Subgroups — #14909131

Order 2816.

578 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13079130	C₂	1	1	yes	maximal, characteristic
2	#13070861	C₂	1	4	yes	maximal, complemented
3	#13070843	C₂	1	4	yes	maximal, complemented
4	#13070336	C₂	1	4	yes	maximal
5	C₄×#66876	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₁×#494	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₁₁×Q₁₆	—	2	1	no	—
8	C₂²×C₁₁×Q₁₆	C₂²	1	2	yes	—
9	C₂×C₁₁×C₄⋊Q₈	—	2	1	no	—
10	#10623001	—	2	1	no	—
11	C₁₁×#494	—	2	1	no	—
12	C₁₁×#494	—	2	1	no	—
13	C₁₁×#494	C₂²	1	2	yes	—
14	C₁₁×#494	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
16	#10622719	—	2	1	no	—
17	#10622719	C₂²	1	2	yes	—
18	C₂×#66876	C₂²	1	2	yes	—
19	#10622296	—	2	1	no	—
20	#10622296	C₂²	1	2	yes	—
21	#10622295	C₂²	1	2	yes	—
22	#10622293	—	2	1	no	—
23	#10622202	C₂²	1	16	yes	—
24	#10622099	C₂²	1	2	yes	—
25	C₄×C₁₁⋊C₁₆	C₂²	1	2	yes	—
26	C₂²×Q₈×C₁₁	—	2	2	no	—
27	C₂×C₁₁×Q₁₆	—	2	2	no	—
28	C₂×C₁₁×Q₁₆	—	2	2	no	—
29	C₂×C₁₁×Q₁₆	—	2	2	no	—
30	C₂×C₁₁×Q₁₆	—	2	8	no	—
31	C₂×C₁₁×Q₁₆	—	4	4	no	—
32	C₂×C₁₁×Q₁₆	C₂³	1	4	yes	—
33	C₂²×C₈×C₁₁	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
35	C₁₁×C₄⋊Q₈	—	2	2	no	—
36	C₁₁×C₄⋊Q₈	—	2	2	no	—
37	C₁₁×C₄⋊Q₈	—	4	2	no	—
38	C₂×C₁₁×C₄⋊C₄	—	4	2	no	—
39	C₂×C₄²×C₁₁	D₄	1	1	yes	characteristic
40	#66979	—	2	2	no	—
41	#66977	—	4	2	no	—
42	#66923	—	2	2	no	—
43	#66922	—	2	2	no	—
44	C₄×Dic₂₂	—	4	2	no	—
45	C₄×C₈×C₁₁	D₄	1	1	yes	characteristic
46	C₄×C₈×C₁₁	D₄	1	1	yes	characteristic
47	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
48	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
49	#66882	—	2	2	no	—
50	#66882	C₂³	1	4	yes	—
51	#66881	—	2	2	no	—
52	#66881	—	2	2	no	—
53	#66876	—	2	4	no	—
54	C₂×C₁₁⋊C₁₆	C₂³	1	4	yes	—
55	#33705	—	11	1	no	maximal, Sylow
56	C₂×Q₈×C₁₁	—	2	4	no	—
57	C₂×Q₈×C₁₁	—	4	2	no	—
58	C₂×Q₈×C₁₁	—	4	4	no	—
59	C₂×Q₈×C₁₁	—	4	4	no	—
60	C₂²×C₄×C₁₁	—	4	2	no	—
61	C₂²×C₄×C₁₁	D₈	1	2	yes	—
62	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
63	C₂×Dic₂₂	—	4	2	no	—
64	C₁₁×Q₁₆	—	2	8	no	—
65	C₁₁×Q₁₆	—	4	4	no	—
66	C₁₁×Q₁₆	—	4	4	no	—
67	C₁₁×Q₁₆	—	4	4	no	—
68	C₂×C₈×C₁₁	SD₁₆	1	4	yes	—
69	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
70	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
72	C₂×C₈×C₁₁	#41	1	4	yes	—
73	C₂×C₈×C₁₁	C₂⁴	1	1	yes	characteristic
74	C₁₁×C₄⋊C₄	—	4	4	no	—
75	C₁₁×C₄⋊C₄	—	4	4	no	—
76	C₄²×C₁₁	D₈	1	2	yes	—
77	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
78	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
79	C₂²⋊Dic₁₁	—	4	2	no	—
80	C₄⋊Dic₁₁	—	2	4	no	—
81	C₄⋊Dic₁₁	—	4	2	no	—
