Subgroups — #14643232

Order 2816.

718 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10622538	C₂	1	2	yes	maximal
2	#13066451	C₂	1	4	yes	maximal, complemented
3	C₂²×C₄×C₈×C₁₁	C₂	1	1	yes	maximal, characteristic
4	C₂³×C₈×C₁₁	C₄	1	2	yes	—
5	C₂²×C₄²×C₁₁	C₂²	1	1	yes	characteristic
6	C₂×C₄×C₈×C₁₁	—	2	4	no	—
7	C₂×C₄×C₈×C₁₁	C₄	1	2	yes	—
8	C₂×C₄×C₈×C₁₁	C₂²	1	2	yes	—
9	C₂×C₂²⋊(C₁₁⋊C₈)	—	2	2	no	—
10	#10622538	—	2	4	no	—
11	#10622538	C₄	1	8	yes	—
12	C₂×C₄⋊(C₁₁⋊C₈)	C₂²	1	4	yes	—
13	C₂×C₄×C₁₁⋊C₈	—	2	2	no	—
14	C₂³×C₄×C₁₁	C₂×C₄	1	1	yes	characteristic
15	C₂³×C₄×C₁₁	D₄	1	1	yes	characteristic
16	C₂³×C₄×C₁₁	Q₈	1	1	yes	characteristic
17	C₂²×C₈×C₁₁	—	2	4	no	—
18	C₂²×C₈×C₁₁	—	2	4	no	—
19	C₂²×C₈×C₁₁	C₈	1	8	yes	complemented
20	C₂²×C₈×C₁₁	C₂×C₄	1	4	yes	—
21	C₂×C₄²×C₁₁	C₂×C₄	1	4	yes	—
22	C₂×C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₁₁	D₄	1	1	yes	characteristic
24	C₂×C₄²×C₁₁	D₄	1	4	yes	—
25	C₂×C₄²×C₁₁	Q₈	1	1	yes	characteristic
26	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
27	C₂²×C₁₁⋊C₈	—	2	4	no	—
28	C₄×C₈×C₁₁	—	2	4	no	—
29	C₄×C₈×C₁₁	—	2	4	no	—
30	C₂²⋊(C₁₁⋊C₈)	—	2	8	no	—
31	C₄⋊(C₁₁⋊C₈)	—	2	4	no	—
32	C₄⋊(C₁₁⋊C₈)	C₂×C₄	1	8	yes	—
33	C₄×C₁₁⋊C₈	—	2	8	no	—
34	#7872	—	11	1	no	maximal, Sylow
35	C₂⁴×C₁₁	C₄⋊C₄	1	1	yes	characteristic
36	C₂²×C₄×C₁₁	—	2	4	no	—
37	C₂²×C₄×C₁₁	—	2	4	no	—
38	C₂²×C₄×C₁₁	C₄²	1	2	yes	—
39	C₂²×C₄×C₁₁	C₂²⋊C₄	1	4	yes	—
40	C₂²×C₄×C₁₁	C₂²⋊C₄	1	2	yes	—
41	C₂²×C₄×C₁₁	C₄⋊C₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₁	C₄⋊C₄	1	4	yes	—
43	C₂²×C₄×C₁₁	C₄⋊C₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₁	C₄⋊C₄	1	1	yes	characteristic
45	C₂²×C₄×C₁₁	C₂×C₈	1	4	yes	—
46	C₂²×C₄×C₁₁	M₁₆	1	4	yes	—
47	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₁	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₈×C₁₁	—	2	16	no	—
51	C₂×C₈×C₁₁	—	2	4	no	—
