Subgroups — C₁₃×#3174 #13112232

Order 1664.

362 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#541	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₃×#540	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#499	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#492	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#491	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#490	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#468	C₂	1	2	yes	maximal, complemented
8	C₁₃×#465	C₂	1	2	yes	maximal, complemented
9	C₁₃×#440	C₂	1	2	yes	maximal, complemented
10	C₄×C₁₃×SD₁₆	C₂	1	2	yes	maximal, complemented
11	C₁₃×#405	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₁₃×SD₁₆	C₂²	1	2	yes	—
13	C₂×C₁₃×SD₁₆	C₂²	1	2	yes	—
14	C₂×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
15	C₂²×C₈×C₁₃	C₂²	1	1	yes	characteristic
16	C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₁₃×#127	C₂²	1	1	yes	characteristic
18	C₁₃×#126	—	2	2	no	—
19	C₁₃×#125	—	2	2	no	—
20	C₁₃×#125	C₂²	1	1	yes	characteristic
21	C₁₃×#125	C₂²	1	1	yes	characteristic
22	C₁₃×#123	—	2	2	no	—
23	C₁₃×C₂²⋊Q₈	C₂²	1	2	yes	—
24	C₁₃×#121	—	2	2	no	—
25	C₁₃×#121	C₂²	1	2	yes	—
26	C₄×Q₈×C₁₃	C₂²	1	2	yes	—
27	C₄×D₄×C₁₃	C₂²	1	2	yes	—
28	C₁₃×#117	C₂²	1	1	yes	characteristic
29	C₁₃×#107	C₂²	1	2	yes	—
30	C₁₃×#106	C₂²	1	2	yes	—
31	C₁₃×#103	C₂²	1	2	yes	—
32	C₁₃×#103	C₂²	1	2	yes	—
33	C₁₃×#102	C₂²	1	2	yes	—
34	C₁₃×#102	C₂²	1	2	yes	—
35	C₁₃×#97	C₂²	1	1	yes	characteristic
36	C₁₃×#97	C₂²	1	1	yes	characteristic
37	C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
38	C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
39	C₂×Q₈×C₁₃	C₂³	1	2	yes	—
40	C₂×D₄×C₁₃	—	2	2	no	—
41	C₂×D₄×C₁₃	—	2	2	no	—
42	C₂×D₄×C₁₃	C₂³	1	2	yes	—
43	C₂²×C₄×C₁₃	—	2	2	no	—
44	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
45	C₁₃×SD₁₆	—	2	4	no	—
46	C₁₃×SD₁₆	—	2	4	no	—
47	C₁₃×M₁₆	—	2	2	no	—
48	C₂×C₈×C₁₃	—	2	2	no	—
49	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
50	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
51	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
52	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
53	C₁₃×C₄⋊C₄	—	2	2	no	—
54	C₁₃×C₄⋊C₄	—	2	2	no	—
55	C₁₃×C₄⋊C₄	—	2	2	no	—
56	C₁₃×C₄⋊C₄	—	2	2	no	—
57	C₁₃×C₄⋊C₄	—	2	2	no	—
58	C₁₃×C₄⋊C₄	D₄	1	2	yes	—
59	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
60	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
61	C₁₃×C₂²⋊C₄	—	2	2	no	—
62	C₁₃×C₂²⋊C₄	—	2	2	no	—
63	C₁₃×C₂²⋊C₄	—	2	2	no	—
64	C₁₃×C₂²⋊C₄	D₄	1	2	yes	—
65	C₄²×C₁₃	—	2	2	no	—
66	C₄²×C₁₃	C₂³	1	1	yes	characteristic
67	C₄²×C₁₃	C₂³	1	1	yes	characteristic
68	#3174	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
69	C₂³×C₁₃	—	2	2	no	—
70	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
71	Q₈×C₁₃	—	2	2	no	—
72	Q₈×C₁₃	—	2	2	no	—