82	Dic₁₁⋊C₄	—	4	4	no	—
83	C₄×Dic₁₁	—	4	2	no	—
84	C₁₁⋊C₁₆	—	2	4	no	—
85	#3160	—	11	1	no	—
86	C₂×#494	D₁₁	1	1	yes	characteristic
87	#2245	—	11	4	no	—
88	#2241	—	11	4	no	—
89	#2141	—	11	4	no	—
90	C₄×M₃₂	—	11	1	no	—
91	C₂³×C₁₁	C₂×D₈	1	1	yes	characteristic
92	Q₈×C₁₁	—	4	4	no	—
93	Q₈×C₁₁	—	4	4	no	—
94	Q₈×C₁₁	—	8	4	no	—
95	C₂×C₄×C₁₁	—	2	4	no	—
96	C₂×C₄×C₁₁	—	4	4	no	—
97	C₂×C₄×C₁₁	—	8	4	no	—
98	C₂×C₄×C₁₁	#124	1	2	yes	—
99	C₂×C₄×C₁₁	C₂×D₈	1	2	yes	—
100	C₂×C₄×C₁₁	C₂×D₈	1	1	yes	characteristic
101	C₂×C₄×C₁₁	C₂×D₈	1	2	yes	—
102	C₂×C₄×C₁₁	C₂×SD₁₆	1	2	yes	—
103	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
104	C₂×Dic₁₁	—	4	4	no	—
105	Dic₂₂	—	8	4	no	—
106	C₈×C₁₁	#124	1	2	yes	—
107	C₈×C₁₁	C₂×SD₁₆	1	2	yes	—
108	C₈×C₁₁	C₂×SD₁₆	1	2	yes	—
109	C₈×C₁₁	C₂×#41	1	2	yes	—
110	C₂²×Q₁₆	—	2	1	no	—
111	C₂²×Q₁₆	C₂×D₁₁	1	2	yes	—
112	#533	—	22	1	no	—
113	C₂×C₄⋊Q₈	—	2	1	no	—
114	C₂×M₃₂	—	11	2	no	—
115	#500	—	11	2	no	—
116	#499	—	11	2	no	—
117	#499	—	22	1	no	—
118	#498	—	22	1	no	—
119	#494	—	2	1	no	—
120	#494	—	2	1	no	—
121	#494	C₂×D₁₁	1	2	yes	—
122	#494	C₂×D₁₁	1	2	yes	—
123	#427	—	11	2	no	—
124	#427	—	22	1	no	—
125	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
126	#358	—	11	16	no	—
127	#346	—	11	2	no	—
128	C₄×C₁₆	—	11	2	no	—
129	C₂²×C₁₁	—	2	2	no	—
130	C₂²×C₁₁	#486	1	2	yes	—
131	C₂²×C₁₁	C₂²×D₈	1	1	yes	characteristic
132	C₄×C₁₁	—	2	2	no	—
133	C₄×C₁₁	—	8	4	no	—
134	C₄×C₁₁	#486	1	2	yes	—
135	C₄×C₁₁	#510	1	4	yes	—
136	C₄×C₁₁	C₂×#124	1	1	yes	characteristic
137	C₄×C₁₁	C₂²×SD₁₆	1	1	yes	characteristic
138	Dic₁₁	—	8	4	no	—
139	C₂²×Q₈	—	2	2	no	—
140	C₂×Q₁₆	—	2	8	no	—
141	C₂×Q₁₆	—	2	2	no	—
142	C₂×Q₁₆	—	2	2	no	—
143	C₂×Q₁₆	—	2	2	no	—
144	C₂×Q₁₆	—	4	4	no	—
145	C₂×Q₁₆	C₂²×D₁₁	1	4	yes	—
146	C₂²×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
147	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
148	C₄⋊Q₈	—	2	2	no	—
149	C₄⋊Q₈	—	2	2	no	—
150	C₄⋊Q₈	—	4	2	no	—
151	C₄⋊Q₈	—	22	2	no	—
152	#125	—	22	2	no	—
153	C₂²⋊Q₈	—	44	2	no	—
154	C₄×Q₈	—	44	2	no	—
155	#117	—	22	2	no	—
156	C₂×C₄⋊C₄	—	4	2	no	—
157	C₂×C₄²	C₂²⋊D₁₁	1	1	yes	characteristic
158	M₃₂	—	22	4	no	—
159	C₂×C₁₆	—	11	4	no	—
160	#107	—	11	4	no	—
161	#107	—	22	2	no	—
162	#106	—	22	2	no	—
163	#106	—	22	2	no	—
164	C₄×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
165	C₄×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
166	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
167	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
168	C₂×C₁₁	—	2	2	no	—
169	C₂×C₁₁	C₂×#486	1	1	yes	characteristic
170	C₂×C₁₁	C₂×#510	1	2	yes	—