52	C₂×C₈×C₁₁	—	2	4	no	—
53	C₂×C₈×C₁₁	C₂×C₈	1	8	yes	—
54	C₄²×C₁₁	C₄²	1	2	yes	—
55	C₄²×C₁₁	C₂²⋊C₄	1	4	yes	—
56	C₄²×C₁₁	C₂²⋊C₄	1	2	yes	—
57	C₄²×C₁₁	C₄⋊C₄	1	4	yes	—
58	C₄²×C₁₁	C₂²×C₄	1	2	yes	—
59	C₄²×C₁₁	C₂×D₄	1	2	yes	—
60	C₂×C₁₁⋊C₈	—	2	8	no	—
61	C₂×C₁₁⋊C₈	—	4	8	no	—
62	C₂×#431	—	11	2	no	—
63	C₂²×C₄×C₈	D₁₁	1	1	yes	characteristic
64	#1277	—	11	4	no	—
65	C₂³×C₁₁	—	2	2	no	—
66	C₂³×C₁₁	—	2	2	no	—
67	C₂³×C₁₁	#95	1	2	yes	—
68	C₂³×C₁₁	C₄⋊C₈	1	4	yes	—
69	C₂³×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₁₁	—	2	2	no	—
71	C₂×C₄×C₁₁	—	2	2	no	—
72	C₂×C₄×C₁₁	—	2	8	no	—
73	C₂×C₄×C₁₁	—	2	8	no	—
74	C₂×C₄×C₁₁	#95	1	2	yes	—
75	C₂×C₄×C₁₁	#95	1	2	yes	—
76	C₂×C₄×C₁₁	#95	1	2	yes	—
77	C₂×C₄×C₁₁	C₄×C₈	1	4	yes	—
78	C₂×C₄×C₁₁	#97	1	4	yes	—
79	C₂×C₄×C₁₁	C₂²⋊C₈	1	8	yes	—
80	C₂×C₄×C₁₁	C₄⋊C₈	1	4	yes	—
81	C₂×C₄×C₁₁	#108	1	4	yes	—
82	C₂×C₄×C₁₁	C₂×C₄²	1	1	yes	characteristic
83	C₂×C₄×C₁₁	C₂×C₂²⋊C₄	1	2	yes	—
84	C₂×C₄×C₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	2	yes	—
86	C₂×C₄×C₁₁	C₂²×C₈	1	2	yes	—
87	C₂×C₄×C₁₁	C₂×M₁₆	1	2	yes	—
88	C₂×C₄×C₁₁	#135	1	4	yes	—
89	C₈×C₁₁	—	2	8	no	—
90	C₁₁⋊C₈	—	4	8	no	—
91	C₂³×C₈	Dic₁₁	1	2	yes	—
92	C₂²×C₄²	C₂×D₁₁	1	1	yes	characteristic
93	#431	—	11	8	no	—
94	#431	—	22	4	no	—
95	C₂×C₄⋊C₈	—	11	4	no	—
96	C₂×C₂²⋊C₈	—	22	2	no	—
97	C₂×C₄×C₈	—	2	4	no	—
98	C₂×C₄×C₈	—	22	2	no	—
99	C₂×C₄×C₈	Dic₁₁	1	2	yes	—
100	C₂×C₄×C₈	C₂×D₁₁	1	2	yes	—
101	C₂²×C₁₁	—	2	8	no	—
102	C₂²×C₁₁	—	2	2	no	—
103	C₂²×C₁₁	—	2	2	no	—
104	C₂²×C₁₁	#336	1	4	yes	—
105	C₂²×C₁₁	C₂×#95	1	1	yes	characteristic
106	C₂²×C₁₁	C₂×C₄⋊C₈	1	2	yes	—
107	C₂²×C₁₁	#427	1	2	yes	—
108	C₂²×C₁₁	C₂×#108	1	2	yes	—
109	C₄×C₁₁	—	2	2	no	—
110	C₄×C₁₁	—	2	2	no	—
111	C₄×C₁₁	#336	1	4	yes	—
112	C₄×C₁₁	#341	1	4	yes	—
113	C₄×C₁₁	C₂×C₄×C₈	1	1	yes	characteristic
114	C₄×C₁₁	C₂×#97	1	1	yes	characteristic
115	C₄×C₁₁	C₂×C₂²⋊C₈	1	2	yes	—
116	C₄×C₁₁	#416	1	4	yes	—
