Subgroups — C₁₃×#3397 #13112455

Order 1664.

406 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#556	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₃×#550	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#496	C₂	1	2	yes	maximal, complemented
4	C₁₃×#492	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#463	C₂	1	2	yes	maximal, complemented
6	C₁₃×#461	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#460	C₂	1	2	yes	maximal, complemented
8	C₁₃×#459	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₃×#446	C₂	1	1	yes	maximal, characteristic, complemented
10	C₁₃×#440	C₂	1	2	yes	maximal, complemented
11	C₄×C₁₃×D₈	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₁₃×#41	—	2	1	no	—
13	C₂×C₁₃×SD₁₆	C₂²	1	2	yes	—
14	C₂×C₁₃×SD₁₆	C₂²	1	2	yes	—
15	C₂×C₁₃×SD₁₆	C₂²	1	2	yes	—
16	C₂×C₁₃×D₈	C₂²	1	1	yes	characteristic
17	C₂×C₁₃×D₈	C₂²	1	2	yes	—
18	C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
19	C₁₃×#127	C₂²	1	1	yes	characteristic
20	C₁₃×#127	C₂²	1	2	yes	—
21	C₁₃×#126	—	2	2	no	—
22	C₁₃×#124	C₂²	1	2	yes	—
23	C₁₃×C₂²⋊Q₈	—	2	1	no	—
24	C₁₃×#121	—	2	1	no	—
25	C₁₃×#121	—	2	2	no	—
26	C₄×Q₈×C₁₃	C₂²	1	2	yes	—
27	C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
28	C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
29	C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
30	C₄×D₄×C₁₃	C₂²	1	2	yes	—
31	C₁₃×#117	—	2	1	no	—
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	2	yes	—
36	C₁₃×#102	C₂²	1	1	yes	characteristic
37	C₁₃×#102	C₂²	1	1	yes	characteristic
38	C₁₃×#102	C₂²	1	2	yes	—
39	C₁₃×#97	C₂²	1	2	yes	—
40	C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
41	C₁₃×#41	—	4	4	no	—
42	C₂×Q₈×C₁₃	D₄	1	1	yes	characteristic
43	C₂×Q₈×C₁₃	C₂³	1	2	yes	—
44	C₂×D₄×C₁₃	—	2	2	no	—
45	C₂×D₄×C₁₃	—	2	1	no	—
46	C₂×D₄×C₁₃	—	2	2	no	—
47	C₂×D₄×C₁₃	—	2	1	no	—
48	C₂×D₄×C₁₃	C₂³	1	1	yes	characteristic
49	C₂×D₄×C₁₃	C₂³	1	1	yes	characteristic
50	C₂×D₄×C₁₃	C₂³	1	2	yes	—
51	C₂²×C₄×C₁₃	—	2	1	no	—
52	C₂²×C₄×C₁₃	—	2	2	no	—
53	C₂²×C₄×C₁₃	—	2	1	no	—
54	C₁₃×SD₁₆	—	2	4	no	—
55	C₁₃×SD₁₆	—	4	2	no	—
56	C₁₃×SD₁₆	—	4	2	no	—
57	C₁₃×D₈	—	2	2	no	—
58	C₁₃×D₈	—	4	2	no	—
59	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
60	C₂×C₈×C₁₃	C₂³	1	2	yes	—
61	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
62	C₁₃×C₄⋊C₄	—	2	1	no	—
63	C₁₃×C₄⋊C₄	—	2	2	no	—
64	C₁₃×C₄⋊C₄	—	2	1	no	—
65	C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
66	C₁₃×C₄⋊C₄	D₄	1	2	yes	—
67	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
68	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
69	C₁₃×C₂²⋊C₄	—	2	2	no	—
70	C₁₃×C₂²⋊C₄	—	2	2	no	—
71	C₁₃×C₂²⋊C₄	—	2	1	no	—
72	C₁₃×C₂²⋊C₄	—	2	1	no	—
73	C₁₃×C₂²⋊C₄	—	2	2	no	—
74	C₄²×C₁₃	—	2	2	no	—
75	C₄²×C₁₃	C₂³	1	1	yes	characteristic
76	C₄²×C₁₃	C₂³	1	2	yes	—
77	#3397	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
78	C₂³×C₁₃	—	2	1	no	—
79	C₂³×C₁₃	—	2	1	no	—
80	C₂³×C₁₃	—	2	2	no	—
81	Q₈×C₁₃	—	2	2	no	—
82	Q₈×C₁₃	—	2	2	no	—
83	Q₈×C₁₃	—	2	2	no	—
84	D₄×C₁₃	—	2	1	no	—
85	D₄×C₁₃	—	2	1	no	—
86	D₄×C₁₃	—	2	2	no	—
87	D₄×C₁₃	—	2	1	no	—
88	D₄×C₁₃	—	2	2	no	—
89	D₄×C₁₃	—	2	1	no	—
90	D₄×C₁₃	—	4	4	no	—
91	D₄×C₁₃	—	4	2	no	—
92	D₄×C₁₃	—	8	2	no	—
93	D₄×C₁₃	—	8	1	no	—
94	C₂×C₄×C₁₃	—	2	1	no	—
95	C₂×C₄×C₁₃	—	2	2	no	—
