Subgroups — #15850577

Order 3328.

814 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13125391	C₂	1	4	yes	maximal, complemented
2	#13125350	C₂	1	1	yes	maximal, characteristic, complemented
3	#13114435	C₂	1	2	yes	maximal, complemented
4	#13114430	C₂	1	2	yes	maximal, complemented
5	#13114425	C₂	1	2	yes	maximal, complemented
6	#13114422	C₂	1	2	yes	maximal, complemented
7	C₂×C₈×#1055	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₃×#3147	C₂	1	1	yes	maximal, characteristic
9	C₂×C₁₃×#135	—	2	2	no	—
10	C₁₃×#532	C₂²	1	2	yes	—
11	#11719175	C₂²	1	2	yes	—
12	C₁₃×#495	—	2	2	no	—
13	C₁₃×#494	C₂²	1	1	yes	characteristic
14	C₁₃×#493	C₂²	1	1	yes	characteristic
15	C₁₃×#492	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
17	#11718767	—	2	2	no	—
18	#11718754	C₂²	1	4	yes	—
19	#11718740	C₂²	1	4	yes	—
20	#11718737	C₂²	1	4	yes	—
21	#11718731	C₂²	1	4	yes	—
22	#11718702	—	2	8	no	—
23	C₄×#6105	—	2	8	no	—
24	C₂×C₄×#1055	C₂²	1	2	yes	—
25	#11718344	—	2	2	no	—
26	#11718343	C₂²	1	2	yes	—
27	#11718341	C₂²	1	2	yes	—
28	C₈×#1055	C₂²	1	1	yes	characteristic
29	C₈×#1055	C₂²	1	2	yes	—
30	C₈×#1055	C₂²	1	1	yes	characteristic
31	C₂×C₁₃×#41	—	2	2	no	—
32	C₁₃×#135	—	4	8	no	—
33	C₂×C₁₃×Q₁₆	—	2	2	no	—
34	C₂×C₁₃×SD₁₆	—	2	4	no	—
35	C₂×C₁₃×D₈	—	2	2	no	—
36	C₂²×C₈×C₁₃	D₄	1	2	yes	—
37	C₁₃×C₄⋊Q₈	C₂³	1	2	yes	—
38	C₁₃×#127	C₂³	1	2	yes	—
39	C₁₃×#124	—	2	2	no	—
40	C₁₃×#121	—	2	4	no	—
41	C₄×D₄×C₁₃	—	2	4	no	—
42	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
43	C₂×#6105	—	2	8	no	—
44	#88180	—	2	2	no	—
45	#88178	—	2	4	no	—
46	C₂²×#1055	D₄	1	4	yes	—
47	#88124	—	2	2	no	—
48	#88123	C₂³	1	4	yes	—
49	C₄×Dic₂₆	—	2	4	no	—
50	C₄×C₈×C₁₃	D₄	1	2	yes	—
51	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
52	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
53	#88072	—	2	8	no	—
54	#88058	—	2	4	no	—
55	#88057	—	2	4	no	—
56	#88050	—	2	8	no	—
57	#88048	—	2	8	no	—
58	C₄×#1055	D₄	1	4	yes	—
59	C₄×#1055	C₂³	1	2	yes	—
60	C₄×#1055	C₂³	1	2	yes	—
61	#19596	—	13	1	no	maximal, Sylow
62	C₁₃×#41	—	4	8	no	—
63	C₂×Q₈×C₁₃	—	2	2	no	—
64	C₂×D₄×C₁₃	—	2	4	no	—
65	C₂×D₄×C₁₃	—	2	2	no	—
66	C₂×D₄×C₁₃	—	4	2	no	—
67	C₂²×C₄×C₁₃	—	4	2	no	—
68	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
70	C₂×Dic₂₆	—	2	4	no	—
71	C₁₃×Q₁₆	—	4	4	no	—
72	C₁₃×SD₁₆	—	4	8	no	—
73	C₁₃×D₈	—	4	4	no	—
74	C₂×C₈×C₁₃	SD₁₆	1	8	yes	complemented
75	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
76	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
77	C₁₃×C₄⋊C₄	—	2	4	no	—
78	C₁₃×C₂²⋊C₄	—	4	4	no	—
79	C₄²×C₁₃	C₂×D₄	1	2	yes	—
80	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
81	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
82	C₂²⋊Dic₁₃	—	4	2	no	—
83	#6105	—	4	16	no	—
