Subgroups — C₁₃×#3354 #13112412

Order 1664.

386 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#555	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₃×#547	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#498	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#482	C₂	1	2	yes	maximal, complemented
5	C₁₃×#481	C₂	1	2	yes	maximal, complemented
6	C₁₃×#476	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#474	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₃×#469	C₂	1	2	yes	maximal, complemented
9	C₄×C₁₃×D₈	C₂	1	1	yes	maximal, characteristic, complemented
10	C₁₃×#436	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₃×#428	C₂	1	2	yes	maximal, complemented
12	C₂×C₁₃×#41	—	2	1	no	—
13	C₂×C₁₃×D₈	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×M₁₆	C₂²	1	2	yes	—
15	C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₁₃×#124	—	2	2	no	—
18	C₁₃×#123	—	2	2	no	—
19	C₁₃×C₂²⋊Q₈	—	2	1	no	—
20	C₁₃×#121	—	2	1	no	—
21	C₁₃×#121	C₂²	1	2	yes	—
22	C₁₃×#121	C₂²	1	2	yes	—
23	C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
24	C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
25	C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
26	C₁₃×#117	—	2	1	no	—
27	C₁₃×#117	C₂²	1	2	yes	—
28	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
29	C₁₃×#107	C₂²	1	1	yes	characteristic
30	C₁₃×#107	C₂²	1	2	yes	—
31	C₁₃×#106	C₂²	1	2	yes	—
32	C₁₃×#106	C₂²	1	2	yes	—
33	C₁₃×#106	C₂²	1	2	yes	—
34	C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
35	C₁₃×#102	C₂²	1	1	yes	characteristic
36	C₁₃×#102	C₂²	1	1	yes	characteristic
37	C₁₃×#102	C₂²	1	2	yes	—
38	C₁₃×#102	C₂²	1	2	yes	—
39	C₁₃×C₂²⋊C₈	C₂²	1	2	yes	—
40	C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
41	C₁₃×#41	—	4	4	no	—
42	C₂×Q₈×C₁₃	—	2	1	no	—
43	C₂×Q₈×C₁₃	—	2	1	no	—
44	C₂×D₄×C₁₃	—	2	2	no	—
45	C₂×D₄×C₁₃	—	2	2	no	—
46	C₂×D₄×C₁₃	D₄	1	1	yes	characteristic
47	C₂×D₄×C₁₃	C₂³	1	1	yes	characteristic
48	C₂×D₄×C₁₃	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₁₃	—	2	2	no	—
50	C₂²×C₄×C₁₃	—	2	1	no	—
51	C₂²×C₄×C₁₃	—	2	1	no	—
52	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
53	C₁₃×D₈	—	2	2	no	—
54	C₁₃×M₁₆	—	2	4	no	—
55	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
56	C₂×C₈×C₁₃	C₂³	1	2	yes	—
57	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
58	C₁₃×C₄⋊C₄	—	2	2	no	—
59	C₁₃×C₄⋊C₄	—	2	1	no	—
60	C₁₃×C₄⋊C₄	—	2	1	no	—
61	C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
62	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
63	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
64	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
65	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
66	C₁₃×C₂²⋊C₄	—	2	2	no	—
67	C₁₃×C₂²⋊C₄	—	2	2	no	—
68	C₁₃×C₂²⋊C₄	—	2	1	no	—
69	C₁₃×C₂²⋊C₄	—	2	1	no	—
70	C₁₃×C₂²⋊C₄	—	2	2	no	—
71	C₁₃×C₂²⋊C₄	D₄	1	2	yes	—
72	C₄²×C₁₃	—	2	2	no	—
73	C₄²×C₁₃	C₂³	1	1	yes	characteristic
74	#3354	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
75	C₂³×C₁₃	—	2	1	no	—
76	C₂³×C₁₃	—	2	1	no	—
77	C₂³×C₁₃	C₂×D₄	1	2	yes	—
78	Q₈×C₁₃	—	4	2	no	—
79	Q₈×C₁₃	—	8	1	no	—
80	D₄×C₁₃	—	2	2	no	—
81	D₄×C₁₃	—	2	1	no	—
82	D₄×C₁₃	—	2	1	no	—
