Subgroups — C₁₃×#3522 #13112580

Order 1664.

1104 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	C₁₃×#559	C₂	1	3	yes	maximal, complemented
2	Q₈²×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#554	C₂	1	3	yes	maximal, complemented
4	C₁₃×#550	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#548	C₂	1	2	yes	maximal, complemented
6	C₁₃×#547	C₂	1	6	yes	maximal, complemented
7	C₁₃×#542	C₂	1	6	yes	maximal, complemented
8	C₁₃×#532	C₂	1	3	yes	maximal, complemented
9	C₁₃×#529	C₂	1	3	yes	maximal, complemented
10	C₁₃×#520	C₂	1	2	yes	maximal, complemented
11	C₂×C₄×Q₈×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₁₃×#41	C₂²	1	2	yes	—
13	C₂×C₁₃×#41	C₂²	1	6	yes	—
14	C₂²×Q₈×C₁₃	C₂²	1	1	yes	characteristic
15	C₁₃×C₄⋊Q₈	C₂²	1	3	yes	—
16	C₁₃×C₄⋊Q₈	C₂²	1	6	yes	—
17	C₁₃×#127	C₂²	1	3	yes	—
18	C₁₃×#126	C₂²	1	6	yes	—
19	C₁₃×#125	C₂²	1	3	yes	—
20	C₁₃×#124	C₂²	1	3	yes	—
21	C₁₃×#124	C₂²	1	6	yes	—
22	C₁₃×#124	C₂²	1	6	yes	—
23	C₁₃×#123	C₂²	1	6	yes	—
24	C₁₃×#123	C₂²	1	12	yes	—
25	C₁₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
26	C₁₃×C₂²⋊Q₈	C₂²	1	12	yes	—
27	C₁₃×C₂²⋊Q₈	C₂²	1	6	yes	—
28	C₁₃×#121	C₂²	1	3	yes	—
29	C₁₃×#121	C₂²	1	6	yes	—
30	C₁₃×#121	C₂²	1	6	yes	—
31	C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
32	C₄×Q₈×C₁₃	C₂²	1	3	yes	—
33	C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
34	C₄×Q₈×C₁₃	C₂²	1	3	yes	—
35	C₄×Q₈×C₁₃	C₂²	1	2	yes	—
36	C₄×Q₈×C₁₃	C₂²	1	3	yes	—
37	C₄×D₄×C₁₃	C₂²	1	6	yes	—
38	C₄×D₄×C₁₃	C₂²	1	6	yes	—
39	C₄×D₄×C₁₃	C₂²	1	6	yes	—
40	C₄×D₄×C₁₃	C₂²	1	3	yes	—
41	C₄×D₄×C₁₃	C₂²	1	6	yes	—
42	C₁₃×#117	C₂²	1	6	yes	—
43	C₂×C₁₃×C₄⋊C₄	C₂²	1	3	yes	—
44	C₂×C₁₃×C₄⋊C₄	C₂²	1	6	yes	—
45	C₂×C₄²×C₁₃	C₂²	1	3	yes	—
46	C₁₃×#41	—	2	8	no	—
47	C₁₃×#41	—	4	12	no	—
48	C₂×Q₈×C₁₃	—	2	4	no	—
49	C₂×Q₈×C₁₃	C₂³	1	1	yes	characteristic
50	C₂×Q₈×C₁₃	C₂³	1	3	yes	—
51	C₂×Q₈×C₁₃	C₂³	1	6	yes	—
52	C₂×Q₈×C₁₃	C₂³	1	1	yes	characteristic
53	C₂×D₄×C₁₃	C₂³	1	3	yes	—
54	C₂×D₄×C₁₃	C₂³	1	6	yes	—
55	C₂×D₄×C₁₃	C₂³	1	6	yes	—
56	C₂×D₄×C₁₃	C₂³	1	6	yes	—
57	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
58	C₂²×C₄×C₁₃	C₂³	1	3	yes	—
59	C₂²×C₄×C₁₃	C₂³	1	3	yes	—
60	C₂²×C₄×C₁₃	C₂³	1	6	yes	—
61	C₂²×C₄×C₁₃	C₂³	1	12	yes	—
62	C₁₃×C₄⋊C₄	C₂³	1	3	yes	—
63	C₁₃×C₄⋊C₄	C₂³	1	6	yes	—
64	C₁₃×C₄⋊C₄	C₂³	1	6	yes	—
65	C₁₃×C₄⋊C₄	C₂³	1	3	yes	—
66	C₁₃×C₄⋊C₄	C₂³	1	6	yes	—
67	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
