Subgroups — #14652816

Order 2816.

570 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13077014	C₂	1	1	yes	maximal, characteristic
2	#13068768	C₂	1	1	yes	maximal, characteristic
3	C₂×#10622120	C₂	1	4	yes	maximal
4	C₄×C₁₁×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₂×#66922	—	2	1	no	—
6	C₂×#66922	—	2	1	no	—
7	C₂×#66922	C₂²	1	2	yes	—
8	C₂×C₁₁×C₄⋊C₈	C₂²	1	2	yes	—
9	C₂×C₄×C₈×C₁₁	—	2	1	no	—
10	C₄³×C₁₁	C₂²	1	1	yes	characteristic
11	#10622548	—	2	2	no	—
12	#10622548	C₄	1	4	yes	—
13	C₂×C₄⋊(C₁₁⋊C₈)	—	2	2	no	—
14	C₂×C₄⋊(C₁₁⋊C₈)	C₂²	1	2	yes	—
15	C₁₁×#336	—	2	1	no	—
16	#10622120	—	2	8	no	—
17	C₂²×Dic₂₂	—	2	2	no	—
18	C₂²×C₈×C₁₁	—	2	2	no	—
19	C₂×C₄²×C₁₁	C₂×C₄	1	2	yes	—
20	C₂×C₄²×C₁₁	D₄	1	2	yes	—
21	C₂×C₄²×C₁₁	D₄	1	1	yes	characteristic
22	C₂×C₄²×C₁₁	D₄	1	1	yes	characteristic
23	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
24	C₂×C₄⋊Dic₁₁	—	2	4	no	—
25	C₂×C₄⋊Dic₁₁	—	2	4	no	—
26	C₂×C₄⋊Dic₁₁	—	2	2	no	—
27	C₂×C₄⋊Dic₁₁	C₂×C₄	1	4	yes	—
28	C₂²×C₁₁⋊C₈	—	2	2	no	—
29	#66922	—	2	4	no	—
30	#66922	—	2	4	no	—
31	#66922	—	4	4	no	—
32	#66922	—	4	2	no	—
33	#66922	—	4	2	no	—
34	C₁₁×C₄⋊C₈	—	2	2	no	—
35	C₁₁×C₄⋊C₈	D₄	1	4	yes	—
36	C₄×C₈×C₁₁	—	4	2	no	—
37	C₄⋊(C₁₁⋊C₈)	—	2	4	no	—
38	C₄⋊(C₁₁⋊C₈)	—	4	4	no	—
39	#9694	—	11	1	no	maximal, Sylow
40	C₂²×C₄×C₁₁	C₂²⋊C₄	1	2	yes	—
41	C₂²×C₄×C₁₁	C₂²⋊C₄	1	2	yes	—
42	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
45	C₂²×Dic₁₁	—	2	4	no	—
46	C₂×Dic₂₂	—	4	4	no	—
47	C₂×Dic₂₂	—	4	2	no	—
48	C₂×Dic₂₂	—	4	2	no	—
49	C₂×Dic₂₂	—	4	4	no	—
50	C₂×C₈×C₁₁	—	2	4	no	—
51	C₂×C₈×C₁₁	—	4	2	no	—
52	C₂×C₈×C₁₁	—	4	2	no	—
53	C₄²×C₁₁	—	2	4	no	—
54	C₄²×C₁₁	—	2	2	no	—
55	C₄²×C₁₁	D₈	1	2	yes	—
56	C₄²×C₁₁	SD₁₆	1	2	yes	—
57	C₄²×C₁₁	SD₁₆	1	4	yes	—
58	C₄²×C₁₁	Q₁₆	1	4	yes	—
59	C₄²×C₁₁	C₂×D₄	1	2	yes	—
60	C₄²×C₁₁	#41	1	2	yes	—
61	C₄⋊Dic₁₁	—	2	8	no	—
62	C₄⋊Dic₁₁	—	4	8	no	—
63	C₄⋊Dic₁₁	—	4	8	no	—
64	C₄⋊Dic₁₁	—	4	2	no	—
65	C₄⋊Dic₁₁	—	4	2	no	—
66	C₂×C₁₁⋊C₈	—	4	2	no	—
67	C₂×C₁₁⋊C₈	—	4	4	no	—
68	#2868	—	11	1	no	—
69	#1842	—	11	1	no	—
70	C₄×C₄⋊C₈	D₁₁	1	1	yes	characteristic
71	C₂×#332	—	11	4	no	—
72	C₂³×C₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₄×C₁₁	—	2	2	no	—
74	C₂×C₄×C₁₁	—	2	2	no	—
75	C₂×C₄×C₁₁	—	2	2	no	—
76	C₂×C₄×C₁₁	#101	1	4	yes	—
77	C₂×C₄×C₁₁	#102	1	4	yes	—
78	C₂×C₄×C₁₁	#103	1	4	yes	—
79	C₂×C₄×C₁₁	#103	1	4	yes	—
80	C₂×C₄×C₁₁	C₄×D₄	1	2	yes	—
