Subgroups — #15773861

Order 3328.

378 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13116635	C₂	1	1	yes	maximal, characteristic, complemented
2	#13114964	C₂	1	1	yes	maximal, characteristic, complemented
3	#13112845	C₂	1	4	yes	maximal, complemented
4	C₁₃×#3169	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#532	—	2	1	no	—
6	C₁₃×#500	C₄	1	1	yes	characteristic, complemented
7	C₁₃×#493	C₄	1	1	yes	characteristic, complemented
8	C₁₃×#486	—	2	1	no	—
9	C₁₃×#466	—	2	1	no	—
10	C₁₃×#466	C₂²	1	2	yes	—
11	C₄×C₁₃×D₈	—	2	1	no	—
12	C₄×C₁₃×D₈	C₄	1	2	yes	complemented
13	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
14	#11718596	—	2	1	no	—
15	#11718143	—	2	1	no	—
16	#11718143	C₂²	1	2	yes	—
17	C₂²⋊#6090	C₂²	1	2	yes	—
18	C₄⋊#6090	—	2	1	no	—
19	C₄×#6090	—	2	1	no	—
20	C₂×C₁₃×#41	—	4	1	no	—
21	C₂×C₁₃×D₈	—	2	1	no	—
22	C₂×C₁₃×D₈	C₂×C₄	1	2	yes	—
23	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
25	C₁₃×C₄⋊Q₈	—	2	1	no	—
26	C₁₃×#127	—	2	1	no	—
27	C₁₃×#124	—	4	1	no	—
28	C₁₃×#121	—	2	2	no	—
29	C₁₃×#121	—	4	1	no	—
30	C₄×D₄×C₁₃	—	2	2	no	—
31	C₄×D₄×C₁₃	—	4	1	no	—
32	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
33	C₂×C₄⋊Dic₁₃	—	2	2	no	—
34	C₂×C₄⋊Dic₁₃	—	4	1	no	—
35	C₂×C₄⋊Dic₁₃	—	4	1	no	—
36	C₁₃×#107	—	2	1	no	—
37	C₁₃×#107	C₂×C₄	1	2	yes	—
38	C₁₃×#102	—	2	2	no	—
39	C₁₃×#102	—	2	2	no	—
40	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
41	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
43	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
44	C₂×#6090	—	2	2	no	—
45	#11856	—	13	1	no	maximal, Sylow
46	C₁₃×#41	—	8	2	no	—
47	C₁₃×#41	—	8	2	no	—
48	C₂×Q₈×C₁₃	—	4	1	no	—
49	C₂×D₄×C₁₃	—	2	2	no	—
50	C₂×D₄×C₁₃	—	4	2	no	—
51	C₂×D₄×C₁₃	—	4	1	no	—
52	C₂×D₄×C₁₃	—	4	1	no	—
53	C₂²×C₄×C₁₃	—	4	2	no	—
54	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
55	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
56	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×Dic₁₃	—	4	2	no	—
58	C₁₃×D₈	—	4	2	no	—
59	C₁₃×D₈	—	4	2	no	—
60	C₂×C₈×C₁₃	—	4	1	no	—
61	C₂×C₈×C₁₃	—	4	1	no	—
62	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
66	C₁₃×C₄⋊C₄	—	2	2	no	—
67	C₁₃×C₄⋊C₄	—	4	1	no	—
68	C₁₃×C₂²⋊C₄	—	4	2	no	—
69	C₁₃×C₂²⋊C₄	—	4	2	no	—
70	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
71	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
72	C₄²×C₁₃	SD₁₆	1	1	yes	characteristic
73	C₄²×C₁₃	SD₁₆	1	1	yes	characteristic
74	C₄⋊Dic₁₃	—	4	2	no	—
75	C₄⋊Dic₁₃	—	4	2	no	—
76	C₄⋊Dic₁₃	—	8	2	no	—
77	C₄⋊Dic₁₃	—	8	2	no	—
78	#6090	—	4	2	no	—
79	#3169	D₁₃	1	1	yes	characteristic
80	#2162	—	13	1	no	—
81	#1834	—	13	1	no	—
82	#1347	—	13	4	no	—
83	C₂³×C₁₃	—	4	2	no	—
84	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
85	Q₈×C₁₃	—	8	2	no	—
86	D₄×C₁₃	—	4	2	no	—
87	D₄×C₁₃	—	4	2	no	—
88	D₄×C₁₃	—	8	2	no	—
89	D₄×C₁₃	—	8	2	no	—
90	D₄×C₁₃	—	8	2	no	—
91	D₄×C₁₃	—	8	2	no	—
92	C₂×C₄×C₁₃	—	2	2	no	—
93	C₂×C₄×C₁₃	—	2	1	no	—
94	C₂×C₄×C₁₃	—	2	1	no	—
