₂²	1	2	yes	—
10	C₂×#66882	C₂²	1	2	yes	—
11	C₂×#66881	—	2	1	no	—
12	#10622549	—	2	1	no	—
13	#10622295	—	2	2	no	—
14	#10622295	C₄	1	4	yes	complemented
15	C₁₁×#346	—	2	2	no	—
16	C₂²×Dic₂₂	—	4	1	no	—
17	C₂²×C₈×C₁₁	D₄	1	1	yes	characteristic
18	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
19	C₂×C₄²×C₁₁	D₄	1	1	yes	characteristic
20	C₂×C₄⋊Dic₁₁	—	2	2	no	—
21	C₂×C₄⋊Dic₁₁	—	2	2	no	—
22	C₂×C₄⋊Dic₁₁	—	4	1	no	—
23	#66922	—	2	4	no	—
24	#66922	—	4	2	no	—
25	C₂×C₁₁×C₁₆	—	2	2	no	—
26	C₂×C₁₁×C₁₆	—	2	2	no	—
27	C₂×C₁₁×C₁₆	Q₈	1	4	yes	complemented
28	C₄×C₈×C₁₁	C₂×C₄	1	2	yes	—
29	C₄×C₈×C₁₁	D₄	1	2	yes	—
30	C₁₁⋊#95	—	4	1	no	—
31	C₁₁⋊#95	—	4	1	no	—
32	#66882	—	2	4	no	—
33	#66882	—	2	2	no	—
34	#66882	—	2	4	no	—
35	#66882	C₂×C₄	1	4	yes	—
36	#66881	—	4	2	no	—
37	#13283	—	11	1	no	maximal, Sylow
38	C₂²×C₄×C₁₁	D₈	1	1	yes	characteristic
39	C₂²×C₄×C₁₁	D₈	1	1	yes	characteristic
40	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
41	C₂²×Dic₁₁	—	4	2	no	—
42	C₂²×Dic₁₁	—	4	2	no	—
43	C₂×Dic₂₂	—	4	4	no	—
44	C₂×Dic₂₂	—	8	4	no	—
45	C₂×C₈×C₁₁	C₂²⋊C₄	1	2	yes	—
46	C₂×C₈×C₁₁	SD₁₆	1	2	yes	—
47	C₂×C₈×C₁₁	Q₁₆	1	2	yes	—
48	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
50	C₂×C₈×C₁₁	C₂×Q₈	1	2	yes	—
51	C₂×C₈×C₁₁	#41	1	2	yes	—
52	C₄²×C₁₁	C₂²⋊C₄	1	2	yes	—
53	C₄²×C₁₁	SD₁₆	1	2	yes	—
54	C₄⋊Dic₁₁	—	2	4	no	—
55	C₄⋊Dic₁₁	—	4	4	no	—
56	C₄⋊Dic₁₁	—	4	2	no	—
57	C₄⋊Dic₁₁	—	4	2	no	—
58	C₄⋊Dic₁₁	—	4	2	no	—
59	C₁₁×C₁₆	—	2	4	no	—
60	C₂×#500	—	11	1	no	—
61	C₂×#346	D₁₁	1	1	yes	characteristic
62	#1846	—	11	1	no	—
63	#1380	—	11	4	no	—
64	C₂³×C₁₁	C₂×D₈	1	1	yes	characteristic
65	C₂×C₄×C₁₁	—	2	2	no	—
66	C₂×C₄×C₁₁	—	2	2	no	—
67	C₂×C₄×C₁₁	#102	1	2	yes	—
68	C₂×C₄×C₁₁	#102	1	2	yes	—
69	C₂×C₄×C₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₁₁	#124	1	1	yes	characteristic
71	C₂×C₄×C₁₁	C₄⋊Q₈	1	1	yes	characteristic
72	C₂×C₄×C₁₁	C₂×SD₁₆	1	1	yes	characteristic
73	C₂×C₄×C₁₁	C₂×SD₁₆	1	1	yes	characteristic
74	C₂×C₄×C₁₁	C₂×Q₁₆	1	1	yes	characteristic
75	C₂×Dic₁₁	—	4	4	no	—
76	C₂×Dic₁₁	—	8	4	no	—
77	C₂×Dic₁₁	—	8	2	no	—
78	C₂×Dic₁₁	—	8	2	no	—
79	C₂×Dic₁₁	—	8	2	no	—
80	Dic₂₂	—	8	8	no	—
81	C₈×C₁₁	#103	1	4	yes	—
82	C₈×C₁₁	C₄×Q₈	1	2	yes	—
83	C₈×C₁₁	C₂²⋊Q₈	1	2	yes	—
84	C₂×C₄⋊Q₈	—	22	1	no	—
85	C₂²×C₁₆	C₂×D₁₁	1	2	yes	—
86	#500	—	11	4	no	—
87	#500	—	22	2	no	—
88	C₂×#107	—	11	2	no	—
89	C₂×#107	—	11	2	no	—
90	C₂×#107	—	22	1	no	—
91	C₂×#106	—	22	1	no	—
92	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
93	#382	—	22	1	no	—
94	#346	—	2	2	no	—
95	C₂²×C₁₁	C₂×#102	1	1	yes	characteristic
96	C₂²×C₁₁	#486	1	1	yes	characteristic
97	C₂²×C₁₁	#500	1	1	yes	characteristic
98	C₂²×C₁₁	#509	1	2	yes	—
99	C₂²×C₁₁	#510	1	2	yes	—
100	C₄×C₁₁	—	2	2	no	—
101	C₄×C₁₁	—	2	2	no	—
102	C₄×C₁₁	#391	1	1	yes	characteristic
103	C₄×C₁₁	C₂×#103	1	1	yes	characteristic
104	C₄×C₁₁	#487	1	1	yes	characteristic
105	C₄×C₁₁	#498	1	1	yes	characteristic
106	Dic₁₁	—	8	4	no	—
107	Dic₁₁	—	16	2	no	—
108	C₂²×Q₈	—	44	1	no	—
109	C₂²×C₈	D₄₄	1	1	yes	characteristic
110	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
111	C₄⋊Q₈	—	22	4	no	—
112	C₄⋊Q₈	—	44	2	no	—
