Subgroups — #14672387

Order 2816.

350 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×C₄×C₁₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	#13068788	C₂	1	1	yes	maximal, characteristic
3	#13066711	C₂	1	4	yes	maximal
4	C₁₁×#3160	C₂	1	1	yes	maximal, characteristic
5	C₁₁×#533	—	2	1	no	—
6	C₁₁×#500	C₄	1	2	yes	—
7	C₁₁×#499	—	2	1	no	—
8	C₁₁×#499	C₄	1	2	yes	—
9	C₁₁×#498	—	2	1	no	—
10	C₁₁×#427	—	2	1	no	—
11	C₁₁×#427	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
13	C₂×#66883	—	2	1	no	—
14	C₂×#66883	C₂²	1	2	yes	—
15	C₂²×C₁₁⋊C₁₆	C₂²	1	2	yes	—
16	#10622537	—	2	1	no	—
17	C₄×C₁₁⋊C₁₆	—	2	1	no	—
18	C₄×C₁₁⋊C₁₆	—	2	1	no	—
19	C₂²×C₈×C₁₁	D₄	1	1	yes	characteristic
20	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
21	C₁₁×C₄⋊Q₈	—	2	2	no	—
22	C₁₁×#125	—	2	2	no	—
23	C₁₁×C₂²⋊Q₈	—	4	2	no	—
24	C₄×Q₈×C₁₁	—	4	2	no	—
25	C₁₁×#117	—	2	2	no	—
26	C₂×C₄²×C₁₁	D₄	1	1	yes	characteristic
27	C₂×#4318	—	2	2	no	—
28	C₁₁×#107	—	2	2	no	—
29	C₁₁×#107	C₂×C₄	1	4	yes	—
30	C₁₁×#106	—	2	2	no	—
31	C₁₁×#106	—	2	2	no	—
32	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
34	C₄×C₈×C₁₁	D₄	1	1	yes	characteristic
35	C₄×C₈×C₁₁	D₄	1	1	yes	characteristic
36	#66883	—	2	4	no	—
37	#66883	—	4	2	no	—
38	C₂×C₁₁⋊C₁₆	—	2	2	no	—
39	C₂×C₁₁⋊C₁₆	—	2	2	no	—
40	C₂×C₁₁⋊C₁₆	Q₈	1	4	yes	complemented
41	C₄⋊(C₁₁⋊C₈)	—	4	2	no	—
42	#13285	—	11	1	no	maximal, Sylow
43	C₂×Q₈×C₁₁	—	4	2	no	—
44	C₂²×C₄×C₁₁	SD₁₆	1	2	yes	—
45	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
46	C₂×C₈×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₈×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₈×C₁₁	SD₁₆	1	2	yes	—
49	C₂×C₈×C₁₁	Q₁₆	1	2	yes	—
50	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₁₁	C₂×Q₈	1	2	yes	—
53	C₂×C₈×C₁₁	#41	1	2	yes	—
54	C₁₁×C₄⋊C₄	—	2	4	no	—
55	C₁₁×C₄⋊C₄	—	4	4	no	—
56	C₁₁×C₄⋊C₄	—	4	2	no	—
57	C₁₁×C₂²⋊C₄	—	4	2	no	—
58	C₄²×C₁₁	—	4	2	no	—
59	C₄²×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
60	C₄²×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
61	C₄²×C₁₁	D₈	1	2	yes	—
62	#4318	—	4	2	no	—
63	#4318	—	4	2	no	—
64	C₂×C₁₁⋊C₈	—	4	2	no	—
65	C₁₁⋊C₁₆	—	2	4	no	—
66	#3160	D₁₁	1	1	yes	characteristic
67	C₂×C₄×C₁₆	—	11	1	no	—
68	#1848	—	11	1	no	—
69	#1382	—	11	4	no	—
70	C₂³×C₁₁	C₂×SD₁₆	1	1	yes	characteristic
