Subgroups — #13123540

Order 1664.

492 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#525	C₂	1	1	yes	maximal, characteristic
2	C₂×#88202	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#6099	C₂	1	1	yes	maximal, characteristic, complemented
4	#11718895	C₂	1	1	yes	maximal, characteristic, complemented
5	#11718868	C₂	1	1	yes	maximal, characteristic, complemented
6	#11718673	C₂	1	4	yes	maximal, complemented
7	#11718672	C₂	1	4	yes	maximal, complemented
8	#11718660	C₂	1	1	yes	maximal, characteristic, complemented
9	#11718658	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₁₃×#41	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₁₃	C₂²	1	1	yes	characteristic
12	C₁₃×#123	C₄	1	2	yes	—
13	C₁₃×C₂²⋊Q₈	C₄	1	2	yes	—
14	C₁₃×#121	C₄	1	2	yes	—
15	C₁₃×#120	C₄	1	2	yes	—
16	C₄×D₄×C₁₃	C₂²	1	4	yes	—
17	C₁₃×#117	C₂²	1	1	yes	characteristic
18	#88202	—	2	4	no	—
19	C₂×#6099	—	2	4	no	—
20	C₂×#6099	C₂²	1	1	yes	characteristic
21	C₂×#6099	C₂²	1	1	yes	characteristic
22	C₂×#6099	C₂²	1	2	yes	—
23	C₂×#6099	C₂²	1	2	yes	—
24	C₂×#6099	C₂²	1	2	yes	complemented
25	C₂²×#1055	C₂²	1	1	yes	characteristic
26	C₂²×#1055	C₂²	1	1	yes	characteristic
27	C₂²×#1055	C₂²	1	2	yes	complemented
28	C₂²⋊#1055	C₂²	1	4	yes	—
29	C₂²⋊#1055	C₂²	1	4	yes	complemented
30	C₄⋊#1055	C₂²	1	4	yes	complemented
31	#88043	C₂²	1	2	yes	complemented
32	C₄×#1055	C₂²	1	2	yes	complemented
33	C₂⁴×C₁₃	C₂×C₄	1	1	yes	characteristic
34	C₁₃×#41	—	2	2	no	—
35	C₁₃×#41	—	2	2	no	—
36	C₂×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
37	C₂×D₄×C₁₃	C₂×C₄	1	2	yes	—
38	C₂×D₄×C₁₃	C₂×C₄	1	1	yes	characteristic
39	C₂×D₄×C₁₃	C₂×C₄	1	4	yes	—
40	C₂²×C₄×C₁₃	—	2	2	no	—
41	C₂²×C₄×C₁₃	—	2	2	no	—
42	C₂²×C₄×C₁₃	C₂×C₄	1	2	yes	—
43	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
45	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
46	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
47	C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
48	C₁₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
49	C₁₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
50	C₁₃×C₂²⋊C₄	C₂×C₄	1	4	yes	—
51	C₁₃×C₂²⋊C₄	C₂×C₄	1	4	yes	—
52	C₄²×C₁₃	C₂³	1	2	yes	—
53	#6099	—	2	2	no	—
54	#6099	—	2	2	no	—
55	#6099	—	2	4	no	—
56	#6099	—	2	4	no	—
57	#6099	—	2	4	no	—
58	C₂×#1055	—	2	2	no	—
59	C₂×#1055	—	2	4	no	—
60	C₂×#1055	D₄	1	4	yes	complemented
61	C₂×#1055	C₂³	1	2	yes	—
62	C₂×#1055	C₂³	1	2	yes	—
63	C₂×#1055	C₂³	1	2	yes	—
64	C₂×#1055	C₂³	1	1	yes	characteristic
65	C₂×#1055	C₂³	1	1	yes	characteristic
66	#2930	—	13	1	no	maximal, Sylow
67	C₂³×C₁₃	—	2	2	no	—
68	C₂³×C₁₃	—	4	2	no	—
69	C₂³×C₁₃	C₂²×C₄	1	2	yes	—
70	C₂³×C₁₃	C₂²×C₄	1	2	yes	—
71	C₂³×C₁₃	C₂²×C₄	1	1	yes	characteristic
