Subgroups — #13124698

Order 1664.

376 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#520	C₂	1	1	yes	maximal, characteristic
2	#11719224	C₂	1	1	yes	maximal, characteristic, complemented
3	#11718894	C₂	1	1	yes	maximal, characteristic
4	#11718893	C₂	1	1	yes	maximal, characteristic
5	#11718683	C₂	1	2	yes	maximal, complemented
6	#11718681	C₂	1	2	yes	maximal, complemented
7	#11718670	C₂	1	2	yes	maximal, complemented
8	#11718669	C₂	1	2	yes	maximal, complemented
9	#11718660	C₂	1	1	yes	maximal, characteristic, complemented
10	#11718625	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#88048	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₁₃×#41	C₂²	1	1	yes	characteristic
13	C₄×Q₈×C₁₃	C₂²	1	2	yes	—
14	C₄×D₄×C₁₃	—	2	2	no	—
15	C₄×D₄×C₁₃	C₂²	1	2	yes	—
16	C₁₃×#117	—	2	1	no	—
17	C₁₃×#117	C₂²	1	1	yes	characteristic
18	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
19	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	#88180	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
22	C₂×#6099	C₂²	1	1	yes	characteristic
23	C₂²×#1055	C₂²	1	1	yes	characteristic
24	#88144	—	2	1	no	—
25	#88142	—	2	1	no	—
26	#88132	—	2	2	no	—
27	#88124	C₂²	1	2	yes	—
28	#88123	C₂²	1	2	yes	—
29	#88075	C₂²	1	1	yes	characteristic
30	#88075	C₂²	1	2	yes	—
31	#88075	C₂²	1	1	yes	characteristic
32	#88072	C₂²	1	1	yes	characteristic
33	#88072	C₂²	1	2	yes	—
34	#88072	C₂²	1	1	yes	characteristic
35	#88048	C₂²	1	2	yes	—
36	#88048	C₂²	1	2	yes	—
37	#88048	C₂²	1	2	yes	—
38	#88047	C₂²	1	2	yes	—
39	C₄⋊#1055	C₂²	1	2	yes	—
40	C₄⋊#1055	C₂²	1	2	yes	—
41	C₁₃×#41	—	2	2	no	—
42	C₁₃×#41	Q₈	1	4	yes	complemented
43	C₂×Q₈×C₁₃	C₂³	1	1	yes	characteristic
44	C₂×D₄×C₁₃	—	2	1	no	—
45	C₂×D₄×C₁₃	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₁₃	—	2	1	no	—
47	C₂²×C₄×C₁₃	—	2	1	no	—
48	C₂²×C₄×C₁₃	—	2	2	no	—
49	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
50	C₂²×Dic₁₃	—	2	1	no	—
51	C₂×Dic₂₆	—	2	2	no	—
52	C₁₃×C₄⋊C₄	—	2	1	no	—
53	C₁₃×C₄⋊C₄	—	2	2	no	—
54	C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
55	C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
56	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
57	C₁₃×C₂²⋊C₄	—	2	2	no	—
58	C₁₃×C₂²⋊C₄	D₄	1	2	yes	—
59	C₄²×C₁₃	—	2	2	no	—
60	C₄²×C₁₃	C₂³	1	2	yes	—
61	C₂²⋊Dic₁₃	—	2	1	no	—
62	C₄⋊Dic₁₃	—	2	2	no	—
63	C₄⋊Dic₁₃	—	2	1	no	—
64	C₄⋊Dic₁₃	C₂³	1	1	yes	characteristic
65	C₄⋊Dic₁₃	C₂³	1	2	yes	—
66	C₄⋊Dic₁₃	C₂³	1	1	yes	characteristic
67	#6101	—	2	2	no	—
68	#6101	—	2	2	no	—
69	C₄×Dic₁₃	—	2	1	no	—
70	#6099	—	2	2	no	—
71	C₂×#1055	—	2	2	no	—
72	C₂×#1055	C₂³	1	1	yes	characteristic
73	C₂×#1055	C₂³	1	1	yes	characteristic
74	C₂×#1055	C₂³	1	2	yes	—
75	#3084	—	13	1	no	maximal, Sylow
76	C₂³×C₁₃	—	2	1	no	—
77	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
78	Q₈×C₁₃	—	2	1	no	—