68	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
69	C₁₃×C₄⋊C₄	C₂³	1	3	yes	—
70	C₁₃×C₄⋊C₄	C₂³	1	6	yes	—
71	C₁₃×C₄⋊C₄	C₂³	1	6	yes	—
72	C₁₃×C₂²⋊C₄	C₂³	1	6	yes	—
73	C₁₃×C₂²⋊C₄	C₂³	1	6	yes	—
74	C₁₃×C₂²⋊C₄	C₂³	1	2	yes	—
75	C₁₃×C₂²⋊C₄	C₂³	1	6	yes	—
76	C₁₃×C₂²⋊C₄	C₂³	1	12	yes	—
77	C₁₃×C₂²⋊C₄	C₂³	1	6	yes	—
78	C₄²×C₁₃	C₂³	1	3	yes	—
79	C₄²×C₁₃	C₂³	1	3	yes	—
80	C₄²×C₁₃	C₂³	1	3	yes	—
81	C₄²×C₁₃	C₂³	1	3	yes	—
82	C₄²×C₁₃	C₂³	1	6	yes	—
83	#3522	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
84	C₂³×C₁₃	C₂⁴	1	6	yes	—
85	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
86	Q₈×C₁₃	—	2	6	no	—
87	Q₈×C₁₃	—	2	2	no	—
88	Q₈×C₁₃	—	4	6	no	—
89	Q₈×C₁₃	#41	1	4	yes	complemented
90	D₄×C₁₃	—	2	12	no	—
91	D₄×C₁₃	—	2	12	no	—
92	D₄×C₁₃	—	2	6	no	—
93	D₄×C₁₃	—	4	6	no	—
94	C₂×C₄×C₁₃	—	2	6	no	—
95	C₂×C₄×C₁₃	—	2	6	no	—
96	C₂×C₄×C₁₃	—	2	12	no	—
97	C₂×C₄×C₁₃	—	2	2	no	—
98	C₂×C₄×C₁₃	—	4	12	no	—
99	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
100	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
101	C₂×C₄×C₁₃	C₂⁴	1	3	yes	—
102	C₂×C₄×C₁₃	C₂⁴	1	3	yes	—
103	C₂×C₄×C₁₃	C₂⁴	1	3	yes	—
104	C₂×C₄×C₁₃	C₂⁴	1	6	yes	—
105	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
106	C₂×C₄×C₁₃	C₂⁴	1	3	yes	—
107	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
108	#559	C₂×C₁₃	1	3	yes	complemented
109	Q₈²	C₂×C₁₃	1	1	yes	characteristic, complemented
110	#554	C₂×C₁₃	1	3	yes	complemented
111	#550	C₂×C₁₃	1	1	yes	characteristic, complemented
112	#548	C₂×C₁₃	1	2	yes	complemented
113	#547	C₂×C₁₃	1	6	yes	complemented
114	#542	C₂×C₁₃	1	6	yes	complemented
115	#532	C₂×C₁₃	1	3	yes	complemented
116	#529	C₂×C₁₃	1	3	yes	complemented
117	#520	C₂×C₁₃	1	2	yes	complemented
118	C₂×C₄×Q₈	C₂×C₁₃	1	1	yes	characteristic, complemented
119	C₂²×C₁₃	—	2	1	no	—
120	C₂²×C₁₃	—	2	1	no	—
121	C₂²×C₁₃	—	2	6	no	—
122	C₂²×C₁₃	—	2	6	no	—
123	C₂²×C₁₃	—	2	6	no	—
124	C₂²×C₁₃	H(5,𝔽₂)	1	2	yes	—
125	C₂²×C₁₃	C₂⁵	1	1	yes	characteristic
126	C₄×C₁₃	—	2	2	no	—
127	C₄×C₁₃	—	2	3	no	—
128	C₄×C₁₃	—	2	3	no	—
129	C₄×C₁₃	—	2	3	no	—
130	C₄×C₁₃	—	2	6	no	—
131	C₄×C₁₃	—	2	1	no	—
132	C₄×C₁₃	—	2	2	no	—
133	C₄×C₁₃	C₂×#41	1	6	yes	—
134	C₄×C₁₃	2₋¹⁺⁴	1	2	yes	—
135	C₂×#41	C₂²×C₁₃	1	2	yes	—
136	C₂×#41	C₂²×C₁₃	1	6	yes	—
137	C₂²×Q₈	C₂²×C₁₃	1	1	yes	characteristic
138	C₄⋊Q₈	C₂²×C₁₃	1	3	yes	—
139	C₄⋊Q₈	C₂²×C₁₃	1	6	yes	—
