Subgroups — C₁₃×#3518 #13112576

Order 1664.

1010 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈²×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₃×#556	C₂	1	6	yes	maximal, complemented
3	C₁₃×#556	C₂	1	3	yes	maximal, complemented
4	C₁₃×#555	C₂	1	3	yes	maximal, complemented
5	D₄×Q₈×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#548	C₂	1	2	yes	maximal, complemented
7	C₁₃×#542	C₂	1	6	yes	maximal, complemented
8	C₁₃×#533	C₂	1	3	yes	maximal, complemented
9	C₁₃×#529	C₂	1	3	yes	maximal, complemented
10	C₁₃×#519	C₂	1	2	yes	maximal, complemented
11	C₄×C₁₃×#41	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₁₃×#41	C₂²	1	1	yes	characteristic
13	C₂²×Q₈×C₁₃	C₂²	1	2	yes	—
14	C₁₃×C₄⋊Q₈	C₂²	1	3	yes	—
15	C₁₃×C₄⋊Q₈	C₂²	1	3	yes	—
16	C₁₃×C₄⋊Q₈	C₂²	1	6	yes	—
17	C₁₃×#126	C₂²	1	6	yes	—
18	C₁₃×#126	C₂²	1	12	yes	—
19	C₁₃×#125	C₂²	1	3	yes	—
20	C₁₃×#124	C₂²	1	1	yes	characteristic
21	C₁₃×#124	C₂²	1	6	yes	—
22	C₁₃×#124	C₂²	1	12	yes	—
23	C₁₃×#123	C₂²	1	6	yes	—
24	C₁₃×C₂²⋊Q₈	C₂²	1	3	yes	—
25	C₁₃×C₂²⋊Q₈	C₂²	1	6	yes	—
26	C₁₃×C₂²⋊Q₈	C₂²	1	6	yes	—
27	C₁₃×C₂²⋊Q₈	C₂²	1	6	yes	—
28	C₁₃×#121	C₂²	1	3	yes	—
29	C₁₃×#121	C₂²	1	6	yes	—
30	C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
31	C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
32	C₄×Q₈×C₁₃	C₂²	1	6	yes	—
33	C₄×Q₈×C₁₃	C₂²	1	2	yes	—
34	C₄×Q₈×C₁₃	C₂²	1	6	yes	—
35	C₄×Q₈×C₁₃	C₂²	1	3	yes	—
36	C₄×Q₈×C₁₃	C₂²	1	6	yes	—
37	C₄×D₄×C₁₃	C₂²	1	3	yes	—
38	C₄×D₄×C₁₃	C₂²	1	3	yes	—
39	C₄×D₄×C₁₃	C₂²	1	6	yes	—
40	C₄×D₄×C₁₃	C₂²	1	3	yes	—
41	C₁₃×#117	C₂²	1	3	yes	—
42	C₁₃×#117	C₂²	1	6	yes	—
43	C₁₃×#117	C₂²	1	6	yes	—
44	C₁₃×#117	C₂²	1	6	yes	—
45	C₂×C₄²×C₁₃	C₂²	1	3	yes	—
46	C₁₃×#41	—	2	4	no	—
47	C₂×Q₈×C₁₃	—	2	8	no	—
48	C₂×Q₈×C₁₃	C₂³	1	1	yes	characteristic
49	C₂×Q₈×C₁₃	C₂³	1	6	yes	—
50	C₂×Q₈×C₁₃	C₂³	1	6	yes	—
51	C₂×Q₈×C₁₃	C₂³	1	1	yes	characteristic
52	C₂×D₄×C₁₃	C₂³	1	1	yes	characteristic
53	C₂×D₄×C₁₃	C₂³	1	3	yes	—
54	C₂×D₄×C₁₃	C₂³	1	6	yes	—
55	C₂²×C₄×C₁₃	C₂³	1	3	yes	—
56	C₂²×C₄×C₁₃	C₂³	1	3	yes	—
57	C₂²×C₄×C₁₃	C₂³	1	6	yes	—
58	C₂²×C₄×C₁₃	C₂³	1	3	yes	—
59	C₁₃×C₄⋊C₄	C₂³	1	3	yes	—
60	C₁₃×C₄⋊C₄	C₂³	1	3	yes	—
61	C₁₃×C₄⋊C₄	C₂³	1	3	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	6	yes	—
66	C₁₃×C₄⋊C₄	C₂³	1	6	yes	—
67	C₁₃×C₄⋊C₄	C₂³	1	6	yes	—
68	C₁₃×C₄⋊C₄	C₂³	1	6	yes	—
69	C₁₃×C₂²⋊C₄	C₂³	1	6	yes	—
70	C₁₃×C₂²⋊C₄	C₂³	1	2	yes	—
71	C₁₃×C₂²⋊C₄	C₂³	1	6	yes	—
72	C₁₃×C₂²⋊C₄	C₂³	1	12	yes	—
73	C₁₃×C₂²⋊C₄	C₂³	1	2	yes	—
74	C₁₃×C₂²⋊C₄	C₂³	1	6	yes	—
75	C₁₃×C₂²⋊C₄	C₂³	1	6	yes	—
76	C₄²×C₁₃	C₂³	1	1	yes	characteristic
