Subgroups — #14754406

Order 2816.

718 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13077254	C₂	1	2	yes	maximal
2	#13068791	C₂	1	2	yes	maximal, complemented
3	#13068754	C₂	1	2	yes	maximal, complemented
4	#13068658	C₂	1	2	yes	maximal
5	#13068654	C₂	1	2	yes	maximal
6	#13068522	C₂	1	2	yes	maximal, complemented
7	C₄×#66883	C₂	1	2	yes	maximal, complemented
8	C₂×C₁₁×#405	C₂	1	1	yes	maximal, characteristic
9	C₂²×C₁₁×M₁₆	C₂²	1	1	yes	characteristic
10	C₂³×C₈×C₁₁	C₂²	1	1	yes	characteristic
11	C₂×C₁₁×#117	C₂²	1	1	yes	characteristic
12	C₁₁×#405	C₄	1	8	yes	—
13	C₂×C₁₁×#97	C₂²	1	1	yes	characteristic
14	C₂×C₁₁×#97	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
17	#10622820	C₂²	1	4	yes	—
18	C₂×#66883	C₂²	1	4	yes	—
19	C₂×#66883	C₂²	1	4	yes	—
20	#10622613	—	2	8	no	—
21	#10622537	C₂²	1	4	yes	—
22	C₄×#4318	C₂²	1	4	yes	—
23	#10622190	C₂²	1	4	yes	—
24	#10622190	C₂²	1	4	yes	—
25	C₂³×C₄×C₁₁	C₂³	1	1	yes	characteristic
26	C₂×C₁₁×M₁₆	—	2	4	no	—
27	C₂×C₁₁×M₁₆	C₂×C₄	1	4	yes	—
28	C₂²×C₈×C₁₁	—	2	4	no	—
29	C₂²×C₈×C₁₁	C₂×C₄	1	4	yes	—
30	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
31	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
32	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
33	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
34	C₁₁×#117	C₂×C₄	1	4	yes	—
35	C₁₁×#117	D₄	1	4	yes	—
36	C₂×C₁₁×C₄⋊C₄	Q₈	1	2	yes	—
37	C₂×C₁₁×C₂²⋊C₄	Q₈	1	2	yes	—
38	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
39	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
40	C₂×#4318	C₂³	1	8	yes	—
41	C₁₁×#97	C₂×C₄	1	4	yes	—
42	C₁₁×#97	C₂×C₄	1	4	yes	—
43	C₄×C₈×C₁₁	C₂×C₄	1	4	yes	—
44	C₄×C₈×C₁₁	C₂×C₄	1	4	yes	—
45	C₂²⋊(C₁₁⋊C₈)	—	2	8	no	—
46	#66883	—	2	8	no	—
47	#66883	—	2	8	no	—
48	C₄⋊(C₁₁⋊C₈)	—	2	8	no	—
49	#66867	—	2	4	no	—
50	C₄×C₁₁⋊C₈	—	2	4	no	—
51	#20829	—	11	1	no	maximal, Sylow
52	C₂⁴×C₁₁	C₂×Q₈	1	1	yes	characteristic
53	C₂²×C₄×C₁₁	—	2	2	no	—
54	C₂²×C₄×C₁₁	—	2	2	no	—
55	C₂²×C₄×C₁₁	C₂²×C₄	1	2	yes	—
56	C₂²×C₄×C₁₁	C₂×D₄	1	2	yes	—
57	C₂²×C₄×C₁₁	C₂×Q₈	1	1	yes	characteristic
58	C₂²×C₄×C₁₁	C₂×Q₈	1	2	yes	—
59	C₂²×C₄×C₁₁	C₂×Q₈	1	2	yes	—
60	C₂²×C₄×C₁₁	C₂⁴	1	1	yes	characteristic
61	C₁₁×M₁₆	—	2	8	no	—
62	C₂×C₈×C₁₁	—	2	8	no	—
63	C₂×C₈×C₁₁	—	2	2	no	—
64	C₂×C₈×C₁₁	—	2	2	no	—
65	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
66	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
67	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
68	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
69	C₂×C₈×C₁₁	#41	1	4	yes	—
70	C₂×C₈×C₁₁	#41	1	4	yes	—
71	C₁₁×C₄⋊C₄	C₄⋊C₄	1	8	yes	—
72	C₁₁×C₂²⋊C₄	C₄⋊C₄	1	8	yes	—
73	C₄²×C₁₁	C₂²×C₄	1	2	yes	—
74	C₄²×C₁₁	C₂²×C₄	1	2	yes	—
75	C₄²×C₁₁	C₂×D₄	1	2	yes	—
76	C₄²×C₁₁	C₂×D₄	1	2	yes	—
77	#4318	—	2	8	no	—
78	#4318	—	2	8	no	—
