Subgroups — #14672313

Order 2816.

350 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13070372	C₂	1	1	yes	maximal, characteristic
2	C₄×#66876	C₂	1	1	yes	maximal, characteristic
3	#13066700	C₂	1	4	yes	maximal
4	C₂×C₁₁×#446	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×Q₈×C₁₁	C₄	1	1	yes	characteristic
6	C₁₁×#446	—	2	1	no	—
7	C₁₁×#446	—	2	1	no	—
8	C₁₁×#446	—	2	1	no	—
9	C₁₁×#446	—	2	1	no	—
10	C₂×C₁₁×C₄⋊C₈	C₄	1	2	yes	—
11	C₂×C₁₁×C₄⋊C₈	C₄	1	1	yes	characteristic
12	C₂×C₁₁×#97	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
14	C₂×#66876	C₂²	1	2	yes	—
15	C₂×#66876	C₂²	1	2	yes	—
16	C₄⋊(C₁₁⋊C₁₆)	—	2	1	no	—
17	C₄⋊(C₁₁⋊C₁₆)	—	2	1	no	—
18	#10622099	—	2	1	no	—
19	C₄×C₁₁⋊C₁₆	—	2	1	no	—
20	C₂²×Q₈×C₁₁	C₈	1	1	yes	characteristic
21	C₂²×C₈×C₁₁	C₂×C₄	1	2	yes	—
22	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
24	C₄×Q₈×C₁₁	—	2	1	no	—
25	C₄×Q₈×C₁₁	—	2	1	no	—
26	C₄×Q₈×C₁₁	—	2	1	no	—
27	C₄×Q₈×C₁₁	—	2	1	no	—
28	C₂×C₁₁×C₄⋊C₄	C₈	1	1	yes	characteristic
29	C₂×C₁₁×C₄⋊C₄	C₈	1	2	yes	—
30	C₂×C₄²×C₁₁	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
32	C₁₁×C₄⋊C₈	—	2	2	no	—
33	C₁₁×C₄⋊C₈	—	2	2	no	—
34	C₁₁×C₄⋊C₈	—	2	1	no	—
35	C₁₁×C₄⋊C₈	—	2	1	no	—
36	C₁₁×#97	—	2	2	no	—
37	C₁₁×#97	—	2	2	no	—
38	C₄×C₈×C₁₁	D₄	1	1	yes	characteristic
39	C₄×C₈×C₁₁	D₄	1	1	yes	characteristic
40	C₄×C₈×C₁₁	D₄	1	1	yes	characteristic
41	C₄×C₈×C₁₁	D₄	1	1	yes	characteristic
42	#66876	—	2	2	no	—
43	#66876	—	2	4	no	—
44	#66876	Q₈	1	4	yes	complemented
45	C₂×C₁₁⋊C₁₆	—	2	2	no	—
46	C₂×C₁₁⋊C₁₆	—	2	2	no	—
47	#13268	—	11	1	no	maximal, Sylow
48	C₂×Q₈×C₁₁	—	2	1	no	—
49	C₂×Q₈×C₁₁	—	2	1	no	—
50	C₂×Q₈×C₁₁	—	4	2	no	—
51	C₂²×C₄×C₁₁	C₂×C₈	1	1	yes	characteristic
52	C₂²×C₄×C₁₁	C₂×C₈	1	2	yes	—
53	C₂²×C₄×C₁₁	C₂×C₈	1	2	yes	—
54	C₂²×C₄×C₁₁	C₂×C₈	1	1	yes	characteristic
55	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₈×C₁₁	—	2	2	no	—
57	C₂×C₈×C₁₁	—	2	2	no	—
58	C₂×C₈×C₁₁	—	2	4	no	—
59	C₂×C₈×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
60	C₂×C₈×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
63	C₂×C₈×C₁₁	C₂×Q₈	1	2	yes	—
64	C₂×C₈×C₁₁	#41	1	2	yes	—
65	C₁₁×C₄⋊C₄	—	2	1	no	—
66	C₁₁×C₄⋊C₄	—	2	1	no	—
67	C₁₁×C₄⋊C₄	—	2	2	no	—
68	C₁₁×C₄⋊C₄	—	2	2	no	—
69	C₄²×C₁₁	—	2	2	no	—
70	C₄²×C₁₁	—	2	2	no	—
71	C₄²×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
72	C₄²×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
73	C₄²×C₁₁	M₁₆	1	1	yes	characteristic
74	C₄²×C₁₁	M₁₆	1	1	yes	characteristic
75	C₁₁⋊C₁₆	—	2	4	no	—
76	C₁₁⋊C₁₆	—	4	2	no	—
77	C₂×#446	D₁₁	1	1	yes	characteristic
78	#2150	—	11	1	no	—
79	C₄×M₃₂	—	11	1	no	—
80	#1377	—	11	4	no	—
81	C₂³×C₁₁	C₂²×C₈	1	1	yes	characteristic
82	Q₈×C₁₁	—	4	2	no	—
