Subgroups — #14909290

Order 2816.

474 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13079155	C₂	1	2	yes	maximal
2	#13070902	C₂	1	4	yes	maximal, complemented
3	#13070898	C₂	1	2	yes	maximal, complemented
4	#13070895	C₂	1	2	yes	maximal, complemented
5	#13070404	C₂	1	2	yes	maximal
6	#13070400	C₂	1	1	yes	maximal, characteristic
7	C₂×#10622143	C₂	1	1	yes	maximal, characteristic
8	C₁₁×#2108	C₂	1	1	yes	maximal, characteristic
9	#10623050	—	2	2	no	—
10	C₂×C₁₁×M₃₂	C₂²	1	1	yes	characteristic
11	C₂²×C₁₁×C₁₆	C₂²	1	1	yes	characteristic
12	C₁₁×#405	C₂²	1	1	yes	characteristic
13	#10622734	—	2	2	no	—
14	#10622719	C₂²	1	2	yes	—
15	C₂×#66882	C₂²	1	2	yes	—
16	#10622315	—	2	4	no	—
17	#10622296	C₂²	1	4	yes	—
18	#10622295	C₂²	1	4	yes	—
19	C₁₁×#346	C₂²	1	2	yes	—
20	C₄×C₁₁×C₁₆	C₂²	1	2	yes	—
21	#10622143	C₂²	1	1	yes	characteristic
22	#10622143	C₂²	1	2	yes	—
23	#10622143	C₂²	1	1	yes	characteristic
24	#10622143	C₂²	1	4	yes	—
25	#10622143	C₂²	1	2	yes	—
26	#10622142	C₂²	1	4	yes	—
27	#10622142	C₂²	1	2	yes	—
28	C₂×C₁₁×M₁₆	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
30	C₁₁×#117	D₄	1	1	yes	characteristic
31	#66979	—	2	2	no	—
32	C₂×C₄⋊Dic₁₁	—	2	2	no	—
33	#66943	—	4	2	no	—
34	#66941	—	4	2	no	—
35	#66931	—	4	4	no	—
36	#66923	—	2	4	no	—
37	#66922	—	2	4	no	—
38	C₁₁×M₃₂	Q₈	1	4	yes	complemented
39	C₂×C₁₁×C₁₆	Q₈	1	4	yes	complemented
40	C₂×C₁₁×C₁₆	C₂³	1	1	yes	characteristic
41	C₂×C₁₁×C₁₆	C₂³	1	1	yes	characteristic
42	C₂×C₁₁×C₁₆	C₂³	1	2	yes	—
43	C₁₁×#97	D₄	1	2	yes	—
44	C₄×C₈×C₁₁	C₂³	1	2	yes	—
45	#66882	—	2	4	no	—
46	#66882	—	2	2	no	—
47	#66882	C₂³	1	2	yes	—
48	#66882	C₂³	1	4	yes	—
49	#66882	C₂³	1	2	yes	—
50	#66881	—	2	4	no	—
51	#66881	—	2	2	no	—
52	#33730	—	11	1	no	maximal, Sylow
53	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
54	C₂²×Dic₁₁	—	4	2	no	—
55	C₂×Dic₂₂	—	4	4	no	—
56	C₁₁×M₁₆	Q₁₆	1	4	yes	—
57	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈×C₁₁	C₂×D₄	1	2	yes	—
59	C₂×C₈×C₁₁	C₂×Q₈	1	2	yes	—
60	C₂×C₈×C₁₁	C₂×Q₈	1	2	yes	—
61	C₂×C₈×C₁₁	C₂⁴	1	1	yes	characteristic
62	C₁₁×C₄⋊C₄	D₈	1	2	yes	—
63	C₁₁×C₂²⋊C₄	D₈	1	2	yes	—
64	C₄²×C₁₁	C₂×D₄	1	2	yes	—
65	C₂²⋊Dic₁₁	—	4	2	no	—
66	C₄⋊Dic₁₁	—	2	2	no	—
67	C₄⋊Dic₁₁	—	2	2	no	—
68	C₄⋊Dic₁₁	—	2	4	no	—
69	C₄⋊Dic₁₁	—	4	4	no	—
70	C₄⋊Dic₁₁	—	4	2	no	—
71	Dic₁₁⋊C₄	—	4	4	no	—
72	Dic₁₁⋊C₄	—	4	4	no	—
73	C₄×Dic₁₁	—	4	2	no	—
