Subgroups — #14669829

Order 2816.

430 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	C₂×C₄⋊(C₁₁⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
2	C₂×#10622202	C₂	1	2	yes	maximal
3	#13068780	C₂	1	1	yes	maximal, characteristic
4	#13066707	C₂	1	2	yes	maximal
5	C₂×C₄×C₁₁×Q₁₆	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₁×Q₁₆	C₂²	1	1	yes	characteristic
7	C₂×C₄×Q₈×C₁₁	—	2	1	no	—
8	C₄×C₁₁×Q₁₆	—	2	2	no	—
9	C₄×C₁₁×Q₁₆	C₄	1	4	yes	complemented
10	C₂×C₁₁×#107	C₂²	1	1	yes	characteristic
11	C₂×C₁₁×#103	—	2	1	no	—
12	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
13	C₂×#66882	C₂²	1	2	yes	—
14	C₂×#66881	—	2	1	no	—
15	C₂²×C₁₁⋊C₁₆	C₂²	1	2	yes	—
16	#10622549	—	2	1	no	—
17	#10622202	—	2	4	no	—
18	C₄⋊(C₁₁⋊C₁₆)	—	2	2	no	—
19	C₂²×Q₈×C₁₁	—	2	1	no	—
20	C₂×C₁₁×Q₁₆	—	2	1	no	—
21	C₂×C₁₁×Q₁₆	—	2	1	no	—
22	C₂×C₁₁×Q₁₆	—	2	1	no	—
23	C₂×C₁₁×Q₁₆	C₂×C₄	1	2	yes	—
24	C₂×C₁₁×Q₁₆	D₄	1	2	yes	—
25	C₂×C₁₁×Q₁₆	Q₈	1	2	yes	—
26	C₂²×C₈×C₁₁	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
28	C₄×Q₈×C₁₁	—	2	4	no	—
29	C₄×Q₈×C₁₁	—	4	2	no	—
30	C₂×C₁₁×C₄⋊C₄	—	2	1	no	—
31	C₂×C₁₁×C₄⋊C₄	—	4	1	no	—
32	C₂×C₄²×C₁₁	—	4	1	no	—
33	C₂×C₄²×C₁₁	D₄	1	1	yes	characteristic
34	C₂×C₄⋊Dic₁₁	—	2	2	no	—
35	C₁₁×#107	—	2	1	no	—
36	C₁₁×#107	C₂×C₄	1	2	yes	—
37	C₁₁×#103	—	2	4	no	—
38	C₄×C₈×C₁₁	C₂×C₄	1	2	yes	—
39	C₄×C₈×C₁₁	D₄	1	2	yes	—
40	C₁₁⋊#95	—	4	1	no	—
41	C₁₁⋊#95	—	4	1	no	—
42	#66882	—	2	2	no	—
43	#66882	—	2	2	no	—
44	#66881	—	4	2	no	—
45	C₂×C₁₁⋊C₁₆	—	2	2	no	—
46	C₂×C₁₁⋊C₁₆	—	2	2	no	—
47	C₂×C₁₁⋊C₁₆	—	2	2	no	—
48	#12814	—	11	1	no	maximal, Sylow
49	C₂×Q₈×C₁₁	—	2	2	no	—
50	C₂×Q₈×C₁₁	—	2	2	no	—
51	C₂×Q₈×C₁₁	—	2	2	no	—
52	C₂×Q₈×C₁₁	—	4	1	no	—
53	C₂×Q₈×C₁₁	—	4	1	no	—
54	C₂×Q₈×C₁₁	—	4	1	no	—
55	C₂²×C₄×C₁₁	—	4	1	no	—
56	C₂²×C₄×C₁₁	—	4	1	no	—
57	C₂²×C₄×C₁₁	D₈	1	1	yes	characteristic
58	C₂²×C₄×C₁₁	D₈	1	1	yes	characteristic
59	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
60	C₂²×Dic₁₁	—	4	2	no	—
61	C₁₁×Q₁₆	—	2	2	no	—
62	C₁₁×Q₁₆	—	2	2	no	—
63	C₁₁×Q₁₆	—	2	2	no	—
64	C₁₁×Q₁₆	C₄⋊C₄	1	4	yes	—
65	C₂×C₈×C₁₁	—	2	1	no	—
66	C₂×C₈×C₁₁	—	2	1	no	—
67	C₂×C₈×C₁₁	C₂²⋊C₄	1	2	yes	—
