Subgroups — #14783744

Order 2816.

664 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10622738	C₂	1	1	yes	maximal, characteristic
2	#13070191	C₂	1	8	yes	maximal, complemented
3	C₂×#10622197	C₂	1	2	yes	maximal
4	#13068326	C₂	1	2	yes	maximal
5	C₂×#10622195	C₂	1	1	yes	maximal, characteristic
6	C₁₁×#2905	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₁₁×M₁₆	C₂²	1	1	yes	characteristic
8	C₂²×C₁₁×M₁₆	C₂²	1	1	yes	characteristic
9	C₂×C₁₁×#117	C₂²	1	1	yes	characteristic
10	C₂×#66979	C₂²	1	2	yes	—
11	C₂²×#4318	C₂²	1	2	yes	—
12	C₁₁×#424	—	2	2	no	—
13	C₁₁×#424	—	2	2	no	—
14	C₁₁×#423	C₂²	1	4	yes	—
15	#10622738	—	2	2	no	—
16	#10622738	C₂²	1	4	yes	—
17	C₂×#66883	—	2	2	no	—
18	#10622197	C₂²	1	8	yes	—
19	#10622195	C₂²	1	4	yes	—
20	#10622161	C₂²	1	8	yes	—
21	C₂³×C₄×C₁₁	C₂³	1	1	yes	characteristic
22	C₂×C₁₁×M₁₆	—	2	2	no	—
23	C₂×C₁₁×M₁₆	—	2	2	no	—
24	C₂×C₁₁×M₁₆	—	2	1	no	—
25	C₂×C₁₁×M₁₆	—	2	2	no	—
26	C₂×C₁₁×M₁₆	—	2	2	no	—
27	C₂×C₁₁×M₁₆	D₄	1	2	yes	—
28	C₂×C₁₁×M₁₆	C₂³	1	2	yes	—
29	C₂×C₁₁×M₁₆	C₂³	1	2	yes	—
30	C₂²×C₈×C₁₁	—	2	1	no	—
31	C₂²×C₈×C₁₁	D₄	1	2	yes	—
32	C₁₁×#117	—	2	1	no	—
33	C₁₁×#117	—	2	1	no	—
34	C₁₁×#117	Q₈	1	2	yes	—
35	C₁₁×#117	Q₈	1	2	yes	—
36	C₂×C₁₁×C₄⋊C₄	—	2	1	no	—
37	C₂×C₁₁×C₂²⋊C₄	—	2	1	no	—
38	C₂×C₄²×C₁₁	C₂³	1	2	yes	—
39	C₂×C₂²⋊Dic₁₁	—	2	2	no	—
40	#66979	—	2	2	no	—
41	#66979	—	2	2	no	—
42	#66979	—	2	2	no	—
43	#66979	C₂³	1	4	yes	—
44	C₂×C₄⋊Dic₁₁	—	2	2	no	—
45	C₂×C₄×Dic₁₁	—	2	2	no	—
46	C₂×#4318	—	2	2	no	—
47	C₂×#4318	—	2	2	no	—
48	C₂×#4318	—	2	2	no	—
49	C₂×#4318	—	2	2	no	—
50	C₂×#4318	—	2	2	no	—
51	C₂×#4318	C₂³	1	4	yes	—
52	C₂²×C₁₁⋊C₈	—	2	2	no	—
53	C₁₁×C₄⋊C₈	—	2	8	no	—
54	#66883	—	2	8	no	—
55	#23751	—	11	1	no	maximal, Sylow
56	C₂⁴×C₁₁	#41	1	1	yes	characteristic
57	C₂²×C₄×C₁₁	—	2	2	no	—
58	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₁	C₂×Q₈	1	1	yes	characteristic
61	C₂²×C₄×C₁₁	C₂×Q₈	1	1	yes	characteristic
62	C₂²×C₄×C₁₁	#41	1	1	yes	characteristic
63	C₂²×C₄×C₁₁	#41	1	2	yes	—
64	C₂²×C₄×C₁₁	#41	1	2	yes	—
65	C₂²×C₄×C₁₁	#41	1	2	yes	—
66	C₂²×C₄×C₁₁	#41	1	2	yes	—
67	C₂²×C₄×C₁₁	C₂⁴	1	1	yes	characteristic
68	C₂²×Dic₁₁	—	4	2	no	—
69	C₁₁×M₁₆	—	2	4	no	—
70	C₁₁×M₁₆	—	2	4	no	—
71	C₁₁×M₁₆	—	2	4	no	—
72	C₁₁×M₁₆	—	4	1	no	—
73	C₁₁×M₁₆	—	4	1	no	—
74	C₂×C₈×C₁₁	—	2	2	no	—
75	C₂×C₈×C₁₁	—	2	2	no	—
76	C₂×C₈×C₁₁	—	2	2	no	—
77	C₂×C₈×C₁₁	—	4	2	no	—
78	C₂×C₈×C₁₁	C₂×D₄	1	4	yes	—