71	Q₈×C₁₁	—	8	4	no	—
72	C₂×C₄×C₁₁	—	2	4	no	—
73	C₂×C₄×C₁₁	—	4	4	no	—
74	C₂×C₄×C₁₁	#102	1	2	yes	—
75	C₂×C₄×C₁₁	#102	1	2	yes	—
76	C₂×C₄×C₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₁₁	#124	1	1	yes	characteristic
78	C₂×C₄×C₁₁	C₄⋊Q₈	1	1	yes	characteristic
79	C₂×C₄×C₁₁	C₂×D₈	1	1	yes	characteristic
80	C₂×C₄×C₁₁	C₂×SD₁₆	1	1	yes	characteristic
81	C₂×C₄×C₁₁	C₂×Q₁₆	1	1	yes	characteristic
82	C₈×C₁₁	#103	1	2	yes	—
83	C₈×C₁₁	#103	1	2	yes	—
84	C₈×C₁₁	C₄×Q₈	1	2	yes	—
85	C₈×C₁₁	C₂²⋊Q₈	1	2	yes	—
86	C₁₁⋊C₈	—	8	2	no	—
87	#533	—	2	1	no	—
88	C₂²×C₁₆	—	11	2	no	—
89	#500	Dic₁₁	1	2	yes	—
90	#499	—	2	1	no	—
91	#499	Dic₁₁	1	2	yes	—
92	#498	—	2	1	no	—
93	C₂×#108	—	11	2	no	—
94	C₂×#108	—	22	1	no	—
95	#427	—	2	1	no	—
96	#427	C₂×D₁₁	1	2	yes	—
97	#423	—	22	1	no	—
98	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
99	C₄×C₁₆	—	22	1	no	—
100	C₄×C₁₆	—	22	1	no	—
101	C₂²×C₁₁	—	2	2	no	—
102	C₂²×C₁₁	C₂×#102	1	1	yes	characteristic
103	C₂²×C₁₁	#487	1	1	yes	characteristic
104	C₂²×C₁₁	#498	1	1	yes	characteristic
105	C₄×C₁₁	—	2	2	no	—
106	C₄×C₁₁	—	8	4	no	—
107	C₄×C₁₁	#359	1	4	yes	—
108	C₄×C₁₁	#391	1	1	yes	characteristic
109	C₄×C₁₁	C₂×#103	1	1	yes	characteristic
110	C₄×C₁₁	#486	1	1	yes	characteristic
111	C₄×C₁₁	#500	1	1	yes	characteristic
112	C₂×M₁₆	—	22	2	no	—
113	C₂²×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
114	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
115	C₄⋊Q₈	—	2	2	no	—
116	#125	—	2	2	no	—
117	C₂²⋊Q₈	—	4	2	no	—
118	C₄×Q₈	—	4	2	no	—
119	#117	—	2	2	no	—
120	C₂×C₄²	C₂²⋊D₁₁	1	1	yes	characteristic
121	C₂×C₁₆	—	11	4	no	—
122	C₂×C₁₆	—	22	2	no	—
123	C₂×C₁₆	—	22	2	no	—
124	#108	—	22	4	no	—
125	#108	—	44	2	no	—
126	#107	—	2	2	no	—
127	#107	C₂×Dic₁₁	1	4	yes	—
128	#106	—	2	2	no	—
129	#106	—	2	2	no	—
130	C₄⋊C₈	—	44	2	no	—
131	C₄×C₈	C₂×Dic₁₁	1	1	yes	characteristic
132	C₄×C₈	C₂×Dic₁₁	1	1	yes	characteristic
133	C₄×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
134	C₄×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
135	C₂×C₁₁	—	2	1	no	—
136	C₂×C₁₁	—	2	1	no	—
137	C₂×C₁₁	#1990	1	1	yes	characteristic
138	C₂×C₁₁	C₂×#359	1	2	yes	—
139	C₂×Q₈	—	4	2	no	—
140	C₂²×C₄	#4324	1	2	yes	—
141	C₂²×C₄	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
142	M₁₆	—	44	2	no	—