171	C₂×Q₈	—	2	4	no	—
172	C₂×Q₈	—	4	2	no	—
173	C₂×Q₈	—	4	4	no	—
174	C₂×Q₈	—	4	4	no	—
175	C₂×Q₈	—	44	2	no	—
176	C₂²×C₄	—	4	2	no	—
177	C₂²×C₄	#4323	1	2	yes	—
178	C₂²×C₄	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
179	Q₁₆	—	2	8	no	—
180	Q₁₆	—	4	4	no	—
181	Q₁₆	—	4	4	no	—
182	Q₁₆	—	4	4	no	—
183	C₂×C₈	#4324	1	4	yes	—
184	C₂×C₈	#4341	1	4	yes	—
185	C₂×C₈	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
186	C₂×C₈	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
187	C₂×C₈	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
188	C₂×C₈	C₂³×D₁₁	1	1	yes	characteristic
189	C₄⋊C₄	—	4	4	no	—
190	C₄⋊C₄	—	4	4	no	—
191	C₄⋊C₄	—	22	4	no	—
192	C₄⋊C₄	—	44	4	no	—
193	C₄⋊C₄	—	44	2	no	—
194	C₂²⋊C₄	—	44	2	no	—
195	C₄²	—	44	2	no	—
196	C₄²	#4323	1	2	yes	—
197	C₄²	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
198	C₄²	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
199	C₁₆	—	22	4	no	—
200	C₁₁	#33705	1	1	yes	characteristic, complemented, Sylow
201	C₂³	C₂×#4323	1	1	yes	characteristic
202	Q₈	—	4	4	no	—
203	Q₈	—	4	4	no	—
204	Q₈	—	8	4	no	—
205	Q₈	—	88	4	no	—
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	—	4	4	no	—
208	C₂×C₄	—	8	4	no	—
209	C₂×C₄	—	44	4	no	—
210	C₂×C₄	C₂×#4323	1	2	yes	—
211	C₂×C₄	C₂×#4323	1	2	yes	—
212	C₂×C₄	C₂×#4323	1	1	yes	characteristic
213	C₂×C₄	C₂×#4324	1	2	yes	—
214	C₂×C₄	#66989	1	2	yes	—
215	C₂×C₄	C₂²×C₂²⋊D₁₁	1	1	yes	characteristic
216	C₈	C₂×#4324	1	2	yes	—
217	C₈	C₂×#4324	1	2	yes	—
218	C₈	#66989	1	2	yes	—
219	C₈	C₂×#4341	1	2	yes	—
220	C₂²	—	2	2	no	—
221	C₂²	#10622683	1	2	yes	—
222	C₂²	C₂²×#4323	1	1	yes	characteristic
223	C₄	—	2	2	no	—
224	C₄	—	8	4	no	—
225	C₄	—	88	4	no	—
226	C₄	#10622683	1	2	yes	—
227	C₄	#10622792	1	4	yes	—
228	C₄	C₂²×#4324	1	1	yes	characteristic
229	C₄	C₂×#66989	1	1	yes	characteristic
230	C₂	—	2	2	no	—
231	C₂	C₂×#10622683	1	1	yes	characteristic
232	C₂	C₂×#10622792	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×#4323
commutator, cosocle	C₂×C₈×C₁₁	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₁
Fitting	C₂×C₁₁×#494	C₂
socle	C₂²×C₁₁	C₂²×D₈
prime radical	C₂³×C₁₁	C₂×D₈
hypercenter	C₂×#494	D₁₁
hypocenter	C₁₁	#33705
LCS	G ▷ C₂×C₈×C₁₁ ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#494

Complements

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

N	H ≅ G/N	type
2. #13070861	C₂	semidirect
3. #13070843	C₂	semidirect
200. C₁₁	#33705	semidirect

Maximal subgroups

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

subgroup	index
#13079130	2
#13070861	2
#13070843	2
#13070336	2
C₄×#66876	2
C₂×C₁₁×#494	2
#33705	11