84	C₄⋊Dic₁₃	—	2	4	no	—
85	C₄⋊Dic₁₃	—	2	4	no	—
86	#6101	—	4	4	no	—
87	C₄×Dic₁₃	—	4	2	no	—
88	C₂×#1055	—	2	8	no	—
89	C₂×#1055	C₂×D₄	1	4	yes	—
90	C₂×#1055	C₂×D₄	1	4	yes	—
91	#3167	—	13	4	no	—
92	#3160	—	13	1	no	—
93	#3147	D₁₃	1	1	yes	characteristic
94	#1708	—	13	2	no	—
95	#1707	—	13	2	no	—
96	#1706	—	13	2	no	—
97	#1705	—	13	2	no	—
98	C₂×C₈²	—	13	1	no	—
99	C₂³×C₁₃	—	4	2	no	—
100	C₂³×C₁₃	#127	1	1	yes	characteristic
101	Q₈×C₁₃	—	4	4	no	—
102	D₄×C₁₃	—	4	4	no	—
103	D₄×C₁₃	—	4	8	no	—
104	D₄×C₁₃	—	8	4	no	—
105	C₂×C₄×C₁₃	—	2	4	no	—
106	C₂×C₄×C₁₃	—	4	2	no	—
107	C₂×C₄×C₁₃	—	8	4	no	—
108	C₂×C₄×C₁₃	#127	1	2	yes	—
109	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
110	C₂×C₄×C₁₃	C₂×SD₁₆	1	4	yes	—
111	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
112	C₂×C₄×C₁₃	C₂²×D₄	1	2	yes	—
113	C₂×Dic₁₃	—	4	4	no	—
114	Dic₂₆	—	4	8	no	—
115	C₈×C₁₃	C₂×SD₁₆	1	4	yes	—
116	C₈×C₁₃	C₂×SD₁₆	1	4	yes	—
117	#1055	—	2	4	no	—
118	#1055	#135	1	8	yes	complemented
119	C₂×#135	—	2	2	no	—
120	#533	—	13	2	no	—
121	#532	C₂×D₁₃	1	2	yes	—
122	#500	—	13	2	no	—
123	#499	—	26	2	no	—
124	#498	—	13	4	no	—
125	#498	—	13	2	no	—
126	#495	—	2	2	no	—
127	#494	C₂×D₁₃	1	1	yes	characteristic
128	#493	C₂×D₁₃	1	1	yes	characteristic
129	#492	—	13	4	no	—
130	#492	C₂×D₁₃	1	2	yes	—
131	#487	—	13	4	no	—
132	#486	—	13	4	no	—
133	#465	—	26	8	no	—
134	C₄×SD₁₆	—	26	8	no	—
135	#427	—	26	2	no	—
136	C₂×C₄×C₈	—	13	2	no	—
137	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
138	C₈²	—	13	1	no	—
139	C₈²	—	13	2	no	—
140	C₈²	—	13	1	no	—
141	C₂²×C₁₃	—	2	2	no	—
142	C₂²×C₁₃	—	8	2	no	—
143	C₂²×C₁₃	—	8	4	no	—
144	C₂²×C₁₃	#492	1	2	yes	—
145	C₂²×C₁₃	C₂×#127	1	1	yes	characteristic
146	C₄×C₁₃	—	2	2	no	—
147	C₄×C₁₃	—	8	2	no	—
148	C₄×C₁₃	#492	1	2	yes	—
149	C₄×C₁₃	C₂²×SD₁₆	1	2	yes	—
150	C₄×C₁₃	C₂×#135	1	4	yes	—
151	Dic₁₃	—	8	4	no	—
152	C₂×#41	—	2	2	no	—
153	#135	—	4	8	no	—
154	C₂×Q₁₆	—	2	2	no	—
155	C₂×SD₁₆	—	2	4	no	—
156	C₂×SD₁₆	—	26	8	no	—
157	C₂×D₈	—	2	2	no	—
158	C₂²×C₈	—	13	4	no	—
159	C₂²×C₈	C₂²⋊D₁₃	1	2	yes	—
160	C₄⋊Q₈	—	13	4	no	—
161	C₄⋊Q₈	C₂²×D₁₃	1	2	yes	—
162	#127	C₂²×D₁₃	1	2	yes	—
163	#125	—	26	2	no	—
164	#124	—	2	2	no	—
165	C₂²⋊Q₈	—	26	4	no	—
166	#121	—	2	4	no	—
167	C₄×Q₈	—	26	4	no	—
168	C₄×D₄	—	2	4	no	—
169	#117	—	26	2	no	—
170	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
171	#107	—	26	4	no	—
172	#106	—	26	8	no	—
173	#106	—	26	4	no	—
174	#103	—	26	8	no	—
175	#102	—	26	8	no	—
176	C₄×C₈	—	13	4	no	—
177	C₄×C₈	—	13	2	no	—
178	C₄×C₈	—	13	2	no	—
179	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
180	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
181	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
182	C₂×C₁₃	—	2	2	no	—
183	C₂×C₁₃	—	16	2	no	—
