Subgroups — #14669830

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×Dic₄₄	C₂	1	1	yes	maximal, characteristic
2	C₂×#10622141	C₂	1	2	yes	maximal
3	#13068780	C₂	1	1	yes	maximal, characteristic
4	#13066708	C₂	1	2	yes	maximal
5	C₂×C₁₁×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₄₄	C₂²	1	1	yes	characteristic
7	C₂×C₄×Dic₂₂	—	2	1	no	—
8	C₂²×C₁₁×C₁₆	C₂²	1	2	yes	—
9	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
10	C₂×#66882	C₂²	1	2	yes	—
11	C₂×#66882	C₂²	1	1	yes	characteristic
12	C₂×#66881	—	2	1	no	—
13	C₂×#66880	—	2	1	no	—
14	#10622549	—	2	1	no	—
15	C₄×Dic₄₄	—	2	2	no	—
16	C₄×Dic₄₄	C₄	1	4	yes	complemented
17	C₁₁×C₄⋊C₁₆	—	2	2	no	—
18	#10622141	—	2	4	no	—
19	C₂²×Dic₂₂	—	2	1	no	—
20	C₂²×C₈×C₁₁	D₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
22	C₂×C₄²×C₁₁	D₄	1	1	yes	characteristic
23	C₂×C₄⋊Dic₁₁	—	2	1	no	—
24	C₂×C₄⋊Dic₁₁	—	2	2	no	—
25	C₂×Dic₁₁⋊C₄	—	4	1	no	—
26	C₂×C₄×Dic₁₁	—	4	1	no	—
27	C₂×Dic₄₄	—	2	1	no	—
28	C₂×Dic₄₄	—	2	1	no	—
29	C₂×Dic₄₄	—	2	1	no	—
30	C₂×Dic₄₄	C₂×C₄	1	2	yes	—
31	C₂×Dic₄₄	D₄	1	2	yes	—
32	C₂×Dic₄₄	Q₈	1	2	yes	—
33	C₄×Dic₂₂	—	2	4	no	—
34	C₄×Dic₂₂	—	4	2	no	—
35	C₂×C₁₁×C₁₆	—	2	2	no	—
36	C₂×C₁₁×C₁₆	—	2	2	no	—
37	C₂×C₁₁×C₁₆	—	2	2	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	1	no	—
43	#66882	—	2	2	no	—
44	#66882	—	2	2	no	—
45	#66882	C₂×C₄	1	2	yes	—
46	#66881	—	4	2	no	—
47	#66880	—	2	4	no	—
48	#12814	—	11	1	no	maximal, Sylow
49	C₂²×C₄×C₁₁	D₈	1	1	yes	characteristic
50	C₂²×C₄×C₁₁	D₈	1	1	yes	characteristic
51	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
52	C₂²×Dic₁₁	—	4	1	no	—
53	C₂²×Dic₁₁	—	4	1	no	—
54	C₂²×Dic₁₁	—	4	2	no	—
55	C₂×Dic₂₂	—	2	2	no	—
56	C₂×Dic₂₂	—	2	2	no	—
57	C₂×Dic₂₂	—	2	2	no	—
58	C₂×Dic₂₂	—	4	1	no	—
59	C₂×Dic₂₂	—	4	1	no	—
60	C₂×Dic₂₂	—	4	1	no	—
61	C₂×C₈×C₁₁	—	2	1	no	—
62	C₂×C₈×C₁₁	—	2	1	no	—
63	C₂×C₈×C₁₁	C₂²⋊C₄	1	2	yes	—
64	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
66	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
67	C₂×C₈×C₁₁	C₂×Q₈	1	1	yes	characteristic
68	C₂×C₈×C₁₁	#41	1	1	yes	characteristic
69	C₂×C₈×C₁₁	#41	1	1	yes	characteristic
70	C₄²×C₁₁	C₂²⋊C₄	1	2	yes	—
71	C₄²×C₁₁	SD₁₆	1	2	yes	—
72	C₄⋊Dic₁₁	—	2	2	no	—
73	C₄⋊Dic₁₁	—	4	2	no	—
74	C₄⋊Dic₁₁	—	4	1	no	—
75	C₄⋊Dic₁₁	—	4	2	no	—
76	Dic₁₁⋊C₄	—	4	4	no	—
77	C₄×Dic₁₁	—	4	4	no	—
78	Dic₄₄	—	2	2	no	—
79	Dic₄₄	—	2	2	no	—
80	Dic₄₄	—	2	2	no	—
81	Dic₄₄	C₄⋊C₄	1	4	yes	—
82	C₁₁×C₁₆	—	4	2	no	—
83	C₂×C₄×Q₁₆	—	11	1	no	—
84	C₂×C₄⋊C₁₆	D₁₁	1	1	yes	characteristic