79	Q₈×C₁₃	C₂×Q₈	1	2	yes	—
80	D₄×C₁₃	—	2	4	no	—
81	D₄×C₁₃	—	2	1	no	—
82	D₄×C₁₃	C₂×Q₈	1	2	yes	—
83	C₂×C₄×C₁₃	—	2	1	no	—
84	C₂×C₄×C₁₃	—	2	4	no	—
85	C₂×C₄×C₁₃	—	2	2	no	—
86	C₂×C₄×C₁₃	—	8	2	no	—
87	C₂×C₄×C₁₃	—	8	1	no	—
88	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
89	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
90	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
91	C₂×C₄×C₁₃	C₂×Q₈	1	2	yes	—
92	C₂×C₄×C₁₃	#41	1	2	yes	—
93	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
94	C₂×Dic₁₃	—	2	1	no	—
95	C₂×Dic₁₃	—	2	1	no	—
96	C₂×Dic₁₃	—	2	2	no	—
97	C₂×Dic₁₃	—	8	1	no	—
98	Dic₂₆	—	8	2	no	—
99	#1055	—	2	1	no	—
100	#1055	—	2	1	no	—
101	#1055	—	2	2	no	—
102	#544	—	13	1	no	—
103	#520	D₁₃	1	1	yes	characteristic
104	#479	—	13	2	no	—
105	#478	—	13	2	no	—
106	#476	—	13	2	no	—
107	#475	—	13	2	no	—
108	#428	—	13	1	no	—
109	C₂×#106	—	13	1	no	—
110	#424	—	13	1	no	—
111	#417	—	13	1	no	—
112	#416	—	13	1	no	—
113	C₂²×C₁₃	—	2	1	no	—
114	C₂²×C₁₃	—	2	1	no	—
115	C₂²×C₁₃	—	2	2	no	—
116	C₂²×C₁₃	—	4	1	no	—
117	C₂²×C₁₃	—	4	1	no	—
118	C₂²×C₁₃	C₂²⋊Q₈	1	2	yes	—
119	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
120	C₄×C₁₃	—	2	2	no	—
121	C₄×C₁₃	—	2	2	no	—
122	C₄×C₁₃	—	2	2	no	—
123	C₄×C₁₃	—	4	2	no	—
124	C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
125	C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
126	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
127	Dic₁₃	—	4	1	no	—
128	Dic₁₃	—	4	2	no	—
129	Dic₁₃	—	4	1	no	—
130	C₂×#41	C₂×D₁₃	1	1	yes	characteristic
131	C₂×M₁₆	—	13	1	no	—
132	C₂²×C₈	—	13	1	no	—
133	C₄⋊Q₈	—	13	2	no	—
134	C₄⋊Q₈	—	26	1	no	—
135	#125	—	13	2	no	—
136	#125	—	26	1	no	—
137	C₂²⋊Q₈	—	26	2	no	—
138	C₄×Q₈	C₂×D₁₃	1	2	yes	—
139	C₄×D₄	—	2	2	no	—
140	C₄×D₄	C₂×D₁₃	1	2	yes	—
141	#117	—	2	1	no	—
142	#117	—	13	1	no	—
143	#117	C₂×D₁₃	1	1	yes	characteristic
144	C₂×C₄⋊C₄	—	2	1	no	—
145	C₂×C₄⋊C₄	—	13	1	no	—
146	C₂×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
147	#107	—	13	2	no	—
148	#106	—	13	2	no	—
149	#106	—	13	2	no	—
150	#106	—	13	2	no	—
151	C₄⋊C₈	—	13	2	no	—
152	C₄⋊C₈	—	13	2	no	—
153	#103	—	13	1	no	—
154	#103	—	13	1	no	—
155	#103	—	13	2	no	—
156	#102	—	13	1	no	—
157	#102	—	13	1	no	—
158	#102	—	13	2	no	—
159	C₂×C₁₃	—	2	2	no	—
160	C₂×C₁₃	—	4	2	no	—
161	C₂×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
162	C₂×C₁₃	#577	1	1	yes	characteristic
163	C₂×C₁₃	#577	1	1	yes	characteristic
164	#41	—	2	2	no	—
165	#41	Dic₂₆	1	4	yes	complemented
166	C₂×Q₈	—	26	2	no	—
167	C₂×Q₈	C₂²×D₁₃	1	1	yes	characteristic
168	C₂×D₄	—	2	1	no	—
169	C₂×D₄	C₂²×D₁₃	1	1	yes	characteristic
170	C₂²×C₄	—	2	1	no	—
171	C₂²×C₄	—	2	1	no	—
172	C₂²×C₄	—	2	2	no	—
173	C₂²×C₄	—	26	1	no	—
174	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
175	M₁₆	—	26	2	no	—
176	C₂×C₈	—	13	1	no	—
177	C₂×C₈	—	13	1	no	—