81	C₂×C₄×C₁₁	#121	1	2	yes	—
82	C₂×C₄×C₁₁	#124	1	1	yes	characteristic
83	C₂×C₄×C₁₁	#127	1	1	yes	characteristic
84	C₂×C₄×C₁₁	C₂×D₈	1	1	yes	characteristic
85	C₂×C₄×C₁₁	C₂×SD₁₆	1	2	yes	—
86	C₂×C₄×C₁₁	C₂×SD₁₆	1	1	yes	characteristic
87	C₂×C₄×C₁₁	C₂×Q₁₆	1	2	yes	—
88	C₂×C₄×C₁₁	AGL(1,ℤ/8)	1	2	yes	—
89	C₂×Dic₁₁	—	4	4	no	—
90	C₂×Dic₁₁	—	4	4	no	—
91	C₂×Dic₁₁	—	4	4	no	—
92	Dic₂₂	—	8	4	no	—
93	Dic₂₂	—	8	4	no	—
94	C₈×C₁₁	—	4	4	no	—
95	C₁₁⋊C₈	—	8	2	no	—
96	C₂×C₄⋊Q₈	—	11	2	no	—
97	C₂×C₄⋊Q₈	—	22	1	no	—
98	C₂×C₄⋊Q₈	—	22	1	no	—
99	C₂×C₄⋊C₈	—	11	2	no	—
100	C₂×C₄⋊C₈	—	22	2	no	—
101	C₂×C₄⋊C₈	C₂×D₁₁	1	2	yes	—
102	C₂×C₄×C₈	—	2	1	no	—
103	#384	—	11	4	no	—
104	#384	—	22	2	no	—
105	C₄³	C₂×D₁₁	1	1	yes	characteristic
106	#336	—	2	1	no	—
107	#332	—	22	8	no	—
108	C₂²×C₁₁	#390	1	1	yes	characteristic
109	C₂²×C₁₁	C₂×#101	1	2	yes	—
110	C₂²×C₁₁	C₂×#102	1	1	yes	characteristic
111	C₂²×C₁₁	C₂×#103	1	2	yes	—
112	C₂²×C₁₁	#418	1	1	yes	characteristic
113	C₄×C₁₁	—	2	4	no	—
114	C₄×C₁₁	—	2	2	no	—
115	C₄×C₁₁	#332	1	8	yes	complemented
116	C₄×C₁₁	#459	1	2	yes	—
117	C₄×C₁₁	#460	1	2	yes	—
118	C₄×C₁₁	#487	1	2	yes	—
119	C₄×C₁₁	#492	1	1	yes	characteristic
120	C₄×C₁₁	#494	1	1	yes	characteristic
121	Dic₁₁	—	8	4	no	—
122	C₂²×Q₈	—	22	2	no	—
123	C₂²×C₈	—	2	2	no	—
124	C₂²×C₈	—	22	2	no	—
125	C₄⋊Q₈	—	22	4	no	—
126	C₄⋊Q₈	—	22	4	no	—
127	C₄⋊Q₈	—	44	4	no	—
128	C₄⋊Q₈	—	44	2	no	—
129	C₄⋊Q₈	—	44	2	no	—
130	C₂×C₄⋊C₄	—	11	4	no	—
131	C₂×C₄⋊C₄	—	22	4	no	—
132	C₂×C₄⋊C₄	—	22	4	no	—
133	C₂×C₄⋊C₄	—	22	2	no	—
134	C₂×C₄²	C₄×D₁₁	1	2	yes	—
135	C₂×C₄²	D₄₄	1	2	yes	—
136	C₂×C₄²	C₂²⋊D₁₁	1	1	yes	characteristic
137	C₂×C₄²	C₂²⋊D₁₁	1	1	yes	characteristic
138	C₂×C₄²	C₂²×D₁₁	1	1	yes	characteristic
139	C₄⋊C₈	—	2	2	no	—
140	C₄⋊C₈	—	22	4	no	—
141	C₄⋊C₈	—	44	4	no	—
142	C₄⋊C₈	D₄₄	1	4	yes	—
143	C₄×C₈	—	4	2	no	—
144	C₂×C₁₁	C₂×#332	1	2	yes	—
145	C₂×C₁₁	#1549	1	2	yes	—
146	C₂×C₁₁	#1956	1	1	yes	characteristic
147	C₂×C₁₁	#1966	1	1	yes	characteristic
148	C₂×C₁₁	#1982	1	1	yes	characteristic
149	C₂×Q₈	—	44	4	no	—
150	C₂×Q₈	—	44	2	no	—
151	C₂×Q₈	—	44	2	no	—
152	C₂×Q₈	—	44	4	no	—
153	C₂²×C₄	—	22	4	no	—
154	C₂²×C₄	(C₂×D₁₁)⋊C₄	1	2	yes	—
155	C₂²×C₄	(C₂×D₁₁)⋊C₄	1	2	yes	—
156	C₂²×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
157	C₂²×C₄	C₂×D₄₄	1	1	yes	characteristic
158	C₂²×C₄	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