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	C₂×C₄×C₁₁	—	2	2	no	—
90	C₂×C₄×C₁₁	—	2	2	no	—
91	C₂×C₄×C₁₁	#102	1	2	yes	—
92	C₂×C₄×C₁₁	#102	1	2	yes	—
93	C₂×C₄×C₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₁₁	#121	1	1	yes	characteristic
95	C₂×C₄×C₁₁	C₂²⋊Q₈	1	1	yes	characteristic
96	C₂×C₄×C₁₁	C₂×SD₁₆	1	1	yes	characteristic
97	C₂×C₄×C₁₁	AGL(1,ℤ/8)	1	1	yes	characteristic
98	C₂×C₄×C₁₁	#137	1	1	yes	characteristic
99	C₂×Dic₁₁	—	4	2	no	—
100	C₂×Dic₁₁	—	4	2	no	—
101	C₂×Dic₁₁	—	4	2	no	—
102	C₂×Dic₁₁	—	4	2	no	—
103	C₂×Dic₁₁	—	8	2	no	—
104	C₂×Dic₁₁	—	8	2	no	—
105	C₂×Dic₁₁	—	8	2	no	—
106	C₂×Dic₁₁	—	8	2	no	—
107	Dic₂₂	—	2	4	no	—
108	Dic₂₂	—	4	2	no	—
109	Dic₂₂	—	4	2	no	—
110	Dic₂₂	—	4	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	—	11	1	no	—
117	C₂×C₄×Q₈	—	22	1	no	—
118	C₂²×C₁₆	C₂×D₁₁	1	2	yes	—
119	C₄×Q₁₆	—	11	4	no	—
120	C₄×Q₁₆	—	22	2	no	—
121	C₂×#107	—	11	1	no	—
122	C₂×#107	—	11	2	no	—
123	C₂×#106	—	22	1	no	—
124	C₂×#103	—	22	1	no	—
125	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
126	#382	—	22	1	no	—
127	C₄⋊C₁₆	—	2	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	#380	1	1	yes	characteristic
137	C₄×C₁₁	#417	1	1	yes	characteristic
138	C₄×C₁₁	#461	1	1	yes	characteristic
139	C₄×C₁₁	#476	1	1	yes	characteristic
140	Dic₁₁	—	4	4	no	—
141	Dic₁₁	—	8	2	no	—
142	Dic₁₁	—	16	2	no	—
143	C₂²×Q₈	—	22	1	no	—
144	C₂×Q₁₆	—	11	2	no	—
145	C₂×Q₁₆	—	11	2	no	—
146	C₂×Q₁₆	—	11	2	no	—
147	C₂×Q₁₆	—	22	1	no	—
148	C₂×Q₁₆	—	22	1	no	—
149	C₂×Q₁₆	—	22	1	no	—
150	C₂²×C₈	D₄₄	1	1	yes	characteristic
151	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
152	C₄×Q₈	—	22	4	no	—
153	C₄×Q₈	—	44	2	no	—
154	C₂×C₄⋊C₄	—	22	1	no	—
155	C₂×C₄⋊C₄	—	22	2	no	—
156	C₂×C₄⋊C₄	—	44	1	no	—
157	C₂×C₄²	—	44	1	no	—
158	C₂×C₄²	D₄₄	1	1	yes	characteristic
159	C₂×C₁₆	—	2	2	no	—
160	C₂×C₁₆	—	2	2	no	—
161	C₂×C₁₆	—	2	2	no	—
162	#107	—	11	2	no	—
163	#107	—	22	1	no	—
164	#107	—	22	2	no	—
165	#107	—	22	2	no	—
166	#106	—	44	2	no	—
167	#103	—	22	4	no	—
168	C₄×C₈	C₄×D₁₁	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	—	22	2	no	—
178	C₂×Q₈	—	22	2	no	—
179	C₂×Q₈	—	22	2	no	—
180	C₂×Q₈	—	44	1	no	—
181	C₂×Q₈	—	44	1	no	—
182	C₂×Q₈	—	44	1	no	—
183	C₂²×C₄	—	44	1	no	—
184	C₂²×C₄	—	44	1	no	—
185	C₂²×C₄	—	44	2	no	—
186	C₂²×C₄	D₈₈	1	1	yes	characteristic
187	C₂²×C₄	D₈₈	1	1	yes	characteristic
188	C₂²×C₄	C₂×D₄₄	1	1	yes	characteristic
189	Q₁₆	—	11	4	no	—
190	Q₁₆	—	22	2	no	—
191	Q₁₆	—	22	2	no	—
192	Q₁₆	—	22	2	no	—
193	C₂×C₈	—	2	1	no	—
194	C₂×C₈	—	2	1	no	—
195	C₂×C₈	(C₂×D₁₁)⋊C₄	1	2	yes	—
