Subgroups — #14672374

Order 2816.

414 conjugacy classes of subgroups.

The subgroup lattice has 216 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	#13068776	C₂	1	1	yes	maximal, characteristic
3	#13066708	C₂	1	4	yes	maximal
4	C₂×C₄×C₁₁×C₁₆	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	—	2	1	no	—
10	C₂×#66882	C₂²	1	2	yes	—
11	C₂×#66882	C₂²	1	2	yes	—
12	#10622548	—	2	1	no	—
13	#10622295	—	2	2	no	—
14	#10622295	C₄	1	4	yes	complemented
15	C₄×C₁₁×C₁₆	—	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	C₂×C₄⋊Dic₁₁	—	4	1	no	—
24	C₂×C₄⋊Dic₁₁	—	4	1	no	—
25	#66922	—	2	4	no	—
26	#66922	—	4	2	no	—
27	C₂×C₁₁×C₁₆	—	2	2	no	—
28	C₂×C₁₁×C₁₆	—	2	2	no	—
29	C₂×C₁₁×C₁₆	Q₈	1	4	yes	complemented
30	C₄×C₈×C₁₁	C₂×C₄	1	2	yes	—
31	C₄×C₈×C₁₁	D₄	1	2	yes	—
32	#66882	—	2	4	no	—
33	#66882	—	2	2	no	—
34	#66882	—	2	4	no	—
35	#66882	—	4	2	no	—
36	#66882	C₂×C₄	1	4	yes	—
37	#13282	—	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	2	no	—
45	C₂×Dic₂₂	—	8	2	no	—
46	C₂×C₈×C₁₁	C₂²⋊C₄	1	2	yes	—
47	C₂×C₈×C₁₁	SD₁₆	1	2	yes	—
48	C₂×C₈×C₁₁	Q₁₆	1	2	yes	—
49	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈×C₁₁	C₂×Q₈	1	2	yes	—
52	C₂×C₈×C₁₁	#41	1	2	yes	—
53	C₄²×C₁₁	C₂²⋊C₄	1	2	yes	—
54	C₄²×C₁₁	SD₁₆	1	2	yes	—
55	C₄⋊Dic₁₁	—	2	4	no	—
56	C₄⋊Dic₁₁	—	4	4	no	—
57	C₄⋊Dic₁₁	—	4	2	no	—
58	C₄⋊Dic₁₁	—	4	2	no	—
59	C₄⋊Dic₁₁	—	4	2	no	—
60	C₄⋊Dic₁₁	—	8	2	no	—
61	C₄⋊Dic₁₁	—	8	2	no	—
62	C₁₁×C₁₆	—	2	4	no	—
63	C₂×#500	—	11	1	no	—
64	C₂×C₄×C₁₆	D₁₁	1	1	yes	characteristic
65	#1845	—	11	1	no	—
66	#1380	—	11	4	no	—
67	C₂³×C₁₁	C₂×D₈	1	1	yes	characteristic
68	C₂×C₄×C₁₁	#102	1	2	yes	—
69	C₂×C₄×C₁₁	#102	1	2	yes	—
70	C₂×C₄×C₁₁	D₁₆	1	2	yes	—
71	C₂×C₄×C₁₁	SD₃₂	1	2	yes	—
72	C₂×C₄×C₁₁	SD₃₂	1	2	yes	—
73	C₂×C₄×C₁₁	Q₃₂	1	2	yes	—
74	C₂×C₄×C₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₁₁	#124	1	1	yes	characteristic
76	C₂×C₄×C₁₁	C₄⋊Q₈	1	1	yes	characteristic
77	C₂×C₄×C₁₁	C₂×SD₁₆	1	1	yes	characteristic
78	C₂×C₄×C₁₁	C₂×SD₁₆	1	1	yes	characteristic
79	C₂×C₄×C₁₁	C₂×Q₁₆	1	1	yes	characteristic
80	C₂×Dic₁₁	—	4	4	no	—
81	C₂×Dic₁₁	—	8	2	no	—
82	C₂×Dic₁₁	—	8	2	no	—
83	C₂×Dic₁₁	—	8	2	no	—
84	C₂×Dic₁₁	—	8	2	no	—
85	C₂×Dic₁₁	—	8	2	no	—
86	Dic₂₂	—	8	8	no	—
87	C₈×C₁₁	#103	1	4	yes	—
88	C₈×C₁₁	C₄×Q₈	1	2	yes	—
89	C₈×C₁₁	C₂²⋊Q₈	1	2	yes	—
90	C₂×C₄⋊Q₈	—	22	1	no	—