96	C₂×C₄×C₁₃	—	2	1	no	—
97	C₂×C₄×C₁₃	—	4	4	no	—
98	C₂×C₄×C₁₃	—	4	2	no	—
99	C₂×C₄×C₁₃	—	4	2	no	—
100	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
101	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
102	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
103	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
104	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
105	C₈×C₁₃	—	2	2	no	—
106	C₈×C₁₃	—	2	1	no	—
107	C₈×C₁₃	#41	1	2	yes	complemented
108	#556	C₂×C₁₃	1	1	yes	characteristic, complemented
109	#550	C₂×C₁₃	1	1	yes	characteristic, complemented
110	#496	C₂×C₁₃	1	2	yes	complemented
111	#492	C₂×C₁₃	1	1	yes	characteristic, complemented
112	#463	C₂×C₁₃	1	2	yes	complemented
113	#461	C₂×C₁₃	1	1	yes	characteristic, complemented
114	#460	C₂×C₁₃	1	2	yes	complemented
115	#459	C₂×C₁₃	1	1	yes	characteristic, complemented
116	#446	C₂×C₁₃	1	1	yes	characteristic, complemented
117	#440	C₂×C₁₃	1	2	yes	complemented
118	C₄×D₈	C₂×C₁₃	1	1	yes	characteristic, complemented
119	C₂²×C₁₃	—	4	1	no	—
120	C₂²×C₁₃	—	4	1	no	—
121	C₂²×C₁₃	—	4	2	no	—
122	C₂²×C₁₃	—	4	2	no	—
123	C₂²×C₁₃	—	4	1	no	—
124	C₂²×C₁₃	—	4	1	no	—
125	C₂²×C₁₃	—	4	1	no	—
126	C₂²×C₁₃	—	4	2	no	—
127	C₂²×C₁₃	—	4	1	no	—
128	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
129	C₄×C₁₃	—	2	2	no	—
130	C₄×C₁₃	—	2	2	no	—
131	C₄×C₁₃	—	2	1	no	—
132	C₄×C₁₃	—	4	2	no	—
133	C₄×C₁₃	—	4	1	no	—
134	C₄×C₁₃	—	4	1	no	—
135	C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
136	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
137	C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
138	C₂×#41	—	2	1	no	—
139	C₂×SD₁₆	C₂²×C₁₃	1	2	yes	—
140	C₂×SD₁₆	C₂²×C₁₃	1	2	yes	—
141	C₂×SD₁₆	C₂²×C₁₃	1	2	yes	—
142	C₂×D₈	C₂²×C₁₃	1	1	yes	characteristic
143	C₂×D₈	C₂²×C₁₃	1	2	yes	—
144	C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
145	#127	C₂²×C₁₃	1	1	yes	characteristic
146	#127	C₂²×C₁₃	1	2	yes	—
147	#126	—	2	2	no	—
148	#124	C₂²×C₁₃	1	2	yes	—
149	C₂²⋊Q₈	—	2	1	no	—
150	#121	—	2	1	no	—
151	#121	—	2	2	no	—
152	C₄×Q₈	C₂²×C₁₃	1	2	yes	—
153	C₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic
154	C₄×D₄	C₂²×C₁₃	1	1	yes	characteristic
155	C₄×D₄	C₂²×C₁₃	1	2	yes	—
156	C₄×D₄	C₂²×C₁₃	1	1	yes	characteristic
157	#117	—	2	1	no	—
158	#107	C₂²×C₁₃	1	1	yes	characteristic
159	C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
160	C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
161	#103	C₂²×C₁₃	1	2	yes	—
162	#102	C₂²×C₁₃	1	1	yes	characteristic
163	#102	C₂²×C₁₃	1	1	yes	characteristic
164	#102	C₂²×C₁₃	1	2	yes	—
165	#97	C₂²×C₁₃	1	2	yes	—
166	C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
167	C₂×C₁₃	—	8	1	no	—
168	C₂×C₁₃	—	8	2	no	—
169	C₂×C₁₃	—	8	1	no	—
170	C₂×C₁₃	#550	1	1	yes	characteristic
171	C₂×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
172	C₂×C₁₃	#577	1	1	yes	characteristic
173	#41	—	4	4	no	—
174	C₂×Q₈	D₄×C₁₃	1	1	yes	characteristic
175	C₂×Q₈	C₂³×C₁₃	1	2	yes	—
176	C₂×D₄	—	2	2	no	—
177	C₂×D₄	—	2	1	no	—
178	C₂×D₄	—	2	2	no	—
179	C₂×D₄	—	2	1	no	—
180	C₂×D₄	C₂³×C₁₃	1	1	yes	characteristic
181	C₂×D₄	C₂³×C₁₃	1	1	yes	characteristic
182	C₂×D₄	C₂³×C₁₃	1	2	yes	—
183	C₂²×C₄	—	2	1	no	—
184	C₂²×C₄	—	2	2	no	—
185	C₂²×C₄	—	2	1	no	—
186	SD₁₆	—	2	4	no	—
187	SD₁₆	—	4	2	no	—
