Subgroups — #14670218

Order 2816.

350 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10622143	C₂	1	2	yes	maximal
2	C₂×#10622142	C₂	1	2	yes	maximal
3	#13068780	C₂	1	1	yes	maximal, characteristic
4	#13068776	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₁×#346	C₂	1	1	yes	maximal, characteristic
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	1	yes	characteristic
10	C₂×#66882	C₂²	1	2	yes	—
11	C₂×#66882	C₂²	1	1	yes	characteristic
12	C₂×#66881	—	2	1	no	—
13	#10622549	—	2	1	no	—
14	#10622548	—	2	1	no	—
15	C₁₁×#346	C₄	1	4	yes	complemented
16	#10622143	—	2	4	no	—
17	#10622142	—	2	4	no	—
18	C₂²×C₈×C₁₁	D₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
20	C₂×C₄²×C₁₁	D₄	1	1	yes	characteristic
21	C₂×C₄⋊Dic₁₁	—	2	2	no	—
22	C₂×C₄⋊Dic₁₁	—	2	1	no	—
23	C₂×C₄⋊Dic₁₁	—	2	1	no	—
24	C₂×C₄⋊Dic₁₁	—	4	2	no	—
25	C₂×C₁₁×C₁₆	C₂×C₄	1	4	yes	—
26	C₂×C₁₁×C₁₆	D₄	1	4	yes	—
27	C₂×C₁₁×C₁₆	Q₈	1	4	yes	—
28	C₄×C₈×C₁₁	C₂×C₄	1	2	yes	—
29	C₄×C₈×C₁₁	Q₈	1	2	yes	—
30	C₁₁⋊#95	—	4	2	no	—
31	#66882	—	2	2	no	—
32	#66882	—	2	2	no	—
33	#66882	—	2	2	no	—
34	#66882	—	2	2	no	—
35	#66882	—	4	2	no	—
36	#66881	—	4	2	no	—
37	#12908	—	11	1	no	maximal, Sylow
38	C₂²×C₄×C₁₁	D₈	1	2	yes	—
39	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
40	C₂²×Dic₁₁	—	4	1	no	—
41	C₂²×Dic₁₁	—	4	1	no	—
42	C₂²×Dic₁₁	—	4	2	no	—
43	C₂×C₈×C₁₁	C₄⋊C₄	1	2	yes	—
44	C₂×C₈×C₁₁	D₈	1	2	yes	—
45	C₂×C₈×C₁₁	Q₁₆	1	2	yes	—
46	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₈×C₁₁	C₂×D₄	1	2	yes	—
48	C₂×C₈×C₁₁	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₈×C₁₁	C₂×Q₈	1	2	yes	—
50	C₄²×C₁₁	C₄⋊C₄	1	2	yes	—
51	C₄²×C₁₁	Q₁₆	1	2	yes	—
52	C₄⋊Dic₁₁	—	4	1	no	—
53	C₄⋊Dic₁₁	—	4	2	no	—
54	C₄⋊Dic₁₁	—	4	1	no	—
55	C₄⋊Dic₁₁	—	4	1	no	—
56	C₄⋊Dic₁₁	—	4	1	no	—
57	C₄⋊Dic₁₁	—	4	2	no	—
58	C₄⋊Dic₁₁	—	8	4	no	—
59	C₁₁×C₁₆	C₄⋊C₄	1	8	yes	complemented
60	C₂×#367	—	11	2	no	—
61	C₂×#366	—	11	2	no	—
62	C₂×#346	D₁₁	1	1	yes	characteristic
63	#1846	—	11	1	no	—
64	#1845	—	11	1	no	—
65	C₂³×C₁₁	C₂×D₈	1	1	yes	characteristic
66	C₂×C₄×C₁₁	—	2	2	no	—
67	C₂×C₄×C₁₁	—	2	2	no	—
68	C₂×C₄×C₁₁	#107	1	4	yes	—
69	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₁₁	#127	1	1	yes	characteristic
71	C₂×C₄×C₁₁	C₄⋊Q₈	1	1	yes	characteristic
72	C₂×C₄×C₁₁	C₂×D₈	1	1	yes	characteristic
73	C₂×C₄×C₁₁	C₂×Q₁₆	1	1	yes	characteristic
74	C₂×C₄×C₁₁	C₂×Q₁₆	1	1	yes	characteristic
75	C₂×Dic₁₁	—	8	1	no	—
76	C₂×Dic₁₁	—	8	2	no	—
77	C₂×Dic₁₁	—	8	1	no	—
78	C₂×Dic₁₁	—	8	4	no	—
79	C₂×Dic₁₁	—	8	2	no	—
80	C₂×Dic₁₁	—	8	2	no	—
81	C₈×C₁₁	#107	1	4	yes	—
82	C₈×C₁₁	C₂×C₄⋊C₄	1	2	yes	—
83	C₈×C₁₁	C₄⋊Q₈	1	2	yes	—
84	C₂²×C₁₆	C₂×D₁₁	1	2	yes	—
85	C₂×#107	—	11	2	no	—
86	C₂×#107	—	11	1	no	—
87	C₂×#107	—	11	1	no	—
88	C₂×#107	—	22	1	no	—
89	C₂×#106	—	22	1	no	—
90	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
91	#384	—	22	1	no	—
92	#382	—	22	1	no	—
93	#367	—	22	4	no	—
94	#366	—	22	4	no	—
95	#346	Dic₁₁	1	4	yes	complemented
96	C₂²×C₁₁	C₂×#107	1	1	yes	characteristic
97	C₂²×C₁₁	#493	1	1	yes	characteristic
98	C₂²×C₁₁	#500	1	1	yes	characteristic
99	C₂²×C₁₁	#509	1	2	yes	—
100	C₂²×C₁₁	#510	1	2	yes	—
101	C₄×C₁₁	—	2	4	no	—
102	C₄×C₁₁	#384	1	1	yes	characteristic