68	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
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×Q₈	1	1	yes	characteristic
72	C₂×C₈×C₁₁	#41	1	1	yes	characteristic
73	C₂×C₈×C₁₁	#41	1	1	yes	characteristic
74	C₁₁×C₄⋊C₄	—	2	2	no	—
75	C₁₁×C₄⋊C₄	—	4	4	no	—
76	C₁₁×C₄⋊C₄	—	4	1	no	—
77	C₄²×C₁₁	—	4	4	no	—
78	C₄²×C₁₁	C₂²⋊C₄	1	2	yes	—
79	C₄²×C₁₁	SD₁₆	1	2	yes	—
80	C₄⋊Dic₁₁	—	4	2	no	—
81	C₄⋊Dic₁₁	—	4	2	no	—
82	C₁₁⋊C₁₆	—	4	2	no	—
83	C₂×C₄×Q₁₆	D₁₁	1	1	yes	characteristic
84	C₂×C₄⋊C₁₆	—	11	1	no	—
85	C₂×#358	—	11	2	no	—
86	#1846	—	11	1	no	—
87	#1380	—	11	2	no	—
88	C₂³×C₁₁	C₂×D₈	1	1	yes	characteristic
89	Q₈×C₁₁	—	2	4	no	—
90	Q₈×C₁₁	—	4	2	no	—
91	Q₈×C₁₁	—	4	2	no	—
92	Q₈×C₁₁	—	4	2	no	—
93	C₂×C₄×C₁₁	—	2	2	no	—
94	C₂×C₄×C₁₁	—	2	2	no	—
95	C₂×C₄×C₁₁	—	4	2	no	—
96	C₂×C₄×C₁₁	—	4	2	no	—
97	C₂×C₄×C₁₁	—	4	2	no	—
98	C₂×C₄×C₁₁	—	4	2	no	—
99	C₂×C₄×C₁₁	—	8	2	no	—
100	C₂×C₄×C₁₁	#102	1	2	yes	—
101	C₂×C₄×C₁₁	#102	1	2	yes	—
102	C₂×C₄×C₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₁₁	#121	1	1	yes	characteristic
104	C₂×C₄×C₁₁	C₂²⋊Q₈	1	1	yes	characteristic
105	C₂×C₄×C₁₁	C₂×SD₁₆	1	1	yes	characteristic
106	C₂×C₄×C₁₁	AGL(1,ℤ/8)	1	1	yes	characteristic
107	C₂×C₄×C₁₁	#137	1	1	yes	characteristic
108	C₂×Dic₁₁	—	8	2	no	—
109	C₂×Dic₁₁	—	8	2	no	—
110	C₂×Dic₁₁	—	8	2	no	—
111	C₈×C₁₁	—	2	2	no	—
112	C₈×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₈×C₁₁	#117	1	1	yes	characteristic
114	C₈×C₁₁	#121	1	1	yes	characteristic
115	C₈×C₁₁	C₂²⋊Q₈	1	1	yes	characteristic
116	C₂²×Q₁₆	C₂×D₁₁	1	1	yes	characteristic
117	C₂×C₄×Q₈	—	2	1	no	—
118	C₂²×C₁₆	—	11	2	no	—
119	C₄×Q₁₆	—	2	2	no	—
120	C₄×Q₁₆	Dic₁₁	1	4	yes	complemented
121	C₂×#107	—	11	2	no	—
122	C₂×#107	C₂×D₁₁	1	1	yes	characteristic
123	C₂×#106	—	22	1	no	—
124	C₂×#103	—	2	1	no	—
125	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
126	#382	—	22	1	no	—
127	C₄⋊C₁₆	—	22	2	no	—
128	#358	—	22	4	no	—
129	C₂²×C₁₁	C₂×#102	1	1	yes	characteristic
130	C₂²×C₁₁	#459	1	1	yes	characteristic
131	C₂²×C₁₁	#474	1	1	yes	characteristic
132	C₂²×C₁₁	#508	1	2	yes	—
133	C₂²×C₁₁	#510	1	2	yes	—
134	C₄×C₁₁	—	2	2	no	—
135	C₄×C₁₁	—	2	2	no	—