83	Q₈×C₁₁	—	4	2	no	—
84	C₂×C₄×C₁₁	—	2	2	no	—
85	C₂×C₄×C₁₁	—	2	2	no	—
86	C₂×C₄×C₁₁	—	2	2	no	—
87	C₂×C₄×C₁₁	—	4	1	no	—
88	C₂×C₄×C₁₁	—	4	2	no	—
89	C₂×C₄×C₁₁	—	4	2	no	—
90	C₂×C₄×C₁₁	C₂²⋊C₈	1	1	yes	characteristic
91	C₂×C₄×C₁₁	C₂²⋊C₈	1	1	yes	characteristic
92	C₂×C₄×C₁₁	C₂²⋊C₈	1	1	yes	characteristic
93	C₂×C₄×C₁₁	C₂²⋊C₈	1	1	yes	characteristic
94	C₂×C₄×C₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₁₁	C₄×Q₈	1	2	yes	—
96	C₂×C₄×C₁₁	C₂×M₁₆	1	1	yes	characteristic
97	C₂×C₄×C₁₁	#131	1	2	yes	—
98	C₈×C₁₁	—	2	1	no	—
99	C₈×C₁₁	—	2	1	no	—
100	C₈×C₁₁	—	4	2	no	—
101	C₈×C₁₁	C₂²⋊Q₈	1	2	yes	—
102	C₈×C₁₁	#124	1	1	yes	characteristic
103	C₈×C₁₁	C₄⋊Q₈	1	1	yes	characteristic
104	C₂×C₄×Q₈	Dic₁₁	1	1	yes	characteristic
105	C₂×M₃₂	—	11	2	no	—
106	C₂×M₃₂	—	11	2	no	—
107	#446	—	2	1	no	—
108	#446	—	2	1	no	—
109	#446	—	2	1	no	—
110	#446	—	2	1	no	—
111	C₂×C₄⋊C₈	Dic₁₁	1	2	yes	—
112	C₂×C₄⋊C₈	Dic₁₁	1	1	yes	characteristic
113	C₂×#97	C₂×D₁₁	1	2	yes	—
114	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
115	C₄⋊C₁₆	—	22	1	no	—
116	C₄⋊C₁₆	—	22	1	no	—
117	#346	—	22	1	no	—
118	C₄×C₁₆	—	22	1	no	—
119	C₂²×C₁₁	—	2	1	no	—
120	C₂²×C₁₁	—	2	1	no	—
121	C₂²×C₁₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
122	C₂²×C₁₁	C₈×Q₈	1	2	yes	—
123	C₄×C₁₁	—	2	1	no	—
124	C₄×C₁₁	—	2	1	no	—
125	C₄×C₁₁	—	2	1	no	—
126	C₄×C₁₁	—	4	2	no	—
127	C₄×C₁₁	—	4	2	no	—
128	C₄×C₁₁	#349	1	2	yes	—
129	C₄×C₁₁	#391	1	1	yes	characteristic
130	C₄×C₁₁	#408	1	1	yes	characteristic
131	C₄×C₁₁	#446	1	2	yes	—
132	C₂²×Q₈	C₁₁⋊C₈	1	1	yes	characteristic
133	C₂²×C₈	C₂×Dic₁₁	1	2	yes	—
134	C₂²×C₈	C₂×Dic₁₁	1	1	yes	characteristic
135	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
136	C₄×Q₈	—	2	1	no	—
137	C₄×Q₈	—	2	1	no	—
138	C₄×Q₈	—	2	1	no	—
139	C₄×Q₈	—	2	1	no	—
140	C₂×C₄⋊C₄	C₁₁⋊C₈	1	1	yes	characteristic
141	C₂×C₄⋊C₄	C₁₁⋊C₈	1	2	yes	—
142	C₂×C₄²	C₂×Dic₁₁	1	2	yes	—
143	C₂×C₄²	C₂×Dic₁₁	1	1	yes	characteristic
144	M₃₂	—	11	4	no	—
145	M₃₂	—	22	2	no	—
146	M₃₂	—	22	4	no	—
147	C₂×C₁₆	—	22	2	no	—
148	C₂×C₁₆	—	22	2	no	—
149	C₄⋊C₈	—	2	2	no	—
150	C₄⋊C₈	—	2	2	no	—
151	C₄⋊C₈	—	2	1	no	—
152	C₄⋊C₈	—	2	1	no	—
153	#97	—	2	2	no	—
154	#97	—	2	2	no	—
155	C₄×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
156	C₄×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
157	C₄×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
158	C₄×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
159	C₂×C₁₁	—	2	2	no	—
160	C₂×C₁₁	#1984	1	1	yes	characteristic
161	C₂×C₁₁	C₂×#349	1	1	yes	characteristic
162	C₂×C₁₁	#2115	1	1	yes	characteristic
163	C₂×Q₈	—	2	1	no	—
164	C₂×Q₈	—	2	1	no	—
165	C₂×Q₈	—	4	2	no	—
166	C₂²×C₄	C₂×C₁₁⋊C₈	1	1	yes	characteristic
167	C₂²×C₄	C₂×C₁₁⋊C₈	1	2	yes	—
