Subgroups — C₁₃×#3298 #13112356

Order 1664.

420 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₄×Q₈×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₃×#547	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#487	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#483	C₂	1	2	yes	maximal, complemented
5	C₁₃×#475	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#465	C₂	1	2	yes	maximal, complemented
7	C₁₃×#461	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₃×#448	C₂	1	2	yes	maximal, complemented
9	C₄×C₁₃×SD₁₆	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈×D₄×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₁₃×#103	C₂	1	2	yes	maximal, complemented
12	C₂×C₁₃×#41	—	2	1	no	—
13	C₂²×Q₈×C₁₃	C₂²	1	2	yes	—
14	C₂×C₁₃×SD₁₆	C₂²	1	1	yes	characteristic
15	C₂×C₁₃×SD₁₆	C₂²	1	2	yes	—
16	C₂²×C₈×C₁₃	C₂²	1	2	yes	—
17	C₁₃×C₄⋊Q₈	—	2	1	no	—
18	C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
19	C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
20	C₁₃×#123	—	2	2	no	—
21	C₁₃×C₂²⋊Q₈	—	2	1	no	—
22	C₁₃×C₂²⋊Q₈	—	2	2	no	—
23	C₁₃×C₂²⋊Q₈	C₂²	1	2	yes	—
24	C₁₃×#121	C₂²	1	2	yes	—
25	C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
26	C₄×D₄×C₁₃	—	2	1	no	—
27	C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
28	C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
29	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
30	C₁₃×#106	C₂²	1	1	yes	characteristic
31	C₁₃×#106	C₂²	1	2	yes	—
32	C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
33	C₁₃×#103	C₂²	1	1	yes	characteristic
34	C₁₃×#103	C₂²	1	2	yes	—
35	C₁₃×#103	C₂²	1	2	yes	—
36	C₁₃×#103	C₂²	1	2	yes	—
37	C₁₃×#103	C₂²	1	2	yes	—
38	C₁₃×#102	C₂²	1	1	yes	characteristic
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	2	no	—
43	C₂×Q₈×C₁₃	—	2	1	no	—
44	C₂×Q₈×C₁₃	—	2	4	no	—
45	C₂×Q₈×C₁₃	—	4	2	no	—
46	C₂×Q₈×C₁₃	C₂³	1	2	yes	—
47	C₂×Q₈×C₁₃	C₂³	1	1	yes	characteristic
48	C₂×D₄×C₁₃	—	2	2	no	—
49	C₂×D₄×C₁₃	D₄	1	1	yes	characteristic
50	C₂×D₄×C₁₃	C₂³	1	1	yes	characteristic
51	C₂²×C₄×C₁₃	—	2	2	no	—
52	C₂²×C₄×C₁₃	—	2	2	no	—
53	C₂²×C₄×C₁₃	—	2	1	no	—
54	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
55	C₁₃×SD₁₆	—	2	2	no	—
56	C₁₃×SD₁₆	—	4	2	no	—
57	C₂×C₈×C₁₃	—	4	2	no	—
58	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
59	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
60	C₂×C₈×C₁₃	C₂³	1	2	yes	—
61	C₁₃×C₄⋊C₄	—	2	2	no	—
62	C₁₃×C₄⋊C₄	—	2	1	no	—
63	C₁₃×C₄⋊C₄	—	2	1	no	—
64	C₁₃×C₄⋊C₄	—	2	1	no	—
65	C₁₃×C₄⋊C₄	—	2	2	no	—
66	C₁₃×C₄⋊C₄	—	2	1	no	—
67	C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
68	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
69	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
70	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
71	C₁₃×C₂²⋊C₄	—	2	2	no	—
72	C₁₃×C₂²⋊C₄	—	2	1	no	—
73	C₁₃×C₂²⋊C₄	—	2	2	no	—
74	C₁₃×C₂²⋊C₄	D₄	1	2	yes	—
75	C₄²×C₁₃	—	2	1	no	—
76	C₄²×C₁₃	C₂³	1	1	yes	characteristic
77	#3298	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
78	C₂³×C₁₃	—	2	1	no	—
79	C₂³×C₁₃	C₂×D₄	1	2	yes	—
80	Q₈×C₁₃	—	2	1	no	—
81	Q₈×C₁₃	—	2	2	no	—
82	Q₈×C₁₃	—	2	2	no	—
83	Q₈×C₁₃	—	4	4	no	—
84	Q₈×C₁₃	—	4	2	no	—
85	Q₈×C₁₃	#41	1	2	yes	complemented
86	D₄×C₁₃	—	2	1	no	—
87	D₄×C₁₃	—	2	1	no	—
