Subgroups — #15863242

Order 3328.

778 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13114981	C₂	1	2	yes	maximal, complemented
2	#13114956	C₂	1	2	yes	maximal, complemented
3	#13114907	C₂	1	2	yes	maximal
4	#13114820	C₂	1	2	yes	maximal
5	C₄×C₄⋊#1055	C₂	1	2	yes	maximal, complemented
6	C₄×C₂²⋊#1055	C₂	1	2	yes	maximal
7	#13114582	C₂	1	2	yes	maximal, complemented
8	C₄×C₁₃×#117	C₂	1	1	yes	maximal, characteristic
9	C₂×C₁₃×#117	C₂²	1	1	yes	characteristic
10	C₂×C₁₃×#117	C₂²	1	1	yes	characteristic
11	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
12	C₄×C₁₃×C₄⋊C₄	C₄	1	4	yes	—
13	C₄×C₁₃×C₂²⋊C₄	C₄	1	4	yes	—
14	C₁₃×#376	C₂²	1	1	yes	characteristic
15	C₁₃×#376	C₂²	1	1	yes	characteristic
16	C₄³×C₁₃	C₂²	1	1	yes	characteristic
17	C₄³×C₁₃	C₂²	1	1	yes	characteristic
18	C₂×C₂²⋊#1055	C₂²	1	4	yes	—
19	C₂×C₄⋊#1055	C₂²	1	4	yes	—
20	C₂×C₄×#1055	C₂²	1	4	yes	—
21	#11718116	C₂²	1	4	yes	—
22	#11718116	C₂²	1	4	yes	—
23	#11718116	C₂²	1	4	yes	—
24	#11718116	C₂²	1	4	yes	—
25	C₂³×C₄×C₁₃	C₂×C₄	1	2	yes	—
26	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
27	C₁₃×#117	—	2	4	no	—
28	C₁₃×#117	—	2	4	no	—
29	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
30	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
31	C₂×C₁₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
32	C₂×C₁₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₁₃	—	2	4	no	—
34	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
35	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
36	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
38	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
39	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
40	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
41	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
42	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
43	C₂²×#1055	C₂³	1	8	yes	—
44	C₁₃×#95	C₂×C₄	1	4	yes	—
45	C₁₃×#95	C₂×C₄	1	4	yes	—
46	C₂²⋊#1055	—	2	8	no	—
47	C₄⋊#1055	—	2	8	no	—
48	C₄×#1055	—	2	8	no	—
49	#20782	—	13	1	no	maximal, Sylow
50	C₂⁴×C₁₃	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₁₃	—	2	2	no	—
52	C₂²×C₄×C₁₃	—	2	2	no	—
53	C₂²×C₄×C₁₃	—	2	4	no	—
54	C₂²×C₄×C₁₃	—	2	8	no	—
55	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
56	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
57	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
58	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
59	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
60	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
62	C₂²×C₄×C₁₃	#41	1	4	yes	—
63	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
64	C₁₃×C₄⋊C₄	—	2	4	no	—
65	C₁₃×C₄⋊C₄	—	2	4	no	—
66	C₁₃×C₂²⋊C₄	—	2	4	no	—
67	C₁₃×C₂²⋊C₄	—	2	4	no	—
68	C₄²×C₁₃	—	2	4	no	—
69	C₄²×C₁₃	—	2	8	no	—
70	C₄²×C₁₃	—	2	8	no	—
71	C₄²×C₁₃	—	2	2	no	—
72	C₄²×C₁₃	—	2	2	no	—
73	C₄²×C₁₃	—	2	2	no	—
74	C₄²×C₁₃	—	2	2	no	—
75	C₄²×C₁₃	#41	1	4	yes	—
76	C₄²×C₁₃	#41	1	4	yes	—
77	C₂×#1055	—	2	8	no	—
78	C₂×#1055	—	2	8	no	—
79	C₂×#1055	—	2	8	no	—
