Subgroups — #15761568

Order 3328.

382 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	#13115595	C₂	1	2	yes	maximal
2	#13114996	C₂	1	1	yes	maximal, characteristic
3	C₈×C₄⋊Dic₁₃	C₂	1	1	yes	maximal, characteristic
4	#13112720	C₂	1	2	yes	maximal
5	C₂×C₁₃×#335	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×#88058	—	2	1	no	—
9	C₂×#88058	—	2	1	no	—
10	C₂×#88058	—	2	2	no	—
11	C₂×C₈×Dic₁₃	—	2	1	no	—
12	#11718596	C₂²	1	2	yes	—
13	C₄×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊#1055	C₂²	1	1	yes	characteristic
15	C₁₃×#335	C₄	1	4	yes	complemented
16	#11718143	—	2	2	no	—
17	#11718142	—	2	1	no	—
18	C₂²×C₈×C₁₃	—	2	2	no	—
19	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
20	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
21	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
22	C₂×C₄⋊Dic₁₃	—	2	2	no	—
23	C₂×C₄⋊Dic₁₃	—	2	2	no	—
24	C₂×C₄⋊Dic₁₃	—	2	2	no	—
25	C₂×C₄⋊Dic₁₃	D₄	1	1	yes	characteristic
26	C₂×C₄⋊Dic₁₃	D₄	1	1	yes	characteristic
27	C₂×C₄×Dic₁₃	—	2	1	no	—
28	C₂²×#1055	—	2	1	no	—
29	C₁₃×C₄⋊C₈	—	2	2	no	—
30	C₁₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
31	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
32	C₄×C₈×C₁₃	Q₈	1	2	yes	—
33	#88071	—	2	1	no	—
34	#88058	—	4	2	no	—
35	#88058	—	4	2	no	—
36	#88058	—	4	4	no	—
37	C₈×Dic₁₃	—	4	2	no	—
38	C₄⋊#1055	—	2	1	no	—
39	C₄⋊#1055	Q₈	1	2	yes	—
40	#9614	—	13	1	no	maximal, Sylow
41	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
44	C₂²×Dic₁₃	—	2	1	no	—
45	C₂²×Dic₁₃	—	2	1	no	—
46	C₂²×Dic₁₃	—	4	2	no	—
47	C₂×C₈×C₁₃	—	2	1	no	—
48	C₂×C₈×C₁₃	—	2	1	no	—
49	C₂×C₈×C₁₃	—	2	4	no	—
50	C₂×C₈×C₁₃	—	4	2	no	—
51	C₂×C₈×C₁₃	—	4	2	no	—
52	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
53	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
54	C₂×C₈×C₁₃	D₈	1	2	yes	—
55	C₂×C₈×C₁₃	Q₁₆	1	2	yes	—
56	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
57	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
58	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
59	C₄²×C₁₃	#41	1	1	yes	characteristic
60	C₄⋊Dic₁₃	—	2	1	no	—
61	C₄⋊Dic₁₃	—	2	1	no	—
62	C₄⋊Dic₁₃	—	4	2	no	—
63	C₄⋊Dic₁₃	—	4	2	no	—
64	C₄⋊Dic₁₃	—	4	2	no	—
65	C₄⋊Dic₁₃	—	4	2	no	—
66	C₄⋊Dic₁₃	—	8	2	no	—
67	C₄⋊Dic₁₃	D₈	1	2	yes	—
68	C₄⋊Dic₁₃	Q₁₆	1	2	yes	—
69	C₄×Dic₁₃	—	4	1	no	—
70	C₄×Dic₁₃	—	4	1	no	—
71	C₂×#1055	—	2	2	no	—
72	C₂×#1055	—	4	1	no	—
73	C₂×#1055	—	4	1	no	—
74	#1947	—	13	2	no	—
75	#1845	—	13	1	no	—
76	C₈×C₄⋊C₄	—	13	1	no	—
77	C₂×#335	D₁₃	1	1	yes	characteristic
78	#1296	—	13	2	no	—
79	C₂³×C₁₃	#121	1	1	yes	characteristic
80	C₂×C₄×C₁₃	—	2	1	no	—
81	C₂×C₄×C₁₃	—	2	1	no	—
82	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
84	C₂×C₄×C₁₃	C₄×Q₈	1	1	yes	characteristic
85	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
86	C₂×C₄×C₁₃	#125	1	1	yes	characteristic
87	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
88	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
89	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
90	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
91	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