140	#127	C₂²×C₁₃	1	3	yes	—
141	#126	C₂²×C₁₃	1	6	yes	—
142	#125	C₂²×C₁₃	1	3	yes	—
143	#124	C₂²×C₁₃	1	3	yes	—
144	#124	C₂²×C₁₃	1	6	yes	—
145	#124	C₂²×C₁₃	1	6	yes	—
146	#123	C₂²×C₁₃	1	6	yes	—
147	#123	C₂²×C₁₃	1	12	yes	—
148	C₂²⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
149	C₂²⋊Q₈	C₂²×C₁₃	1	12	yes	—
150	C₂²⋊Q₈	C₂²×C₁₃	1	6	yes	—
151	#121	C₂²×C₁₃	1	3	yes	—
152	#121	C₂²×C₁₃	1	6	yes	—
153	#121	C₂²×C₁₃	1	6	yes	—
154	C₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic
155	C₄×Q₈	C₂²×C₁₃	1	3	yes	—
156	C₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic
157	C₄×Q₈	C₂²×C₁₃	1	3	yes	—
158	C₄×Q₈	C₂²×C₁₃	1	2	yes	—
159	C₄×Q₈	C₂²×C₁₃	1	3	yes	—
160	C₄×D₄	C₂²×C₁₃	1	6	yes	—
161	C₄×D₄	C₂²×C₁₃	1	6	yes	—
162	C₄×D₄	C₂²×C₁₃	1	6	yes	—
163	C₄×D₄	C₂²×C₁₃	1	3	yes	—
164	C₄×D₄	C₂²×C₁₃	1	6	yes	—
165	#117	C₂²×C₁₃	1	6	yes	—
166	C₂×C₄⋊C₄	C₂²×C₁₃	1	3	yes	—
167	C₂×C₄⋊C₄	C₂²×C₁₃	1	6	yes	—
168	C₂×C₄²	C₂²×C₁₃	1	3	yes	—
169	C₂×C₁₃	—	2	2	no	—
170	C₂×C₁₃	—	4	6	no	—
171	C₂×C₁₃	C₂²×#41	1	1	yes	characteristic
172	C₂×C₁₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
173	C₂×C₁₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
174	#41	—	2	8	no	—
175	#41	—	4	12	no	—
176	C₂×Q₈	—	2	4	no	—
177	C₂×Q₈	C₂³×C₁₃	1	1	yes	characteristic
178	C₂×Q₈	C₂³×C₁₃	1	3	yes	—
179	C₂×Q₈	C₂³×C₁₃	1	6	yes	—
180	C₂×Q₈	C₂³×C₁₃	1	1	yes	characteristic
181	C₂×D₄	C₂³×C₁₃	1	3	yes	—
182	C₂×D₄	C₂³×C₁₃	1	6	yes	—
183	C₂×D₄	C₂³×C₁₃	1	6	yes	—
184	C₂×D₄	C₂³×C₁₃	1	6	yes	—
185	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
186	C₂²×C₄	C₂³×C₁₃	1	3	yes	—
187	C₂²×C₄	C₂³×C₁₃	1	3	yes	—
188	C₂²×C₄	C₂³×C₁₃	1	6	yes	—
189	C₂²×C₄	C₂³×C₁₃	1	12	yes	—
190	C₄⋊C₄	C₂³×C₁₃	1	3	yes	—
191	C₄⋊C₄	C₂³×C₁₃	1	6	yes	—
192	C₄⋊C₄	C₂³×C₁₃	1	6	yes	—
193	C₄⋊C₄	C₂³×C₁₃	1	3	yes	—
194	C₄⋊C₄	C₂³×C₁₃	1	6	yes	—
195	C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
196	C₄⋊C₄	C₂³×C₁₃	1	1	yes	characteristic
197	C₄⋊C₄	C₂³×C₁₃	1	3	yes	—
198	C₄⋊C₄	C₂³×C₁₃	1	6	yes	—
199	C₄⋊C₄	C₂³×C₁₃	1	6	yes	—
200	C₂²⋊C₄	C₂³×C₁₃	1	6	yes	—
201	C₂²⋊C₄	C₂³×C₁₃	1	6	yes	—
202	C₂²⋊C₄	C₂³×C₁₃	1	2	yes	—
203	C₂²⋊C₄	C₂³×C₁₃	1	6	yes	—
204	C₂²⋊C₄	C₂³×C₁₃	1	12	yes	—
205	C₂²⋊C₄	C₂³×C₁₃	1	6	yes	—
206	C₄²	C₂³×C₁₃	1	3	yes	—
207	C₄²	C₂³×C₁₃	1	3	yes	—
208	C₄²	C₂³×C₁₃	1	3	yes	—
209	C₄²	C₂³×C₁₃	1	3	yes	—
210	C₄²	C₂³×C₁₃	1	6	yes	—
211	C₁₃	#3522	1	1	yes	characteristic, direct product factor, Sylow
