Subgroups — #15761861

Order 3328.

434 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13113584	C₂	1	1	yes	maximal, characteristic
2	#13113231	C₂	1	2	yes	maximal
3	#13113160	C₂	1	2	yes	maximal
4	#13113128	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×C₁₃×Q₁₆	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₃×Q₁₆	C₄	1	1	yes	characteristic
7	C₂×C₄×Q₈×C₁₃	C₂²	1	2	yes	—
8	C₄×C₁₃×Q₁₆	—	2	1	no	—
9	C₄×C₁₃×Q₁₆	—	2	1	no	—
10	C₄×C₁₃×Q₁₆	—	2	1	no	—
11	C₄×C₁₃×Q₁₆	—	2	1	no	—
12	C₂×C₁₃×#107	C₄	1	1	yes	characteristic
13	C₂×C₁₃×#103	C₄	1	2	yes	—
14	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
15	#11718586	C₂²	1	2	yes	—
16	#11718586	C₂²	1	1	yes	characteristic
17	#11718586	C₂²	1	1	yes	characteristic
18	Q₈⋊#1055	—	2	4	no	—
19	#11718120	—	2	4	no	—
20	#11718050	—	2	1	no	—
21	#11718049	—	2	1	no	—
22	#11718046	—	2	1	no	—
23	#11718046	—	2	1	no	—
24	C₂²×Q₈×C₁₃	C₂×C₄	1	2	yes	—
25	C₂×C₁₃×Q₁₆	—	2	4	no	—
26	C₂×C₁₃×Q₁₆	—	2	1	no	—
27	C₂×C₁₃×Q₁₆	—	2	1	no	—
28	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
30	C₄×Q₈×C₁₃	—	2	2	no	—
31	C₄×Q₈×C₁₃	—	2	2	no	—
32	C₄×Q₈×C₁₃	—	2	2	no	—
33	C₄×Q₈×C₁₃	—	2	2	no	—
34	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
35	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₁₃	—	2	2	no	—
37	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
38	C₁₃×#107	—	2	1	no	—
39	C₁₃×#107	—	2	1	no	—
40	C₁₃×#103	—	2	2	no	—
41	C₁₃×#103	—	2	2	no	—
42	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
43	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
44	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
45	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
46	C₂²⋊#1055	—	2	1	no	—
47	C₂²⋊#1055	—	2	2	no	—
48	C₂²⋊#1055	D₄	1	2	yes	—
49	C₄⋊#1055	—	2	1	no	—
50	C₄⋊#1055	—	2	2	no	—
51	C₄⋊#1055	D₄	1	2	yes	—
52	C₄×#1055	—	2	1	no	—
53	C₄×#1055	—	2	1	no	—
54	C₄×#1055	—	2	2	no	—
55	#9671	—	13	1	no	maximal, Sylow
56	C₂×Q₈×C₁₃	—	2	2	no	—
57	C₂×Q₈×C₁₃	—	2	2	no	—
58	C₂×Q₈×C₁₃	—	2	2	no	—
59	C₂×Q₈×C₁₃	—	2	2	no	—
60	C₂×Q₈×C₁₃	—	2	2	no	—
61	C₂×Q₈×C₁₃	M₁₆	1	4	yes	—
62	C₂²×C₄×C₁₃	—	2	2	no	—
63	C₂²×C₄×C₁₃	—	2	2	no	—
64	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
67	C₁₃×Q₁₆	—	2	4	no	—
68	C₁₃×Q₁₆	—	2	4	no	—
69	C₂×C₈×C₁₃	—	2	2	no	—
70	C₂×C₈×C₁₃	—	2	2	no	—
71	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
75	C₁₃×C₄⋊C₄	—	2	2	no	—
76	C₁₃×C₄⋊C₄	—	2	2	no	—
77	C₁₃×C₄⋊C₄	—	4	2	no	—
78	C₁₃×C₄⋊C₄	—	4	2	no	—
79	C₄²×C₁₃	—	4	2	no	—
80	C₄²×C₁₃	—	4	2	no	—
81	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
82	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₁₃	#41	1	1	yes	characteristic
85	C₂×#1055	—	2	2	no	—
86	C₂×#1055	—	2	2	no	—
87	C₂×#1055	—	4	2	no	—
88	C₂×C₄×Q₁₆	D₁₃	1	1	yes	characteristic
89	#1564	—	13	1	no	—
90	#1479	—	13	2	no	—
91	#1461	—	13	2	no	—
