Subgroups — C₁₃×#3400 #13112458

Order 1664.

362 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#559	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#557	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#497	C₂	1	2	yes	maximal, complemented
4	C₁₃×#492	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#464	C₂	1	2	yes	maximal, complemented
6	C₁₃×#462	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#461	C₂	1	2	yes	maximal, complemented
8	C₁₃×#460	C₂	1	1	yes	maximal, characteristic
9	C₁₃×#446	C₂	1	1	yes	maximal, characteristic, complemented
10	C₁₃×#440	C₂	1	2	yes	maximal, complemented
11	C₄×C₁₃×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×Q₁₆	C₂²	1	2	yes	—
14	C₂×C₁₃×SD₁₆	C₂²	1	2	yes	—
15	C₂×C₁₃×SD₁₆	C₂²	1	2	yes	—
16	C₂×C₁₃×SD₁₆	C₂²	1	2	yes	—
17	C₁₃×C₄⋊Q₈	C₂²	1	2	yes	—
18	C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
19	C₁₃×#127	C₂²	1	1	yes	characteristic
20	C₁₃×#125	—	2	1	no	—
21	C₁₃×#125	—	2	2	no	—
22	C₁₃×#124	—	2	1	no	—
23	C₁₃×#124	C₂²	1	2	yes	—
24	C₁₃×#123	—	2	2	no	—
25	C₄×Q₈×C₁₃	—	2	1	no	—
26	C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
27	C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
28	C₄×Q₈×C₁₃	C₂²	1	2	yes	—
29	C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
30	C₄×D₄×C₁₃	—	2	1	no	—
31	C₄×D₄×C₁₃	C₂²	1	2	yes	—
32	C₁₃×#107	C₂²	1	1	yes	characteristic
33	C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
34	C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
35	C₁₃×#103	C₂²	1	1	yes	characteristic
36	C₁₃×#103	C₂²	1	1	yes	characteristic
37	C₁₃×#103	C₂²	1	2	yes	—
38	C₁₃×#102	C₂²	1	2	yes	—
39	C₁₃×#97	C₂²	1	2	yes	—
40	C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
41	C₂×Q₈×C₁₃	D₄	1	1	yes	characteristic
42	C₂×Q₈×C₁₃	C₂³	1	1	yes	characteristic
43	C₂×Q₈×C₁₃	C₂³	1	2	yes	—
44	C₂×Q₈×C₁₃	C₂³	1	1	yes	characteristic
45	C₂×D₄×C₁₃	—	2	1	no	—
46	C₂×D₄×C₁₃	—	2	1	no	—
47	C₂×D₄×C₁₃	C₂³	1	2	yes	—
48	C₂²×C₄×C₁₃	—	2	2	no	—
49	C₁₃×Q₁₆	—	2	2	no	—
50	C₁₃×Q₁₆	—	4	2	no	—
51	C₁₃×SD₁₆	—	2	4	no	—
52	C₁₃×SD₁₆	—	4	2	no	—
53	C₁₃×SD₁₆	—	4	2	no	—
54	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
55	C₂×C₈×C₁₃	C₂³	1	2	yes	—
56	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
57	C₁₃×C₄⋊C₄	—	2	2	no	—
58	C₁₃×C₄⋊C₄	—	2	1	no	—
59	C₁₃×C₄⋊C₄	—	2	1	no	—
60	C₁₃×C₄⋊C₄	—	2	2	no	—
61	C₁₃×C₄⋊C₄	—	2	1	no	—
62	C₁₃×C₄⋊C₄	—	2	2	no	—
63	C₁₃×C₄⋊C₄	—	2	1	no	—
64	C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
65	C₁₃×C₄⋊C₄	D₄	1	2	yes	—
66	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
67	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
68	C₁₃×C₂²⋊C₄	—	2	2	no	—
69	C₁₃×C₂²⋊C₄	—	2	2	no	—
70	C₁₃×C₂²⋊C₄	—	2	2	no	—
71	C₄²×C₁₃	—	2	1	no	—
72	C₄²×C₁₃	—	2	2	no	—
73	C₄²×C₁₃	—	2	1	no	—
74	C₄²×C₁₃	C₂³	1	1	yes	characteristic
75	C₄²×C₁₃	C₂³	1	2	yes	—
76	#3400	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
77	C₂³×C₁₃	—	2	2	no	—
78	Q₈×C₁₃	—	2	2	no	—
79	Q₈×C₁₃	—	2	2	no	—
80	Q₈×C₁₃	—	2	1	no	—
81	Q₈×C₁₃	—	2	1	no	—