212	C₂³	C₂⁴×C₁₃	1	6	yes	—
213	C₂³	C₂⁴×C₁₃	1	1	yes	characteristic
214	Q₈	—	2	6	no	—
215	Q₈	—	2	2	no	—
216	Q₈	—	4	6	no	—
217	Q₈	C₁₃×#41	1	4	yes	complemented
218	D₄	—	2	12	no	—
219	D₄	—	2	12	no	—
220	D₄	—	2	6	no	—
221	D₄	—	4	6	no	—
222	C₂×C₄	—	2	6	no	—
223	C₂×C₄	—	2	6	no	—
224	C₂×C₄	—	2	12	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	4	12	no	—
227	C₂×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
228	C₂×C₄	C₂⁴×C₁₃	1	2	yes	—
229	C₂×C₄	C₂⁴×C₁₃	1	3	yes	—
230	C₂×C₄	C₂⁴×C₁₃	1	3	yes	—
231	C₂×C₄	C₂⁴×C₁₃	1	3	yes	—
232	C₂×C₄	C₂⁴×C₁₃	1	6	yes	—
233	C₂×C₄	C₂⁴×C₁₃	1	2	yes	—
234	C₂×C₄	C₂⁴×C₁₃	1	3	yes	—
235	C₂×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
236	C₂²	—	2	1	no	—
237	C₂²	—	2	1	no	—
238	C₂²	—	2	6	no	—
239	C₂²	—	2	6	no	—
240	C₂²	—	2	6	no	—
241	C₂²	C₁₃×H(5,𝔽₂)	1	2	yes	—
242	C₂²	C₂⁵×C₁₃	1	1	yes	characteristic
243	C₄	—	2	2	no	—
244	C₄	—	2	3	no	—
245	C₄	—	2	3	no	—
246	C₄	—	2	3	no	—
247	C₄	—	2	6	no	—
248	C₄	—	2	1	no	—
249	C₄	—	2	2	no	—
250	C₄	C₂×C₁₃×#41	1	6	yes	—
251	C₄	C₁₃×2₋¹⁺⁴	1	2	yes	—
252	C₂	—	2	2	no	—
253	C₂	—	4	6	no	—
254	C₂	C₂²×C₁₃×#41	1	1	yes	characteristic
255	C₂	C₂×C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
256	C₂	C₂×C₁₃×2₋¹⁺⁴	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂⁵
commutator, Frattini, cosocle	C₂²	C₂⁵×C₁₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₃×#559	C₂	semidirect
2. Q₈²×C₁₃	C₂	semidirect
3. C₁₃×#554	C₂	semidirect
4. C₁₃×#550	C₂	semidirect
5. C₁₃×#548	C₂	semidirect
6. C₁₃×#547	C₂	semidirect
7. C₁₃×#542	C₂	semidirect
8. C₁₃×#532	C₂	semidirect
9. C₁₃×#529	C₂	semidirect
10. C₁₃×#520	C₂	semidirect
11. C₂×C₄×Q₈×C₁₃	C₂	semidirect
83. #3522	C₁₃	direct (N × H)
89. Q₈×C₁₃	#41	semidirect
108. #559	C₂×C₁₃	semidirect
109. Q₈²	C₂×C₁₃	semidirect
110. #554	C₂×C₁₃	semidirect
111. #550	C₂×C₁₃	semidirect
112. #548	C₂×C₁₃	semidirect
113. #547	C₂×C₁₃	semidirect
114. #542	C₂×C₁₃	semidirect
115. #532	C₂×C₁₃	semidirect
116. #529	C₂×C₁₃	semidirect
117. #520	C₂×C₁₃	semidirect
118. C₂×C₄×Q₈	C₂×C₁₃	semidirect
211. C₁₃	#3522	direct (N × H)
217. Q₈	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×#559	2
Q₈²×C₁₃	2
C₁₃×#554	2
C₁₃×#550	2
C₁₃×#548	2
C₁₃×#547	2
C₁₃×#542	2
C₁₃×#532	2
C₁₃×#529	2
C₁₃×#520	2
C₂×C₄×Q₈×C₁₃	2
#3522	13

Subgroups — C13×#3522 #13112580

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#3522 #13112580