79	C₂×C₁₁⋊C₈	—	2	8	no	—
80	#2921	—	11	2	no	—
81	C₂×#405	D₁₁	1	1	yes	characteristic
82	#1849	—	11	2	no	—
83	#1837	—	11	2	no	—
84	#1811	—	11	2	no	—
85	#1810	—	11	2	no	—
86	#1778	—	11	2	no	—
87	C₄×#108	—	11	2	no	—
88	C₂³×C₁₁	—	2	2	no	—
89	C₂³×C₁₁	—	4	2	no	—
90	C₂³×C₁₁	C₂×C₄⋊C₄	1	2	yes	—
91	C₂³×C₁₁	C₂²×Q₈	1	1	yes	characteristic
92	C₂×C₄×C₁₁	—	2	2	no	—
93	C₂×C₄×C₁₁	—	2	2	no	—
94	C₂×C₄×C₁₁	—	2	2	no	—
95	C₂×C₄×C₁₁	—	2	4	no	—
96	C₂×C₄×C₁₁	—	4	2	no	—
97	C₂×C₄×C₁₁	—	4	2	no	—
98	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	4	yes	—
99	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	4	yes	—
100	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	2	yes	—
101	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
102	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
103	C₂×C₄×C₁₁	C₂×#41	1	2	yes	—
104	C₂×C₄×C₁₁	C₂×#41	1	2	yes	—
105	C₈×C₁₁	—	2	2	no	—
106	C₈×C₁₁	—	2	2	no	—
107	C₈×C₁₁	#117	1	4	yes	—
108	C₈×C₁₁	#117	1	4	yes	—
109	C₁₁⋊C₈	—	4	8	no	—
110	C₂²×M₁₆	C₂×D₁₁	1	1	yes	characteristic
111	C₂³×C₈	C₂×D₁₁	1	1	yes	characteristic
112	C₂×#117	C₂×D₁₁	1	1	yes	characteristic
113	#433	—	22	8	no	—
114	C₂×#108	—	11	4	no	—
115	C₂×#108	—	11	4	no	—
116	#423	—	11	4	no	—
117	#407	—	11	4	no	—
118	#405	Dic₁₁	1	8	yes	—
119	C₄×M₁₆	—	11	4	no	—
120	C₂×#97	C₂×D₁₁	1	1	yes	characteristic
121	C₂×#97	C₂×D₁₁	1	1	yes	characteristic
122	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
123	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
124	#341	—	11	4	no	—
125	#341	—	11	4	no	—
126	C₂²×C₁₁	—	2	2	no	—
127	C₂²×C₁₁	—	2	4	no	—
128	C₂²×C₁₁	—	2	2	no	—
129	C₂²×C₁₁	—	4	4	no	—
130	C₂²×C₁₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
131	C₂²×C₁₁	#533	1	2	yes	—
132	C₄×C₁₁	—	2	4	no	—
133	C₄×C₁₁	—	2	4	no	—
134	C₄×C₁₁	—	2	2	no	—
135	C₄×C₁₁	C₂×#117	1	1	yes	characteristic
136	C₄×C₁₁	C₂×#117	1	1	yes	characteristic
137	C₄×C₁₁	#532	1	2	yes	—
138	C₂³×C₄	C₂²×D₁₁	1	1	yes	characteristic
139	C₂×M₁₆	—	2	4	no	—
140	C₂×M₁₆	—	11	8	no	—
141	C₂×M₁₆	C₂×Dic₁₁	1	4	yes	—
142	C₂²×C₈	—	2	4	no	—
143	C₂²×C₈	C₂×Dic₁₁	1	4	yes	—
144	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
145	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
146	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
147	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
148	#117	D₄₄	1	4	yes	—
149	#117	C₂×Dic₁₁	1	4	yes	—
150	C₂×C₄⋊C₄	Dic₂₂	1	2	yes	—
151	C₂×C₂²⋊C₄	Dic₂₂	1	2	yes	—
152	C₂×C₄²	C₂²×D₁₁	1	1	yes	characteristic
153	C₂×C₄²	C₂²×D₁₁	1	1	yes	characteristic
154	#108	—	22	8	no	—
155	#108	—	22	8	no	—
156	C₄⋊C₈	—	22	8	no	—
157	C₂²⋊C₈	—	22	8	no	—
158	#97	—	22	4	no	—
159	#97	C₂×Dic₁₁	1	4	yes	—
160	#97	C₂×Dic₁₁	1	4	yes	—
161	C₄×C₈	—	22	4	no	—
162	C₄×C₈	C₂×Dic₁₁	1	4	yes	—
163	C₄×C₈	C₂×Dic₁₁	1	4	yes	—
164	C₂×C₁₁	—	2	2	no	—