83	D₄×C₁₃	—	2	1	no	—
84	D₄×C₁₃	—	2	1	no	—
85	D₄×C₁₃	—	4	4	no	—
86	D₄×C₁₃	—	8	2	no	—
87	C₂×C₄×C₁₃	—	2	2	no	—
88	C₂×C₄×C₁₃	—	2	2	no	—
89	C₂×C₄×C₁₃	—	2	1	no	—
90	C₂×C₄×C₁₃	—	2	2	no	—
91	C₂×C₄×C₁₃	—	2	1	no	—
92	C₂×C₄×C₁₃	—	2	2	no	—
93	C₂×C₄×C₁₃	—	4	4	no	—
94	C₂×C₄×C₁₃	—	4	2	no	—
95	C₂×C₄×C₁₃	—	4	2	no	—
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	2	no	—
101	C₈×C₁₃	—	2	1	no	—
102	C₈×C₁₃	#41	1	2	yes	complemented
103	#555	C₂×C₁₃	1	1	yes	characteristic, complemented
104	#547	C₂×C₁₃	1	1	yes	characteristic, complemented
105	#498	C₂×C₁₃	1	1	yes	characteristic, complemented
106	#482	C₂×C₁₃	1	2	yes	complemented
107	#481	C₂×C₁₃	1	2	yes	complemented
108	#476	C₂×C₁₃	1	1	yes	characteristic, complemented
109	#474	C₂×C₁₃	1	1	yes	characteristic, complemented
110	#469	C₂×C₁₃	1	2	yes	complemented
111	C₄×D₈	C₂×C₁₃	1	1	yes	characteristic, complemented
112	#436	C₂×C₁₃	1	1	yes	characteristic, complemented
113	#428	C₂×C₁₃	1	2	yes	complemented
114	C₂²×C₁₃	—	2	2	no	—
115	C₂²×C₁₃	—	2	2	no	—
116	C₂²×C₁₃	—	2	2	no	—
117	C₂²×C₁₃	—	4	1	no	—
118	C₂²×C₁₃	—	4	1	no	—
119	C₂²×C₁₃	—	4	1	no	—
120	C₂²×C₁₃	—	4	1	no	—
121	C₂²×C₁₃	—	4	1	no	—
122	C₂²×C₁₃	—	4	1	no	—
123	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
124	C₄×C₁₃	—	2	2	no	—
125	C₄×C₁₃	—	2	1	no	—
126	C₄×C₁₃	—	4	1	no	—
127	C₄×C₁₃	—	4	1	no	—
128	C₄×C₁₃	—	4	2	no	—
129	C₄×C₁₃	—	4	2	no	—
130	C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
131	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
132	C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
133	C₂×#41	—	2	1	no	—
134	C₂×D₈	C₂²×C₁₃	1	1	yes	characteristic
135	C₂×M₁₆	C₂²×C₁₃	1	2	yes	—
136	C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
137	C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
138	#124	—	2	2	no	—
139	#123	—	2	2	no	—
140	C₂²⋊Q₈	—	2	1	no	—
141	#121	—	2	1	no	—
142	#121	C₂²×C₁₃	1	2	yes	—
143	#121	C₂²×C₁₃	1	2	yes	—
144	C₄×D₄	C₂²×C₁₃	1	1	yes	characteristic
145	C₄×D₄	C₂²×C₁₃	1	1	yes	characteristic
146	C₄×D₄	C₂²×C₁₃	1	1	yes	characteristic
147	#117	—	2	1	no	—
148	#117	C₂²×C₁₃	1	2	yes	—
149	C₂×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
150	#107	C₂²×C₁₃	1	1	yes	characteristic
151	#107	C₂²×C₁₃	1	2	yes	—
152	#106	C₂²×C₁₃	1	2	yes	—
153	#106	C₂²×C₁₃	1	2	yes	—
154	#106	C₂²×C₁₃	1	2	yes	—
155	C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
156	#102	C₂²×C₁₃	1	1	yes	characteristic
157	#102	C₂²×C₁₃	1	1	yes	characteristic
158	#102	C₂²×C₁₃	1	2	yes	—
159	#102	C₂²×C₁₃	1	2	yes	—
160	C₂²⋊C₈	C₂²×C₁₃	1	2	yes	—
161	C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
162	C₂×C₁₃	—	4	2	no	—
163	C₂×C₁₃	—	8	1	no	—
164	C₂×C₁₃	—	8	1	no	—
165	C₂×C₁₃	#547	1	1	yes	characteristic
166	C₂×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
167	C₂×C₁₃	#577	1	1	yes	characteristic
168	#41	—	4	4	no	—
169	C₂×Q₈	—	2	1	no	—
170	C₂×Q₈	—	2	1	no	—
171	C₂×D₄	—	2	2	no	—
172	C₂×D₄	—	2	2	no	—
173	C₂×D₄	D₄×C₁₃	1	1	yes	characteristic
174	C₂×D₄	C₂³×C₁₃	1	1	yes	characteristic
175	C₂×D₄	C₂³×C₁₃	1	1	yes	characteristic
176	C₂²×C₄	—	2	2	no	—
177	C₂²×C₄	—	2	1	no	—
178	C₂²×C₄	—	2	1	no	—
179	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