136	C₄×C₁₁	—	4	4	no	—
137	C₄×C₁₁	—	8	2	no	—
138	C₄×C₁₁	#380	1	1	yes	characteristic
139	C₄×C₁₁	#417	1	1	yes	characteristic
140	C₄×C₁₁	#461	1	1	yes	characteristic
141	C₄×C₁₁	#476	1	1	yes	characteristic
142	Dic₁₁	—	16	2	no	—
143	C₂²×Q₈	—	2	1	no	—
144	C₂×Q₁₆	—	2	1	no	—
145	C₂×Q₁₆	—	2	1	no	—
146	C₂×Q₁₆	—	2	1	no	—
147	C₂×Q₁₆	Dic₂₂	1	2	yes	—
148	C₂×Q₁₆	D₄₄	1	2	yes	—
149	C₂×Q₁₆	C₂×Dic₁₁	1	2	yes	—
150	C₂²×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
151	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
152	C₄×Q₈	—	2	4	no	—
153	C₄×Q₈	—	4	2	no	—
154	C₂×C₄⋊C₄	—	2	1	no	—
155	C₂×C₄⋊C₄	—	4	1	no	—
156	C₂×C₄⋊C₄	—	22	2	no	—
157	C₂×C₄²	—	4	1	no	—
158	C₂×C₄²	C₂²⋊D₁₁	1	1	yes	characteristic
159	C₂×C₁₆	—	22	2	no	—
160	C₂×C₁₆	—	22	2	no	—
161	C₂×C₁₆	—	22	2	no	—
162	#107	—	2	1	no	—
163	#107	—	22	2	no	—
164	#107	—	22	2	no	—
165	#107	C₂×Dic₁₁	1	2	yes	—
166	#106	—	44	2	no	—
167	#103	—	2	4	no	—
168	C₄×C₈	C₂×Dic₁₁	1	2	yes	—
169	C₄×C₈	C₂²⋊D₁₁	1	2	yes	—
170	#95	—	44	1	no	—
171	#95	—	44	1	no	—
172	C₂×C₁₁	#1802	1	1	yes	characteristic
173	C₂×C₁₁	#2138	1	1	yes	characteristic
174	C₂×C₁₁	#2141	1	1	yes	characteristic
175	C₂×C₁₁	#2211	1	2	yes	—
176	C₂×C₁₁	#2229	1	2	yes	—
177	C₂×Q₈	—	2	2	no	—
178	C₂×Q₈	—	2	2	no	—
179	C₂×Q₈	—	2	2	no	—
180	C₂×Q₈	—	4	1	no	—
181	C₂×Q₈	—	4	1	no	—
182	C₂×Q₈	—	4	1	no	—
183	C₂²×C₄	—	4	1	no	—
184	C₂²×C₄	—	4	1	no	—
185	C₂²×C₄	—	44	2	no	—
186	C₂²×C₄	#4323	1	1	yes	characteristic
187	C₂²×C₄	#4323	1	1	yes	characteristic
188	C₂²×C₄	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
189	Q₁₆	—	2	2	no	—
190	Q₁₆	—	2	2	no	—
191	Q₁₆	—	2	2	no	—
192	Q₁₆	C₄⋊Dic₁₁	1	4	yes	—
193	C₂×C₈	—	2	1	no	—
194	C₂×C₈	—	2	1	no	—
195	C₂×C₈	C₂²⋊Dic₁₁	1	2	yes	—
196	C₂×C₈	C₂×Dic₂₂	1	1	yes	characteristic
197	C₂×C₈	C₂×D₄₄	1	1	yes	characteristic
198	C₂×C₈	#4339	1	1	yes	characteristic
199	C₂×C₈	#4339	1	1	yes	characteristic
200	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
201	C₂×C₈	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
202	C₄⋊C₄	—	2	2	no	—
203	C₄⋊C₄	—	4	4	no	—
204	C₄⋊C₄	—	4	1	no	—
205	C₄⋊C₄	—	44	2	no	—
206	C₄⋊C₄	—	44	2	no	—
207	C₄²	—	4	4	no	—
