Subgroups — #15756568

Order 3328.

310 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13114522	C₂	1	1	yes	maximal, characteristic
2	#13113584	C₂	1	1	yes	maximal, characteristic
3	#13112680	C₂	1	2	yes	maximal
4	#13112664	C₂	1	2	yes	maximal
5	C₁₃×#1564	C₂	1	1	yes	maximal, characteristic
6	C₁₃×#431	C₂²	1	2	yes	—
7	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×#88055	C₄	1	1	yes	characteristic
9	C₂×#88055	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₁₃	C₄	1	1	yes	characteristic
11	C₂×C₈×Dic₁₃	C₄	1	1	yes	characteristic
12	#11718593	C₂²	1	1	yes	characteristic
13	#11718586	C₂²	1	2	yes	—
14	C₂×#88043	C₂²	1	1	yes	characteristic
15	C₁₃×#335	—	2	1	no	—
16	C₁₃×#334	—	2	1	no	—
17	#11718050	—	2	1	no	—
18	#11718049	—	2	1	no	—
19	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
22	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
25	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
26	C₁₃×C₄⋊C₈	—	2	2	no	—
27	C₁₃×C₂²⋊C₈	—	2	2	no	—
28	C₄×C₈×C₁₃	—	2	2	no	—
29	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
30	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
31	C₄×C₈×C₁₃	Q₈	1	1	yes	characteristic
32	C₄×C₈×C₁₃	Q₈	1	1	yes	characteristic
33	#88071	—	2	1	no	—
34	#88071	—	2	1	no	—
35	C₂²⋊#1055	—	2	2	no	—
36	#88055	—	2	2	no	—
37	#88055	C₈	1	4	yes	complemented
38	C₈×Dic₁₃	—	2	2	no	—
39	C₈×Dic₁₃	—	2	2	no	—
40	C₄⋊#1055	—	2	2	no	—
41	#88043	—	2	1	no	—
42	#88043	—	2	1	no	—
43	C₄×#1055	—	2	2	no	—
44	#8781	—	13	1	no	maximal, Sylow
45	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
47	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
48	C₂²×Dic₁₃	—	2	1	no	—
49	C₂²×Dic₁₃	—	2	1	no	—
50	C₂×C₈×C₁₃	—	2	1	no	—
51	C₂×C₈×C₁₃	—	2	1	no	—
52	C₂×C₈×C₁₃	—	4	2	no	—
53	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
58	C₂×C₈×C₁₃	M₁₆	1	2	yes	—
59	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
60	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
61	C₄²×C₁₃	#41	1	1	yes	characteristic
62	C₄²×C₁₃	#41	1	1	yes	characteristic
63	C₄×Dic₁₃	—	2	1	no	—
64	C₄×Dic₁₃	—	2	1	no	—
65	C₄×Dic₁₃	C₂×C₈	1	2	yes	—
66	C₄×Dic₁₃	M₁₆	1	2	yes	—
67	C₂×#1055	—	2	1	no	—
68	C₂×#1055	—	2	1	no	—
69	C₂×#1055	—	2	1	no	—
70	C₂×#1055	—	2	1	no	—
71	C₂×#1055	—	4	2	no	—
72	C₂×#1055	C₂×C₈	1	2	yes	—
73	C₂×#1055	M₁₆	1	2	yes	—
74	#1726	—	13	1	no	—
75	#1564	—	13	1	no	—
76	#1564	D₁₃	1	1	yes	characteristic
77	#1281	—	13	2	no	—
78	#1274	—	13	2	no	—
79	C₂³×C₁₃	#117	1	1	yes	characteristic
80	C₂×C₄×C₁₃	—	4	1	no	—
81	C₂×C₄×C₁₃	—	4	1	no	—
82	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
84	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
85	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