178	C₂×C₈	—	13	2	no	—
179	C₂×C₈	—	26	2	no	—
180	C₄⋊C₄	—	2	1	no	—
181	C₄⋊C₄	—	2	2	no	—
182	C₄⋊C₄	—	13	2	no	—
183	C₄⋊C₄	—	13	1	no	—
184	C₄⋊C₄	—	13	1	no	—
185	C₄⋊C₄	—	26	1	no	—
186	C₄⋊C₄	—	26	2	no	—
187	C₄⋊C₄	—	26	2	no	—
188	C₄⋊C₄	—	26	2	no	—
189	C₄⋊C₄	C₂²⋊D₁₃	1	1	yes	characteristic
190	C₄⋊C₄	C₂²⋊D₁₃	1	1	yes	characteristic
191	C₄⋊C₄	C₂²×D₁₃	1	2	yes	—
192	C₂²⋊C₄	—	2	2	no	—
193	C₂²⋊C₄	—	26	1	no	—
194	C₂²⋊C₄	C₂²⋊D₁₃	1	2	yes	—
195	C₄²	—	2	2	no	—
196	C₄²	—	26	1	no	—
197	C₄²	C₂²×D₁₃	1	2	yes	—
198	C₁₃	#3084	1	1	yes	characteristic, complemented, Sylow
199	C₂³	—	2	1	no	—
200	C₂³	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
201	Q₈	—	2	1	no	—
202	Q₈	—	104	2	no	—
203	Q₈	C₂×Dic₂₆	1	2	yes	—
204	D₄	—	2	4	no	—
205	D₄	—	2	1	no	—
206	D₄	C₂×Dic₂₆	1	2	yes	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	8	2	no	—
211	C₂×C₄	—	8	1	no	—
212	C₂×C₄	—	26	1	no	—
213	C₂×C₄	—	26	1	no	—
214	C₂×C₄	—	26	2	no	—
215	C₂×C₄	—	104	1	no	—
216	C₂×C₄	C₂×Dic₂₆	1	2	yes	—
217	C₂×C₄	#6127	1	2	yes	—
218	C₂×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
219	C₂×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
220	C₂×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
221	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
222	C₈	—	26	1	no	—
223	C₈	—	26	1	no	—
224	C₈	—	26	2	no	—
225	C₂²	—	2	1	no	—
226	C₂²	—	2	1	no	—
227	C₂²	—	2	2	no	—
228	C₂²	—	4	1	no	—
229	C₂²	—	4	1	no	—
230	C₂²	#88178	1	2	yes	—
231	C₂²	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
232	C₄	—	2	2	no	—
233	C₄	—	2	2	no	—
234	C₄	—	2	2	no	—
235	C₄	—	4	2	no	—
236	C₄	—	52	1	no	—
237	C₄	—	52	2	no	—
238	C₄	—	52	1	no	—
239	C₄	#88178	1	2	yes	—
240	C₄	C₂²×Dic₂₆	1	1	yes	characteristic
241	C₄	C₂×#6127	1	1	yes	characteristic
242	C₂	—	2	2	no	—
243	C₂	—	4	2	no	—
244	C₂	C₂×#88178	1	1	yes	characteristic
245	C₂	#11719541	1	1	yes	characteristic
246	C₂	#11719542	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₃
commutator, cosocle	C₂×C₄×C₁₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃
Fitting	C₁₃×#520	C₂
socle	C₂²×C₁₃	C₂²×D₄
prime radical	C₂×C₁₃×#41	C₂²
hypercenter	#520	D₁₃
hypocenter	C₁₃	#3084
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #520

Complements

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

N	H ≅ G/N	type
2. #11719224	C₂	semidirect
5. #11718683	C₂	semidirect
6. #11718681	C₂	semidirect
7. #11718670	C₂	semidirect
8. #11718669	C₂	semidirect
9. #11718660	C₂	semidirect
10. #11718625	C₂	semidirect
11. C₂×#88048	C₂	semidirect
42. C₁₃×#41	Q₈	semidirect
165. #41	Dic₂₆	semidirect
198. C₁₃	#3084	semidirect

Maximal subgroups

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

subgroup	index
C₁₃×#520	2
#11719224	2
#11718894	2
#11718893	2
#11718683	2
#11718681	2
#11718670	2
#11718669	2
#11718660	2
#11718625	2
C₂×#88048	2
#3084	13