159	C₂×C₈	—	2	4	no	—
160	C₂×C₈	—	4	2	no	—
161	C₂×C₈	—	4	2	no	—
162	C₂×C₈	—	44	2	no	—
163	C₂×C₈	—	44	4	no	—
164	C₄⋊C₄	—	22	8	no	—
165	C₄⋊C₄	—	44	8	no	—
166	C₄⋊C₄	—	44	8	no	—
167	C₄⋊C₄	—	44	2	no	—
168	C₄⋊C₄	—	44	2	no	—
169	C₄²	—	2	4	no	—
170	C₄²	—	2	2	no	—
171	C₄²	#4314	1	4	yes	—
172	C₄²	Dic₄₄	1	4	yes	—
173	C₄²	#4323	1	2	yes	—
174	C₄²	#4325	1	2	yes	—
175	C₄²	C₂×D₄₄	1	2	yes	—
176	C₄²	#4339	1	2	yes	—
177	C₁₁	#9694	1	1	yes	characteristic, complemented, Sylow
178	C₂³	C₂×(C₂×D₁₁)⋊C₄	1	1	yes	characteristic
179	Q₈	—	88	4	no	—
180	Q₈	—	88	4	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	44	4	no	—
185	C₂×C₄	—	44	4	no	—
186	C₂×C₄	—	44	4	no	—
187	C₂×C₄	#66873	1	4	yes	—
188	C₂×C₄	#66880	1	4	yes	—
189	C₂×C₄	#66880	1	4	yes	—
190	C₂×C₄	#66888	1	4	yes	—
191	C₂×C₄	C₄×D₄₄	1	2	yes	—
192	C₂×C₄	#66927	1	1	yes	characteristic
193	C₂×C₄	#66928	1	1	yes	characteristic
194	C₂×C₄	C₂×#4314	1	2	yes	—
195	C₂×C₄	C₂×Dic₄₄	1	2	yes	—
196	C₂×C₄	#66983	1	2	yes	—
197	C₂×C₄	C₂×#4323	1	1	yes	characteristic
198	C₂×C₄	C₂×#4325	1	1	yes	characteristic
199	C₂×C₄	#67002	1	2	yes	—
200	C₈	—	4	4	no	—
201	C₈	—	88	2	no	—
202	C₂²	#10622552	1	1	yes	characteristic
203	C₂²	C₂×#66873	1	1	yes	characteristic
204	C₂²	#10622614	1	1	yes	characteristic
205	C₂²	C₂×#66880	1	2	yes	—
206	C₂²	C₂×#66888	1	2	yes	—
207	C₄	—	2	4	no	—
208	C₄	—	2	2	no	—
209	C₄	—	88	4	no	—
210	C₄	#10622120	1	8	yes	complemented
211	C₄	#10622294	1	2	yes	—
212	C₄	#10622306	1	1	yes	characteristic
213	C₄	#10622312	1	1	yes	characteristic
214	C₄	#10622628	1	2	yes	—
215	C₄	#10622640	1	2	yes	—
216	C₂	C₂×#10622120	1	2	yes	—
217	C₂	#13067300	1	2	yes	—
218	C₂	#13069418	1	1	yes	characteristic
219	C₂	#13069460	1	1	yes	characteristic
220	C₂	#13069543	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×(C₂×D₁₁)⋊C₄
commutator	C₂²×C₄×C₁₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₁
Fitting	C₄×C₁₁×C₄⋊C₈	C₂
socle, prime radical	C₂³×C₁₁	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₁₁	C₂³
hypercenter	C₄×C₄⋊C₈	D₁₁
hypocenter	C₁₁	#9694
LCS	G ▷ C₂²×C₄×C₁₁ ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
115. C₄×C₁₁	#332	semidirect
177. C₁₁	#9694	semidirect
210. C₄	#10622120	semidirect

Maximal subgroups

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

subgroup	index
#13077014	2
#13068768	2
C₂×#10622120	2
C₄×C₁₁×C₄⋊C₈	2
#9694	11