165	C₂×C₁₁	—	4	2	no	—
166	C₂×C₁₁	#2285	1	1	yes	characteristic
167	C₂×C₁₁	#2916	1	2	yes	—
168	C₂⁴	C₂×Dic₂₂	1	1	yes	characteristic
169	C₂²×C₄	—	2	2	no	—
170	C₂²×C₄	—	2	2	no	—
171	C₂²×C₄	C₂×Dic₂₂	1	1	yes	characteristic
172	C₂²×C₄	C₂×Dic₂₂	1	2	yes	—
173	C₂²×C₄	C₂×Dic₂₂	1	2	yes	—
174	C₂²×C₄	C₂×D₄₄	1	2	yes	—
175	C₂²×C₄	C₂²×Dic₁₁	1	2	yes	—
176	C₂²×C₄	C₂³×D₁₁	1	1	yes	characteristic
177	M₁₆	—	2	8	no	—
178	M₁₆	—	22	8	no	—
179	M₁₆	—	22	8	no	—
180	C₂×C₈	—	2	2	no	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	2	8	no	—
183	C₂×C₈	—	22	8	no	—
184	C₂×C₈	#4339	1	4	yes	—
185	C₂×C₈	#4339	1	4	yes	—
186	C₂×C₈	C₂²×Dic₁₁	1	2	yes	—
187	C₂×C₈	C₂²×Dic₁₁	1	2	yes	—
188	C₂×C₈	C₂²×Dic₁₁	1	2	yes	—
189	C₂×C₈	C₂²×Dic₁₁	1	2	yes	—
190	C₄⋊C₄	C₄⋊Dic₁₁	1	8	yes	—
191	C₂²⋊C₄	C₄⋊Dic₁₁	1	8	yes	—
192	C₄²	C₂×D₄₄	1	2	yes	—
193	C₄²	C₂×D₄₄	1	2	yes	—
194	C₄²	C₂²×Dic₁₁	1	2	yes	—
195	C₄²	C₂²×Dic₁₁	1	2	yes	—
196	C₁₁	#20829	1	1	yes	characteristic, complemented, Sylow
197	C₂³	—	2	2	no	—
198	C₂³	—	4	2	no	—
199	C₂³	C₂×C₄⋊Dic₁₁	1	2	yes	—
200	C₂³	C₂²×Dic₂₂	1	1	yes	characteristic
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	4	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	4	2	no	—
206	C₂×C₄	—	4	2	no	—
207	C₂×C₄	C₂×C₄⋊Dic₁₁	1	4	yes	—
208	C₂×C₄	C₂×C₄⋊Dic₁₁	1	4	yes	—
209	C₂×C₄	C₂×C₄⋊Dic₁₁	1	2	yes	—
210	C₂×C₄	C₂²×D₄₄	1	1	yes	characteristic
211	C₂×C₄	C₂×#4339	1	2	yes	—
212	C₂×C₄	C₂×#4339	1	2	yes	—
213	C₂×C₄	C₂³×Dic₁₁	1	1	yes	characteristic
214	C₈	—	2	2	no	—
215	C₈	—	2	2	no	—
216	C₈	—	44	8	no	—
217	C₈	#66979	1	4	yes	—
218	C₈	#66979	1	4	yes	—
219	C₂²	—	2	2	no	—
220	C₂²	—	2	2	no	—
221	C₂²	—	2	4	no	—
222	C₂²	—	4	4	no	—
223	C₂²	#10623001	1	2	yes	—
224	C₂²	C₂²×C₄⋊Dic₁₁	1	1	yes	characteristic
225	C₄	—	2	2	no	—
226	C₄	—	2	4	no	—
227	C₄	—	2	4	no	—
228	C₄	#10623008	1	2	yes	—
229	C₄	C₂×#66979	1	1	yes	characteristic
230	C₄	C₂×#66979	1	1	yes	characteristic
231	C₂	—	2	2	no	—
232	C₂	—	4	2	no	—
233	C₂	#13071023	1	1	yes	characteristic
234	C₂	#13077227	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₄₄
commutator	C₂×C₄×C₁₁	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₁
Fitting	C₂×C₁₁×#405	C₂
socle	C₂²×C₁₁	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₁₁	C₂×Q₈
cosocle	C₂²×C₄×C₁₁	C₂⁴
hypercenter	C₂×#405	D₁₁
hypocenter	C₁₁	#20829
LCS	G ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117 ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
2. #13068791	C₂	semidirect
3. #13068754	C₂	semidirect
6. #13068522	C₂	semidirect
7. C₄×#66883	C₂	semidirect
196. C₁₁	#20829	semidirect

Maximal subgroups

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

subgroup	index
#13077254	2
#13068791	2
#13068754	2
#13068658	2
#13068654	2
#13068522	2
C₄×#66883	2
C₂×C₁₁×#405	2
#20829	11