196	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
197	C₂×C₈	D₄×D₁₁	1	1	yes	characteristic
198	C₂×C₈	#4341	1	1	yes	characteristic
199	C₂×C₈	Q₈×D₁₁	1	1	yes	characteristic
200	C₂×C₈	#4343	1	1	yes	characteristic
201	C₂×C₈	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
202	C₄⋊C₄	—	22	2	no	—
203	C₄⋊C₄	—	44	4	no	—
204	C₄⋊C₄	—	44	1	no	—
205	C₄⋊C₄	—	44	2	no	—
206	C₄⋊C₄	—	44	2	no	—
207	C₄²	—	44	4	no	—
208	C₄²	#4314	1	2	yes	—
209	C₄²	(C₂×D₁₁)⋊C₄	1	2	yes	—
210	C₁₆	—	4	2	no	—
211	C₁₁	#12814	1	1	yes	characteristic, complemented, Sylow
212	C₂³	C₂×D₈₈	1	1	yes	characteristic
213	Q₈	—	22	4	no	—
214	Q₈	—	44	2	no	—
215	Q₈	—	44	2	no	—
216	Q₈	—	44	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	44	2	no	—
220	C₂×C₄	—	44	2	no	—
221	C₂×C₄	—	44	2	no	—
222	C₂×C₄	—	44	2	no	—
223	C₂×C₄	—	88	2	no	—
224	C₂×C₄	—	88	2	no	—
225	C₂×C₄	—	88	2	no	—
226	C₂×C₄	—	88	2	no	—
227	C₂×C₄	#66885	1	2	yes	—
228	C₂×C₄	#66885	1	2	yes	—
229	C₂×C₄	#66948	1	1	yes	characteristic
230	C₂×C₄	#66950	1	1	yes	characteristic
231	C₂×C₄	C₂×#4314	1	1	yes	characteristic
232	C₂×C₄	#66961	1	1	yes	characteristic
233	C₂×C₄	#66962	1	1	yes	characteristic
234	C₂×C₄	C₂×(C₂×D₁₁)⋊C₄	1	1	yes	characteristic
235	C₈	—	2	2	no	—
236	C₈	C₄⋊C₄×D₁₁	1	1	yes	characteristic
237	C₈	#66945	1	1	yes	characteristic
238	C₈	#66991	1	1	yes	characteristic
239	C₈	#66999	1	1	yes	characteristic
240	C₂²	#10622456	1	1	yes	characteristic
241	C₂²	#10622459	1	1	yes	characteristic
242	C₂²	#10622517	1	2	yes	—
243	C₂²	#10622522	1	2	yes	—
244	C₂²	C₂×#66885	1	1	yes	characteristic
245	C₄	—	2	2	no	—
246	C₄	—	2	2	no	—
247	C₄	—	44	4	no	—
248	C₄	—	88	2	no	—
249	C₄	—	176	2	no	—
250	C₄	#10622454	1	1	yes	characteristic
251	C₄	#10622455	1	1	yes	characteristic
252	C₄	#10622600	1	1	yes	characteristic
253	C₄	#10622746	1	1	yes	characteristic
254	C₂	#13068617	1	1	yes	characteristic
255	C₂	#13070324	1	1	yes	characteristic
256	C₂	#13070337	1	1	yes	characteristic
257	C₂	#13070679	1	2	yes	—
258	C₂	#13070781	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₈₈
commutator	C₂×C₈×C₁₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₁
Fitting	C₂×C₁₁×C₄⋊C₁₆	C₂
socle, prime radical	C₂³×C₁₁	C₂×D₈
cosocle	C₂²×C₈×C₁₁	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₁₁
hypocenter	C₁₁	#12814
LCS	G ▷ C₂×C₈×C₁₁ ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
16. C₄×Dic₄₄	C₄	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×Dic₄₄	2
C₂×#10622141	2
#13068780	2
#13066708	2
C₂×C₁₁×C₄⋊C₁₆	2
#12814	11