Subgroups — #15623462

Order 2816.

1762 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13082442	C₂	1	2	yes	maximal
2	#13079168	C₂	1	6	yes	maximal, complemented
3	#13079155	C₂	1	6	yes	maximal, complemented
4	#13079147	C₂	1	6	yes	maximal, complemented
5	#13079130	C₂	1	3	yes	maximal, complemented
6	C₂×#10622296	C₂	1	3	yes	maximal, complemented
7	#13077216	C₂	1	2	yes	maximal
8	#13077211	C₂	1	2	yes	maximal
9	C₄×C₁₁×#131	C₂	1	1	yes	maximal, characteristic
10	C₂×C₁₁×#131	C₂²	1	1	yes	characteristic
11	C₂×C₁₁×#131	C₂²	1	1	yes	characteristic
12	C₄×C₁₁×#41	C₂²	1	1	yes	characteristic
13	#10623358	C₂²	1	4	yes	—
14	#10623100	—	2	4	no	—
15	#10623070	—	2	12	no	—
16	#10623050	C₂²	1	12	yes	—
17	#10623001	C₂²	1	6	yes	—
18	C₂×#66923	C₂²	1	6	yes	—
19	C₁₁×#405	C₂²	1	3	yes	—
20	C₁₁×#405	C₂²	1	3	yes	—
21	C₄×C₁₁×M₁₆	C₂²	1	3	yes	—
22	C₂×C₄×C₈×C₁₁	C₂²	1	3	yes	—
23	#10622734	C₂²	1	12	yes	—
24	#10622734	C₂²	1	12	yes	—
25	#10622719	C₂²	1	6	yes	—
26	#10622719	C₂²	1	6	yes	—
27	C₂×#66882	C₂²	1	6	yes	—
28	C₂×#66881	C₂²	1	6	yes	—
29	#10622315	C₂²	1	24	yes	—
30	#10622296	C₂²	1	3	yes	—
31	#10622296	C₂²	1	6	yes	—
32	#10622296	C₂²	1	6	yes	—
33	#10622296	C₂²	1	6	yes	—
34	#10622296	C₂²	1	1	yes	characteristic
35	#10622296	C₂²	1	6	yes	—
36	#10622295	C₂²	1	6	yes	—
37	#10622293	C₂²	1	6	yes	—
38	C₂×C₁₁×#41	C₂³	1	1	yes	characteristic
39	C₁₁×#131	Q₈	1	8	yes	complemented
40	C₁₁×#131	Q₈	1	8	yes	complemented
41	C₂×C₁₁×M₁₆	C₂³	1	3	yes	—
42	C₂×C₁₁×M₁₆	C₂³	1	3	yes	—
43	C₂²×C₈×C₁₁	C₂³	1	3	yes	—
44	C₂²×C₈×C₁₁	C₂³	1	3	yes	—
45	C₄×Q₈×C₁₁	D₄	1	2	yes	—
46	C₄×D₄×C₁₁	D₄	1	6	yes	—
47	C₁₁×#117	C₂³	1	3	yes	—
48	C₂×C₄²×C₁₁	C₂³	1	3	yes	—
49	Q₈×Dic₁₁	—	2	4	no	—
50	D₄×Dic₁₁	—	2	12	no	—
51	#66979	C₂³	1	12	yes	—
52	C₂×C₄⋊Dic₁₁	C₂³	1	12	yes	—
53	#66977	—	2	12	no	—
54	C₂×Dic₁₁⋊C₄	—	2	12	no	—
55	#66943	—	2	12	no	—
56	#66941	—	2	12	no	—
57	#66931	—	2	24	no	—
58	#66923	—	2	12	no	—
59	#66923	C₂³	1	6	yes	—
60	#66923	C₂³	1	12	yes	—
61	#66923	C₂³	1	2	yes	—
62	#66922	C₂³	1	12	yes	—
63	C₄×Dic₂₂	—	2	12	no	—
64	C₁₁×#97	C₂³	1	6	yes	—
65	C₄×C₈×C₁₁	C₂³	1	3	yes	—
66	C₄×C₈×C₁₁	C₂³	1	3	yes	—
67	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
68	C₄×C₈×C₁₁	C₂³	1	3	yes	—
69	#66882	C₂³	1	12	yes	—