208	C₄²	#4324	1	2	yes	—
209	C₄²	C₂²⋊Dic₁₁	1	2	yes	—
210	C₁₆	—	44	2	no	—
211	C₁₁	#12814	1	1	yes	characteristic, complemented, Sylow
212	C₂³	C₂×#4323	1	1	yes	characteristic
213	Q₈	—	2	4	no	—
214	Q₈	—	4	2	no	—
215	Q₈	—	4	2	no	—
216	Q₈	—	4	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	4	2	no	—
220	C₂×C₄	—	4	2	no	—
221	C₂×C₄	—	4	2	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	—	8	2	no	—
224	C₂×C₄	—	88	2	no	—
225	C₂×C₄	—	88	2	no	—
226	C₂×C₄	—	88	2	no	—
227	C₂×C₄	#66896	1	2	yes	—
228	C₂×C₄	#66896	1	2	yes	—
229	C₂×C₄	#66977	1	1	yes	characteristic
230	C₂×C₄	#66983	1	1	yes	characteristic
231	C₂×C₄	C₂×#4324	1	1	yes	characteristic
232	C₂×C₄	#67002	1	1	yes	characteristic
233	C₂×C₄	#67004	1	1	yes	characteristic
234	C₂×C₄	C₂×C₂²⋊Dic₁₁	1	1	yes	characteristic
235	C₈	—	2	2	no	—
236	C₈	#66977	1	1	yes	characteristic
237	C₈	C₂×C₄⋊Dic₁₁	1	1	yes	characteristic
238	C₈	#66979	1	1	yes	characteristic
239	C₈	#66983	1	1	yes	characteristic
240	C₂²	#10622620	1	1	yes	characteristic
241	C₂²	#10622628	1	1	yes	characteristic
242	C₂²	#10622792	1	2	yes	—
243	C₂²	#10622807	1	2	yes	—
244	C₂²	C₂×#66896	1	1	yes	characteristic
245	C₄	—	2	2	no	—
246	C₄	—	2	2	no	—
247	C₄	—	4	4	no	—
248	C₄	—	8	2	no	—
249	C₄	—	176	2	no	—
250	C₄	#10622621	1	1	yes	characteristic
251	C₄	#10622632	1	1	yes	characteristic
252	C₄	#10622825	1	1	yes	characteristic
253	C₄	#10622845	1	1	yes	characteristic
254	C₂	#13068616	1	1	yes	characteristic
255	C₂	#13070323	1	1	yes	characteristic
256	C₂	#13070336	1	1	yes	characteristic
257	C₂	#13070678	1	2	yes	—
258	C₂	#13070780	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#4323
commutator	C₂×C₈×C₁₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₁
Fitting	C₂×C₄×C₁₁×Q₁₆	C₂
socle, prime radical	C₂³×C₁₁	C₂×D₈
cosocle	C₂²×C₈×C₁₁	C₂³
hypercenter	C₂×C₄×Q₁₆	D₁₁
hypocenter	C₁₁	#12814
LCS	G ▷ C₂×C₈×C₁₁ ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
9. C₄×C₁₁×Q₁₆	C₄	semidirect
120. C₄×Q₁₆	Dic₁₁	semidirect
211. C₁₁	#12814	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊(C₁₁⋊C₁₆)	2
C₂×#10622202	2
#13068780	2
#13066707	2
C₂×C₄×C₁₁×Q₁₆	2
#12814	11