Subgroups — #14640342

Order 2816.

352 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13068768	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊(C₁₁⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₄×C₁₁×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₁×C₄⋊C₈	—	2	1	no	—
5	C₂×C₄×C₈×C₁₁	—	2	1	no	—
6	C₄³×C₁₁	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊(C₁₁⋊C₈)	—	2	1	no	—
8	C₂×C₄⋊(C₁₁⋊C₈)	—	2	1	no	—
9	C₂×C₄⋊(C₁₁⋊C₈)	C₄	1	2	yes	—
10	C₂×C₄⋊(C₁₁⋊C₈)	C₄	1	2	yes	—
11	C₂×C₄×C₁₁⋊C₈	—	2	1	no	—
12	C₁₁×#336	C₄	1	2	yes	—
13	#10622162	—	2	1	no	—
14	C₂²×C₈×C₁₁	—	2	2	no	—
15	C₂×C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄²×C₁₁	D₄	1	1	yes	characteristic
19	C₂×C₄²×C₁₁	D₄	1	1	yes	characteristic
20	C₂×C₄²×C₁₁	D₄	1	1	yes	characteristic
21	C₂×C₄²×C₁₁	Q₈	1	1	yes	characteristic
22	C₂²×C₁₁⋊C₈	—	2	2	no	—
23	C₂²×C₁₁⋊C₈	—	2	2	no	—
24	C₁₁×C₄⋊C₈	—	2	4	no	—
25	C₄×C₈×C₁₁	—	4	1	no	—
26	C₄×C₈×C₁₁	—	4	1	no	—
27	C₄⋊(C₁₁⋊C₈)	—	2	2	no	—
28	C₄⋊(C₁₁⋊C₈)	—	4	1	no	—
29	C₄⋊(C₁₁⋊C₈)	—	4	1	no	—
30	C₄⋊(C₁₁⋊C₈)	—	4	1	no	—
31	C₄⋊(C₁₁⋊C₈)	—	4	1	no	—
32	C₄⋊(C₁₁⋊C₈)	—	4	2	no	—
33	C₄⋊(C₁₁⋊C₈)	C₈	1	4	yes	complemented
34	C₄×C₁₁⋊C₈	—	4	1	no	—
35	C₄×C₁₁⋊C₈	—	4	1	no	—
36	#7031	—	11	1	no	maximal, Sylow
37	C₂²×C₄×C₁₁	C₄²	1	1	yes	characteristic
38	C₂²×C₄×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
39	C₂²×C₄×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
40	C₂²×C₄×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
41	C₂²×C₄×C₁₁	C₄⋊C₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₁	C₄⋊C₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₁	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₁₁	—	2	4	no	—
45	C₂×C₈×C₁₁	—	4	2	no	—
46	C₂×C₈×C₁₁	—	4	2	no	—
47	C₄²×C₁₁	—	2	1	no	—
48	C₄²×C₁₁	—	2	1	no	—
49	C₄²×C₁₁	—	2	2	no	—
50	C₄²×C₁₁	—	2	2	no	—
51	C₄²×C₁₁	—	4	1	no	—
52	C₄²×C₁₁	C₂×C₈	1	2	yes	—
53	C₄²×C₁₁	M₁₆	1	2	yes	—
54	C₄²×C₁₁	D₈	1	1	yes	characteristic
55	C₄²×C₁₁	D₈	1	1	yes	characteristic
56	C₄²×C₁₁	SD₁₆	1	1	yes	characteristic
57	C₄²×C₁₁	SD₁₆	1	1	yes	characteristic
58	C₄²×C₁₁	SD₁₆	1	1	yes	characteristic
59	C₄²×C₁₁	SD₁₆	1	1	yes	characteristic
60	C₄²×C₁₁	Q₁₆	1	1	yes	characteristic
61	C₄²×C₁₁	Q₁₆	1	1	yes	characteristic