103	C₄×C₁₁	C₂×#107	1	1	yes	characteristic
104	C₄×C₁₁	#494	1	1	yes	characteristic
105	C₄×C₁₁	#500	1	1	yes	characteristic
106	Dic₁₁	—	16	2	no	—
107	Dic₁₁	—	16	1	no	—
108	Dic₁₁	—	16	1	no	—
109	C₂²×C₈	D₄₄	1	1	yes	characteristic
110	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
111	C₂×C₄⋊C₄	—	22	1	no	—
112	C₂×C₄⋊C₄	—	22	1	no	—
113	C₂×C₄⋊C₄	—	22	2	no	—
114	C₂×C₄⋊C₄	—	44	2	no	—
115	C₂×C₄²	D₄₄	1	1	yes	characteristic
116	C₂×C₁₆	Dic₂₂	1	4	yes	—
117	C₂×C₁₆	D₄₄	1	4	yes	—
118	C₂×C₁₆	C₂×Dic₁₁	1	4	yes	—
119	#107	—	22	2	no	—
120	#107	—	22	2	no	—
121	#107	—	22	2	no	—
122	#107	—	22	2	no	—
123	#107	—	44	2	no	—
124	#106	—	44	2	no	—
125	C₄×C₈	Dic₂₂	1	2	yes	—
126	C₄×C₈	C₂×Dic₁₁	1	2	yes	—
127	#95	—	44	2	no	—
128	C₂×C₁₁	#1845	1	1	yes	characteristic
129	C₂×C₁₁	#2159	1	2	yes	—
130	C₂×C₁₁	#2250	1	1	yes	characteristic
131	C₂×C₁₁	#2251	1	1	yes	characteristic
132	C₂×C₁₁	#2255	1	2	yes	—
133	C₂²×C₄	—	44	1	no	—
134	C₂²×C₄	—	44	1	no	—
135	C₂²×C₄	—	44	2	no	—
136	C₂²×C₄	D₈₈	1	2	yes	—
137	C₂²×C₄	C₂×D₄₄	1	1	yes	characteristic
138	C₂×C₈	D₈₈	1	2	yes	—
139	C₂×C₈	Dic₄₄	1	2	yes	—
140	C₂×C₈	C₄⋊Dic₁₁	1	2	yes	—
141	C₂×C₈	C₂×Dic₂₂	1	1	yes	characteristic
142	C₂×C₈	C₂×Dic₂₂	1	2	yes	—
143	C₂×C₈	C₂×D₄₄	1	2	yes	—
144	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
145	C₄⋊C₄	—	44	1	no	—
146	C₄⋊C₄	—	44	2	no	—
147	C₄⋊C₄	—	44	1	no	—
148	C₄⋊C₄	—	44	1	no	—
149	C₄⋊C₄	—	44	1	no	—
150	C₄⋊C₄	—	44	2	no	—
151	C₄⋊C₄	—	88	4	no	—
152	C₄²	Dic₄₄	1	2	yes	—
153	C₄²	C₄⋊Dic₁₁	1	2	yes	—
154	C₁₆	C₄⋊Dic₁₁	1	8	yes	complemented
155	C₁₁	#12908	1	1	yes	characteristic, complemented, Sylow
156	C₂³	C₂×D₈₈	1	1	yes	characteristic
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	88	4	no	—
160	C₂×C₄	—	88	2	no	—
161	C₂×C₄	—	88	2	no	—
162	C₂×C₄	—	88	1	no	—
163	C₂×C₄	—	88	2	no	—
164	C₂×C₄	—	88	1	no	—
165	C₂×C₄	#66882	1	4	yes	—
166	C₂×C₄	#66922	1	1	yes	characteristic
167	C₂×C₄	#66927	1	1	yes	characteristic
168	C₂×C₄	C₂×D₈₈	1	1	yes	characteristic
169	C₂×C₄	C₂×Dic₄₄	1	1	yes	characteristic
170	C₂×C₄	C₂×Dic₄₄	1	1	yes	characteristic
171	C₂×C₄	C₂×C₄⋊Dic₁₁	1	1	yes	characteristic
172	C₈	#66882	1	4	yes	—
173	C₈	#66922	1	2	yes	—
174	C₈	C₂×C₄⋊Dic₁₁	1	2	yes	—
175	C₂²	#10622295	1	1	yes	characteristic
176	C₂²	#10622308	1	1	yes	characteristic
177	C₂²	#10622521	1	2	yes	—
178	C₂²	#10622522	1	2	yes	—
179	C₂²	C₂×#66882	1	1	yes	characteristic
180	C₄	—	2	4	no	—
181	C₄	—	176	2	no	—
182	C₄	—	176	1	no	—
183	C₄	—	176	1	no	—
184	C₄	#10622295	1	1	yes	characteristic
185	C₄	#10622312	1	1	yes	characteristic
186	C₄	#10622548	1	1	yes	characteristic
187	C₄	C₂×#66882	1	1	yes	characteristic
188	C₂	#13068776	1	1	yes	characteristic
189	C₂	#13070404	1	2	yes	—
190	C₂	#13070882	1	1	yes	characteristic
191	C₂	#13070887	1	1	yes	characteristic
192	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	#12908
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
15. C₁₁×#346	C₄	semidirect
59. C₁₁×C₁₆	C₄⋊C₄	semidirect
95. #346	Dic₁₁	semidirect
154. C₁₆	C₄⋊Dic₁₁	semidirect
155. C₁₁	#12908	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10622143	2
C₂×#10622142	2
#13068780	2
#13068776	2
C₂×C₁₁×#346	2
#12908	11