188	SD₁₆	—	4	2	no	—
189	D₈	—	2	2	no	—
190	D₈	—	4	2	no	—
191	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
192	C₂×C₈	C₂³×C₁₃	1	2	yes	—
193	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
194	C₄⋊C₄	—	2	1	no	—
195	C₄⋊C₄	—	2	2	no	—
196	C₄⋊C₄	—	2	1	no	—
197	C₄⋊C₄	D₄×C₁₃	1	1	yes	characteristic
198	C₄⋊C₄	D₄×C₁₃	1	2	yes	—
199	C₄⋊C₄	C₂³×C₁₃	1	1	yes	characteristic
200	C₄⋊C₄	C₂³×C₁₃	1	1	yes	characteristic
201	C₂²⋊C₄	—	2	2	no	—
202	C₂²⋊C₄	—	2	2	no	—
203	C₂²⋊C₄	—	2	1	no	—
204	C₂²⋊C₄	—	2	1	no	—
205	C₂²⋊C₄	—	2	2	no	—
206	C₄²	—	2	2	no	—
207	C₄²	C₂³×C₁₃	1	1	yes	characteristic
208	C₄²	C₂³×C₁₃	1	2	yes	—
209	C₁₃	#3397	1	1	yes	characteristic, direct product factor, Sylow
210	C₂³	—	2	1	no	—
211	C₂³	—	2	1	no	—
212	C₂³	—	2	2	no	—
213	Q₈	—	2	2	no	—
214	Q₈	—	2	2	no	—
215	Q₈	—	2	2	no	—
216	D₄	—	2	1	no	—
217	D₄	—	2	1	no	—
218	D₄	—	2	2	no	—
219	D₄	—	2	1	no	—
220	D₄	—	2	2	no	—
221	D₄	—	2	1	no	—
222	D₄	—	4	4	no	—
223	D₄	—	4	2	no	—
224	D₄	—	8	2	no	—
225	D₄	—	8	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	4	4	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
233	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
234	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
235	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
236	C₂×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
237	C₈	—	2	2	no	—
238	C₈	—	2	1	no	—
239	C₈	C₁₃×#41	1	2	yes	complemented
240	C₂²	—	4	1	no	—
241	C₂²	—	4	1	no	—
242	C₂²	—	4	2	no	—
243	C₂²	—	4	2	no	—
244	C₂²	—	4	1	no	—
245	C₂²	—	4	1	no	—
246	C₂²	—	4	1	no	—
247	C₂²	—	4	2	no	—
248	C₂²	—	4	1	no	—
249	C₂²	C₂²×D₄×C₁₃	1	1	yes	characteristic
250	C₄	—	2	2	no	—
251	C₄	—	2	2	no	—
252	C₄	—	2	1	no	—
253	C₄	—	4	2	no	—
254	C₄	—	4	1	no	—
255	C₄	—	4	1	no	—
256	C₄	C₁₃×AGL(1,ℤ/8)	1	2	yes	—
257	C₄	C₂×C₁₃×#41	1	1	yes	characteristic
258	C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
259	C₂	—	8	1	no	—
260	C₂	—	8	2	no	—
261	C₂	—	8	1	no	—
262	C₂	C₁₃×#550	1	1	yes	characteristic
263	C₂	C₂×C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
264	C₂	C₁₃×#577	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₃
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
1. C₁₃×#556	C₂	semidirect
2. C₁₃×#550	C₂	semidirect
3. C₁₃×#496	C₂	semidirect
4. C₁₃×#492	C₂	semidirect
5. C₁₃×#463	C₂	semidirect
6. C₁₃×#461	C₂	semidirect
7. C₁₃×#460	C₂	semidirect
8. C₁₃×#459	C₂	semidirect
9. C₁₃×#446	C₂	semidirect
10. C₁₃×#440	C₂	semidirect
11. C₄×C₁₃×D₈	C₂	semidirect
77. #3397	C₁₃	direct (N × H)
107. C₈×C₁₃	#41	semidirect
108. #556	C₂×C₁₃	semidirect
109. #550	C₂×C₁₃	semidirect
110. #496	C₂×C₁₃	semidirect
111. #492	C₂×C₁₃	semidirect
112. #463	C₂×C₁₃	semidirect
113. #461	C₂×C₁₃	semidirect
114. #460	C₂×C₁₃	semidirect
115. #459	C₂×C₁₃	semidirect
116. #446	C₂×C₁₃	semidirect
117. #440	C₂×C₁₃	semidirect
118. C₄×D₈	C₂×C₁₃	semidirect
209. C₁₃	#3397	direct (N × H)
239. C₈	C₁₃×#41	semidirect

Maximal subgroups

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

subgroup	index
C₁₃×#556	2
C₁₃×#550	2
C₁₃×#496	2
C₁₃×#492	2
C₁₃×#463	2
C₁₃×#461	2
C₁₃×#460	2
C₁₃×#459	2
C₁₃×#446	2
C₁₃×#440	2
C₄×C₁₃×D₈	2
#3397	13

Subgroups — C13×#3397 #13112455

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#3397 #13112455