113	C₂×C₄⋊C₄	—	22	2	no	—
114	C₂×C₄⋊C₄	—	22	2	no	—
115	C₂×C₄⋊C₄	—	44	1	no	—
116	C₂×C₄²	D₄₄	1	1	yes	characteristic
117	C₂×C₁₆	—	2	2	no	—
118	C₂×C₁₆	—	2	2	no	—
119	C₂×C₁₆	Dic₂₂	1	4	yes	complemented
120	#107	—	11	4	no	—
121	#107	—	22	4	no	—
122	#107	—	22	2	no	—
123	#107	—	22	4	no	—
124	#106	—	44	2	no	—
125	C₄×C₈	C₄×D₁₁	1	2	yes	—
126	C₄×C₈	C₂²⋊D₁₁	1	2	yes	—
127	#95	—	44	1	no	—
128	#95	—	44	1	no	—
129	C₂×C₁₁	#1990	1	1	yes	characteristic
130	C₂×C₁₁	#2140	1	1	yes	characteristic
131	C₂×C₁₁	#2141	1	1	yes	characteristic
132	C₂×C₁₁	#2243	1	2	yes	—
133	C₂×C₁₁	#2255	1	2	yes	—
134	C₂×Q₈	—	44	4	no	—
135	C₂×Q₈	—	88	4	no	—
136	C₂²×C₄	—	44	2	no	—
137	C₂²×C₄	—	44	2	no	—
138	C₂²×C₄	D₈₈	1	1	yes	characteristic
139	C₂²×C₄	D₈₈	1	1	yes	characteristic
140	C₂²×C₄	C₂×D₄₄	1	1	yes	characteristic
141	C₂×C₈	#4314	1	2	yes	—
142	C₂×C₈	Dic₄₄	1	2	yes	—
143	C₂×C₈	(C₂×D₁₁)⋊C₄	1	2	yes	—
144	C₂×C₈	C₂×Dic₂₂	1	2	yes	—
145	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
146	C₂×C₈	#4339	1	2	yes	—
147	C₂×C₈	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
148	C₄⋊C₄	—	22	4	no	—
149	C₄⋊C₄	—	44	4	no	—
150	C₄⋊C₄	—	44	2	no	—
151	C₄⋊C₄	—	44	2	no	—
152	C₄⋊C₄	—	44	2	no	—
153	C₄²	#4314	1	2	yes	—
154	C₄²	(C₂×D₁₁)⋊C₄	1	2	yes	—
155	C₁₆	—	2	4	no	—
156	C₁₁	#13283	1	1	yes	characteristic, complemented, Sylow
157	C₂³	C₂×D₈₈	1	1	yes	characteristic
158	Q₈	—	88	8	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	44	4	no	—
162	C₂×C₄	—	88	4	no	—
163	C₂×C₄	—	88	2	no	—
164	C₂×C₄	—	88	2	no	—
165	C₂×C₄	—	88	2	no	—
166	C₂×C₄	#66885	1	2	yes	—
167	C₂×C₄	#66885	1	2	yes	—
168	C₂×C₄	#66922	1	1	yes	characteristic
169	C₂×C₄	#66928	1	1	yes	characteristic
170	C₂×C₄	C₂×#4314	1	1	yes	characteristic
171	C₂×C₄	C₂×#4314	1	1	yes	characteristic
172	C₂×C₄	C₂×Dic₄₄	1	1	yes	characteristic
173	C₂×C₄	C₂×(C₂×D₁₁)⋊C₄	1	1	yes	characteristic
174	C₈	#66880	1	4	yes	—
175	C₈	C₄×Dic₂₂	1	2	yes	—
176	C₈	#66977	1	2	yes	—
177	C₂²	#10622295	1	1	yes	characteristic
178	C₂²	#10622307	1	1	yes	characteristic
179	C₂²	#10622521	1	2	yes	—
180	C₂²	#10622522	1	2	yes	—
181	C₂²	C₂×#66885	1	1	yes	characteristic
182	C₄	—	2	2	no	—
183	C₄	—	2	2	no	—
184	C₄	—	88	4	no	—
185	C₄	—	176	2	no	—
186	C₄	#10622293	1	1	yes	characteristic
187	C₄	#10622294	1	1	yes	characteristic
188	C₄	#10622542	1	1	yes	characteristic
189	C₄	C₂×#66880	1	1	yes	characteristic
190	C₂	#13069577	1	1	yes	characteristic
191	C₂	#13070333	1	1	yes	characteristic
192	C₂	#13070337	1	1	yes	characteristic
193	C₂	#13070853	1	2	yes	—
194	C₂	#13070902	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₈₈
commutator	C₂×C₈×C₁₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₁
Fitting	C₂×C₁₁×#346	C₂
socle, prime radical	C₂³×C₁₁	C₂×D₈
cosocle	C₂²×C₈×C₁₁	C₂³
hypercenter	C₂×#346	D₁₁
hypocenter	C₁₁	#13283
LCS	G ▷ C₂×C₈×C₁₁ ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
14. #10622295	C₄	semidirect
27. C₂×C₁₁×C₁₆	Q₈	semidirect
119. C₂×C₁₆	Dic₂₂	semidirect
156. C₁₁	#13283	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10622295	2
#13068780	2
#13066708	2
C₂×C₁₁×#346	2
#13283	11