86	C₂×C₄×C₁₃	#125	1	1	yes	characteristic
87	C₂×C₄×C₁₃	#125	1	1	yes	characteristic
88	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
89	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
90	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
91	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
92	C₂×Dic₁₃	—	2	2	no	—
93	C₂×Dic₁₃	—	2	2	no	—
94	C₂×Dic₁₃	—	4	1	no	—
95	C₂×Dic₁₃	—	4	1	no	—
96	C₂×Dic₁₃	—	4	1	no	—
97	C₂×Dic₁₃	—	4	1	no	—
98	C₈×C₁₃	—	2	1	no	—
99	C₈×C₁₃	—	2	1	no	—
100	C₈×C₁₃	—	8	2	no	—
101	C₈×C₁₃	C₄⋊C₈	1	2	yes	—
102	C₈×C₁₃	C₄⋊C₈	1	2	yes	—
103	#1055	—	2	2	no	—
104	#1055	—	2	2	no	—
105	#1055	—	8	2	no	—
106	#431	—	13	2	no	—
107	#431	C₂×D₁₃	1	2	yes	—
108	C₂×#97	—	13	1	no	—
109	C₂×#97	—	13	1	no	—
110	C₂×#97	—	13	1	no	—
111	C₂×C₄×C₈	—	13	1	no	—
112	C₂×C₄×C₈	—	13	1	no	—
113	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
114	#376	—	13	1	no	—
115	#335	—	2	1	no	—
116	#335	—	26	1	no	—
117	#334	—	2	1	no	—
118	#334	—	26	1	no	—
119	C₂²×C₁₃	—	2	1	no	—
120	C₂²×C₁₃	—	2	1	no	—
121	C₂²×C₁₃	#382	1	1	yes	characteristic
122	C₂²×C₁₃	#431	1	1	yes	characteristic
123	C₂²×C₁₃	#432	1	1	yes	characteristic
124	C₄×C₁₃	—	2	1	no	—
125	C₄×C₁₃	—	2	1	no	—
126	C₄×C₁₃	—	2	1	no	—
127	C₄×C₁₃	—	2	1	no	—
128	C₄×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
129	C₄×C₁₃	#423	1	1	yes	characteristic
130	C₄×C₁₃	#431	1	1	yes	characteristic
131	C₄×C₁₃	#432	1	1	yes	characteristic
132	Dic₁₃	—	4	2	no	—
133	Dic₁₃	—	4	2	no	—
134	C₂²×C₈	—	13	1	no	—
135	C₂²×C₈	—	13	1	no	—
136	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
137	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
138	C₂×C₄²	—	13	1	no	—
139	C₂×C₄²	—	13	1	no	—
140	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
141	C₄⋊C₈	—	2	2	no	—
142	C₄⋊C₈	—	26	2	no	—
143	C₂²⋊C₈	—	2	2	no	—
144	C₂²⋊C₈	—	26	2	no	—
145	#97	—	13	4	no	—
146	#97	—	26	2	no	—
147	#97	—	26	1	no	—
148	#97	—	26	1	no	—
149	C₄×C₈	—	2	2	no	—
150	C₄×C₈	—	26	2	no	—
151	C₄×C₈	—	26	2	no	—
152	C₄×C₈	—	26	2	no	—
153	C₄×C₈	Dic₂₆	1	1	yes	characteristic
154	C₄×C₈	Dic₂₆	1	1	yes	characteristic
155	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
156	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
157	#95	—	26	1	no	—
158	#95	—	26	1	no	—
159	C₂×C₁₃	—	2	1	no	—
160	C₂×C₁₃	—	2	1	no	—
161	C₂×C₁₃	#1831	1	1	yes	characteristic
162	C₂×C₁₃	#1833	1	1	yes	characteristic
163	C₂×C₁₃	#1833	1	1	yes	characteristic
164	C₂²×C₄	—	26	1	no	—
165	C₂²×C₄	—	26	1	no	—
166	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
167	C₂²×C₄	#6129	1	1	yes	characteristic
168	C₂²×C₄	#6131	1	1	yes	characteristic
169	C₂×C₈	—	2	1	no	—
170	C₂×C₈	—	2	1	no	—