95	C₂×C₄×C₁₃	—	4	2	no	—
96	C₂×C₄×C₁₃	—	8	4	no	—
97	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
98	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
99	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
100	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
101	C₂×C₄×C₁₃	D₁₆	1	2	yes	—
102	C₂×C₄×C₁₃	SD₃₂	1	2	yes	—
103	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
105	C₂×Dic₁₃	—	8	2	no	—
106	C₂×Dic₁₃	—	8	2	no	—
107	C₂×Dic₁₃	—	8	2	no	—
108	C₈×C₁₃	—	2	1	no	—
109	C₈×C₁₃	—	2	1	no	—
110	C₈×C₁₃	—	2	1	no	—
111	C₈×C₁₃	—	2	1	no	—
112	#532	—	2	1	no	—
113	#500	Dic₁₃	1	1	yes	characteristic, complemented
114	#493	Dic₁₃	1	1	yes	characteristic, complemented
115	#486	—	2	1	no	—
116	#466	—	2	1	no	—
117	#466	C₂×D₁₃	1	2	yes	—
118	C₄×D₈	—	2	1	no	—
119	C₄×D₈	Dic₁₃	1	2	yes	complemented
120	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
121	#384	—	26	1	no	—
122	C₄⋊C₁₆	—	26	1	no	—
123	C₂²⋊C₁₆	—	13	2	no	—
124	C₄×C₁₆	—	26	1	no	—
125	#340	—	13	2	no	—
126	#340	—	26	1	no	—
127	C₂²×C₁₃	—	2	1	no	—
128	C₂²×C₁₃	—	2	1	no	—
129	C₂²×C₁₃	—	8	2	no	—
130	C₂²×C₁₃	—	8	2	no	—
131	C₂²×C₁₃	—	8	2	no	—
132	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
133	C₂²×C₁₃	C₂×D₁₆	1	1	yes	characteristic
134	C₂²×C₁₃	C₂×SD₃₂	1	1	yes	characteristic
135	C₄×C₁₃	—	2	1	no	—
136	C₄×C₁₃	—	2	1	no	—
137	C₄×C₁₃	—	8	2	no	—
138	C₄×C₁₃	#357	1	2	yes	—
139	C₄×C₁₃	#357	1	2	yes	—
140	C₄×C₁₃	#360	1	2	yes	—
141	C₄×C₁₃	#410	1	1	yes	characteristic
142	C₄×C₁₃	#421	1	1	yes	characteristic
143	Dic₁₃	—	16	2	no	—
144	C₂×#41	—	4	1	no	—
145	C₂×D₈	—	2	1	no	—
146	C₂×D₈	C₂×Dic₁₃	1	2	yes	—
147	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
148	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
149	C₄⋊Q₈	—	2	1	no	—
150	#127	—	2	1	no	—
151	#124	—	4	1	no	—
152	#121	—	2	2	no	—
153	#121	—	4	1	no	—
154	C₄×D₄	—	2	2	no	—
155	C₄×D₄	—	4	1	no	—
156	C₂×C₄⋊C₄	—	26	2	no	—
157	C₂×C₄⋊C₄	—	52	1	no	—
158	C₂×C₄⋊C₄	—	52	1	no	—
159	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
160	C₂×C₁₆	—	26	2	no	—
161	#107	—	2	1	no	—
162	#107	C₂×Dic₁₃	1	2	yes	—
163	#102	—	2	2	no	—
164	#102	—	2	2	no	—
165	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
166	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
167	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
168	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
169	C₂×C₁₃	—	2	1	no	—
170	C₂×C₁₃	—	2	1	no	—
171	C₂×C₁₃	—	16	2	no	—
172	C₂×C₁₃	#1517	1	1	yes	characteristic
173	C₂×C₁₃	C₂×#357	1	1	yes	characteristic
174	C₂×C₁₃	C₂×#360	1	1	yes	characteristic
175	#41	—	8	2	no	—
176	#41	—	8	2	no	—
177	C₂×Q₈	—	4	1	no	—
178	C₂×D₄	—	2	2	no	—
179	C₂×D₄	—	4	2	no	—
180	C₂×D₄	—	4	1	no	—
181	C₂×D₄	—	4	1	no	—
182	C₂²×C₄	—	4	2	no	—
183	C₂²×C₄	—	52	2	no	—
184	C₂²×C₄	#6104	1	1	yes	characteristic
185	C₂²×C₄	#6104	1	1	yes	characteristic
186	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
187	D₈	—	4	2	no	—
188	D₈	—	4	2	no	—