62	C₂×C₁₁⋊C₈	—	2	4	no	—
63	C₂×C₁₁⋊C₈	—	4	2	no	—
64	C₂×C₁₁⋊C₈	—	4	2	no	—
65	C₂×C₁₁⋊C₈	—	4	2	no	—
66	C₂×C₁₁⋊C₈	—	4	2	no	—
67	C₂×C₁₁⋊C₈	—	4	2	no	—
68	#1842	—	11	1	no	—
69	C₄×C₄⋊C₈	—	11	1	no	—
70	C₄×C₄⋊C₈	D₁₁	1	1	yes	characteristic
71	C₂³×C₁₁	#95	1	1	yes	characteristic
72	C₂×C₄×C₁₁	—	2	1	no	—
73	C₂×C₄×C₁₁	—	2	1	no	—
74	C₂×C₄×C₁₁	—	2	1	no	—
75	C₂×C₄×C₁₁	—	2	1	no	—
76	C₂×C₄×C₁₁	—	2	1	no	—
77	C₂×C₄×C₁₁	—	2	1	no	—
78	C₂×C₄×C₁₁	—	2	1	no	—
79	C₂×C₄×C₁₁	—	2	1	no	—
80	C₂×C₄×C₁₁	—	2	1	no	—
81	C₂×C₄×C₁₁	C₄×C₈	1	1	yes	characteristic
82	C₂×C₄×C₁₁	#97	1	1	yes	characteristic
83	C₂×C₄×C₁₁	C₂²⋊C₈	1	2	yes	—
84	C₂×C₄×C₁₁	#100	1	1	yes	characteristic
85	C₂×C₄×C₁₁	#101	1	1	yes	characteristic
86	C₂×C₄×C₁₁	#102	1	1	yes	characteristic
87	C₂×C₄×C₁₁	#102	1	1	yes	characteristic
88	C₂×C₄×C₁₁	#102	1	1	yes	characteristic
89	C₂×C₄×C₁₁	#102	1	1	yes	characteristic
90	C₂×C₄×C₁₁	#103	1	1	yes	characteristic
91	C₂×C₄×C₁₁	#103	1	1	yes	characteristic
92	C₂×C₄×C₁₁	#103	1	1	yes	characteristic
93	C₂×C₄×C₁₁	#103	1	1	yes	characteristic
94	C₂×C₄×C₁₁	#104	1	2	yes	—
95	C₂×C₄×C₁₁	C₄⋊C₈	1	2	yes	—
96	C₂×C₄×C₁₁	#106	1	1	yes	characteristic
97	C₂×C₄×C₁₁	#106	1	1	yes	characteristic
98	C₂×C₄×C₁₁	#107	1	1	yes	characteristic
99	C₂×C₄×C₁₁	#107	1	1	yes	characteristic
100	C₂×C₄×C₁₁	#108	1	2	yes	—
101	C₈×C₁₁	—	4	4	no	—
102	C₁₁⋊C₈	—	4	4	no	—
103	C₁₁⋊C₈	—	8	2	no	—
104	C₂×C₄⋊C₈	—	2	1	no	—
105	C₂×C₄⋊C₈	—	11	2	no	—
106	C₂×C₄⋊C₈	—	11	2	no	—
107	C₂×C₄⋊C₈	—	22	1	no	—
108	C₂×C₄⋊C₈	—	22	1	no	—
109	C₂×C₄×C₈	—	2	1	no	—
110	C₂×C₄×C₈	—	22	1	no	—
111	C₄³	C₂×D₁₁	1	1	yes	characteristic
112	#336	—	22	1	no	—
113	#336	Dic₁₁	1	2	yes	—
114	C₂²×C₁₁	#336	1	1	yes	characteristic
115	C₂²×C₁₁	#339	1	1	yes	characteristic
116	C₂²×C₁₁	#340	1	1	yes	characteristic
117	C₂²×C₁₁	#340	1	1	yes	characteristic
118	C₂²×C₁₁	#341	1	1	yes	characteristic
119	C₂²×C₁₁	#343	1	1	yes	characteristic
120	C₂²×C₁₁	#344	1	1	yes	characteristic
121	C₄×C₁₁	—	2	1	no	—
122	C₄×C₁₁	—	2	1	no	—
123	C₄×C₁₁	—	2	1	no	—
124	C₄×C₁₁	—	2	1	no	—
125	C₄×C₁₁	—	2	1	no	—
126	C₄×C₁₁	—	2	1	no	—
127	C₄×C₁₁	—	4	1	no	—
128	C₄×C₁₁	D₄⋊C₈	1	2	yes	—
129	C₄×C₁₁	Q₈⋊C₈	1	2	yes	—
130	C₄×C₁₁	#331	1	1	yes	characteristic
131	C₄×C₁₁	#332	1	1	yes	characteristic
132	C₄×C₁₁	#332	1	1	yes	characteristic
