Subgroups — #14909282

Order 2816.

474 conjugacy classes of subgroups.

The subgroup lattice has 220 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	#13070895	C₂	1	2	yes	maximal, complemented
4	#13070891	C₂	1	2	yes	maximal, complemented
5	#13070404	C₂	1	2	yes	maximal
6	#13070400	C₂	1	1	yes	maximal, characteristic
7	C₂×#10622142	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	4	yes	—
22	#10622143	C₂²	1	2	yes	—
23	#10622142	C₂²	1	1	yes	characteristic
24	#10622142	C₂²	1	2	yes	—
25	#10622142	C₂²	1	4	yes	—
26	#10622142	C₂²	1	1	yes	characteristic
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	#33729	—	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	1	yes	characteristic
63	C₁₁×C₄⋊C₄	D₈	1	1	yes	characteristic
64	C₁₁×C₂²⋊C₄	D₈	1	2	yes	—
65	C₄²×C₁₁	C₂×D₄	1	2	yes	—
66	C₂²⋊Dic₁₁	—	4	2	no	—
67	C₄⋊Dic₁₁	—	2	2	no	—
68	C₄⋊Dic₁₁	—	2	2	no	—
69	C₄⋊Dic₁₁	—	2	4	no	—
70	C₄⋊Dic₁₁	—	4	4	no	—
71	C₄⋊Dic₁₁	—	4	2	no	—
72	Dic₁₁⋊C₄	—	4	4	no	—
73	Dic₁₁⋊C₄	—	4	4	no	—
74	C₄×Dic₁₁	—	4	2	no	—
75	C₁₁×C₁₆	C₂×Q₈	1	2	yes	—
76	C₁₁×C₁₆	C₂×Q₈	1	2	yes	—
77	C₁₁×C₁₆	C₂×Q₈	1	4	yes	—
78	#3164	—	11	2	no	—
79	#2255	—	11	4	no	—
80	#2253	—	11	2	no	—
81	#2252	—	11	2	no	—
82	#2159	—	11	2	no	—
83	#2158	—	11	1	no	—
84	C₂×#366	—	11	1	no	—
85	#2108	D₁₁	1	1	yes	characteristic
86	C₂³×C₁₁	C₂×D₈	1	1	yes	characteristic
87	C₂×C₄×C₁₁	C₄⋊Q₈	1	2	yes	—
88	C₂×C₄×C₁₁	C₂×D₈	1	2	yes	—
89	C₂×C₄×C₁₁	C₂×D₈	1	2	yes	—
90	C₂×C₄×C₁₁	C₂×D₈	1	1	yes	characteristic
91	C₂×C₄×C₁₁	C₂×Q₁₆	1	2	yes	—
92	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
93	C₂×Dic₁₁	—	4	2	no	—
94	C₂×Dic₁₁	—	4	2	no	—
95	C₂×Dic₁₁	—	4	4	no	—
96	C₂×Dic₁₁	—	16	2	no	—
97	Dic₂₂	—	16	4	no	—
98	C₈×C₁₁	C₄⋊Q₈	1	2	yes	—
99	C₈×C₁₁	C₂×Q₁₆	1	4	yes	—
100	C₈×C₁₁	C₂²×Q₈	1	1	yes	characteristic
101	C₈×C₁₁	C₂²×Q₈	1	1	yes	characteristic
102	#544	—	22	2	no	—
103	C₂×M₃₂	C₂×D₁₁	1	1	yes	characteristic
104	C₂²×C₁₆	C₂×D₁₁	1	1	yes	characteristic
105	#501	—	22	4	no	—
106	#500	—	11	4	no	—
107	#499	—	11	4	no	—
108	#428	—	22	2	no	—
109	#427	—	11	2	no	—
110	C₂×#107	—	11	2	no	—
111	#405	C₂×D₁₁	1	1	yes	characteristic
112	#367	—	11	2	no	—
113	#367	—	11	4	no	—
114	#366	—	11	2	no	—
115	#366	—	11	2	no	—
116	#366	—	11	1	no	—
117	#366	—	11	1	no	—
118	#366	—	11	4	no	—
119	#346	C₂×D₁₁	1	2	yes	—
120	C₄×C₁₆	C₂×D₁₁	1	2	yes	—
121	C₂²×C₁₁	—	2	1	no	—
122	C₂²×C₁₁	—	2	1	no	—
123	C₂²×C₁₁	#500	1	2	yes	—
124	C₂²×C₁₁	C₂²×D₈	1	1	yes	characteristic
125	C₄×C₁₁	—	2	2	no	—
126	C₄×C₁₁	—	2	2	no	—
127	C₄×C₁₁	#500	1	2	yes	—
128	C₄×C₁₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
129	C₄×C₁₁	C₂²×Q₁₆	1	1	yes	characteristic
130	Dic₁₁	—	8	2	no	—
131	Dic₁₁	—	8	4	no	—
132	Dic₁₁	—	8	2	no	—
133	C₂×M₁₆	D₄₄	1	1	yes	characteristic
134	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
135	C₄⋊Q₈	—	22	4	no	—
136	C₄⋊Q₈	—	44	2	no	—