189	C₂×C₈	—	4	1	no	—
190	C₂×C₈	—	4	1	no	—
191	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
192	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
193	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
194	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
195	C₄⋊C₄	—	2	2	no	—
196	C₄⋊C₄	—	4	1	no	—
197	C₄⋊C₄	—	52	2	no	—
198	C₄⋊C₄	—	52	2	no	—
199	C₄⋊C₄	—	104	2	no	—
200	C₄⋊C₄	—	104	2	no	—
201	C₂²⋊C₄	—	4	2	no	—
202	C₂²⋊C₄	—	4	2	no	—
203	C₄²	#6105	1	1	yes	characteristic
204	C₄²	#6105	1	1	yes	characteristic
205	C₄²	C₂²⋊Dic₁₃	1	1	yes	characteristic
206	C₄²	C₂²⋊Dic₁₃	1	1	yes	characteristic
207	C₁₆	—	52	2	no	—
208	C₁₃	#11856	1	1	yes	characteristic, complemented, Sylow
209	C₂³	—	4	2	no	—
210	C₂³	C₂×#6104	1	1	yes	characteristic
211	Q₈	—	8	2	no	—
212	D₄	—	4	2	no	—
213	D₄	—	4	2	no	—
214	D₄	—	8	2	no	—
215	D₄	—	8	2	no	—
216	D₄	—	8	2	no	—
217	D₄	—	8	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	4	2	no	—
222	C₂×C₄	—	8	4	no	—
223	C₂×C₄	—	104	2	no	—
224	C₂×C₄	—	104	2	no	—
225	C₂×C₄	—	104	2	no	—
226	C₂×C₄	#88066	1	2	yes	—
227	C₂×C₄	#88067	1	2	yes	—
228	C₂×C₄	#88072	1	1	yes	characteristic
229	C₂×C₄	#88072	1	1	yes	characteristic
230	C₂×C₄	#88072	1	1	yes	characteristic
231	C₂×C₄	#88072	1	1	yes	characteristic
232	C₂×C₄	C₂×#6105	1	1	yes	characteristic
233	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
234	C₈	—	2	1	no	—
235	C₈	—	2	1	no	—
236	C₈	—	2	1	no	—
237	C₈	—	2	1	no	—
238	C₂²	—	2	1	no	—
239	C₂²	—	2	1	no	—
240	C₂²	—	8	2	no	—
241	C₂²	—	8	2	no	—
242	C₂²	—	8	2	no	—
243	C₂²	C₂×#88066	1	1	yes	characteristic
244	C₂²	C₂×#88067	1	1	yes	characteristic
245	C₂²	C₂×#88072	1	1	yes	characteristic
246	C₄	—	2	1	no	—
247	C₄	—	2	1	no	—
248	C₄	—	8	2	no	—
249	C₄	—	208	2	no	—
250	C₄	#11718154	1	2	yes	—
251	C₄	#11718154	1	2	yes	—
252	C₄	#11718158	1	2	yes	—
253	C₄	#11718897	1	1	yes	characteristic
254	C₄	#11718901	1	1	yes	characteristic
255	C₂	—	2	1	no	—
256	C₂	—	2	1	no	—
257	C₂	—	16	2	no	—
258	C₂	#13113396	1	1	yes	characteristic
259	C₂	C₂×#11718154	1	1	yes	characteristic
260	C₂	C₂×#11718158	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#88072
commutator	C₂×C₈×C₁₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting, prime radical	C₁₃×#3169	C₂
socle	C₂²×C₁₃	C₂×#102
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	#3169	D₁₃
hypocenter	C₁₃	#11856
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3169

Complements

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

N	H ≅ G/N	type
1. #13116635	C₂	semidirect
2. #13114964	C₂	semidirect
3. #13112845	C₂	semidirect
6. C₁₃×#500	C₄	semidirect
7. C₁₃×#493	C₄	semidirect
12. C₄×C₁₃×D₈	C₄	semidirect
113. #500	Dic₁₃	semidirect
114. #493	Dic₁₃	semidirect
119. C₄×D₈	Dic₁₃	semidirect
208. C₁₃	#11856	semidirect

Maximal subgroups

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

subgroup	index
#13116635	2
#13114964	2
#13112845	2
C₁₃×#3169	2
#11856	13