143	M₁₆	—	44	2	no	—
144	C₂×C₈	—	44	2	no	—
145	C₂×C₈	#4325	1	2	yes	—
146	C₂×C₈	#4326	1	2	yes	—
147	C₂×C₈	C₂²⋊Dic₁₁	1	1	yes	characteristic
148	C₂×C₈	C₂²⋊Dic₁₁	1	1	yes	characteristic
149	C₂×C₈	Q₈×D₁₁	1	2	yes	—
150	C₂×C₈	#4343	1	2	yes	—
151	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
152	C₂×C₈	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
153	C₄⋊C₄	—	2	4	no	—
154	C₄⋊C₄	—	4	4	no	—
155	C₄⋊C₄	—	4	2	no	—
156	C₂²⋊C₄	—	4	2	no	—
157	C₄²	—	4	2	no	—
158	C₄²	#4323	1	2	yes	—
159	C₄²	C₂²⋊Dic₁₁	1	1	yes	characteristic
160	C₄²	C₂²⋊Dic₁₁	1	1	yes	characteristic
161	C₁₆	—	22	4	no	—
162	C₁₁	#13285	1	1	yes	characteristic, complemented, Sylow
163	C₂³	C₂×#4324	1	1	yes	characteristic
164	Q₈	—	8	4	no	—
165	C₂×C₄	—	2	4	no	—
166	C₂×C₄	—	4	4	no	—
167	C₂×C₄	#66896	1	2	yes	—
168	C₂×C₄	#66896	1	2	yes	—
169	C₂×C₄	C₂×#4323	1	1	yes	characteristic
170	C₂×C₄	C₂×#4325	1	1	yes	characteristic
171	C₂×C₄	C₂×#4326	1	1	yes	characteristic
172	C₂×C₄	#66997	1	1	yes	characteristic
173	C₂×C₄	#67000	1	1	yes	characteristic
174	C₂×C₄	C₂×C₂²⋊Dic₁₁	1	1	yes	characteristic
175	C₈	—	88	2	no	—
176	C₈	#66899	1	2	yes	—
177	C₈	#66899	1	2	yes	—
178	C₈	Q₈×Dic₁₁	1	2	yes	—
179	C₈	#66999	1	2	yes	—
180	C₂²	—	2	2	no	—
181	C₂²	#10622692	1	1	yes	characteristic
182	C₂²	#10622706	1	1	yes	characteristic
183	C₂²	C₂×#66896	1	1	yes	characteristic
184	C₄	—	2	2	no	—
185	C₄	—	8	4	no	—
186	C₄	#10622205	1	4	yes	—
187	C₄	#10622691	1	1	yes	characteristic
188	C₄	#10622701	1	1	yes	characteristic
189	C₄	C₂×#66899	1	1	yes	characteristic
190	C₄	#10622843	1	1	yes	characteristic
191	C₂	—	2	1	no	—
192	C₂	—	2	1	no	—
193	C₂	#13069578	1	1	yes	characteristic
194	C₂	C₂×#10622205	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×#4323
commutator	C₂×C₈×C₁₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₁
Fitting	C₁₁×#3160	C₂
socle	C₂²×C₁₁	C₂×#102
prime radical	C₂³×C₁₁	C₂×SD₁₆
cosocle	C₂²×C₈×C₁₁	C₂³
hypercenter	#3160	D₁₁
hypocenter	C₁₁	#13285
LCS	G ▷ C₂×C₈×C₁₁ ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3160

Complements

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

N	H ≅ G/N	type
40. C₂×C₁₁⋊C₁₆	Q₈	semidirect
162. C₁₁	#13285	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
#13068788	2
#13066711	2
C₁₁×#3160	2
#13285	11