72	Q₈×C₁₃	—	2	2	no	—
73	D₄×C₁₃	—	2	4	no	—
74	D₄×C₁₃	—	2	2	no	—
75	D₄×C₁₃	—	4	4	no	—
76	C₂×C₄×C₁₃	—	2	2	no	—
77	C₂×C₄×C₁₃	—	2	4	no	—
78	C₂×C₄×C₁₃	—	2	2	no	—
79	C₂×C₄×C₁₃	—	2	2	no	—
80	C₂×C₄×C₁₃	—	2	2	no	—
81	C₂×C₄×C₁₃	—	2	2	no	—
82	C₂×C₄×C₁₃	C₂²×C₄	1	4	yes	—
83	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
84	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
85	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
87	C₂×C₄×C₁₃	#41	1	2	yes	—
88	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
89	#1055	—	2	2	no	—
90	#1055	—	2	2	no	—
91	#1055	—	2	2	no	—
92	#1055	C₂×D₄	1	2	yes	—
93	#1055	C₂×D₄	1	2	yes	—
94	C₂×#131	—	13	1	no	—
95	C₂²×M₁₆	—	13	1	no	—
96	#525	D₁₃	1	1	yes	characteristic
97	#436	—	13	4	no	—
98	#435	—	13	4	no	—
99	#424	—	13	1	no	—
100	#408	—	13	1	no	—
101	#407	—	13	1	no	—
102	#405	—	13	1	no	—
103	C₂²×C₁₃	—	2	2	no	—
104	C₂²×C₁₃	—	2	2	no	—
105	C₂²×C₁₃	—	2	2	no	—
106	C₂²×C₁₃	—	2	2	no	—
107	C₂²×C₁₃	—	2	4	no	—
108	C₂²×C₁₃	—	2	2	no	—
109	C₂²×C₁₃	—	4	4	no	—
110	C₂²×C₁₃	C₄×D₄	1	2	yes	—
111	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
112	C₄×C₁₃	—	2	2	no	—
113	C₄×C₁₃	—	2	2	no	—
114	C₄×C₁₃	—	2	4	no	—
115	C₄×C₁₃	C₄×D₄	1	2	yes	—
116	C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
117	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
118	C₂×#41	C₂×D₁₃	1	1	yes	characteristic
119	C₂³×C₄	C₂×D₁₃	1	1	yes	characteristic
120	#131	—	26	4	no	—
121	C₂×M₁₆	—	13	2	no	—
122	C₂×M₁₆	—	13	2	no	—
123	C₂×M₁₆	—	13	1	no	—
124	C₂×M₁₆	—	13	2	no	—
125	C₂×M₁₆	—	13	1	no	—
126	C₂×M₁₆	—	26	4	no	—
127	C₂²×C₈	—	13	1	no	—
128	C₂²×C₈	—	13	1	no	—
129	C₂²×C₈	—	13	2	no	—
130	#123	Dic₁₃	1	2	yes	—
131	C₂²⋊Q₈	Dic₁₃	1	2	yes	—
132	#121	Dic₁₃	1	2	yes	—
133	#120	Dic₁₃	1	2	yes	—
134	C₄×D₄	C₂×D₁₃	1	4	yes	—
135	#117	C₂×D₁₃	1	1	yes	characteristic
136	C₄⋊C₈	—	13	4	no	—
137	C₂²⋊C₈	—	13	4	no	—
138	C₂²⋊C₈	—	13	4	no	—
139	#97	—	13	2	no	—
140	C₄×C₈	—	13	2	no	—
141	C₂×C₁₃	—	2	2	no	—
142	C₂×C₁₃	—	4	2	no	—
143	C₂×C₁₃	—	4	2	no	—
144	C₂×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
145	C₂×C₁₃	#568	1	2	yes	—
146	C₂⁴	C₂×Dic₁₃	1	1	yes	characteristic
147	#41	—	2	2	no	—
148	#41	—	2	2	no	—
149	C₂×Q₈	C₂×Dic₁₃	1	1	yes	characteristic
150	C₂×D₄	C₂×Dic₁₃	1	2	yes	—
151	C₂×D₄	C₂×Dic₁₃	1	1	yes	characteristic
152	C₂×D₄	C₂×Dic₁₃	1	4	yes	—
153	C₂²×C₄	—	2	2	no	—
154	C₂²×C₄	—	2	2	no	—
155	C₂²×C₄	C₂×Dic₁₃	1	2	yes	—
156	C₂²×C₄	C₂×Dic₁₃	1	1	yes	characteristic
157	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
158	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
159	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
160	M₁₆	—	26	4	no	—
161	M₁₆	—	26	4	no	—
162	M₁₆	—	26	4	no	—