92	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
93	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
94	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
95	C₂×C₄×C₁₃	#137	1	1	yes	characteristic
96	C₂×Dic₁₃	—	2	2	no	—
97	C₂×Dic₁₃	—	2	2	no	—
98	C₂×Dic₁₃	—	4	1	no	—
99	C₂×Dic₁₃	—	4	1	no	—
100	C₂×Dic₁₃	—	4	1	no	—
101	C₂×Dic₁₃	—	4	1	no	—
102	C₂×Dic₁₃	—	8	2	no	—
103	C₂×Dic₁₃	—	8	2	no	—
104	C₂×Dic₁₃	—	8	2	no	—
105	C₈×C₁₃	—	2	2	no	—
106	C₈×C₁₃	—	4	4	no	—
107	C₈×C₁₃	#107	1	4	yes	complemented
108	#1055	—	4	2	no	—
109	C₂×#107	—	26	2	no	—
110	C₂×#107	—	26	1	no	—
111	C₂×#107	—	26	1	no	—
112	C₂×C₄⋊C₈	—	13	1	no	—
113	C₂×C₄⋊C₈	C₂×D₁₃	1	2	yes	—
114	C₂×C₄×C₈	—	26	1	no	—
115	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
116	#384	—	13	2	no	—
117	C₄×C₄⋊C₄	—	13	1	no	—
118	#340	—	26	2	no	—
119	#336	—	26	1	no	—
120	#335	Dic₁₃	1	4	yes	complemented
121	C₂²×C₁₃	#389	1	1	yes	characteristic
122	C₂²×C₁₃	#459	1	1	yes	characteristic
123	C₂²×C₁₃	#462	1	1	yes	characteristic
124	C₂²×C₁₃	#466	1	1	yes	characteristic
125	C₂²×C₁₃	#467	1	1	yes	characteristic
126	C₂²×C₁₃	#472	1	1	yes	characteristic
127	C₂²×C₁₃	#473	1	1	yes	characteristic
128	C₄×C₁₃	—	2	1	no	—
129	C₄×C₁₃	—	2	1	no	—
130	C₄×C₁₃	C₂×#107	1	1	yes	characteristic
131	C₄×C₁₃	#428	1	1	yes	characteristic
132	C₄×C₁₃	C₄×D₈	1	1	yes	characteristic
133	C₄×C₁₃	C₄×Q₁₆	1	1	yes	characteristic
134	C₄×C₁₃	#474	1	1	yes	characteristic
135	C₄×C₁₃	#477	1	1	yes	characteristic
136	C₄×C₁₃	#499	1	1	yes	characteristic
137	C₄×C₁₃	#500	1	1	yes	characteristic
138	Dic₁₃	—	4	2	no	—
139	Dic₁₃	—	4	2	no	—
140	Dic₁₃	—	16	2	no	—
141	C₂²×C₈	—	2	2	no	—
142	C₂²×C₈	—	26	1	no	—
143	C₂²×C₈	D₅₂	1	1	yes	characteristic
144	C₂²×C₈	D₅₂	1	1	yes	characteristic
145	C₂×C₄⋊C₄	—	13	1	no	—
146	C₂×C₄⋊C₄	—	13	1	no	—
147	C₂×C₄⋊C₄	—	26	2	no	—
148	C₂×C₄⋊C₄	—	26	2	no	—
149	C₂×C₄⋊C₄	—	26	2	no	—
150	C₂×C₄²	—	26	1	no	—
151	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
152	#107	—	52	4	no	—
153	#107	—	52	2	no	—
154	#107	—	52	2	no	—
155	C₄⋊C₈	—	2	2	no	—
156	C₄⋊C₈	—	13	2	no	—
157	C₄⋊C₈	—	26	1	no	—
158	C₄⋊C₈	C₂×Dic₁₃	1	4	yes	—
159	C₄×C₈	—	52	2	no	—
160	C₄×C₈	Dic₂₆	1	2	yes	—
161	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
162	#95	—	26	1	no	—
163	C₂×C₁₃	#1667	1	1	yes	characteristic
164	C₂×C₁₃	#1669	1	1	yes	characteristic
165	C₂×C₁₃	#1674	1	1	yes	characteristic
166	C₂×C₁₃	#1676	1	1	yes	characteristic
167	C₂×C₁₃	#1942	1	1	yes	characteristic
168	C₂×C₁₃	#1947	1	1	yes	characteristic
169	C₂×C₁₃	#1952	1	1	yes	characteristic
170	C₂²×C₄	—	26	1	no	—
171	C₂²×C₄	—	26	1	no	—
172	C₂²×C₄	—	52	2	no	—
173	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
174	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
175	C₂²×C₄	#6131	1	1	yes	characteristic
176	C₂×C₈	—	2	1	no	—
177	C₂×C₈	—	2	1	no	—
178	C₂×C₈	—	2	4	no	—
179	C₂×C₈	—	4	2	no	—
180	C₂×C₈	—	4	2	no	—
181	C₂×C₈	—	26	2	no	—
182	C₂×C₈	—	52	1	no	—
183	C₂×C₈	—	52	1	no	—
184	C₂×C₈	D₁₀₄	1	2	yes	—
185	C₂×C₈	Dic₅₂	1	2	yes	—
186	C₂×C₈	C₄⋊Dic₁₃	1	2	yes	—
187	C₂×C₈	C₄⋊Dic₁₃	1	2	yes	—
188	C₄⋊C₄	—	13	2	no	—
189	C₄⋊C₄	—	13	2	no	—