184	C₂×C₁₃	C₂×#492	1	1	yes	characteristic
185	C₂×C₁₃	#3147	1	2	yes	—
186	#41	—	4	8	no	—
187	C₂×Q₈	—	2	2	no	—
188	C₂×Q₈	—	26	4	no	—
189	C₂×D₄	—	2	4	no	—
190	C₂×D₄	—	2	2	no	—
191	C₂×D₄	—	4	2	no	—
192	C₂²×C₄	—	4	2	no	—
193	C₂²×C₄	D₄×D₁₃	1	2	yes	—
194	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
195	Q₁₆	—	4	4	no	—
196	SD₁₆	—	4	8	no	—
197	SD₁₆	—	52	16	no	—
198	D₈	—	4	4	no	—
199	C₂×C₈	—	13	4	no	—
200	C₂×C₈	—	13	4	no	—
201	C₂×C₈	—	26	8	no	—
202	C₂×C₈	#6105	1	8	yes	complemented
203	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
204	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
205	C₄⋊C₄	—	2	4	no	—
206	C₄⋊C₄	—	26	4	no	—
207	C₄⋊C₄	—	26	4	no	—
208	C₄⋊C₄	—	52	4	no	—
209	C₂²⋊C₄	—	4	4	no	—
210	C₂²⋊C₄	—	52	2	no	—
211	C₄²	—	52	2	no	—
212	C₄²	D₄×D₁₃	1	2	yes	—
213	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
214	C₄²	C₂³×D₁₃	1	1	yes	characteristic
215	C₁₃	#19596	1	1	yes	characteristic, complemented, Sylow
216	C₂³	—	4	2	no	—
217	C₂³	#88194	1	1	yes	characteristic
218	Q₈	—	4	4	no	—
219	Q₈	—	52	8	no	—
220	D₄	—	4	8	no	—
221	D₄	—	4	4	no	—
222	D₄	—	8	4	no	—
223	C₂×C₄	—	2	4	no	—
224	C₂×C₄	—	4	2	no	—
225	C₂×C₄	—	8	4	no	—
226	C₂×C₄	—	52	4	no	—
227	C₂×C₄	C₂×#6105	1	4	yes	—
228	C₂×C₄	#88194	1	2	yes	—
229	C₂×C₄	#88194	1	1	yes	characteristic
230	C₂×C₄	C₂×D₄×D₁₃	1	2	yes	—
231	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
232	C₈	—	13	8	no	—
233	C₈	—	26	4	no	—
234	C₈	C₂×#6105	1	4	yes	—
235	C₈	C₂×#6105	1	4	yes	—
236	C₂²	—	2	2	no	—
237	C₂²	—	8	2	no	—
238	C₂²	—	8	4	no	—
239	C₂²	#11718737	1	2	yes	—
240	C₂²	C₂×#88194	1	1	yes	characteristic
241	C₄	—	2	2	no	—
242	C₄	—	8	2	no	—
243	C₄	—	104	4	no	—
244	C₄	#11718737	1	2	yes	—
245	C₄	D₁₃×#135	1	4	yes	—
246	C₄	C₂²×#6105	1	2	yes	—
247	C₂	—	2	2	no	—
248	C₂	—	16	2	no	—
249	C₂	C₂×#11718737	1	1	yes	characteristic
250	C₂	#13125260	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#88194
commutator, cosocle	C₄²×C₁₃	C₂⁴
Frattini	C₄²	C₂³×D₁₃
Fitting, prime radical	C₁₃×#3147	C₂
socle	C₂²×C₁₃	C₂×#127
hypercenter	#3147	D₁₃
hypocenter	C₁₃	#19596
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3147

Complements

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

N	H ≅ G/N	type
1. #13125391	C₂	semidirect
2. #13125350	C₂	semidirect
3. #13114435	C₂	semidirect
4. #13114430	C₂	semidirect
5. #13114425	C₂	semidirect
6. #13114422	C₂	semidirect
7. C₂×C₈×#1055	C₂	semidirect
74. C₂×C₈×C₁₃	SD₁₆	semidirect
118. #1055	#135	semidirect
202. C₂×C₈	#6105	semidirect
215. C₁₃	#19596	semidirect

Maximal subgroups

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

subgroup	index
#13125391	2
#13125350	2
#13114435	2
#13114430	2
#13114425	2
#13114422	2
C₂×C₈×#1055	2
C₁₃×#3147	2
#19596	13