80	C₄×#117	D₁₃	1	1	yes	characteristic
81	#1840	—	13	2	no	—
82	#1832	—	13	2	no	—
83	#1818	—	13	2	no	—
84	#1798	—	13	2	no	—
85	C₄×C₄⋊C₈	—	13	2	no	—
86	C₄×C₂²⋊C₈	—	13	2	no	—
87	#1744	—	13	2	no	—
88	C₂³×C₁₃	—	2	1	no	—
89	C₂³×C₁₃	—	2	1	no	—
90	C₂³×C₁₃	—	2	4	no	—
91	C₂³×C₁₃	#117	1	2	yes	—
92	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
93	C₂×C₄×C₁₃	—	2	1	no	—
94	C₂×C₄×C₁₃	—	2	1	no	—
95	C₂×C₄×C₁₃	—	2	4	no	—
96	C₂×C₄×C₁₃	—	2	8	no	—
97	C₂×C₄×C₁₃	—	2	2	no	—
98	C₂×C₄×C₁₃	—	2	2	no	—
99	C₂×C₄×C₁₃	—	2	2	no	—
100	C₂×C₄×C₁₃	—	2	2	no	—
101	C₂×C₄×C₁₃	—	2	4	no	—
102	C₂×C₄×C₁₃	—	2	4	no	—
103	C₂×C₄×C₁₃	—	2	4	no	—
104	C₂×C₄×C₁₃	—	2	4	no	—
105	C₂×C₄×C₁₃	—	4	8	no	—
106	C₂×C₄×C₁₃	#117	1	4	yes	—
107	C₂×C₄×C₁₃	#117	1	2	yes	—
108	C₂×C₄×C₁₃	#117	1	4	yes	—
109	C₂×C₄×C₁₃	#131	1	8	yes	—
110	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
111	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
112	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
113	#1055	—	4	8	no	—
114	C₂×#117	C₂×D₁₃	1	1	yes	characteristic
115	C₂×#117	C₂×D₁₃	1	1	yes	characteristic
116	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
117	C₂×C₄⋊C₈	—	13	4	no	—
118	C₂×C₂²⋊C₈	—	13	4	no	—
119	C₂×C₄×C₈	—	13	4	no	—
120	C₄×C₄⋊C₄	Dic₁₃	1	4	yes	—
121	C₄×C₂²⋊C₄	Dic₁₃	1	4	yes	—
122	#376	C₂×D₁₃	1	1	yes	characteristic
123	#376	C₂×D₁₃	1	1	yes	characteristic
124	C₄³	C₂×D₁₃	1	1	yes	characteristic
125	C₄³	C₂×D₁₃	1	1	yes	characteristic
126	#336	—	13	4	no	—
127	#336	—	13	4	no	—
128	#336	—	13	4	no	—
129	#336	—	13	4	no	—
130	C₂²×C₁₃	—	2	4	no	—
131	C₂²×C₁₃	—	2	2	no	—
132	C₂²×C₁₃	—	4	2	no	—
133	C₂²×C₁₃	—	4	2	no	—
134	C₂²×C₁₃	C₂×#117	1	1	yes	characteristic
135	C₂²×C₁₃	C₂×#117	1	1	yes	characteristic
136	C₂²×C₁₃	C₂×#117	1	1	yes	characteristic
137	C₂²×C₁₃	C₂×#131	1	4	yes	—
138	C₄×C₁₃	—	2	4	no	—
139	C₄×C₁₃	—	2	2	no	—
140	C₄×C₁₃	—	4	2	no	—
141	C₄×C₁₃	—	4	2	no	—
142	C₄×C₁₃	—	4	2	no	—
143	C₄×C₁₃	—	4	2	no	—
144	C₄×C₁₃	#433	1	8	yes	—
145	C₄×C₁₃	#529	1	4	yes	—
146	C₂³×C₄	C₂×Dic₁₃	1	2	yes	—
147	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
148	C₂²×C₈	—	13	8	no	—
149	#117	—	2	4	no	—
150	#117	—	2	4	no	—
151	C₂×C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
152	C₂×C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
153	C₂×C₂²⋊C₄	C₂×Dic₁₃	1	2	yes	—
154	C₂×C₂²⋊C₄	C₂×Dic₁₃	1	2	yes	—
155	C₂×C₄²	—	2	4	no	—
156	C₂×C₄²	C₂×Dic₁₃	1	4	yes	—
157	C₂×C₄²	C₂×Dic₁₃	1	4	yes	—
158	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
159	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
160	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
161	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
162	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
163	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
164	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
165	C₄⋊C₈	—	26	8	no	—
166	C₂²⋊C₈	—	26	8	no	—
167	C₄×C₈	—	26	8	no	—
168	#95	C₂×Dic₁₃	1	4	yes	—
169	#95	C₂×Dic₁₃	1	4	yes	—
170	C₂×C₁₃	—	4	2	no	—
171	C₂×C₁₃	#2283	1	1	yes	characteristic