79	C₁₁×C₄⋊C₄	—	2	2	no	—
80	C₁₁×C₄⋊C₄	—	2	2	no	—
81	C₁₁×C₂²⋊C₄	—	2	2	no	—
82	C₁₁×C₂²⋊C₄	—	2	2	no	—
83	C₄²×C₁₁	—	2	2	no	—
84	C₄²×C₁₁	C₂×Q₈	1	4	yes	—
85	C₂²⋊Dic₁₁	—	2	4	no	—
86	C₂²⋊Dic₁₁	—	4	2	no	—
87	C₄⋊Dic₁₁	—	2	4	no	—
88	C₄⋊Dic₁₁	—	4	2	no	—
89	C₄×Dic₁₁	—	2	4	no	—
90	C₄×Dic₁₁	—	4	2	no	—
91	#4318	—	2	4	no	—
92	#4318	—	2	4	no	—
93	#4318	—	4	2	no	—
94	#4318	—	4	2	no	—
95	C₂×C₁₁⋊C₈	—	2	4	no	—
96	C₂×C₁₁⋊C₈	—	4	2	no	—
97	C₂×C₁₁⋊C₈	—	4	2	no	—
98	C₂×#430	—	11	1	no	—
99	#2905	D₁₁	1	1	yes	characteristic
100	#2102	—	11	8	no	—
101	C₂×#344	—	11	2	no	—
102	#1733	—	11	2	no	—
103	C₂×H(3,ℤ/4)	—	11	1	no	—
104	C₂³×C₁₁	—	8	1	no	—
105	C₂³×C₁₁	C₂²⋊Q₈	1	1	yes	characteristic
106	C₂³×C₁₁	C₂²⋊Q₈	1	1	yes	characteristic
107	C₂³×C₁₁	#126	1	2	yes	—
108	C₂³×C₁₁	#126	1	2	yes	—
109	C₂³×C₁₁	C₂×#41	1	1	yes	characteristic
110	C₂×C₄×C₁₁	—	2	4	no	—
111	C₂×C₄×C₁₁	—	8	2	no	—
112	C₂×C₄×C₁₁	C₂²⋊Q₈	1	1	yes	characteristic
113	C₂×C₄×C₁₁	C₂²⋊Q₈	1	1	yes	characteristic
114	C₂×C₄×C₁₁	#123	1	1	yes	characteristic
115	C₂×C₄×C₁₁	#123	1	1	yes	characteristic
116	C₂×C₄×C₁₁	#123	1	2	yes	—
117	C₂×C₄×C₁₁	#124	1	1	yes	characteristic
118	C₂×C₄×C₁₁	#124	1	2	yes	—
119	C₂×C₄×C₁₁	#124	1	1	yes	characteristic
120	C₂×C₄×C₁₁	#125	1	2	yes	—
121	C₂×C₄×C₁₁	#125	1	2	yes	—
122	C₂×C₄×C₁₁	#125	1	2	yes	—
123	C₂×C₄×C₁₁	#125	1	2	yes	—
124	C₂×C₄×C₁₁	#126	1	2	yes	—
125	C₂×C₄×C₁₁	#126	1	2	yes	—
126	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
127	C₂×C₄×C₁₁	C₂²×Q₈	1	1	yes	characteristic
128	C₂×C₄×C₁₁	C₂×#41	1	2	yes	—
129	C₂×C₄×C₁₁	C₂×#41	1	2	yes	—
130	C₂×Dic₁₁	—	4	4	no	—
131	C₂×Dic₁₁	—	8	4	no	—
132	C₈×C₁₁	—	2	4	no	—
133	C₈×C₁₁	—	4	2	no	—
134	C₁₁⋊C₈	—	4	4	no	—
135	C₂²×M₁₆	—	11	2	no	—
136	C₂²×M₁₆	C₂×D₁₁	1	1	yes	characteristic
137	C₂²×M₁₆	C₂×D₁₁	1	1	yes	characteristic
138	C₂×#117	—	11	2	no	—
139	C₂×#117	C₂×D₁₁	1	1	yes	characteristic
140	#430	—	11	4	no	—
141	#430	—	22	2	no	—
142	C₂×#108	—	22	2	no	—
143	#424	—	2	2	no	—
144	#424	—	2	2	no	—
145	#423	C₂×D₁₁	1	4	yes	—
146	#344	—	11	8	no	—
147	#339	—	11	8	no	—
148	H(3,ℤ/4)	—	11	4	no	—
149	C₂²×C₁₁	—	4	1	no	—
150	C₂²×C₁₁	—	4	1	no	—
151	C₂²×C₁₁	—	4	1	no	—
152	C₂²×C₁₁	—	4	2	no	—
153	C₂²×C₁₁	—	4	2	no	—
154	C₂²×C₁₁	#400	1	1	yes	characteristic
155	C₂²×C₁₁	#400	1	1	yes	characteristic
156	C₂²×C₁₁	#400	1	2	yes	—
157	C₂²×C₁₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
