Subgroups — #14670229

Order 2816.

266 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10622144	C₂	1	1	yes	maximal, characteristic
2	#13070400	C₂	1	1	yes	maximal, characteristic
3	#13070389	C₂	1	2	yes	maximal
4	#13068740	C₂	1	2	yes	maximal
5	C₁₁×#2108	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₁×M₃₂	C₂²	1	1	yes	characteristic
7	C₂²×C₁₁×C₁₆	C₂²	1	1	yes	characteristic
8	C₁₁×#405	C₂²	1	1	yes	characteristic
9	C₂×#66883	C₂²	1	2	yes	—
10	#10622719	C₂²	1	2	yes	—
11	C₁₁×#346	C₄	1	2	yes	complemented
12	C₄×C₁₁×C₁₆	C₄	1	2	yes	complemented
13	#10622194	—	2	2	no	—
14	#10622169	—	2	2	no	—
15	#10622151	—	2	4	no	—
16	#10622144	—	2	1	no	—
17	#10622144	—	2	1	no	—
18	#10622143	—	2	1	no	—
19	#10622142	—	2	1	no	—
20	C₂×C₁₁×M₁₆	D₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
22	C₁₁×#117	D₄	1	1	yes	characteristic
23	#66979	—	2	2	no	—
24	C₂×#4318	—	2	2	no	—
25	C₁₁×M₃₂	D₄	1	2	yes	—
26	C₁₁×M₃₂	Q₈	1	2	yes	—
27	C₂×C₁₁×C₁₆	C₂×C₄	1	1	yes	characteristic
28	C₂×C₁₁×C₁₆	C₂×C₄	1	1	yes	characteristic
29	C₂×C₁₁×C₁₆	C₂×C₄	1	2	yes	—
30	C₂×C₁₁×C₁₆	D₄	1	1	yes	characteristic
31	C₂×C₁₁×C₁₆	D₄	1	1	yes	characteristic
32	C₂×C₁₁×C₁₆	Q₈	1	1	yes	characteristic
33	C₂×C₁₁×C₁₆	Q₈	1	1	yes	characteristic
34	C₁₁×#97	Q₈	1	2	yes	—
35	C₄×C₈×C₁₁	C₂×C₄	1	2	yes	—
36	#66883	—	2	2	no	—
37	#66883	—	2	2	no	—
38	#66882	—	2	2	no	—
39	#66881	—	2	2	no	—
40	#12911	—	11	1	no	maximal, Sylow
41	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
42	C₁₁×M₁₆	—	2	1	no	—
43	C₁₁×M₁₆	—	2	1	no	—
44	C₂×C₈×C₁₁	C₄⋊C₄	1	2	yes	—
45	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
47	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
48	C₂×C₈×C₁₁	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₈×C₁₁	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₈×C₁₁	C₂×Q₈	1	1	yes	characteristic
51	C₁₁×C₄⋊C₄	—	2	1	no	—
52	C₁₁×C₂²⋊C₄	—	2	1	no	—
53	C₄²×C₁₁	C₄⋊C₄	1	2	yes	—
54	C₂²⋊Dic₁₁	—	4	2	no	—
55	C₄⋊Dic₁₁	—	4	2	no	—
56	C₄×Dic₁₁	—	4	2	no	—
57	#4318	—	4	2	no	—
58	#4318	—	4	2	no	—
59	C₂×C₁₁⋊C₈	—	4	2	no	—
60	C₁₁×C₁₆	C₄⋊C₄	1	1	yes	characteristic
61	C₁₁×C₁₆	C₄⋊C₄	1	1	yes	characteristic
62	C₁₁×C₁₆	C₄⋊C₄	1	1	yes	characteristic
63	C₁₁×C₁₆	C₄⋊C₄	1	1	yes	characteristic
64	C₁₁×C₁₆	C₄⋊C₄	1	4	yes	—
65	C₂×#368	—	11	1	no	—
66	#2158	—	11	1	no	—
67	#2155	—	11	2	no	—
68	#2108	D₁₁	1	1	yes	characteristic