171	C₂×C₈	—	4	2	no	—
172	C₂×C₈	—	13	2	no	—
173	C₂×C₈	—	13	2	no	—
174	C₂×C₈	—	26	1	no	—
175	C₂×C₈	—	26	1	no	—
176	C₂×C₈	—	26	1	no	—
177	C₂×C₈	—	26	1	no	—
178	C₂×C₈	—	52	2	no	—
179	C₂×C₈	C₈×D₁₃	1	2	yes	—
180	C₂×C₈	#6094	1	2	yes	—
181	C₂×C₈	#6101	1	1	yes	characteristic
182	C₂×C₈	#6101	1	1	yes	characteristic
183	C₂×C₈	#6101	1	1	yes	characteristic
184	C₂×C₈	#6101	1	1	yes	characteristic
185	C₄²	—	13	2	no	—
186	C₄²	—	13	2	no	—
187	C₄²	—	26	1	no	—
188	C₄²	—	26	1	no	—
189	C₄²	C₂×Dic₂₆	1	1	yes	characteristic
190	C₄²	#6129	1	1	yes	characteristic
191	C₄²	#6131	1	1	yes	characteristic
192	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
193	C₁₃	#8781	1	1	yes	characteristic, complemented, Sylow
194	C₂³	#88146	1	1	yes	characteristic
195	C₂×C₄	—	4	1	no	—
196	C₂×C₄	—	4	1	no	—
197	C₂×C₄	—	26	2	no	—
198	C₂×C₄	—	26	2	no	—
199	C₂×C₄	—	52	1	no	—
200	C₂×C₄	—	52	1	no	—
201	C₂×C₄	—	52	1	no	—
202	C₂×C₄	—	52	1	no	—
203	C₂×C₄	#88144	1	1	yes	characteristic
204	C₂×C₄	#88144	1	1	yes	characteristic
205	C₂×C₄	#88146	1	1	yes	characteristic
206	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
207	C₂×C₄	C₂×#6094	1	1	yes	characteristic
208	C₂×C₄	M₁₆×D₁₃	1	1	yes	characteristic
209	C₂×C₄	M₁₆×D₁₃	1	1	yes	characteristic
210	C₂×C₄	C₂×#6101	1	1	yes	characteristic
211	C₂×C₄	#88190	1	1	yes	characteristic
212	C₂×C₄	#88201	1	1	yes	characteristic
213	C₈	—	2	1	no	—
214	C₈	—	2	1	no	—
215	C₈	—	8	2	no	—
216	C₈	—	26	2	no	—
217	C₈	—	26	2	no	—
218	C₈	—	104	2	no	—
219	C₈	#88054	1	2	yes	—
220	C₈	#88054	1	2	yes	—
221	C₂²	—	2	1	no	—
222	C₂²	—	2	1	no	—
223	C₂²	#11718494	1	1	yes	characteristic
224	C₂²	#11718496	1	1	yes	characteristic
225	C₂²	#11718638	1	1	yes	characteristic
226	C₄	—	2	1	no	—
227	C₄	—	2	1	no	—
228	C₄	—	2	1	no	—
229	C₄	—	2	1	no	—
230	C₄	—	52	2	no	—
231	C₄	—	52	2	no	—
232	C₄	#11718494	1	1	yes	characteristic
233	C₄	#11718496	1	1	yes	characteristic
234	C₄	C₂×#88054	1	1	yes	characteristic
235	C₄	#11718779	1	1	yes	characteristic
236	C₂	—	2	1	no	—
237	C₂	—	2	1	no	—
238	C₂	#13114952	1	1	yes	characteristic
239	C₂	#13114959	1	1	yes	characteristic
240	C₂	#13114961	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#88146
commutator	C₂×C₄×C₁₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1564	C₂
socle	C₂²×C₁₃	#382
prime radical	C₂³×C₁₃	#117
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	#1564	D₁₃
hypocenter	C₁₃	#8781
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
37. #88055	C₈	semidirect
193. C₁₃	#8781	semidirect

Maximal subgroups

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

subgroup	index
#13114522	2
#13113584	2
#13112680	2
#13112664	2
C₁₃×#1564	2
#8781	13