73	D₄×C₁₃	—	2	2	no	—
74	D₄×C₁₃	—	2	2	no	—
75	D₄×C₁₃	—	8	2	no	—
76	D₄×C₁₃	—	8	2	no	—
77	C₂×C₄×C₁₃	—	2	1	no	—
78	C₂×C₄×C₁₃	—	2	1	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	—	8	2	no	—
83	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
84	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
85	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
86	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
87	C₈×C₁₃	—	2	1	no	—
88	C₈×C₁₃	—	2	1	no	—
89	C₈×C₁₃	#41	1	2	yes	—
90	C₈×C₁₃	#41	1	2	yes	—
91	#541	C₂×C₁₃	1	1	yes	characteristic, complemented
92	#540	C₂×C₁₃	1	1	yes	characteristic
93	#499	C₂×C₁₃	1	1	yes	characteristic, complemented
94	#492	C₂×C₁₃	1	1	yes	characteristic, complemented
95	#491	C₂×C₁₃	1	1	yes	characteristic, complemented
96	#490	C₂×C₁₃	1	1	yes	characteristic, complemented
97	#468	C₂×C₁₃	1	2	yes	complemented
98	#465	C₂×C₁₃	1	2	yes	complemented
99	#440	C₂×C₁₃	1	2	yes	complemented
100	C₄×SD₁₆	C₂×C₁₃	1	2	yes	complemented
101	#405	C₂×C₁₃	1	1	yes	characteristic, complemented
102	C₂²×C₁₃	—	2	1	no	—
103	C₂²×C₁₃	—	2	2	no	—
104	C₂²×C₁₃	—	4	2	no	—
105	C₂²×C₁₃	—	4	2	no	—
106	C₂²×C₁₃	—	4	2	no	—
107	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
108	C₄×C₁₃	—	2	1	no	—
109	C₄×C₁₃	—	2	2	no	—
110	C₄×C₁₃	—	2	2	no	—
111	C₄×C₁₃	—	4	2	no	—
112	C₄×C₁₃	—	4	2	no	—
113	C₄×C₁₃	—	4	2	no	—
114	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
115	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
116	C₂×SD₁₆	C₂²×C₁₃	1	2	yes	—
117	C₂×SD₁₆	C₂²×C₁₃	1	2	yes	—
118	C₂×M₁₆	C₂²×C₁₃	1	1	yes	characteristic
119	C₂²×C₈	C₂²×C₁₃	1	1	yes	characteristic
120	C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
121	#127	C₂²×C₁₃	1	1	yes	characteristic
122	#126	—	2	2	no	—
123	#125	—	2	2	no	—
124	#125	C₂²×C₁₃	1	1	yes	characteristic
125	#125	C₂²×C₁₃	1	1	yes	characteristic
126	#123	—	2	2	no	—
127	C₂²⋊Q₈	C₂²×C₁₃	1	2	yes	—
128	#121	—	2	2	no	—
129	#121	C₂²×C₁₃	1	2	yes	—
130	C₄×Q₈	C₂²×C₁₃	1	2	yes	—
131	C₄×D₄	C₂²×C₁₃	1	2	yes	—
132	#117	C₂²×C₁₃	1	1	yes	characteristic
133	#107	C₂²×C₁₃	1	2	yes	—
134	#106	C₂²×C₁₃	1	2	yes	—
135	#103	C₂²×C₁₃	1	2	yes	—
136	#103	C₂²×C₁₃	1	2	yes	—
137	#102	C₂²×C₁₃	1	2	yes	—
138	#102	C₂²×C₁₃	1	2	yes	—
139	#97	C₂²×C₁₃	1	1	yes	characteristic
140	#97	C₂²×C₁₃	1	1	yes	characteristic
141	C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
142	C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
143	C₂×C₁₃	—	4	1	no	—
144	C₂×C₁₃	—	8	2	no	—
145	C₂×C₁₃	#532	1	1	yes	characteristic
146	C₂×C₁₃	#577	1	2	yes	—
147	C₂×Q₈	C₂³×C₁₃	1	2	yes	—
148	C₂×D₄	—	2	2	no	—
149	C₂×D₄	—	2	2	no	—
150	C₂×D₄	C₂³×C₁₃	1	2	yes	—
151	C₂²×C₄	—	2	2	no	—
152	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
153	SD₁₆	—	2	4	no	—
154	SD₁₆	—	2	4	no	—
155	M₁₆	—	2	2	no	—
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