69	#1833	—	11	2	no	—
70	C₂³×C₁₁	#135	1	1	yes	characteristic
71	C₂×C₄×C₁₁	—	2	2	no	—
72	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₂×C₄×C₁₁	#127	1	1	yes	characteristic
74	C₂×C₄×C₁₁	C₄⋊Q₈	1	1	yes	characteristic
75	C₂×C₄×C₁₁	#135	1	1	yes	characteristic
76	C₂×C₄×C₁₁	#135	1	1	yes	characteristic
77	C₂×C₄×C₁₁	#135	1	1	yes	characteristic
78	C₂×Dic₁₁	—	8	2	no	—
79	C₈×C₁₁	—	2	2	no	—
80	C₈×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₈×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
82	C₈×C₁₁	C₄⋊Q₈	1	1	yes	characteristic
83	C₈×C₁₁	C₄⋊Q₈	1	1	yes	characteristic
84	C₁₁⋊C₈	—	8	2	no	—
85	C₂×M₃₂	C₂×D₁₁	1	1	yes	characteristic
86	C₂²×C₁₆	C₂×D₁₁	1	1	yes	characteristic
87	C₂×#108	—	11	2	no	—
88	#427	—	11	2	no	—
89	#405	C₂×D₁₁	1	1	yes	characteristic
90	#368	—	22	1	no	—
91	#368	—	22	1	no	—
92	#367	—	22	1	no	—
93	#366	—	22	1	no	—
94	#365	—	22	4	no	—
95	#346	Dic₁₁	1	2	yes	complemented
96	C₄×C₁₆	Dic₁₁	1	2	yes	complemented
97	#344	—	22	2	no	—
98	H(3,ℤ/4)	—	22	2	no	—
99	C₂²×C₁₁	—	4	1	no	—
100	C₂²×C₁₁	#427	1	1	yes	characteristic
101	C₂²×C₁₁	#495	1	1	yes	characteristic
102	C₂²×C₁₁	#499	1	1	yes	characteristic
103	C₄×C₁₁	—	4	2	no	—
104	C₄×C₁₁	#384	1	1	yes	characteristic
105	C₄×C₁₁	#427	1	1	yes	characteristic
106	C₄×C₁₁	#495	1	1	yes	characteristic
107	C₄×C₁₁	#499	1	1	yes	characteristic
108	Dic₁₁	—	16	2	no	—
109	C₂×M₁₆	—	22	2	no	—
110	C₂×M₁₆	D₄₄	1	1	yes	characteristic
111	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
112	#117	—	22	2	no	—
113	#117	D₄₄	1	1	yes	characteristic
114	M₃₂	Dic₂₂	1	2	yes	—
115	M₃₂	D₄₄	1	2	yes	—
116	C₂×C₁₆	Dic₂₂	1	1	yes	characteristic
117	C₂×C₁₆	Dic₂₂	1	1	yes	characteristic
118	C₂×C₁₆	D₄₄	1	1	yes	characteristic
119	C₂×C₁₆	D₄₄	1	1	yes	characteristic
120	C₂×C₁₆	C₂×Dic₁₁	1	1	yes	characteristic
121	C₂×C₁₆	C₂×Dic₁₁	1	2	yes	—
122	C₂×C₁₆	C₂×Dic₁₁	1	1	yes	characteristic
123	#108	—	22	2	no	—
124	#108	—	22	2	no	—
125	#107	—	22	2	no	—
126	#106	—	22	2	no	—
127	#97	Dic₂₂	1	2	yes	—
128	C₄×C₈	C₂×Dic₁₁	1	2	yes	—
129	C₂×C₁₁	—	2	1	no	—
130	C₂×C₁₁	—	2	1	no	—
131	C₂×C₁₁	—	2	1	no	—
132	C₂×C₁₁	#1846	1	1	yes	characteristic
133	C₂²×C₄	C₂×D₄₄	1	1	yes	characteristic
134	M₁₆	—	2	1	no	—
135	M₁₆	—	2	1	no	—
136	M₁₆	—	44	2	no	—
137	M₁₆	—	44	2	no	—
138	C₂×C₈	—	44	2	no	—
139	C₂×C₈	C₄⋊Dic₁₁	1	2	yes	—
140	C₂×C₈	C₂×Dic₂₂	1	1	yes	characteristic