91	C₂²×C₁₆	C₂×D₁₁	1	2	yes	—
92	#500	—	11	4	no	—
93	#500	—	22	2	no	—
94	C₂×#107	—	11	2	no	—
95	C₂×#107	—	11	2	no	—
96	C₂×#107	—	22	1	no	—
97	C₂×#107	—	22	1	no	—
98	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
99	#384	—	22	1	no	—
100	C₄×C₁₆	—	2	2	no	—
101	C₂²×C₁₁	C₂×#102	1	1	yes	characteristic
102	C₂²×C₁₁	#486	1	1	yes	characteristic
103	C₂²×C₁₁	#500	1	1	yes	characteristic
104	C₂²×C₁₁	C₂×D₁₆	1	1	yes	characteristic
105	C₂²×C₁₁	C₂×SD₃₂	1	1	yes	characteristic
106	C₂²×C₁₁	C₂×SD₃₂	1	1	yes	characteristic
107	C₂²×C₁₁	C₂×Q₃₂	1	1	yes	characteristic
108	C₄×C₁₁	#357	1	4	yes	—
109	C₄×C₁₁	#358	1	4	yes	—
110	C₄×C₁₁	#391	1	1	yes	characteristic
111	C₄×C₁₁	C₂×#103	1	1	yes	characteristic
112	C₄×C₁₁	#487	1	1	yes	characteristic
113	C₄×C₁₁	#498	1	1	yes	characteristic
114	Dic₁₁	—	8	4	no	—
115	Dic₁₁	—	16	2	no	—
116	C₂²×Q₈	—	44	1	no	—
117	C₂²×C₈	D₄₄	1	1	yes	characteristic
118	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
119	C₄⋊Q₈	—	22	4	no	—
120	C₄⋊Q₈	—	44	2	no	—
121	C₂×C₄⋊C₄	—	22	2	no	—
122	C₂×C₄⋊C₄	—	22	2	no	—
123	C₂×C₄⋊C₄	—	44	1	no	—
124	C₂×C₄⋊C₄	—	44	1	no	—
125	C₂×C₄⋊C₄	—	44	1	no	—
126	C₂×C₄²	D₄₄	1	1	yes	characteristic
127	C₂×C₁₆	—	2	2	no	—
128	C₂×C₁₆	—	2	2	no	—
129	C₂×C₁₆	Dic₂₂	1	4	yes	complemented
130	#107	—	11	4	no	—
131	#107	—	22	4	no	—
132	#107	—	22	2	no	—
133	#107	—	22	4	no	—
134	#107	—	44	2	no	—
135	C₄×C₈	C₄×D₁₁	1	2	yes	—
136	C₄×C₈	C₂²⋊D₁₁	1	2	yes	—
137	C₂×C₁₁	#1990	1	1	yes	characteristic
138	C₂×C₁₁	C₂×#357	1	1	yes	characteristic
139	C₂×C₁₁	C₂×#358	1	1	yes	characteristic
140	C₂×C₁₁	#2240	1	1	yes	characteristic
141	C₂×C₁₁	#2241	1	1	yes	characteristic
142	C₂×C₁₁	#2252	1	1	yes	characteristic
143	C₂×C₁₁	#2254	1	1	yes	characteristic
144	C₂×Q₈	—	44	4	no	—
145	C₂×Q₈	—	88	2	no	—
146	C₂×Q₈	—	88	2	no	—
147	C₂²×C₄	—	44	2	no	—
148	C₂²×C₄	—	44	2	no	—
149	C₂²×C₄	D₈₈	1	1	yes	characteristic
150	C₂²×C₄	D₈₈	1	1	yes	characteristic
151	C₂²×C₄	C₂×D₄₄	1	1	yes	characteristic
152	C₂×C₈	#4314	1	2	yes	—
153	C₂×C₈	Dic₄₄	1	2	yes	—
154	C₂×C₈	(C₂×D₁₁)⋊C₄	1	2	yes	—
155	C₂×C₈	C₂×Dic₂₂	1	2	yes	—
156	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
157	C₂×C₈	#4339	1	2	yes	—
158	C₂×C₈	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
159	C₄⋊C₄	—	22	4	no	—
160	C₄⋊C₄	—	44	4	no	—
161	C₄⋊C₄	—	44	2	no	—