70	#66882	C₂³	1	12	yes	—
71	#66882	C₂³	1	2	yes	—
72	#66882	C₂³	1	6	yes	—
73	#66881	C₂³	1	12	yes	—
74	#66881	C₂³	1	12	yes	—
75	#66881	C₂³	1	2	yes	—
76	#66881	C₂³	1	6	yes	—
77	#60666	—	11	1	no	maximal, Sylow
78	C₁₁×#41	C₂×Q₈	1	4	yes	—
79	C₁₁×#41	C₂×Q₈	1	4	yes	—
80	C₂×Q₈×C₁₁	C₂×D₄	1	1	yes	characteristic
81	C₂×D₄×C₁₁	C₂×D₄	1	3	yes	—
82	C₂²×C₄×C₁₁	C₂×D₄	1	6	yes	—
83	C₂²×C₄×C₁₁	C₂⁴	1	3	yes	—
84	C₂²×Dic₁₁	—	2	12	no	—
85	C₂×Dic₂₂	—	2	12	no	—
86	C₁₁×M₁₆	C₂×Q₈	1	12	yes	—
87	C₁₁×M₁₆	C₂×Q₈	1	12	yes	—
88	C₂×C₈×C₁₁	C₂×Q₈	1	12	yes	—
89	C₂×C₈×C₁₁	C₂×Q₈	1	12	yes	—
90	C₂×C₈×C₁₁	C₂⁴	1	1	yes	characteristic
91	C₂×C₈×C₁₁	C₂⁴	1	3	yes	—
92	C₂×C₈×C₁₁	C₂⁴	1	3	yes	—
93	C₂×C₈×C₁₁	C₂⁴	1	1	yes	characteristic
94	C₁₁×C₄⋊C₄	C₂×D₄	1	6	yes	—
95	C₁₁×C₂²⋊C₄	C₂×D₄	1	6	yes	—
96	C₄²×C₁₁	C₂×D₄	1	6	yes	—
97	C₄²×C₁₁	C₂⁴	1	1	yes	characteristic
98	C₄²×C₁₁	C₂⁴	1	3	yes	—
99	C₂²⋊Dic₁₁	—	2	12	no	—
100	C₄⋊Dic₁₁	—	2	12	no	—
101	C₄⋊Dic₁₁	—	2	12	no	—
102	C₄⋊Dic₁₁	C₂⁴	1	4	yes	—
103	C₄⋊Dic₁₁	C₂⁴	1	12	yes	—
104	Dic₁₁⋊C₄	—	2	4	no	—
105	Dic₁₁⋊C₄	—	2	12	no	—
106	Dic₁₁⋊C₄	—	2	24	no	—
107	C₄×Dic₁₁	—	2	12	no	—
108	#3504	—	11	2	no	—
109	#3166	—	11	6	no	—
110	#3164	—	11	6	no	—
111	#3163	—	11	6	no	—
112	#3160	—	11	3	no	—
113	C₂×#499	—	11	3	no	—
114	#2913	—	11	2	no	—
115	#2912	—	11	2	no	—
116	C₄×#131	D₁₁	1	1	yes	characteristic
117	C₂³×C₁₁	C₂²×D₄	1	3	yes	—
118	Q₈×C₁₁	C₄⋊Q₈	1	4	yes	—
119	D₄×C₁₁	C₄⋊Q₈	1	12	yes	—
120	C₂×C₄×C₁₁	—	4	6	no	—
121	C₂×C₄×C₁₁	C₂²×D₄	1	2	yes	—
122	C₂×C₄×C₁₁	C₂²×D₄	1	6	yes	—
123	C₂×C₄×C₁₁	C₂²×D₄	1	3	yes	—
124	C₂×C₄×C₁₁	C₂²×Q₈	1	6	yes	—
125	C₂×C₄×C₁₁	C₂²×Q₈	1	6	yes	—
126	C₂×C₄×C₁₁	C₂⁵	1	1	yes	characteristic
127	C₂×Dic₁₁	—	2	4	no	—
128	C₂×Dic₁₁	—	2	12	no	—
129	C₂×Dic₁₁	—	8	12	no	—
130	Dic₂₂	—	8	12	no	—
131	C₈×C₁₁	C₂²×Q₈	1	2	yes	—
132	C₈×C₁₁	C₂²×Q₈	1	2	yes	—
133	C₈×C₁₁	C₂²×Q₈	1	6	yes	—
134	C₈×C₁₁	C₂²×Q₈	1	6	yes	—
135	C₂×#131	C₂×D₁₁	1	1	yes	characteristic
136	C₂×#131	C₂×D₁₁	1	1	yes	characteristic
137	#557	—	22	4	no	—
138	#554	—	22	12	no	—
139	#544	—	11	12	no	—
140	#533	—	11	6	no	—
141	C₂×#125	—	11	6	no	—
142	#520	—	11	4	no	—
143	C₄×#41	C₂×D₁₁	1	1	yes	characteristic
144	#501	—	11	24	no	—
145	#500	—	11	6	no	—
146	#499	—	11	3	no	—
147	#499	—	11	6	no	—
148	#499	—	11	6	no	—
149	#499	—	11	6	no	—