117	C₂³×C₄	Dic₂₂	1	1	yes	characteristic
118	C₂³×C₄	D₄₄	1	1	yes	characteristic
119	C₂³×C₄	C₂×Dic₁₁	1	1	yes	characteristic
120	C₂²×C₈	—	2	4	no	—
121	C₂²×C₈	—	2	4	no	—
122	C₂²×C₈	—	22	4	no	—
123	C₂²×C₈	C₁₁⋊C₈	1	8	yes	complemented
124	C₂²×C₈	C₂×Dic₁₁	1	4	yes	—
125	C₂×C₄²	Dic₂₂	1	1	yes	characteristic
126	C₂×C₄²	C₄×D₁₁	1	4	yes	—
127	C₂×C₄²	D₄₄	1	1	yes	characteristic
128	C₂×C₄²	C₂×Dic₁₁	1	1	yes	characteristic
129	C₂×C₄²	C₂²⋊D₁₁	1	4	yes	—
130	C₂×C₄²	C₂²×D₁₁	1	1	yes	characteristic
131	C₄⋊C₈	—	11	8	no	—
132	C₄⋊C₈	—	22	4	no	—
133	C₂²⋊C₈	—	22	8	no	—
134	C₄×C₈	—	2	4	no	—
135	C₄×C₈	—	2	4	no	—
136	C₄×C₈	—	22	8	no	—
137	C₂×C₁₁	—	2	4	no	—
138	C₂×C₁₁	#1564	1	4	yes	—
139	C₂×C₁₁	C₂×#336	1	1	yes	characteristic
140	C₂×C₁₁	#1735	1	1	yes	characteristic
141	C₂×C₁₁	C₂×#341	1	1	yes	characteristic
142	C₂⁴	C₄⋊Dic₁₁	1	1	yes	characteristic
143	C₂²×C₄	—	2	4	no	—
144	C₂²×C₄	—	2	4	no	—
145	C₂²×C₄	C₂×C₁₁⋊C₈	1	4	yes	—
146	C₂²×C₄	#4318	1	4	yes	—
147	C₂²×C₄	C₄×Dic₁₁	1	2	yes	—
148	C₂²×C₄	Dic₁₁⋊C₄	1	4	yes	—
149	C₂²×C₄	C₄⋊Dic₁₁	1	1	yes	characteristic
150	C₂²×C₄	C₄⋊Dic₁₁	1	1	yes	characteristic
151	C₂²×C₄	C₄⋊Dic₁₁	1	1	yes	characteristic
152	C₂²×C₄	(C₂×D₁₁)⋊C₄	1	4	yes	—
153	C₂²×C₄	C₂²⋊Dic₁₁	1	2	yes	—
154	C₂²×C₄	C₂×Dic₂₂	1	1	yes	characteristic
155	C₂²×C₄	C₂×D₄₄	1	1	yes	characteristic
156	C₂²×C₄	C₂²×Dic₁₁	1	1	yes	characteristic
157	C₂×C₈	—	2	16	no	—
158	C₂×C₈	—	2	4	no	—
159	C₂×C₈	—	2	4	no	—
160	C₂×C₈	—	22	8	no	—
161	C₂×C₈	—	44	8	no	—
162	C₂×C₈	C₂×C₁₁⋊C₈	1	8	yes	—
163	C₄²	C₄×Dic₁₁	1	2	yes	—
164	C₄²	Dic₁₁⋊C₄	1	4	yes	—
165	C₄²	(C₂×D₁₁)⋊C₄	1	4	yes	—
166	C₄²	C₂²⋊Dic₁₁	1	2	yes	—
167	C₄²	C₂×C₄×D₁₁	1	2	yes	—
168	C₄²	C₂×C₂²⋊D₁₁	1	2	yes	—
169	C₁₁	#7872	1	1	yes	characteristic, complemented, Sylow
170	C₂³	—	2	2	no	—
171	C₂³	—	2	2	no	—
172	C₂³	C₄⋊(C₁₁⋊C₈)	1	4	yes	—
173	C₂³	C₁₁⋊#95	1	2	yes	—
174	C₂³	C₂×C₄⋊Dic₁₁	1	1	yes	characteristic