88	D₄×C₁₃	—	4	4	no	—
89	D₄×C₁₃	SD₁₆	1	4	yes	complemented
90	C₂×C₄×C₁₃	—	2	1	no	—
91	C₂×C₄×C₁₃	—	2	2	no	—
92	C₂×C₄×C₁₃	—	2	2	no	—
93	C₂×C₄×C₁₃	—	2	1	no	—
94	C₂×C₄×C₁₃	—	2	2	no	—
95	C₂×C₄×C₁₃	—	2	1	no	—
96	C₂×C₄×C₁₃	—	2	2	no	—
97	C₂×C₄×C₁₃	—	4	4	no	—
98	C₂×C₄×C₁₃	—	4	4	no	—
99	C₂×C₄×C₁₃	—	4	2	no	—
100	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
101	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
102	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
104	C₈×C₁₃	—	2	1	no	—
105	C₈×C₁₃	—	2	1	no	—
106	C₈×C₁₃	—	2	2	no	—
107	D₄×Q₈	C₂×C₁₃	1	1	yes	characteristic, complemented
108	#547	C₂×C₁₃	1	1	yes	characteristic
109	#487	C₂×C₁₃	1	1	yes	characteristic, complemented
110	#483	C₂×C₁₃	1	2	yes	complemented
111	#475	C₂×C₁₃	1	1	yes	characteristic, complemented
112	#465	C₂×C₁₃	1	2	yes	complemented
113	#461	C₂×C₁₃	1	1	yes	characteristic, complemented
114	#448	C₂×C₁₃	1	2	yes	complemented
115	C₄×SD₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
116	C₈×D₄	C₂×C₁₃	1	1	yes	characteristic, complemented
117	C₂×#103	C₂×C₁₃	1	2	yes	complemented
118	C₂²×C₁₃	—	2	2	no	—
119	C₂²×C₁₃	—	2	2	no	—
120	C₂²×C₁₃	—	4	1	no	—
121	C₂²×C₁₃	—	4	1	no	—
122	C₂²×C₁₃	—	4	1	no	—
123	C₂²×C₁₃	C₂×SD₁₆	1	4	yes	—
124	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
125	C₄×C₁₃	—	2	2	no	—
126	C₄×C₁₃	—	2	2	no	—
127	C₄×C₁₃	—	2	1	no	—
128	C₄×C₁₃	—	4	2	no	—
129	C₄×C₁₃	—	4	1	no	—
130	C₄×C₁₃	—	4	2	no	—
131	C₄×C₁₃	—	4	1	no	—
132	C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
133	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
134	C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
135	C₂×#41	—	2	1	no	—
136	C₂²×Q₈	C₂²×C₁₃	1	2	yes	—
137	C₂×SD₁₆	C₂²×C₁₃	1	1	yes	characteristic
138	C₂×SD₁₆	C₂²×C₁₃	1	2	yes	—
139	C₂²×C₈	C₂²×C₁₃	1	2	yes	—
140	C₄⋊Q₈	—	2	1	no	—
141	C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
142	C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
143	#123	—	2	2	no	—
144	C₂²⋊Q₈	—	2	1	no	—
145	C₂²⋊Q₈	—	2	2	no	—
146	C₂²⋊Q₈	C₂²×C₁₃	1	2	yes	—
147	#121	C₂²×C₁₃	1	2	yes	—
148	C₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic
149	C₄×D₄	—	2	1	no	—
150	C₄×D₄	C₂²×C₁₃	1	1	yes	characteristic
151	C₄×D₄	C₂²×C₁₃	1	1	yes	characteristic
152	C₂×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
153	#106	C₂²×C₁₃	1	1	yes	characteristic
154	#106	C₂²×C₁₃	1	2	yes	—
155	C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
156	#103	C₂²×C₁₃	1	1	yes	characteristic
157	#103	C₂²×C₁₃	1	2	yes	—
158	#103	C₂²×C₁₃	1	2	yes	—
159	#103	C₂²×C₁₃	1	2	yes	—
160	#103	C₂²×C₁₃	1	2	yes	—
161	#102	C₂²×C₁₃	1	1	yes	characteristic
162	C₂²⋊C₈	C₂²×C₁₃	1	2	yes	—
163	C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
164	C₂×C₁₃	—	2	4	no	—
165	C₂×C₁₃	—	8	1	no	—
166	C₂×C₁₃	#546	1	1	yes	characteristic
167	C₂×C₁₃	C₂²×SD₁₆	1	1	yes	characteristic
168	C₂×C₁₃	#578	1	1	yes	characteristic
169	#41	—	4	4	no	—
170	C₂×Q₈	—	2	2	no	—
171	C₂×Q₈	—	2	1	no	—
172	C₂×Q₈	—	2	4	no	—
173	C₂×Q₈	—	4	2	no	—
174	C₂×Q₈	C₂³×C₁₃	1	2	yes	—
175	C₂×Q₈	C₂³×C₁₃	1	1	yes	characteristic
176	C₂×D₄	—	2	2	no	—
177	C₂×D₄	D₄×C₁₃	1	1	yes	characteristic
178	C₂×D₄	C₂³×C₁₃	1	1	yes	characteristic
179	C₂²×C₄	—	2	2	no	—
180	C₂²×C₄	—	2	2	no	—
181	C₂²×C₄	—	2	1	no	—