190	C₄⋊C₄	—	26	1	no	—
191	C₄⋊C₄	—	26	1	no	—
192	C₄⋊C₄	—	52	2	no	—
193	C₄⋊C₄	—	52	2	no	—
194	C₄⋊C₄	—	52	2	no	—
195	C₄⋊C₄	—	52	2	no	—
196	C₄⋊C₄	—	104	2	no	—
197	C₄²	—	52	1	no	—
198	C₄²	—	52	1	no	—
199	C₄²	C₂×Dic₂₆	1	1	yes	characteristic
200	C₄²	#6129	1	1	yes	characteristic
201	C₄²	Q₈×D₁₃	1	1	yes	characteristic
202	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
203	C₁₃	#9614	1	1	yes	characteristic, complemented, Sylow
204	C₂³	#88149	1	1	yes	characteristic
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	26	2	no	—
208	C₂×C₄	—	26	2	no	—
209	C₂×C₄	—	52	1	no	—
210	C₂×C₄	—	52	1	no	—
211	C₂×C₄	—	52	1	no	—
212	C₂×C₄	—	52	1	no	—
213	C₂×C₄	—	104	2	no	—
214	C₂×C₄	—	104	2	no	—
215	C₂×C₄	—	104	2	no	—
216	C₂×C₄	#88142	1	1	yes	characteristic
217	C₂×C₄	#88144	1	1	yes	characteristic
218	C₂×C₄	#88151	1	1	yes	characteristic
219	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
220	C₂×C₄	C₂×Dic₅₂	1	1	yes	characteristic
221	C₂×C₄	#88162	1	1	yes	characteristic
222	C₂×C₄	#88163	1	1	yes	characteristic
223	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
224	C₂×C₄	#88166	1	1	yes	characteristic
225	C₂×C₄	Q₁₆×D₁₃	1	1	yes	characteristic
226	C₂×C₄	#88173	1	1	yes	characteristic
227	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
228	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
229	C₂×C₄	Q₈×Dic₁₃	1	1	yes	characteristic
230	C₈	—	2	2	no	—
231	C₈	—	4	4	no	—
232	C₈	—	52	2	no	—
233	C₈	#88058	1	4	yes	complemented
234	C₂²	#11718504	1	1	yes	characteristic
235	C₂²	#11718510	1	1	yes	characteristic
236	C₂²	#11718544	1	1	yes	characteristic
237	C₂²	#11718548	1	1	yes	characteristic
238	C₂²	#11718557	1	1	yes	characteristic
239	C₂²	#11718558	1	1	yes	characteristic
240	C₂²	#11718646	1	1	yes	characteristic
241	C₄	—	2	1	no	—
242	C₄	—	2	1	no	—
243	C₄	—	52	2	no	—
244	C₄	—	52	2	no	—
245	C₄	—	208	2	no	—
246	C₄	#11718507	1	1	yes	characteristic
247	C₄	#11718511	1	1	yes	characteristic
248	C₄	#11718535	1	1	yes	characteristic
249	C₄	#11718539	1	1	yes	characteristic
250	C₄	C₂×#88058	1	1	yes	characteristic
251	C₄	#11718782	1	1	yes	characteristic
252	C₄	D₈×Dic₁₃	1	1	yes	characteristic
253	C₄	Q₁₆×Dic₁₃	1	1	yes	characteristic
254	C₂	#13114192	1	1	yes	characteristic
255	C₂	#13114202	1	1	yes	characteristic
256	C₂	#13114231	1	1	yes	characteristic
257	C₂	#13114241	1	1	yes	characteristic
258	C₂	#13115570	1	1	yes	characteristic
259	C₂	#13115595	1	1	yes	characteristic
260	C₂	#13115622	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#88149
commutator	C₄²×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₂×C₁₃×#335	C₂
socle, prime radical	C₂³×C₁₃	#121
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	C₂×#335	D₁₃
hypocenter	C₁₃	#9614
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
15. C₁₃×#335	C₄	semidirect
107. C₈×C₁₃	#107	semidirect
120. #335	Dic₁₃	semidirect
203. C₁₃	#9614	semidirect
233. C₈	#88058	semidirect

Maximal subgroups

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

subgroup	index
#13115595	2
#13114996	2
C₈×C₄⋊Dic₁₃	2
#13112720	2
C₂×C₁₃×#335	2
#9614	13