158	C₂²×C₁₁	C₂×#126	1	2	yes	—
159	C₄×C₁₁	—	4	4	no	—
160	C₄×C₁₁	#400	1	1	yes	characteristic
161	C₄×C₁₁	#400	1	1	yes	characteristic
162	C₄×C₁₁	#400	1	2	yes	—
163	C₄×C₁₁	C₂×#123	1	1	yes	characteristic
164	C₄×C₁₁	C₂×#124	1	1	yes	characteristic
165	C₄×C₁₁	C₂×#125	1	2	yes	—
166	Dic₁₁	—	8	4	no	—
167	C₂³×C₄	C₂²×D₁₁	1	1	yes	characteristic
168	C₂×M₁₆	—	2	2	no	—
169	C₂×M₁₆	—	2	2	no	—
170	C₂×M₁₆	—	2	1	no	—
171	C₂×M₁₆	—	2	2	no	—
172	C₂×M₁₆	—	2	2	no	—
173	C₂×M₁₆	—	11	4	no	—
174	C₂×M₁₆	—	22	2	no	—
175	C₂×M₁₆	—	22	2	no	—
176	C₂×M₁₆	—	22	2	no	—
177	C₂×M₁₆	—	22	2	no	—
178	C₂×M₁₆	—	22	2	no	—
179	C₂×M₁₆	C₂²⋊D₁₁	1	2	yes	—
180	C₂×M₁₆	C₂²×D₁₁	1	2	yes	—
181	C₂×M₁₆	C₂²×D₁₁	1	2	yes	—
182	C₂²×C₈	—	2	1	no	—
183	C₂²×C₈	—	22	2	no	—
184	C₂²×C₈	C₂²⋊D₁₁	1	2	yes	—
185	#117	—	2	1	no	—
186	#117	—	2	1	no	—
187	#117	—	11	4	no	—
188	#117	—	22	2	no	—
189	#117	—	22	2	no	—
190	#117	—	22	2	no	—
191	#117	Dic₂₂	1	2	yes	—
192	#117	Dic₂₂	1	2	yes	—
193	C₂×C₄⋊C₄	—	2	1	no	—
194	C₂×C₄⋊C₄	—	22	2	no	—
195	C₂×C₂²⋊C₄	—	2	1	no	—
196	C₂×C₂²⋊C₄	—	22	2	no	—
197	C₂×C₄²	—	22	2	no	—
198	C₂×C₄²	C₂²×D₁₁	1	2	yes	—
199	#108	—	22	8	no	—
200	C₄⋊C₈	—	2	8	no	—
201	C₂×C₁₁	—	2	2	no	—
202	C₂×C₁₁	—	2	1	no	—
203	C₂×C₁₁	—	2	1	no	—
204	C₂×C₁₁	—	2	2	no	—
205	C₂×C₁₁	—	2	1	no	—
206	C₂×C₁₁	C₂×#400	1	1	yes	characteristic
207	C₂⁴	#4339	1	1	yes	characteristic
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	—	44	2	no	—
210	C₂²×C₄	C₂×Dic₂₂	1	1	yes	characteristic
211	C₂²×C₄	C₂×Dic₂₂	1	1	yes	characteristic
212	C₂²×C₄	#4339	1	1	yes	characteristic
213	C₂²×C₄	#4341	1	2	yes	—
214	C₂²×C₄	#4341	1	2	yes	—
215	C₂²×C₄	#4343	1	2	yes	—
216	C₂²×C₄	#4343	1	2	yes	—
217	C₂²×C₄	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
218	C₂²×C₄	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
219	C₂²×C₄	C₂³×D₁₁	1	1	yes	characteristic
220	M₁₆	—	2	4	no	—
221	M₁₆	—	2	4	no	—
222	M₁₆	—	2	4	no	—
223	M₁₆	—	4	1	no	—
224	M₁₆	—	4	1	no	—
225	M₁₆	—	22	4	no	—
226	M₁₆	—	22	4	no	—
227	M₁₆	—	44	2	no	—
228	M₁₆	—	44	2	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	4	2	no	—
233	C₂×C₈	—	22	4	no	—
234	C₂×C₈	—	44	2	no	—
235	C₂×C₈	—	44	2	no	—
236	C₂×C₈	C₂×C₂²⋊D₁₁	1	4	yes	—
237	C₄⋊C₄	—	2	2	no	—
238	C₄⋊C₄	—	2	2	no	—
239	C₄⋊C₄	—	22	4	no	—
240	C₄⋊C₄	—	44	2	no	—
241	C₂²⋊C₄	—	2	2	no	—
242	C₂²⋊C₄	—	2	2	no	—
243	C₂²⋊C₄	—	22	4	no	—
244	C₂²⋊C₄	—	44	2	no	—
245	C₄²	—	2	2	no	—