182	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
183	SD₁₆	—	2	2	no	—
184	SD₁₆	—	4	2	no	—
185	C₂×C₈	—	4	2	no	—
186	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
187	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
188	C₂×C₈	C₂³×C₁₃	1	2	yes	—
189	C₄⋊C₄	—	2	2	no	—
190	C₄⋊C₄	—	2	1	no	—
191	C₄⋊C₄	—	2	1	no	—
192	C₄⋊C₄	—	2	1	no	—
193	C₄⋊C₄	—	2	2	no	—
194	C₄⋊C₄	—	2	1	no	—
195	C₄⋊C₄	D₄×C₁₃	1	1	yes	characteristic
196	C₄⋊C₄	C₂³×C₁₃	1	1	yes	characteristic
197	C₄⋊C₄	C₂³×C₁₃	1	1	yes	characteristic
198	C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
199	C₂²⋊C₄	—	2	2	no	—
200	C₂²⋊C₄	—	2	1	no	—
201	C₂²⋊C₄	—	2	2	no	—
202	C₂²⋊C₄	D₄×C₁₃	1	2	yes	—
203	C₄²	—	2	1	no	—
204	C₄²	C₂³×C₁₃	1	1	yes	characteristic
205	C₁₃	#3298	1	1	yes	characteristic, direct product factor, Sylow
206	C₂³	—	2	1	no	—
207	C₂³	C₂×D₄×C₁₃	1	2	yes	—
208	Q₈	—	2	1	no	—
209	Q₈	—	2	2	no	—
210	Q₈	—	2	2	no	—
211	Q₈	—	4	4	no	—
212	Q₈	—	4	2	no	—
213	Q₈	C₁₃×#41	1	2	yes	complemented
214	D₄	—	2	1	no	—
215	D₄	—	2	1	no	—
216	D₄	—	4	4	no	—
217	D₄	C₁₃×SD₁₆	1	4	yes	complemented
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	4	4	no	—
226	C₂×C₄	—	4	4	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
229	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
230	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
231	C₂×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
232	C₈	—	2	1	no	—
233	C₈	—	2	1	no	—
234	C₈	—	2	2	no	—
235	C₂²	—	2	2	no	—
236	C₂²	—	2	2	no	—
237	C₂²	—	4	1	no	—
238	C₂²	—	4	1	no	—
239	C₂²	—	4	1	no	—
240	C₂²	C₂×C₁₃×SD₁₆	1	4	yes	—
241	C₂²	C₂²×D₄×C₁₃	1	1	yes	characteristic
242	C₄	—	2	2	no	—
243	C₄	—	2	2	no	—
244	C₄	—	2	1	no	—
245	C₄	—	4	2	no	—
246	C₄	—	4	1	no	—
247	C₄	—	4	2	no	—
248	C₄	—	4	1	no	—
249	C₄	C₂×C₁₃×SD₁₆	1	2	yes	—
250	C₄	C₂×C₁₃×#41	1	1	yes	characteristic
251	C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
252	C₂	—	2	4	no	—
253	C₂	—	8	1	no	—
254	C₂	C₁₃×#546	1	1	yes	characteristic
255	C₂	C₂²×C₁₃×SD₁₆	1	1	yes	characteristic
256	C₂	C₁₃×#578	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₃
prime radical	C₁₃×#547	C₂
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. D₄×Q₈×C₁₃	C₂	semidirect
3. C₁₃×#487	C₂	semidirect
4. C₁₃×#483	C₂	semidirect
5. C₁₃×#475	C₂	semidirect
6. C₁₃×#465	C₂	semidirect
7. C₁₃×#461	C₂	semidirect
8. C₁₃×#448	C₂	semidirect
9. C₄×C₁₃×SD₁₆	C₂	semidirect
10. C₈×D₄×C₁₃	C₂	semidirect
11. C₂×C₁₃×#103	C₂	semidirect
77. #3298	C₁₃	direct (N × H)
85. Q₈×C₁₃	#41	semidirect
89. D₄×C₁₃	SD₁₆	semidirect
107. D₄×Q₈	C₂×C₁₃	semidirect
109. #487	C₂×C₁₃	semidirect
110. #483	C₂×C₁₃	semidirect
111. #475	C₂×C₁₃	semidirect
112. #465	C₂×C₁₃	semidirect
113. #461	C₂×C₁₃	semidirect
114. #448	C₂×C₁₃	semidirect
115. C₄×SD₁₆	C₂×C₁₃	semidirect
116. C₈×D₄	C₂×C₁₃	semidirect
117. C₂×#103	C₂×C₁₃	semidirect
205. C₁₃	#3298	direct (N × H)
213. Q₈	C₁₃×#41	semidirect
217. D₄	C₁₃×SD₁₆	semidirect

Maximal subgroups

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

subgroup	index
D₄×Q₈×C₁₃	2
C₁₃×#547	2
C₁₃×#487	2
C₁₃×#483	2
C₁₃×#475	2
C₁₃×#465	2
C₁₃×#461	2
C₁₃×#448	2
C₄×C₁₃×SD₁₆	2
C₈×D₄×C₁₃	2
C₂×C₁₃×#103	2
#3298	13

Subgroups — C13×#3298 #13112356

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#3298 #13112356