133	C₄×C₁₁	#333	1	1	yes	characteristic
134	C₄×C₁₁	#334	1	2	yes	—
135	C₄×C₁₁	#335	1	2	yes	—
136	C₂²×C₈	—	2	2	no	—
137	C₂²×C₈	—	22	2	no	—
138	C₂²×C₈	—	22	2	no	—
139	C₂×C₄²	Dic₂₂	1	1	yes	characteristic
140	C₂×C₄²	C₄×D₁₁	1	1	yes	characteristic
141	C₂×C₄²	C₄×D₁₁	1	1	yes	characteristic
142	C₂×C₄²	D₄₄	1	1	yes	characteristic
143	C₂×C₄²	C₂×Dic₁₁	1	1	yes	characteristic
144	C₂×C₄²	C₂²⋊D₁₁	1	1	yes	characteristic
145	C₂×C₄²	C₂²⋊D₁₁	1	1	yes	characteristic
146	C₄⋊C₈	—	2	4	no	—
147	C₄⋊C₈	—	11	4	no	—
148	C₄⋊C₈	—	22	2	no	—
149	C₄⋊C₈	—	44	2	no	—
150	C₄⋊C₈	—	44	1	no	—
151	C₄⋊C₈	—	44	1	no	—
152	C₄⋊C₈	—	44	1	no	—
153	C₄⋊C₈	—	44	1	no	—
154	C₄×C₈	—	4	1	no	—
155	C₄×C₈	—	4	1	no	—
156	C₄×C₈	—	44	1	no	—
157	C₄×C₈	—	44	1	no	—
158	C₂×C₁₁	#1277	1	1	yes	characteristic
159	C₂×C₁₁	#1279	1	1	yes	characteristic
160	C₂×C₁₁	#1283	1	1	yes	characteristic
161	C₂×C₁₁	#1296	1	1	yes	characteristic
162	C₂×C₁₁	#1299	1	1	yes	characteristic
163	C₂×C₁₁	#1301	1	1	yes	characteristic
164	C₂×C₁₁	#1303	1	1	yes	characteristic
165	C₂²×C₄	C₄×Dic₁₁	1	1	yes	characteristic
166	C₂²×C₄	Dic₁₁⋊C₄	1	1	yes	characteristic
167	C₂²×C₄	Dic₁₁⋊C₄	1	1	yes	characteristic
168	C₂²×C₄	C₄⋊Dic₁₁	1	1	yes	characteristic
169	C₂²×C₄	(C₂×D₁₁)⋊C₄	1	1	yes	characteristic
170	C₂²×C₄	(C₂×D₁₁)⋊C₄	1	1	yes	characteristic
171	C₂²×C₄	C₂²⋊Dic₁₁	1	1	yes	characteristic
172	C₂×C₈	—	2	4	no	—
173	C₂×C₈	—	4	2	no	—
174	C₂×C₈	—	4	2	no	—
175	C₂×C₈	—	22	4	no	—
176	C₂×C₈	—	44	2	no	—
177	C₂×C₈	—	44	2	no	—
178	C₂×C₈	—	44	2	no	—
179	C₂×C₈	—	44	2	no	—
180	C₂×C₈	—	44	2	no	—
181	C₄²	—	2	2	no	—
182	C₄²	—	2	2	no	—
183	C₄²	—	2	1	no	—
184	C₄²	—	2	1	no	—
185	C₄²	—	4	1	no	—
186	C₄²	C₈×D₁₁	1	2	yes	—
187	C₄²	#4313	1	2	yes	—
188	C₄²	#4314	1	1	yes	characteristic
189	C₄²	#4314	1	1	yes	characteristic
190	C₄²	D₈₈	1	1	yes	characteristic
191	C₄²	Dic₄₄	1	1	yes	characteristic
192	C₄²	#4323	1	1	yes	characteristic
193	C₄²	#4324	1	1	yes	characteristic
194	C₄²	#4325	1	1	yes	characteristic
195	C₄²	#4326	1	1	yes	characteristic
196	C₁₁	#7031	1	1	yes	characteristic, complemented, Sylow
197	C₂³	C₁₁⋊#95	1	1	yes	characteristic
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	#66869	1	2	yes	—
208	C₂×C₄	#66871	1	1	yes	characteristic
209	C₂×C₄	#66872	1	1	yes	characteristic