158	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
159	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
160	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
161	C₄⋊C₄	—	2	2	no	—
162	C₄⋊C₄	—	2	2	no	—
163	C₄⋊C₄	—	2	2	no	—
164	C₄⋊C₄	—	2	2	no	—
165	C₄⋊C₄	—	2	2	no	—
166	C₄⋊C₄	D₄×C₁₃	1	2	yes	—
167	C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
168	C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
169	C₂²⋊C₄	—	2	2	no	—
170	C₂²⋊C₄	—	2	2	no	—
171	C₂²⋊C₄	—	2	2	no	—
172	C₂²⋊C₄	D₄×C₁₃	1	2	yes	—
173	C₄²	—	2	2	no	—
174	C₄²	C₂³×C₁₃	1	1	yes	characteristic
175	C₄²	C₂³×C₁₃	1	1	yes	characteristic
176	C₁₃	#3174	1	1	yes	characteristic, direct product factor, Sylow
177	C₂³	—	2	2	no	—
178	C₂³	C₂×D₄×C₁₃	1	1	yes	characteristic
179	Q₈	—	2	2	no	—
180	Q₈	—	2	2	no	—
181	D₄	—	2	2	no	—
182	D₄	—	2	2	no	—
183	D₄	—	8	2	no	—
184	D₄	—	8	2	no	—
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	—	2	1	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	8	2	no	—
191	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
192	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
193	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
194	C₂×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
195	C₈	—	2	1	no	—
196	C₈	—	2	1	no	—
197	C₈	C₁₃×#41	1	2	yes	—
198	C₈	C₁₃×#41	1	2	yes	—
199	C₂²	—	2	1	no	—
200	C₂²	—	2	2	no	—
201	C₂²	—	4	2	no	—
202	C₂²	—	4	2	no	—
203	C₂²	—	4	2	no	—
204	C₂²	C₂²×D₄×C₁₃	1	1	yes	characteristic
205	C₄	—	2	1	no	—
206	C₄	—	2	2	no	—
207	C₄	—	2	2	no	—
208	C₄	—	4	2	no	—
209	C₄	—	4	2	no	—
210	C₄	—	4	2	no	—
211	C₄	C₂×C₁₃×#41	1	1	yes	characteristic
212	C₄	C₂×C₁₃×#41	1	1	yes	characteristic
213	C₂	—	4	1	no	—
214	C₂	—	8	2	no	—
215	C₂	C₁₃×#532	1	1	yes	characteristic
216	C₂	C₁₃×#577	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₃
prime radical	C₁₃×#540	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₁₃×#117 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₃×#541	C₂	semidirect
3. C₁₃×#499	C₂	semidirect
4. C₁₃×#492	C₂	semidirect
5. C₁₃×#491	C₂	semidirect
6. C₁₃×#490	C₂	semidirect
7. C₁₃×#468	C₂	semidirect
8. C₁₃×#465	C₂	semidirect
9. C₁₃×#440	C₂	semidirect
10. C₄×C₁₃×SD₁₆	C₂	semidirect
11. C₁₃×#405	C₂	semidirect
68. #3174	C₁₃	direct (N × H)
91. #541	C₂×C₁₃	semidirect
93. #499	C₂×C₁₃	semidirect
94. #492	C₂×C₁₃	semidirect
95. #491	C₂×C₁₃	semidirect
96. #490	C₂×C₁₃	semidirect
97. #468	C₂×C₁₃	semidirect
98. #465	C₂×C₁₃	semidirect
99. #440	C₂×C₁₃	semidirect
100. C₄×SD₁₆	C₂×C₁₃	semidirect
101. #405	C₂×C₁₃	semidirect
176. C₁₃	#3174	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₃×#541	2
C₁₃×#540	2
C₁₃×#499	2
C₁₃×#492	2
C₁₃×#491	2
C₁₃×#490	2
C₁₃×#468	2
C₁₃×#465	2
C₁₃×#440	2
C₄×C₁₃×SD₁₆	2
C₁₃×#405	2
#3174	13

Subgroups — C13×#3174 #13112232

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#3174 #13112232