172	C₂×C₁₃	C₂×#433	1	2	yes	—
173	C₂×C₁₃	#2922	1	2	yes	—
174	C₂×C₁₃	#2922	1	2	yes	—
175	C₂⁴	C₂²×Dic₁₃	1	1	yes	characteristic
176	C₂²×C₄	—	2	2	no	—
177	C₂²×C₄	—	2	2	no	—
178	C₂²×C₄	—	2	4	no	—
179	C₂²×C₄	—	2	8	no	—
180	C₂²×C₄	#6127	1	4	yes	—
181	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
182	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
183	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
184	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
185	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
186	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
187	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
188	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
189	C₂×C₈	—	26	8	no	—
190	C₂×C₈	—	26	8	no	—
191	C₂×C₈	—	26	8	no	—
192	C₄⋊C₄	—	2	4	no	—
193	C₄⋊C₄	—	2	4	no	—
194	C₂²⋊C₄	—	2	4	no	—
195	C₂²⋊C₄	—	2	4	no	—
196	C₄²	—	2	4	no	—
197	C₄²	—	2	8	no	—
198	C₄²	—	2	8	no	—
199	C₄²	—	2	2	no	—
200	C₄²	—	2	2	no	—
201	C₄²	—	2	2	no	—
202	C₄²	—	2	2	no	—
203	C₄²	#6127	1	4	yes	—
204	C₄²	#6127	1	4	yes	—
205	C₁₃	#20782	1	1	yes	characteristic, complemented, Sylow
206	C₂³	—	2	1	no	—
207	C₂³	—	2	1	no	—
208	C₂³	—	2	4	no	—
209	C₂³	#88180	1	2	yes	—
210	C₂³	C₂³×Dic₁₃	1	1	yes	characteristic
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	4	no	—
214	C₂×C₄	—	2	8	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	4	no	—
220	C₂×C₄	—	2	4	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	—	4	8	no	—
224	C₂×C₄	#88180	1	4	yes	—
225	C₂×C₄	#88180	1	2	yes	—
226	C₂×C₄	#88180	1	4	yes	—
227	C₂×C₄	#88202	1	8	yes	—
228	C₂×C₄	C₂×#6127	1	2	yes	—
229	C₂×C₄	C₂×#6127	1	2	yes	—
230	C₂×C₄	C₂×#6127	1	2	yes	—
231	C₈	—	52	8	no	—
232	C₂²	—	2	2	no	—
233	C₂²	—	2	4	no	—
234	C₂²	—	4	2	no	—
235	C₂²	—	4	2	no	—
236	C₂²	C₂×#88180	1	1	yes	characteristic
237	C₂²	C₂×#88180	1	1	yes	characteristic
238	C₂²	C₂×#88180	1	1	yes	characteristic
239	C₂²	C₂×#88202	1	4	yes	—
240	C₄	—	2	2	no	—
241	C₄	—	2	4	no	—
242	C₄	—	4	2	no	—
243	C₄	—	4	2	no	—
244	C₄	—	4	2	no	—
245	C₄	—	4	2	no	—
246	C₄	#11718661	1	8	yes	—
247	C₄	#11719184	1	4	yes	—
248	C₂	—	4	2	no	—
249	C₂	#13117234	1	1	yes	characteristic
250	C₂	C₂×#11718661	1	2	yes	—
251	C₂	#13123480	1	2	yes	—
252	C₂	#13123480	1	2	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂²×D₁₃
commutator, socle	C₂³×C₁₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₄×C₁₃×#117	C₂
prime radical	C₂⁴×C₁₃	C₂²×C₄
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₄×#117	D₁₃
hypocenter	C₁₃	#20782
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄² ◁ C₄×#117

Complements

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

N	H ≅ G/N	type
1. #13114981	C₂	semidirect
2. #13114956	C₂	semidirect
5. C₄×C₄⋊#1055	C₂	semidirect
7. #13114582	C₂	semidirect
205. C₁₃	#20782	semidirect

Maximal subgroups

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

subgroup	index
#13114981	2
#13114956	2
#13114907	2
#13114820	2
C₄×C₄⋊#1055	2
C₄×C₂²⋊#1055	2
#13114582	2
C₄×C₁₃×#117	2
#20782	13