163	M₁₆	—	26	2	no	—
164	M₁₆	—	26	2	no	—
165	C₂×C₈	—	13	4	no	—
166	C₂×C₈	—	13	1	no	—
167	C₂×C₈	—	13	1	no	—
168	C₂×C₈	—	13	2	no	—
169	C₂×C₈	—	13	2	no	—
170	C₂×C₈	—	13	2	no	—
171	C₂×C₈	—	26	4	no	—
172	C₂×C₈	—	26	2	no	—
173	C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
174	C₄⋊C₄	C₂×Dic₁₃	1	4	yes	—
175	C₂²⋊C₄	C₂×Dic₁₃	1	2	yes	—
176	C₂²⋊C₄	C₂×Dic₁₃	1	4	yes	—
177	C₂²⋊C₄	C₂×Dic₁₃	1	4	yes	—
178	C₄²	C₂²×D₁₃	1	2	yes	—
179	C₁₃	#2930	1	1	yes	characteristic, complemented, Sylow
180	C₂³	—	2	2	no	—
181	C₂³	—	4	2	no	—
182	C₂³	C₂²×Dic₁₃	1	2	yes	—
183	C₂³	C₂²×Dic₁₃	1	2	yes	—
184	C₂³	C₂²×Dic₁₃	1	1	yes	characteristic
185	Q₈	—	2	2	no	—
186	D₄	—	2	4	no	—
187	D₄	—	2	2	no	—
188	D₄	—	4	4	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	4	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	D₄×D₁₃	1	2	yes	—
196	C₂×C₄	#6129	1	2	yes	—
197	C₂×C₄	C₂²×Dic₁₃	1	4	yes	—
198	C₂×C₄	C₂²×Dic₁₃	1	2	yes	—
199	C₂×C₄	C₂²×Dic₁₃	1	2	yes	—
200	C₂×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
201	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
202	C₈	—	13	2	no	—
203	C₈	—	13	2	no	—
204	C₈	—	26	2	no	—
205	C₈	—	26	2	no	—
206	C₈	—	26	2	no	—
207	C₂²	—	2	2	no	—
208	C₂²	—	2	2	no	—
209	C₂²	—	2	2	no	—
210	C₂²	—	2	2	no	—
211	C₂²	—	2	4	no	—
212	C₂²	—	2	2	no	—
213	C₂²	—	4	4	no	—
214	C₂²	D₄×Dic₁₃	1	2	yes	—
215	C₂²	C₂³×Dic₁₃	1	1	yes	characteristic
216	C₄	—	2	2	no	—
217	C₄	—	2	2	no	—
218	C₄	—	2	4	no	—
219	C₄	D₄×Dic₁₃	1	2	yes	—
220	C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
221	C₄	C₂×#6129	1	1	yes	characteristic
222	C₂	—	2	2	no	—
223	C₂	—	4	2	no	—
224	C₂	—	4	2	no	—
225	C₂	C₂×D₄×Dic₁₃	1	1	yes	characteristic
226	C₂	#11719524	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₃
commutator, socle	C₂²×C₁₃	C₂³×C₄
Fitting, prime radical	C₁₃×#525	C₂
cosocle	C₂×C₄×C₁₃	C₂⁴
hypercenter	#525	D₁₃
hypocenter	C₁₃	#2930
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #525

Complements

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

N	H ≅ G/N	type
2. C₂×#88202	C₂	semidirect
3. C₂²×#6099	C₂	semidirect
4. #11718895	C₂	semidirect
5. #11718868	C₂	semidirect
6. #11718673	C₂	semidirect
7. #11718672	C₂	semidirect
8. #11718660	C₂	semidirect
9. #11718658	C₂	semidirect
24. C₂×#6099	C₂²	semidirect
27. C₂²×#1055	C₂²	semidirect
29. C₂²⋊#1055	C₂²	semidirect
30. C₄⋊#1055	C₂²	semidirect
31. #88043	C₂²	semidirect
32. C₄×#1055	C₂²	semidirect
60. C₂×#1055	D₄	semidirect
179. C₁₃	#2930	semidirect

Maximal subgroups

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

subgroup	index
C₁₃×#525	2
C₂×#88202	2
C₂²×#6099	2
#11718895	2
#11718868	2
#11718673	2
#11718672	2
#11718660	2
#11718658	2
#2930	13