82	Q₈×C₁₃	—	2	1	no	—
83	Q₈×C₁₃	—	2	2	no	—
84	Q₈×C₁₃	—	2	1	no	—
85	D₄×C₁₃	—	2	2	no	—
86	D₄×C₁₃	—	2	2	no	—
87	D₄×C₁₃	—	4	2	no	—
88	D₄×C₁₃	—	8	1	no	—
89	C₂×C₄×C₁₃	—	2	1	no	—
90	C₂×C₄×C₁₃	—	2	1	no	—
91	C₂×C₄×C₁₃	—	2	1	no	—
92	C₂×C₄×C₁₃	—	2	2	no	—
93	C₂×C₄×C₁₃	—	2	1	no	—
94	C₂×C₄×C₁₃	—	8	2	no	—
95	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
96	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
97	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
99	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
100	C₈×C₁₃	—	2	1	no	—
101	C₈×C₁₃	—	2	2	no	—
102	C₈×C₁₃	#41	1	2	yes	—
103	#559	C₂×C₁₃	1	1	yes	characteristic
104	#557	C₂×C₁₃	1	1	yes	characteristic, complemented
105	#497	C₂×C₁₃	1	2	yes	complemented
106	#492	C₂×C₁₃	1	1	yes	characteristic
107	#464	C₂×C₁₃	1	2	yes	complemented
108	#462	C₂×C₁₃	1	1	yes	characteristic, complemented
109	#461	C₂×C₁₃	1	2	yes	complemented
110	#460	C₂×C₁₃	1	1	yes	characteristic
111	#446	C₂×C₁₃	1	1	yes	characteristic, complemented
112	#440	C₂×C₁₃	1	2	yes	complemented
113	C₄×Q₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
114	C₂²×C₁₃	—	4	2	no	—
115	C₂²×C₁₃	—	4	2	no	—
116	C₂²×C₁₃	—	4	2	no	—
117	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
118	C₄×C₁₃	—	2	2	no	—
119	C₄×C₁₃	—	2	2	no	—
120	C₄×C₁₃	—	2	1	no	—
121	C₄×C₁₃	—	4	1	no	—
122	C₄×C₁₃	—	4	2	no	—
123	C₄×C₁₃	—	4	1	no	—
124	C₄×C₁₃	—	4	1	no	—
125	C₄×C₁₃	—	4	1	no	—
126	C₄×C₁₃	#137	1	2	yes	—
127	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
128	C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
129	C₂×Q₁₆	C₂²×C₁₃	1	1	yes	characteristic
130	C₂×Q₁₆	C₂²×C₁₃	1	2	yes	—
131	C₂×SD₁₆	C₂²×C₁₃	1	2	yes	—
132	C₂×SD₁₆	C₂²×C₁₃	1	2	yes	—
133	C₂×SD₁₆	C₂²×C₁₃	1	2	yes	—
134	C₄⋊Q₈	C₂²×C₁₃	1	2	yes	—
135	C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
136	#127	C₂²×C₁₃	1	1	yes	characteristic
137	#125	—	2	1	no	—
138	#125	—	2	2	no	—
139	#124	—	2	1	no	—
140	#124	C₂²×C₁₃	1	2	yes	—
141	#123	—	2	2	no	—
142	C₄×Q₈	—	2	1	no	—
143	C₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic
144	C₄×Q₈	C₂²×C₁₃	1	2	yes	—
145	C₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic
146	C₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic
147	C₄×D₄	—	2	1	no	—
148	C₄×D₄	C₂²×C₁₃	1	2	yes	—
149	#107	C₂²×C₁₃	1	1	yes	characteristic
150	C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
151	C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
152	#103	C₂²×C₁₃	1	1	yes	characteristic
153	#103	C₂²×C₁₃	1	1	yes	characteristic
154	#103	C₂²×C₁₃	1	2	yes	—
155	#102	C₂²×C₁₃	1	2	yes	—
156	#97	C₂²×C₁₃	1	2	yes	—
157	C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
158	C₂×C₁₃	—	8	2	no	—
159	C₂×C₁₃	#550	1	1	yes	characteristic
160	C₂×C₁₃	C₂×#137	1	1	yes	characteristic
161	C₂×C₁₃	#577	1	1	yes	characteristic
162	C₂×Q₈	D₄×C₁₃	1	1	yes	characteristic
163	C₂×Q₈	C₂³×C₁₃	1	1	yes	characteristic
164	C₂×Q₈	C₂³×C₁₃	1	2	yes	—
165	C₂×Q₈	C₂³×C₁₃	1	1	yes	characteristic
166	C₂×D₄	—	2	1	no	—
167	C₂×D₄	—	2	1	no	—