210	C₂×C₄	#66873	1	1	yes	characteristic
211	C₂×C₄	#66874	1	1	yes	characteristic
212	C₂×C₄	C₈×Dic₁₁	1	1	yes	characteristic
213	C₂×C₄	Dic₁₁⋊C₈	1	2	yes	—
214	C₂×C₄	#66879	1	1	yes	characteristic
215	C₂×C₄	#66880	1	1	yes	characteristic
216	C₂×C₄	#66880	1	1	yes	characteristic
217	C₂×C₄	#66881	1	1	yes	characteristic
218	C₂×C₄	#66882	1	1	yes	characteristic
219	C₂×C₄	(C₂×D₁₁)⋊C₈	1	2	yes	—
220	C₂×C₄	#66885	1	1	yes	characteristic
221	C₂×C₄	#66885	1	1	yes	characteristic
222	C₂×C₄	#66886	1	2	yes	—
223	C₂×C₄	#66887	1	1	yes	characteristic
224	C₂×C₄	#66888	1	1	yes	characteristic
225	C₂×C₄	#66896	1	1	yes	characteristic
226	C₂×C₄	#66899	1	1	yes	characteristic
227	C₈	—	4	4	no	—
228	C₈	—	44	4	no	—
229	C₈	—	88	2	no	—
230	C₂²	#10622161	1	1	yes	characteristic
231	C₂²	#10622167	1	1	yes	characteristic
232	C₂²	#10622171	1	1	yes	characteristic
233	C₂²	#10622188	1	1	yes	characteristic
234	C₂²	#10622189	1	1	yes	characteristic
235	C₂²	#10622192	1	1	yes	characteristic
236	C₂²	#10622196	1	1	yes	characteristic
237	C₄	—	2	1	no	—
238	C₄	—	2	1	no	—
239	C₄	—	2	1	no	—
240	C₄	—	2	1	no	—
241	C₄	—	2	1	no	—
242	C₄	—	2	1	no	—
243	C₄	—	4	1	no	—
244	C₄	#10622094	1	2	yes	—
245	C₄	#10622097	1	2	yes	—
246	C₄	#10622117	1	2	yes	—
247	C₄	#10622118	1	2	yes	—
248	C₄	#10622119	1	1	yes	characteristic
249	C₄	#10622120	1	1	yes	characteristic
250	C₄	#10622123	1	1	yes	characteristic
251	C₄	#10622124	1	1	yes	characteristic
252	C₂	#13066450	1	1	yes	characteristic
253	C₂	#13066456	1	1	yes	characteristic
254	C₂	#13066465	1	1	yes	characteristic
255	C₂	#13066500	1	1	yes	characteristic
256	C₂	#13066506	1	1	yes	characteristic
257	C₂	#13066512	1	1	yes	characteristic
258	C₂	#13066519	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	(C₂×D₁₁)⋊C₄
commutator	C₂×C₄×C₁₁	C₄×C₈
Frattini	C₄³	C₂×D₁₁
Fitting	C₄×C₁₁×C₄⋊C₈	C₂
socle, prime radical	C₂³×C₁₁	#95
cosocle	C₄³×C₁₁	C₂²
hypercenter	C₄×C₄⋊C₈	D₁₁
hypocenter	C₁₁	#7031
LCS	G ▷ C₂×C₄×C₁₁ ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
33. C₄⋊(C₁₁⋊C₈)	C₈	semidirect
196. C₁₁	#7031	semidirect

Maximal subgroups

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

subgroup	index
#13068768	2
C₄×C₄⋊(C₁₁⋊C₈)	2
C₄×C₁₁×C₄⋊C₈	2
#7031	11