150	#499	—	11	1	no	—
151	#499	—	11	6	no	—
152	#498	—	11	6	no	—
153	#428	—	11	12	no	—
154	#428	—	11	12	no	—
155	#427	—	11	6	no	—
156	#427	—	11	6	no	—
157	C₂×#107	—	11	6	no	—
158	C₂×#106	—	11	6	no	—
159	#405	C₂×D₁₁	1	3	yes	—
160	#405	C₂×D₁₁	1	3	yes	—
161	C₄×M₁₆	C₂×D₁₁	1	3	yes	—
162	C₂×C₄×C₈	C₂×D₁₁	1	3	yes	—
163	C₂²×C₁₁	—	2	6	no	—
164	C₂²×C₁₁	C₂×C₄⋊Q₈	1	6	yes	—
165	C₂²×C₁₁	C₂³×D₄	1	1	yes	characteristic
166	C₄×C₁₁	—	2	2	no	—
167	C₄×C₁₁	—	2	6	no	—
168	C₄×C₁₁	C₂×C₄⋊Q₈	1	6	yes	—
169	C₄×C₁₁	C₂³×Q₈	1	1	yes	characteristic
170	C₄×C₁₁	C₂³×Q₈	1	1	yes	characteristic
171	Dic₁₁	—	4	4	no	—
172	Dic₁₁	—	4	12	no	—
173	C₂×#41	C₂²×D₁₁	1	1	yes	characteristic
174	#131	Dic₂₂	1	8	yes	complemented
175	#131	Dic₂₂	1	8	yes	complemented
176	C₂×M₁₆	C₂²×D₁₁	1	3	yes	—
177	C₂×M₁₆	C₂²×D₁₁	1	3	yes	—
178	C₂²×C₈	C₂²×D₁₁	1	3	yes	—
179	C₂²×C₈	C₂²×D₁₁	1	3	yes	—
180	C₄⋊Q₈	—	11	12	no	—
181	C₄⋊Q₈	—	22	12	no	—
182	#125	—	11	6	no	—
183	#125	—	11	12	no	—
184	#125	—	11	2	no	—
185	#125	—	22	12	no	—
186	#125	—	22	12	no	—
187	C₂²⋊Q₈	—	22	12	no	—
188	C₂²⋊Q₈	—	22	24	no	—
189	C₄×Q₈	—	22	4	no	—
190	C₄×Q₈	—	22	12	no	—
191	C₄×Q₈	D₄₄	1	2	yes	—
192	C₄×D₄	—	22	12	no	—
193	C₄×D₄	D₄₄	1	6	yes	—
194	#117	—	11	12	no	—
195	#117	C₂²×D₁₁	1	3	yes	—
196	C₂×C₄⋊C₄	—	11	12	no	—
197	C₂×C₄⋊C₄	—	22	12	no	—
198	C₂×C₄²	C₂²×D₁₁	1	3	yes	—
199	#107	—	11	6	no	—
200	#107	—	11	12	no	—
201	#107	—	11	2	no	—
202	#107	—	11	12	no	—
203	#106	—	11	6	no	—
204	#106	—	11	12	no	—
205	#106	—	11	2	no	—
206	#106	—	11	12	no	—
207	#97	C₂²×D₁₁	1	6	yes	—
208	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
209	C₄×C₈	C₂²×D₁₁	1	3	yes	—
210	C₄×C₈	C₂²×D₁₁	1	3	yes	—
211	C₄×C₈	C₂²×D₁₁	1	3	yes	—
212	C₂×C₁₁	—	2	6	no	—
213	C₂×C₁₁	C₂²×C₄⋊Q₈	1	1	yes	characteristic
214	C₂×C₁₁	#3584	1	2	yes	—
215	#41	C₂×Dic₂₂	1	4	yes	—
216	#41	C₂×Dic₂₂	1	4	yes	—
217	C₂×Q₈	—	22	12	no	—
218	C₂×Q₈	C₂×D₄₄	1	1	yes	characteristic
219	C₂×D₄	C₂×D₄₄	1	3	yes	—
220	C₂²×C₄	—	22	12	no	—
221	C₂²×C₄	C₂×D₄₄	1	6	yes	—
222	C₂²×C₄	C₂³×D₁₁	1	3	yes	—
223	M₁₆	C₂×Dic₂₂	1	12	yes	—
224	M₁₆	C₂×Dic₂₂	1	12	yes	—
225	C₂×C₈	C₂×Dic₂₂	1	12	yes	—
226	C₂×C₈	C₂×Dic₂₂	1	12	yes	—
227	C₂×C₈	C₂³×D₁₁	1	1	yes	characteristic
228	C₂×C₈	C₂³×D₁₁	1	1	yes	characteristic
229	C₂×C₈	C₂³×D₁₁	1	3	yes	—