137	#125	—	22	4	no	—
138	#125	—	44	2	no	—
139	C₂²⋊Q₈	—	44	4	no	—
140	#117	—	22	2	no	—
141	#117	D₄₄	1	1	yes	characteristic
142	C₂×C₄⋊C₄	—	22	2	no	—
143	M₃₂	Dic₂₂	1	4	yes	complemented
144	C₂×C₁₆	Dic₂₂	1	4	yes	complemented
145	C₂×C₁₆	C₂²×D₁₁	1	1	yes	characteristic
146	C₂×C₁₆	C₂²×D₁₁	1	2	yes	—
147	C₂×C₁₆	C₂²×D₁₁	1	1	yes	characteristic
148	#107	—	11	4	no	—
149	#107	—	11	2	no	—
150	#107	—	11	2	no	—
151	#107	—	22	2	no	—
152	#107	—	22	4	no	—
153	#106	—	22	2	no	—
154	#106	—	22	4	no	—
155	#97	D₄₄	1	2	yes	—
156	C₄×C₈	C₂²×D₁₁	1	2	yes	—
157	C₂×C₁₁	—	2	2	no	—
158	C₂×C₁₁	C₂×#500	1	1	yes	characteristic
159	C₂×C₁₁	#3415	1	1	yes	characteristic
160	C₂×C₁₁	#3417	1	1	yes	characteristic
161	C₂×Q₈	—	44	4	no	—
162	C₂²×C₄	—	44	2	no	—
163	C₂²×C₄	C₂×D₄₄	1	1	yes	characteristic
164	M₁₆	Dic₄₄	1	4	yes	—
165	C₂×C₈	C₂×Dic₂₂	1	2	yes	—
166	C₂×C₈	C₂×Dic₂₂	1	2	yes	—
167	C₂×C₈	C₂×D₄₄	1	1	yes	characteristic
168	C₂×C₈	C₂×D₄₄	1	2	yes	—
169	C₂×C₈	C₂³×D₁₁	1	1	yes	characteristic
170	C₄⋊C₄	—	22	2	no	—
171	C₄⋊C₄	—	22	4	no	—
172	C₄⋊C₄	—	22	2	no	—
173	C₄⋊C₄	—	44	2	no	—
174	C₄⋊C₄	—	44	4	no	—
175	C₄⋊C₄	—	44	4	no	—
176	C₄⋊C₄	—	44	4	no	—
177	C₄⋊C₄	D₈₈	1	1	yes	characteristic
178	C₄⋊C₄	D₈₈	1	1	yes	characteristic
179	C₂²⋊C₄	—	44	2	no	—
180	C₂²⋊C₄	D₈₈	1	2	yes	—
181	C₄²	—	44	2	no	—
182	C₄²	C₂×D₄₄	1	2	yes	—
183	C₁₆	C₂×Dic₂₂	1	2	yes	—
184	C₁₆	C₂×Dic₂₂	1	2	yes	—
185	C₁₆	C₂×Dic₂₂	1	4	yes	—
186	C₁₁	#33729	1	1	yes	characteristic, complemented, Sylow
187	C₂³	C₂×D₈₈	1	1	yes	characteristic
188	Q₈	—	176	4	no	—
189	C₂×C₄	—	44	2	no	—
190	C₂×C₄	—	44	2	no	—
191	C₂×C₄	—	44	4	no	—
192	C₂×C₄	—	176	2	no	—
193	C₂×C₄	#66922	1	2	yes	—
194	C₂×C₄	C₂×D₈₈	1	2	yes	—
195	C₂×C₄	C₂×D₈₈	1	2	yes	—
196	C₂×C₄	C₂×D₈₈	1	1	yes	characteristic
197	C₂×C₄	C₂×Dic₄₄	1	2	yes	—
198	C₂×C₄	C₂²×D₄₄	1	1	yes	characteristic
199	C₈	#66922	1	2	yes	—
200	C₈	C₂×Dic₄₄	1	4	yes	—
201	C₈	C₂²×Dic₂₂	1	1	yes	characteristic
202	C₈	C₂²×Dic₂₂	1	1	yes	characteristic
203	C₂²	—	2	1	no	—
204	C₂²	—	2	1	no	—
205	C₂²	#10622295	1	2	yes	—
206	C₂²	C₂²×D₈₈	1	1	yes	characteristic
207	C₄	—	2	2	no	—
208	C₄	—	2	2	no	—
209	C₄	—	88	4	no	—
210	C₄	—	88	2	no	—
211	C₄	—	88	2	no	—
212	C₄	#10622295	1	2	yes	—
213	C₄	C₂×#66922	1	1	yes	characteristic
214	C₄	C₂²×Dic₄₄	1	1	yes	characteristic
215	C₂	—	2	2	no	—
216	C₂	C₂×#10622295	1	1	yes	characteristic
217	C₂	#13081757	1	1	yes	characteristic
218	C₂	#13081769	1	1	yes	characteristic

	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	#33729
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. #13070895	C₂	semidirect
4. #13070891	C₂	semidirect
38. C₁₁×M₃₂	Q₈	semidirect
39. C₂×C₁₁×C₁₆	Q₈	semidirect
143. M₃₂	Dic₂₂	semidirect
144. C₂×C₁₆	Dic₂₂	semidirect
186. C₁₁	#33729	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
#13070895	2
#13070891	2
#13070404	2
#13070400	2
C₂×#10622142	2
C₁₁×#2108	2
#33729	11