92	#1452	—	13	1	no	—
93	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
94	Q₈×C₁₃	—	2	4	no	—
95	Q₈×C₁₃	—	2	4	no	—
96	Q₈×C₁₃	—	4	2	no	—
97	Q₈×C₁₃	—	4	2	no	—
98	C₂×C₄×C₁₃	—	2	2	no	—
99	C₂×C₄×C₁₃	—	2	4	no	—
100	C₂×C₄×C₁₃	—	4	1	no	—
101	C₂×C₄×C₁₃	—	4	2	no	—
102	C₂×C₄×C₁₃	—	4	2	no	—
103	C₂×C₄×C₁₃	—	4	2	no	—
104	C₂×C₄×C₁₃	—	8	2	no	—
105	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
107	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
108	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
109	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
110	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
111	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
112	C₂×C₄×C₁₃	#131	1	2	yes	—
113	C₈×C₁₃	—	2	1	no	—
114	C₈×C₁₃	—	2	1	no	—
115	C₈×C₁₃	—	2	1	no	—
116	C₈×C₁₃	—	2	1	no	—
117	#1055	—	4	2	no	—
118	#1055	—	4	2	no	—
119	#1055	—	8	2	no	—
120	C₂²×Q₁₆	Dic₁₃	1	1	yes	characteristic
121	C₂×C₄×Q₈	C₂×D₁₃	1	2	yes	—
122	C₄×Q₁₆	—	2	1	no	—
123	C₄×Q₁₆	—	2	1	no	—
124	C₄×Q₁₆	—	2	1	no	—
125	C₄×Q₁₆	—	2	1	no	—
126	#431	—	13	1	no	—
127	#431	—	13	1	no	—
128	#431	—	13	2	no	—
129	C₂×#107	Dic₁₃	1	1	yes	characteristic
130	C₂×#103	Dic₁₃	1	2	yes	—
131	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
132	#335	—	26	1	no	—
133	#334	—	26	1	no	—
134	Q₈⋊C₈	—	26	4	no	—
135	#324	—	26	4	no	—
136	#322	—	26	1	no	—
137	#322	—	26	1	no	—
138	C₂²×C₁₃	—	2	1	no	—
139	C₂²×C₁₃	—	2	1	no	—
140	C₂²×C₁₃	#390	1	1	yes	characteristic
141	C₂²×C₁₃	#435	1	2	yes	—
142	C₄×C₁₃	—	2	1	no	—
143	C₄×C₁₃	—	2	1	no	—
144	C₄×C₁₃	—	2	1	no	—
145	C₄×C₁₃	—	4	4	no	—
146	C₄×C₁₃	—	8	2	no	—
147	C₄×C₁₃	#407	1	1	yes	characteristic
148	C₄×C₁₃	#408	1	1	yes	characteristic
149	C₄×C₁₃	#435	1	2	yes	—
150	C₄×C₁₃	#473	1	2	yes	—
151	C₂²×Q₈	C₂×Dic₁₃	1	2	yes	—
152	C₂×Q₁₆	—	2	4	no	—
153	C₂×Q₁₆	—	2	1	no	—
154	C₂×Q₁₆	—	2	1	no	—
155	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
156	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
157	C₄×Q₈	—	2	2	no	—
158	C₄×Q₈	—	2	2	no	—
159	C₄×Q₈	—	2	2	no	—
160	C₄×Q₈	—	2	2	no	—
161	C₂×C₄⋊C₄	—	2	2	no	—
162	C₂×C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
163	C₂×C₄²	—	2	2	no	—
164	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
165	#107	—	2	1	no	—
166	#107	—	2	1	no	—
167	C₄⋊C₈	—	13	2	no	—
168	C₄⋊C₈	—	26	1	no	—
169	C₄⋊C₈	—	26	2	no	—
170	#103	—	2	2	no	—
171	#103	—	2	2	no	—
172	C₂²⋊C₈	—	13	2	no	—
173	C₂²⋊C₈	—	26	1	no	—
174	C₂²⋊C₈	—	26	2	no	—
175	C₄×C₈	—	26	2	no	—
176	C₄×C₈	—	26	1	no	—
177	C₄×C₈	—	26	1	no	—
178	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
179	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
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	#1955	1	1	yes	characteristic
184	C₂×C₁₃	#1976	1	1	yes	characteristic
185	C₂×C₁₃	#1979	1	1	yes	characteristic