246	C₄²	—	22	4	no	—
247	C₄²	—	44	2	no	—
248	C₄²	C₂×Dic₂₂	1	4	yes	—
249	C₁₁	#23751	1	1	yes	characteristic, complemented, Sylow
250	C₂³	—	8	1	no	—
251	C₂³	#66951	1	2	yes	—
252	C₂³	#66951	1	2	yes	—
253	C₂³	#66977	1	1	yes	characteristic
254	C₂³	#66977	1	1	yes	characteristic
255	C₂³	C₂×#4339	1	1	yes	characteristic
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	8	2	no	—
258	C₂×C₄	—	44	4	no	—
259	C₂×C₄	—	88	4	no	—
260	C₂×C₄	#66943	1	2	yes	—
261	C₂×C₄	#66943	1	2	yes	—
262	C₂×C₄	#66943	1	2	yes	—
263	C₂×C₄	#66943	1	2	yes	—
264	C₂×C₄	#66951	1	2	yes	—
265	C₂×C₄	#66951	1	2	yes	—
266	C₂×C₄	#66977	1	1	yes	characteristic
267	C₂×C₄	#66977	1	1	yes	characteristic
268	C₂×C₄	#66988	1	1	yes	characteristic
269	C₂×C₄	#66988	1	1	yes	characteristic
270	C₂×C₄	#66988	1	2	yes	—
271	C₂×C₄	#67000	1	1	yes	characteristic
272	C₂×C₄	#67000	1	2	yes	—
273	C₂×C₄	#67000	1	1	yes	characteristic
274	C₂×C₄	C₂²×Dic₂₂	1	1	yes	characteristic
275	C₂×C₄	C₂×#4341	1	2	yes	—
276	C₂×C₄	C₂×#4343	1	2	yes	—
277	C₂×C₄	C₂²×C₂²⋊D₁₁	1	1	yes	characteristic
278	C₈	—	2	4	no	—
279	C₈	—	4	2	no	—
280	C₈	—	44	4	no	—
281	C₂²	—	4	1	no	—
282	C₂²	—	4	2	no	—
283	C₂²	—	4	2	no	—
284	C₂²	—	4	1	no	—
285	C₂²	—	4	1	no	—
286	C₂²	#10622599	1	1	yes	characteristic
287	C₂²	#10622599	1	1	yes	characteristic
288	C₂²	#10622599	1	2	yes	—
289	C₂²	C₂×#66951	1	2	yes	—
290	C₂²	C₂×#66977	1	1	yes	characteristic
291	C₄	—	4	4	no	—
292	C₄	—	88	4	no	—
293	C₄	#10622599	1	1	yes	characteristic
294	C₄	#10622599	1	2	yes	—
295	C₄	#10622599	1	1	yes	characteristic
296	C₄	C₂×#66943	1	2	yes	—
297	C₄	C₂×#66988	1	1	yes	characteristic
298	C₄	C₂×#67000	1	1	yes	characteristic
299	C₂	—	2	1	no	—
300	C₂	—	2	2	no	—
301	C₂	—	2	1	no	—
302	C₂	—	2	2	no	—
303	C₂	—	2	1	no	—
304	C₂	C₂×#10622599	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#67000
commutator, cosocle	C₂²×C₄×C₁₁	C₂⁴
Frattini	C₂²×C₄	C₂³×D₁₁
Fitting	C₁₁×#2905	C₂
socle	C₂×C₁₁	C₂×#400
prime radical	C₂⁴×C₁₁	#41
hypercenter	#2905	D₁₁
hypocenter	C₁₁	#23751
LCS	G ▷ C₂²×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ #2905

Complements

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

N	H ≅ G/N	type
2. #13070191	C₂	semidirect
249. C₁₁	#23751	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10622738	2
#13070191	2
C₂×#10622197	2
#13068326	2
C₂×#10622195	2
C₁₁×#2905	2
#23751	11