141	C₂×C₈	C₂×Dic₂₂	1	1	yes	characteristic
142	C₂×C₈	C₂×Dic₂₂	1	1	yes	characteristic
143	C₂×C₈	C₂×D₄₄	1	1	yes	characteristic
144	C₂×C₈	C₂×D₄₄	1	1	yes	characteristic
145	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
146	C₄⋊C₄	—	2	1	no	—
147	C₄⋊C₄	—	44	2	no	—
148	C₂²⋊C₄	—	2	1	no	—
149	C₂²⋊C₄	—	44	2	no	—
150	C₄²	—	44	2	no	—
151	C₄²	C₄⋊Dic₁₁	1	2	yes	—
152	C₁₆	C₄⋊Dic₁₁	1	1	yes	characteristic
153	C₁₆	C₄⋊Dic₁₁	1	1	yes	characteristic
154	C₁₆	C₄⋊Dic₁₁	1	1	yes	characteristic
155	C₁₆	C₄⋊Dic₁₁	1	1	yes	characteristic
156	C₁₆	C₄⋊Dic₁₁	1	4	yes	—
157	C₁₁	#12911	1	1	yes	characteristic, complemented, Sylow
158	C₂³	#66957	1	1	yes	characteristic
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	88	2	no	—
161	C₂×C₄	#66922	1	1	yes	characteristic
162	C₂×C₄	#66927	1	1	yes	characteristic
163	C₂×C₄	#66957	1	1	yes	characteristic
164	C₂×C₄	#66957	1	1	yes	characteristic
165	C₂×C₄	#66957	1	1	yes	characteristic
166	C₂×C₄	C₂×C₄⋊Dic₁₁	1	1	yes	characteristic
167	C₈	—	2	2	no	—
168	C₈	—	88	2	no	—
169	C₈	#66922	1	1	yes	characteristic
170	C₈	#66922	1	1	yes	characteristic
171	C₈	C₂×C₄⋊Dic₁₁	1	1	yes	characteristic
172	C₈	C₂×C₄⋊Dic₁₁	1	1	yes	characteristic
173	C₂²	—	4	1	no	—
174	C₂²	#10622296	1	1	yes	characteristic
175	C₂²	#10622309	1	1	yes	characteristic
176	C₂²	#10622719	1	1	yes	characteristic
177	C₄	—	4	2	no	—
178	C₄	—	176	2	no	—
179	C₄	#10622296	1	1	yes	characteristic
180	C₄	#10622309	1	1	yes	characteristic
181	C₄	#10622548	1	1	yes	characteristic
182	C₄	#10622719	1	1	yes	characteristic
183	C₂	—	2	1	no	—
184	C₂	—	2	1	no	—
185	C₂	—	2	1	no	—
186	C₂	#13068780	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#10622719
commutator	C₂×C₈×C₁₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₁
Fitting	C₁₁×#2108	C₂
socle	C₂×C₁₁	#1846
prime radical	C₂³×C₁₁	#135
cosocle	C₂²×C₈×C₁₁	C₂³
hypercenter	#2108	D₁₁
hypocenter	C₁₁	#12911
LCS	G ▷ C₂×C₈×C₁₁ ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₄ ◁ #117 ◁ #405 ◁ #2108

Complements

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

N	H ≅ G/N	type
11. C₁₁×#346	C₄	semidirect
12. C₄×C₁₁×C₁₆	C₄	semidirect
95. #346	Dic₁₁	semidirect
96. C₄×C₁₆	Dic₁₁	semidirect
157. C₁₁	#12911	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10622144	2
#13070400	2
#13070389	2
#13068740	2
C₁₁×#2108	2
#12911	11