186	C₂×Q₈	—	2	2	no	—
187	C₂×Q₈	—	2	2	no	—
188	C₂×Q₈	—	2	2	no	—
189	C₂×Q₈	—	2	2	no	—
190	C₂×Q₈	—	2	2	no	—
191	C₂×Q₈	#6099	1	4	yes	—
192	C₂²×C₄	—	2	2	no	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
195	C₂²×C₄	#6129	1	1	yes	characteristic
196	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
197	Q₁₆	—	2	4	no	—
198	Q₁₆	—	2	4	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	26	2	no	—
202	C₂×C₈	—	26	2	no	—
203	C₂×C₈	—	52	2	no	—
204	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
205	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
206	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
207	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
208	C₄⋊C₄	—	2	2	no	—
209	C₄⋊C₄	—	2	2	no	—
210	C₄⋊C₄	—	4	2	no	—
211	C₄⋊C₄	—	4	2	no	—
212	C₄²	—	4	2	no	—
213	C₄²	—	4	2	no	—
214	C₄²	D₄×D₁₃	1	1	yes	characteristic
215	C₄²	#6129	1	1	yes	characteristic
216	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
217	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
218	C₁₃	#9671	1	1	yes	characteristic, complemented, Sylow
219	C₂³	D₄×Dic₁₃	1	1	yes	characteristic
220	Q₈	—	2	4	no	—
221	Q₈	—	2	4	no	—
222	Q₈	—	4	2	no	—
223	Q₈	—	4	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	4	no	—
226	C₂×C₄	—	4	1	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	8	2	no	—
231	C₂×C₄	C₂×#6099	1	2	yes	—
232	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
233	C₂×C₄	#88190	1	1	yes	characteristic
234	C₂×C₄	#88192	1	1	yes	characteristic
235	C₂×C₄	#88192	1	1	yes	characteristic
236	C₂×C₄	#88194	1	1	yes	characteristic
237	C₂×C₄	#88202	1	2	yes	—
238	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
239	C₈	—	2	1	no	—
240	C₈	—	2	1	no	—
241	C₈	—	2	1	no	—
242	C₈	—	2	1	no	—
243	C₈	—	52	2	no	—
244	C₈	—	52	2	no	—
245	C₈	—	104	2	no	—
246	C₂²	—	2	1	no	—
247	C₂²	—	2	1	no	—
248	C₂²	#11718672	1	2	yes	—
249	C₂²	#11718882	1	1	yes	characteristic
250	C₄	—	2	1	no	—
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	4	4	no	—
254	C₄	—	8	2	no	—
255	C₄	#11718672	1	2	yes	—
256	C₄	#11718853	1	2	yes	—
257	C₄	#11718868	1	1	yes	characteristic
258	C₄	#11718895	1	1	yes	characteristic
259	C₂	—	2	2	no	—
260	C₂	#13115635	1	1	yes	characteristic
261	C₂	#13115735	1	1	yes	characteristic
262	C₂	#13115752	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#88192
commutator	C₂²×C₄×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₂×C₄×C₁₃×Q₁₆	C₂
socle	C₂²×C₁₃	#390
prime radical	C₂³×C₁₃	C₄×D₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	C₂×C₄×Q₁₆	D₁₃
hypocenter	C₁₃	#9671
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
218. C₁₃	#9671	semidirect

Maximal subgroups

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

subgroup	index
#13113584	2
#13113231	2
#13113160	2
#13113128	2
C₂×C₄×C₁₃×Q₁₆	2
#9671	13