230	C₂×C₈	C₂³×D₁₁	1	3	yes	—
231	C₄⋊C₄	—	11	12	no	—
232	C₄⋊C₄	—	11	4	no	—
233	C₄⋊C₄	—	22	12	no	—
234	C₄⋊C₄	—	22	12	no	—
235	C₄⋊C₄	—	22	24	no	—
236	C₄⋊C₄	—	22	4	no	—
237	C₄⋊C₄	—	22	12	no	—
238	C₄⋊C₄	C₂×D₄₄	1	6	yes	—
239	C₂²⋊C₄	—	22	12	no	—
240	C₂²⋊C₄	C₂×D₄₄	1	6	yes	—
241	C₄²	—	22	12	no	—
242	C₄²	C₂×D₄₄	1	6	yes	—
243	C₄²	C₂³×D₁₁	1	1	yes	characteristic
244	C₄²	C₂³×D₁₁	1	3	yes	—
245	C₁₁	#60666	1	1	yes	characteristic, complemented, Sylow
246	C₂³	C₂²×D₄₄	1	3	yes	—
247	Q₈	—	88	12	no	—
248	Q₈	#66922	1	4	yes	—
249	D₄	#66922	1	12	yes	—
250	C₂×C₄	—	4	6	no	—
251	C₂×C₄	—	22	4	no	—
252	C₂×C₄	—	22	12	no	—
253	C₂×C₄	—	88	12	no	—
254	C₂×C₄	C₂²×Dic₂₂	1	6	yes	—
255	C₂×C₄	C₂²×Dic₂₂	1	6	yes	—
256	C₂×C₄	C₂²×D₄₄	1	2	yes	—
257	C₂×C₄	C₂²×D₄₄	1	6	yes	—
258	C₂×C₄	C₂²×D₄₄	1	3	yes	—
259	C₂×C₄	C₂⁴×D₁₁	1	1	yes	characteristic
260	C₈	C₂²×Dic₂₂	1	2	yes	—
261	C₈	C₂²×Dic₂₂	1	2	yes	—
262	C₈	C₂²×Dic₂₂	1	6	yes	—
263	C₈	C₂²×Dic₂₂	1	6	yes	—
264	C₂²	—	2	6	no	—
265	C₂²	C₂×#66922	1	6	yes	—
266	C₂²	C₂³×D₄₄	1	1	yes	characteristic
267	C₄	—	2	2	no	—
268	C₄	—	2	6	no	—
269	C₄	—	44	12	no	—
270	C₄	—	44	4	no	—
271	C₄	C₂×#66922	1	6	yes	—
272	C₄	C₂³×Dic₂₂	1	1	yes	characteristic
273	C₄	C₂³×Dic₂₂	1	1	yes	characteristic
274	C₂	—	2	6	no	—
275	C₂	C₂²×#66922	1	1	yes	characteristic
276	C₂	#13083390	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂³×D₄₄
commutator, cosocle	C₂×C₄×C₁₁	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₁₁
Fitting	C₄×C₁₁×#131	C₂
socle	C₂²×C₁₁	C₂³×D₄
prime radical	C₂×C₁₁×#41	C₂³
hypercenter	C₄×#131	D₁₁
hypocenter	C₁₁	#60666
LCS	G ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₄×#131

Complements

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

N	H ≅ G/N	type
2. #13079168	C₂	semidirect
3. #13079155	C₂	semidirect
4. #13079147	C₂	semidirect
5. #13079130	C₂	semidirect
6. C₂×#10622296	C₂	semidirect
39. C₁₁×#131	Q₈	semidirect
40. C₁₁×#131	Q₈	semidirect
174. #131	Dic₂₂	semidirect
175. #131	Dic₂₂	semidirect
245. C₁₁	#60666	semidirect

Maximal subgroups

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

subgroup	index
#13082442	2
#13079168	2
#13079155	2
#13079147	2
#13079130	2
C₂×#10622296	2
#13077216	2
#13077211	2
C₄×C₁₁×#131	2
#60666	11