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	8	no	—
178	C₂×C₄	—	2	8	no	—
179	C₂×C₄	C₄×C₁₁⋊C₈	1	4	yes	—
180	C₂×C₄	#66867	1	4	yes	—
181	C₂×C₄	C₄⋊(C₁₁⋊C₈)	1	4	yes	—
182	C₂×C₄	#66883	1	4	yes	—
183	C₂×C₄	C₂²⋊(C₁₁⋊C₈)	1	8	yes	—
184	C₂×C₄	C₁₁⋊#95	1	2	yes	—
185	C₂×C₄	C₁₁⋊#95	1	2	yes	—
186	C₂×C₄	C₁₁⋊#95	1	2	yes	—
187	C₂×C₄	#66957	1	4	yes	—
188	C₂×C₄	C₂²×C₁₁⋊C₈	1	2	yes	—
189	C₂×C₄	C₂×#4318	1	2	yes	—
190	C₂×C₄	C₂×C₄×Dic₁₁	1	1	yes	characteristic
191	C₂×C₄	C₂×Dic₁₁⋊C₄	1	2	yes	—
192	C₂×C₄	C₂×(C₂×D₁₁)⋊C₄	1	2	yes	—
193	C₂×C₄	C₂×C₂²⋊Dic₁₁	1	1	yes	characteristic
194	C₈	—	2	8	no	—
195	C₈	—	44	8	no	—
196	C₂²	—	2	2	no	—
197	C₂²	—	2	2	no	—
198	C₂²	—	2	8	no	—
199	C₂²	#10622162	1	4	yes	—
200	C₂²	C₂×C₄⋊(C₁₁⋊C₈)	1	2	yes	—
201	C₂²	#10622719	1	2	yes	—
202	C₂²	C₂×#66883	1	2	yes	—
203	C₂²	C₂×C₁₁⋊#95	1	1	yes	characteristic
204	C₄	—	2	2	no	—
205	C₄	—	2	2	no	—
206	C₄	#10622162	1	4	yes	—
207	C₄	#10622190	1	4	yes	—
208	C₄	C₂×C₄×C₁₁⋊C₈	1	1	yes	characteristic
209	C₄	C₂×#66867	1	1	yes	characteristic
210	C₄	#10622726	1	4	yes	—
211	C₄	C₂×C₂²⋊(C₁₁⋊C₈)	1	2	yes	—
212	C₂	—	2	4	no	—
213	C₂	#13067364	1	4	yes	—
214	C₂	C₂×#10622162	1	1	yes	characteristic
215	C₂	#13068333	1	1	yes	characteristic
216	C₂	C₂×#10622190	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄²	D₄₄
commutator	C₄×C₁₁	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₁₁
Fitting	C₂²×C₄×C₈×C₁₁	C₂
socle	C₂³×C₁₁	C₂×C₄⋊C₄
prime radical	C₂⁴×C₁₁	C₄⋊C₄
cosocle	C₂×C₄²×C₁₁	C₂³
hypercenter	C₂²×C₄×C₈	D₁₁
hypocenter	C₁₁	#7872
LCS	G ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₄² ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #13066451	C₂	semidirect
19. C₂²×C₈×C₁₁	C₈	semidirect
123. C₂²×C₈	C₁₁⋊C₈	semidirect
169. C₁₁	#7872	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10622538	2
#13066451	2
C₂²×C₄×C₈×C₁₁	2
#7872	11