74	C₁₁×C₁₆	C₂×Q₈	1	2	yes	—
75	C₁₁×C₁₆	C₂×Q₈	1	2	yes	—
76	C₁₁×C₁₆	C₂×Q₈	1	4	yes	—
77	#3164	—	11	2	no	—
78	#2255	—	11	4	no	—
79	#2254	—	11	2	no	—
80	#2253	—	11	2	no	—
81	#2159	—	11	2	no	—
82	#2158	—	11	1	no	—
83	C₂×#367	—	11	1	no	—
84	#2108	D₁₁	1	1	yes	characteristic
85	C₂³×C₁₁	C₂×D₈	1	1	yes	characteristic
86	C₂×C₄×C₁₁	C₄⋊Q₈	1	2	yes	—
87	C₂×C₄×C₁₁	C₂×D₈	1	4	yes	—
88	C₂×C₄×C₁₁	C₂×D₈	1	1	yes	characteristic
89	C₂×C₄×C₁₁	C₂×Q₁₆	1	2	yes	—
90	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
91	C₂×Dic₁₁	—	4	2	no	—
92	C₂×Dic₁₁	—	4	2	no	—
93	C₂×Dic₁₁	—	4	4	no	—
94	C₂×Dic₁₁	—	16	2	no	—
95	Dic₂₂	—	16	4	no	—
96	C₈×C₁₁	C₄⋊Q₈	1	2	yes	—
97	C₈×C₁₁	C₂×Q₁₆	1	4	yes	—
98	C₈×C₁₁	C₂²×Q₈	1	1	yes	characteristic
99	C₈×C₁₁	C₂²×Q₈	1	1	yes	characteristic
100	#544	—	22	2	no	—
101	C₂×M₃₂	C₂×D₁₁	1	1	yes	characteristic
102	C₂²×C₁₆	C₂×D₁₁	1	1	yes	characteristic
103	#501	—	22	4	no	—
104	#500	—	11	4	no	—
105	#499	—	11	4	no	—
106	#428	—	22	2	no	—
107	#427	—	11	2	no	—
108	C₂×#107	—	11	2	no	—
109	#405	C₂×D₁₁	1	1	yes	characteristic
110	#367	—	11	2	no	—
111	#367	—	11	2	no	—
112	#367	—	11	1	no	—
113	#367	—	11	1	no	—
114	#367	—	11	4	no	—
115	#366	—	11	2	no	—
116	#366	—	11	4	no	—
117	#346	C₂×D₁₁	1	2	yes	—
118	C₄×C₁₆	C₂×D₁₁	1	2	yes	—
119	C₂²×C₁₁	—	2	2	no	—
120	C₂²×C₁₁	#500	1	2	yes	—
121	C₂²×C₁₁	C₂²×D₈	1	1	yes	characteristic
122	C₄×C₁₁	—	2	4	no	—
123	C₄×C₁₁	#500	1	2	yes	—
124	C₄×C₁₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
125	C₄×C₁₁	C₂²×Q₁₆	1	1	yes	characteristic
126	Dic₁₁	—	8	2	no	—
127	Dic₁₁	—	8	4	no	—
128	Dic₁₁	—	8	2	no	—
129	C₂×M₁₆	D₄₄	1	1	yes	characteristic
130	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
131	C₄⋊Q₈	—	22	4	no	—
132	C₄⋊Q₈	—	44	2	no	—
133	#125	—	22	4	no	—
134	#125	—	44	2	no	—
135	C₂²⋊Q₈	—	44	4	no	—
136	#117	—	22	2	no	—
137	#117	D₄₄	1	1	yes	characteristic
138	C₂×C₄⋊C₄	—	22	2	no	—
139	M₃₂	Dic₂₂	1	4	yes	complemented
140	C₂×C₁₆	Dic₂₂	1	4	yes	complemented
141	C₂×C₁₆	C₂²×D₁₁	1	1	yes	characteristic
142	C₂×C₁₆	C₂²×D₁₁	1	2	yes	—
143	C₂×C₁₆	C₂²×D₁₁	1	1	yes	characteristic
144	#107	—	11	2	no	—
145	#107	—	11	4	no	—
146	#107	—	11	2	no	—
147	#107	—	22	2	no	—
148	#107	—	22	4	no	—
149	#106	—	22	2	no	—
150	#106	—	22	4	no	—
151	#97	D₄₄	1	2	yes	—
152	C₄×C₈	C₂²×D₁₁	1	2	yes	—
153	C₂×C₁₁	—	2	2	no	—
154	C₂×C₁₁	C₂×#500	1	1	yes	characteristic
155	C₂×C₁₁	#3416	1	2	yes	—
156	C₂×Q₈	—	44	4	no	—