168	C₂×D₄	C₂³×C₁₃	1	2	yes	—
169	C₂²×C₄	—	2	2	no	—
170	Q₁₆	—	2	2	no	—
171	Q₁₆	—	4	2	no	—
172	SD₁₆	—	2	4	no	—
173	SD₁₆	—	4	2	no	—
174	SD₁₆	—	4	2	no	—
175	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
176	C₂×C₈	C₂³×C₁₃	1	2	yes	—
177	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
178	C₄⋊C₄	—	2	2	no	—
179	C₄⋊C₄	—	2	1	no	—
180	C₄⋊C₄	—	2	1	no	—
181	C₄⋊C₄	—	2	2	no	—
182	C₄⋊C₄	—	2	1	no	—
183	C₄⋊C₄	—	2	2	no	—
184	C₄⋊C₄	—	2	1	no	—
185	C₄⋊C₄	D₄×C₁₃	1	1	yes	characteristic
186	C₄⋊C₄	D₄×C₁₃	1	2	yes	—
187	C₄⋊C₄	C₂³×C₁₃	1	1	yes	characteristic
188	C₄⋊C₄	C₂³×C₁₃	1	1	yes	characteristic
189	C₂²⋊C₄	—	2	2	no	—
190	C₂²⋊C₄	—	2	2	no	—
191	C₂²⋊C₄	—	2	2	no	—
192	C₄²	—	2	1	no	—
193	C₄²	—	2	2	no	—
194	C₄²	—	2	1	no	—
195	C₄²	C₂³×C₁₃	1	1	yes	characteristic
196	C₄²	C₂³×C₁₃	1	2	yes	—
197	C₁₃	#3400	1	1	yes	characteristic, direct product factor, Sylow
198	C₂³	—	2	2	no	—
199	Q₈	—	2	2	no	—
200	Q₈	—	2	2	no	—
201	Q₈	—	2	1	no	—
202	Q₈	—	2	1	no	—
203	Q₈	—	2	1	no	—
204	Q₈	—	2	2	no	—
205	Q₈	—	2	1	no	—
206	D₄	—	2	2	no	—
207	D₄	—	2	2	no	—
208	D₄	—	4	2	no	—
209	D₄	—	8	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	8	2	no	—
216	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
217	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
218	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
219	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
220	C₂×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
221	C₈	—	2	1	no	—
222	C₈	—	2	2	no	—
223	C₈	C₁₃×#41	1	2	yes	—
224	C₂²	—	4	2	no	—
225	C₂²	—	4	2	no	—
226	C₂²	—	4	2	no	—
227	C₂²	C₂²×D₄×C₁₃	1	1	yes	characteristic
228	C₄	—	2	2	no	—
229	C₄	—	2	2	no	—
230	C₄	—	2	1	no	—
231	C₄	—	4	1	no	—
232	C₄	—	4	2	no	—
233	C₄	—	4	1	no	—
234	C₄	—	4	1	no	—
235	C₄	—	4	1	no	—
236	C₄	C₁₃×#137	1	2	yes	—
237	C₄	C₂×C₁₃×#41	1	1	yes	characteristic
238	C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
239	C₂	—	8	2	no	—
240	C₂	C₁₃×#550	1	1	yes	characteristic
241	C₂	C₂×C₁₃×#137	1	1	yes	characteristic
242	C₂	C₁₃×#577	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₃
prime radical	C₁₃×#127	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₄×Q₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₃×#557	C₂	semidirect
3. C₁₃×#497	C₂	semidirect
5. C₁₃×#464	C₂	semidirect
6. C₁₃×#462	C₂	semidirect
7. C₁₃×#461	C₂	semidirect
9. C₁₃×#446	C₂	semidirect
10. C₁₃×#440	C₂	semidirect
11. C₄×C₁₃×Q₁₆	C₂	semidirect
76. #3400	C₁₃	direct (N × H)
104. #557	C₂×C₁₃	semidirect
105. #497	C₂×C₁₃	semidirect
107. #464	C₂×C₁₃	semidirect
108. #462	C₂×C₁₃	semidirect
109. #461	C₂×C₁₃	semidirect
111. #446	C₂×C₁₃	semidirect
112. #440	C₂×C₁₃	semidirect
113. C₄×Q₁₆	C₂×C₁₃	semidirect
197. C₁₃	#3400	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×#559	2
C₁₃×#557	2
C₁₃×#497	2
C₁₃×#492	2
C₁₃×#464	2
C₁₃×#462	2
C₁₃×#461	2
C₁₃×#460	2
C₁₃×#446	2
C₁₃×#440	2
C₄×C₁₃×Q₁₆	2
#3400	13

Subgroups — C13×#3400 #13112458

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#3400 #13112458