162	C₄⋊C₄	—	44	2	no	—
163	C₄⋊C₄	—	44	2	no	—
164	C₄⋊C₄	—	88	2	no	—
165	C₄⋊C₄	—	88	2	no	—
166	C₄²	#4314	1	2	yes	—
167	C₄²	(C₂×D₁₁)⋊C₄	1	2	yes	—
168	C₁₆	—	2	4	no	—
169	C₁₁	#13282	1	1	yes	characteristic, complemented, Sylow
170	C₂³	C₂×D₈₈	1	1	yes	characteristic
171	Q₈	—	88	8	no	—
172	C₂×C₄	—	44	4	no	—
173	C₂×C₄	—	88	2	no	—
174	C₂×C₄	—	88	2	no	—
175	C₂×C₄	—	88	2	no	—
176	C₂×C₄	—	88	2	no	—
177	C₂×C₄	—	88	2	no	—
178	C₂×C₄	D₁₇₆	1	2	yes	—
179	C₂×C₄	#66864	1	2	yes	—
180	C₂×C₄	#66864	1	2	yes	—
181	C₂×C₄	Dic₈₈	1	2	yes	—
182	C₂×C₄	#66885	1	2	yes	—
183	C₂×C₄	#66885	1	2	yes	—
184	C₂×C₄	#66922	1	1	yes	characteristic
185	C₂×C₄	#66928	1	1	yes	characteristic
186	C₂×C₄	C₂×#4314	1	1	yes	characteristic
187	C₂×C₄	C₂×#4314	1	1	yes	characteristic
188	C₂×C₄	C₂×Dic₄₄	1	1	yes	characteristic
189	C₂×C₄	C₂×(C₂×D₁₁)⋊C₄	1	1	yes	characteristic
190	C₈	#66880	1	4	yes	—
191	C₈	C₄×Dic₂₂	1	2	yes	—
192	C₈	#66977	1	2	yes	—
193	C₂²	#10622295	1	1	yes	characteristic
194	C₂²	#10622307	1	1	yes	characteristic
195	C₂²	C₂×D₁₇₆	1	1	yes	characteristic
196	C₂²	C₂×#66864	1	1	yes	characteristic
197	C₂²	C₂×#66864	1	1	yes	characteristic
198	C₂²	C₂×Dic₈₈	1	1	yes	characteristic
199	C₂²	C₂×#66885	1	1	yes	characteristic
200	C₄	—	88	4	no	—
201	C₄	—	176	2	no	—
202	C₄	#10622141	1	4	yes	—
203	C₄	#10622147	1	4	yes	—
204	C₄	#10622293	1	1	yes	characteristic
205	C₄	#10622294	1	1	yes	characteristic
206	C₄	#10622542	1	1	yes	characteristic
207	C₄	C₂×#66880	1	1	yes	characteristic
208	C₂	#13069577	1	1	yes	characteristic
209	C₂	C₂×#10622147	1	1	yes	characteristic
210	C₂	C₂×#10622141	1	1	yes	characteristic
211	C₂	#13070840	1	1	yes	characteristic
212	C₂	#13070844	1	1	yes	characteristic
213	C₂	#13070891	1	1	yes	characteristic
214	C₂	#13070898	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₈₈
commutator	C₂×C₈×C₁₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₁
Fitting	C₂×C₄×C₁₁×C₁₆	C₂
socle, prime radical	C₂³×C₁₁	C₂×D₈
cosocle	C₂²×C₈×C₁₁	C₂³
hypercenter	C₂×C₄×C₁₆	D₁₁
hypocenter	C₁₁	#13282
LCS	G ▷ C₂×C₈×C₁₁ ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
14. #10622295	C₄	semidirect
29. C₂×C₁₁×C₁₆	Q₈	semidirect
129. C₂×C₁₆	Dic₂₂	semidirect
169. C₁₁	#13282	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
#13068776	2
#13066708	2
C₂×C₄×C₁₁×C₁₆	2
#13282	11