157	C₂²×C₄	—	44	2	no	—
158	C₂²×C₄	C₂×D₄₄	1	1	yes	characteristic
159	M₁₆	Dic₄₄	1	4	yes	—
160	C₂×C₈	C₂×Dic₂₂	1	2	yes	—
161	C₂×C₈	C₂×Dic₂₂	1	2	yes	—
162	C₂×C₈	C₂×D₄₄	1	1	yes	characteristic
163	C₂×C₈	C₂×D₄₄	1	2	yes	—
164	C₂×C₈	C₂³×D₁₁	1	1	yes	characteristic
165	C₄⋊C₄	—	22	2	no	—
166	C₄⋊C₄	—	22	4	no	—
167	C₄⋊C₄	—	22	2	no	—
168	C₄⋊C₄	—	44	2	no	—
169	C₄⋊C₄	—	44	4	no	—
170	C₄⋊C₄	—	44	4	no	—
171	C₄⋊C₄	—	44	4	no	—
172	C₄⋊C₄	D₈₈	1	2	yes	—
173	C₂²⋊C₄	—	44	2	no	—
174	C₂²⋊C₄	D₈₈	1	2	yes	—
175	C₄²	—	44	2	no	—
176	C₄²	C₂×D₄₄	1	2	yes	—
177	C₁₆	C₂×Dic₂₂	1	2	yes	—
178	C₁₆	C₂×Dic₂₂	1	2	yes	—
179	C₁₆	C₂×Dic₂₂	1	4	yes	—
180	C₁₁	#33730	1	1	yes	characteristic, complemented, Sylow
181	C₂³	C₂×D₈₈	1	1	yes	characteristic
182	Q₈	—	176	4	no	—
183	C₂×C₄	—	44	2	no	—
184	C₂×C₄	—	44	2	no	—
185	C₂×C₄	—	44	4	no	—
186	C₂×C₄	—	176	2	no	—
187	C₂×C₄	#66922	1	2	yes	—
188	C₂×C₄	C₂×D₈₈	1	4	yes	—
189	C₂×C₄	C₂×D₈₈	1	1	yes	characteristic
190	C₂×C₄	C₂×Dic₄₄	1	2	yes	—
191	C₂×C₄	C₂²×D₄₄	1	1	yes	characteristic
192	C₈	#66922	1	2	yes	—
193	C₈	C₂×Dic₄₄	1	4	yes	—
194	C₈	C₂²×Dic₂₂	1	1	yes	characteristic
195	C₈	C₂²×Dic₂₂	1	1	yes	characteristic
196	C₂²	—	2	2	no	—
197	C₂²	#10622295	1	2	yes	—
198	C₂²	C₂²×D₈₈	1	1	yes	characteristic
199	C₄	—	2	4	no	—
200	C₄	—	88	4	no	—
201	C₄	—	88	2	no	—
202	C₄	—	88	2	no	—
203	C₄	#10622295	1	2	yes	—
204	C₄	C₂×#66922	1	1	yes	characteristic
205	C₄	C₂²×Dic₄₄	1	1	yes	characteristic
206	C₂	—	2	2	no	—
207	C₂	C₂×#10622295	1	1	yes	characteristic
208	C₂	#13081764	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×D₈₈
commutator, cosocle	C₂×C₈×C₁₁	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₁
Fitting	C₁₁×#2108	C₂
socle	C₂²×C₁₁	C₂²×D₈
prime radical	C₂³×C₁₁	C₂×D₈
hypercenter	#2108	D₁₁
hypocenter	C₁₁	#33730
LCS	G ▷ C₂×C₈×C₁₁ ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #2108

Complements

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

N	H ≅ G/N	type
2. #13070902	C₂	semidirect
3. #13070898	C₂	semidirect
4. #13070895	C₂	semidirect
38. C₁₁×M₃₂	Q₈	semidirect
39. C₂×C₁₁×C₁₆	Q₈	semidirect
139. M₃₂	Dic₂₂	semidirect
140. C₂×C₁₆	Dic₂₂	semidirect
180. C₁₁	#33730	semidirect

Maximal subgroups

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

subgroup	index
#13079155	2
#13070902	2
#13070898	2
#13070895	2
#13070404	2
#13070400	2
C₂×#10622143	2
C₁₁×#2108	2
#33730	11