180	D₈	—	2	2	no	—
181	M₁₆	—	2	4	no	—
182	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
183	C₂×C₈	C₂³×C₁₃	1	2	yes	—
184	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
185	C₄⋊C₄	—	2	2	no	—
186	C₄⋊C₄	—	2	1	no	—
187	C₄⋊C₄	—	2	1	no	—
188	C₄⋊C₄	D₄×C₁₃	1	1	yes	characteristic
189	C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
190	C₄⋊C₄	C₂³×C₁₃	1	1	yes	characteristic
191	C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
192	C₄⋊C₄	C₂³×C₁₃	1	1	yes	characteristic
193	C₂²⋊C₄	—	2	2	no	—
194	C₂²⋊C₄	—	2	2	no	—
195	C₂²⋊C₄	—	2	1	no	—
196	C₂²⋊C₄	—	2	1	no	—
197	C₂²⋊C₄	—	2	2	no	—
198	C₂²⋊C₄	D₄×C₁₃	1	2	yes	—
199	C₄²	—	2	2	no	—
200	C₄²	C₂³×C₁₃	1	1	yes	characteristic
201	C₁₃	#3354	1	1	yes	characteristic, direct product factor, Sylow
202	C₂³	—	2	1	no	—
203	C₂³	—	2	1	no	—
204	C₂³	C₂×D₄×C₁₃	1	2	yes	—
205	Q₈	—	4	2	no	—
206	Q₈	—	8	1	no	—
207	D₄	—	2	2	no	—
208	D₄	—	2	1	no	—
209	D₄	—	2	1	no	—
210	D₄	—	2	1	no	—
211	D₄	—	2	1	no	—
212	D₄	—	4	4	no	—
213	D₄	—	8	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	4	4	no	—
221	C₂×C₄	—	4	2	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
224	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
225	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
226	C₂×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
227	C₈	—	2	2	no	—
228	C₈	—	2	1	no	—
229	C₈	C₁₃×#41	1	2	yes	complemented
230	C₂²	—	2	2	no	—
231	C₂²	—	2	2	no	—
232	C₂²	—	2	2	no	—
233	C₂²	—	4	1	no	—
234	C₂²	—	4	1	no	—
235	C₂²	—	4	1	no	—
236	C₂²	—	4	1	no	—
237	C₂²	—	4	1	no	—
238	C₂²	—	4	1	no	—
239	C₂²	C₂²×D₄×C₁₃	1	1	yes	characteristic
240	C₄	—	2	2	no	—
241	C₄	—	2	1	no	—
242	C₄	—	4	1	no	—
243	C₄	—	4	1	no	—
244	C₄	—	4	2	no	—
245	C₄	—	4	2	no	—
246	C₄	C₁₃×AGL(1,ℤ/8)	1	2	yes	—
247	C₄	C₂×C₁₃×#41	1	1	yes	characteristic
248	C₄	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
249	C₂	—	4	2	no	—
250	C₂	—	8	1	no	—
251	C₂	—	8	1	no	—
252	C₂	C₁₃×#547	1	1	yes	characteristic
253	C₂	C₂×C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
254	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×D₄×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₃×#555	C₂	semidirect
2. C₁₃×#547	C₂	semidirect
3. C₁₃×#498	C₂	semidirect
4. C₁₃×#482	C₂	semidirect
5. C₁₃×#481	C₂	semidirect
6. C₁₃×#476	C₂	semidirect
7. C₁₃×#474	C₂	semidirect
8. C₁₃×#469	C₂	semidirect
9. C₄×C₁₃×D₈	C₂	semidirect
10. C₁₃×#436	C₂	semidirect
11. C₁₃×#428	C₂	semidirect
74. #3354	C₁₃	direct (N × H)
102. C₈×C₁₃	#41	semidirect
103. #555	C₂×C₁₃	semidirect
104. #547	C₂×C₁₃	semidirect
105. #498	C₂×C₁₃	semidirect
106. #482	C₂×C₁₃	semidirect
107. #481	C₂×C₁₃	semidirect
108. #476	C₂×C₁₃	semidirect
109. #474	C₂×C₁₃	semidirect
110. #469	C₂×C₁₃	semidirect
111. C₄×D₈	C₂×C₁₃	semidirect
112. #436	C₂×C₁₃	semidirect
113. #428	C₂×C₁₃	semidirect
201. C₁₃	#3354	direct (N × H)
229. 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×#555	2
C₁₃×#547	2
C₁₃×#498	2
C₁₃×#482	2
C₁₃×#481	2
C₁₃×#476	2
C₁₃×#474	2
C₁₃×#469	2
C₄×C₁₃×D₈	2
C₁₃×#436	2
C₁₃×#428	2
#3354	13

Subgroups — C13×#3354 #13112412

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#3354 #13112412