168	C₂²×C₄	C₂×C₁₁⋊C₈	1	2	yes	—
169	C₂²×C₄	C₂×C₁₁⋊C₈	1	1	yes	characteristic
170	C₂²×C₄	C₂²×Dic₁₁	1	1	yes	characteristic
171	C₂×C₈	—	2	2	no	—
172	C₂×C₈	—	2	4	no	—
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	C₂²⋊Dic₁₁	1	1	yes	characteristic
175	C₂×C₈	C₂²⋊Dic₁₁	1	1	yes	characteristic
176	C₂×C₈	Q₈×D₁₁	1	2	yes	—
177	C₂×C₈	#4343	1	2	yes	—
178	C₂×C₈	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
179	C₂×C₈	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
180	C₄⋊C₄	—	2	1	no	—
181	C₄⋊C₄	—	2	1	no	—
182	C₄⋊C₄	—	2	2	no	—
183	C₄⋊C₄	—	2	2	no	—
184	C₄²	—	2	2	no	—
185	C₄²	—	2	2	no	—
186	C₄²	#4318	1	1	yes	characteristic
187	C₄²	#4318	1	1	yes	characteristic
188	C₄²	C₂²⋊Dic₁₁	1	1	yes	characteristic
189	C₄²	C₂²⋊Dic₁₁	1	1	yes	characteristic
190	C₁₆	—	22	4	no	—
191	C₁₆	—	44	2	no	—
192	C₁₁	#13268	1	1	yes	characteristic, complemented, Sylow
193	C₂³	C₂²×C₁₁⋊C₈	1	1	yes	characteristic
194	Q₈	—	4	2	no	—
195	Q₈	—	4	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	4	1	no	—
200	C₂×C₄	—	4	2	no	—
201	C₂×C₄	—	4	2	no	—
202	C₂×C₄	C₂²⋊(C₁₁⋊C₈)	1	1	yes	characteristic
203	C₂×C₄	C₂²⋊(C₁₁⋊C₈)	1	1	yes	characteristic
204	C₂×C₄	C₂²⋊(C₁₁⋊C₈)	1	1	yes	characteristic
205	C₂×C₄	C₂²⋊(C₁₁⋊C₈)	1	1	yes	characteristic
206	C₂×C₄	C₂×#4318	1	1	yes	characteristic
207	C₂×C₄	Q₈×Dic₁₁	1	2	yes	—
208	C₂×C₄	#67001	1	2	yes	—
209	C₂×C₄	C₂×C₂²⋊Dic₁₁	1	1	yes	characteristic
210	C₈	—	2	1	no	—
211	C₈	—	2	1	no	—
212	C₈	—	4	2	no	—
213	C₈	#66997	1	1	yes	characteristic
214	C₈	#66999	1	2	yes	—
215	C₈	#67000	1	1	yes	characteristic
216	C₂²	—	2	1	no	—
217	C₂²	—	2	1	no	—
218	C₂²	Q₈×C₁₁⋊C₈	1	2	yes	—
219	C₂²	C₂×C₂²⋊(C₁₁⋊C₈)	1	1	yes	characteristic
220	C₄	—	2	1	no	—
221	C₄	—	2	1	no	—
222	C₄	—	2	1	no	—
223	C₄	—	4	2	no	—
224	C₄	—	4	2	no	—
225	C₄	#10622191	1	2	yes	—
226	C₄	#10622637	1	2	yes	—
227	C₄	#10622843	1	1	yes	characteristic
228	C₄	#10622847	1	1	yes	characteristic
229	C₂	—	2	2	no	—
230	C₂	#13069553	1	1	yes	characteristic
231	C₂	C₂×#10622191	1	1	yes	characteristic
232	C₂	#13070237	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊Dic₁₁
commutator, prime radical	C₂³×C₁₁	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₁₁
Fitting	C₂×C₁₁×#446	C₂
socle	C₂²×C₁₁	C₂×C₂²⋊C₈
cosocle	C₂²×C₈×C₁₁	C₂³
hypercenter	C₂×#446	D₁₁
hypocenter	C₁₁	#13268
LCS	G ▷ C₂³×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#446

Complements

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

N	H ≅ G/N	type
44. #66876	Q₈	semidirect
192. C₁₁	#13268	semidirect

Maximal subgroups

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

subgroup	index
#13070372	2
C₄×#66876	2
#13066700	2
C₂×C₁₁×#446	2
#13268	11