77	C₄²×C₁₃	C₂³	1	3	yes	—
78	C₄²×C₁₃	C₂³	1	3	yes	—
79	C₄²×C₁₃	C₂³	1	3	yes	—
80	C₄²×C₁₃	C₂³	1	12	yes	—
81	C₄²×C₁₃	C₂³	1	6	yes	—
82	#3518	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
83	C₂³×C₁₃	C₂⁴	1	2	yes	—
84	C₂³×C₁₃	C₂⁴	1	3	yes	—
85	Q₈×C₁₃	—	2	12	no	—
86	Q₈×C₁₃	—	2	2	no	—
87	Q₈×C₁₃	—	4	6	no	—
88	Q₈×C₁₃	#41	1	4	yes	—
89	D₄×C₁₃	—	2	6	no	—
90	D₄×C₁₃	—	2	2	no	—
91	D₄×C₁₃	—	4	6	no	—
92	C₂×C₄×C₁₃	—	2	12	no	—
93	C₂×C₄×C₁₃	—	2	6	no	—
94	C₂×C₄×C₁₃	—	2	6	no	—
95	C₂×C₄×C₁₃	—	4	3	no	—
96	C₂×C₄×C₁₃	C₂⁴	1	3	yes	—
97	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
98	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
99	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
100	C₂×C₄×C₁₃	C₂⁴	1	6	yes	—
101	C₂×C₄×C₁₃	C₂⁴	1	6	yes	—
102	C₂×C₄×C₁₃	C₂⁴	1	3	yes	—
103	C₂×C₄×C₁₃	C₂⁴	1	3	yes	—
104	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
105	Q₈²	C₂×C₁₃	1	1	yes	characteristic, complemented
106	#556	C₂×C₁₃	1	6	yes	complemented
107	#556	C₂×C₁₃	1	3	yes	complemented
108	#555	C₂×C₁₃	1	3	yes	complemented
109	D₄×Q₈	C₂×C₁₃	1	1	yes	characteristic, complemented
110	#548	C₂×C₁₃	1	2	yes	complemented
111	#542	C₂×C₁₃	1	6	yes	complemented
112	#533	C₂×C₁₃	1	3	yes	complemented
113	#529	C₂×C₁₃	1	3	yes	complemented
114	#519	C₂×C₁₃	1	2	yes	complemented
115	C₄×#41	C₂×C₁₃	1	1	yes	characteristic, complemented
116	C₂²×C₁₃	—	2	2	no	—
117	C₂²×C₁₃	—	2	3	no	—
118	C₂²×C₁₃	—	2	3	no	—
119	C₂²×C₁₃	—	2	2	no	—
120	C₂²×C₁₃	—	2	3	no	—
121	C₂²×C₁₃	—	2	2	no	—
122	C₂²×C₁₃	C₂⁵	1	1	yes	characteristic
123	C₄×C₁₃	—	2	1	no	—
124	C₄×C₁₃	—	2	6	no	—
125	C₄×C₁₃	—	2	3	no	—
126	C₄×C₁₃	—	2	2	no	—
127	C₄×C₁₃	—	2	1	no	—
128	C₄×C₁₃	—	2	3	no	—
129	C₄×C₁₃	—	2	6	no	—
130	C₄×C₁₃	C₂×#41	1	6	yes	—
131	C₄×C₁₃	2₋¹⁺⁴	1	2	yes	—
132	C₂×#41	C₂²×C₁₃	1	1	yes	characteristic
133	C₂²×Q₈	C₂²×C₁₃	1	2	yes	—
134	C₄⋊Q₈	C₂²×C₁₃	1	3	yes	—
135	C₄⋊Q₈	C₂²×C₁₃	1	3	yes	—
136	C₄⋊Q₈	C₂²×C₁₃	1	6	yes	—
137	#126	C₂²×C₁₃	1	6	yes	—
138	#126	C₂²×C₁₃	1	12	yes	—
139	#125	C₂²×C₁₃	1	3	yes	—
140	#124	C₂²×C₁₃	1	1	yes	characteristic
141	#124	C₂²×C₁₃	1	6	yes	—
142	#124	C₂²×C₁₃	1	12	yes	—
143	#123	C₂²×C₁₃	1	6	yes	—
144	C₂²⋊Q₈	C₂²×C₁₃	1	3	yes	—
145	C₂²⋊Q₈	C₂²×C₁₃	1	6	yes	—
146	C₂²⋊Q₈	C₂²×C₁₃	1	6	yes	—
147	C₂²⋊Q₈	C₂²×C₁₃	1	6	yes	—
148	#121	C₂²×C₁₃	1	3	yes	—
149	#121	C₂²×C₁₃	1	6	yes	—
150	C₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic
151	C₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic
152	C₄×Q₈	C₂²×C₁₃	1	6	yes	—
153	C₄×Q₈	C₂²×C₁₃	1	2	yes	—
154	C₄×Q₈	C₂²×C₁₃	1	6	yes	—
155	C₄×Q₈	C₂²×C₁₃	1	3	yes	—
156	C₄×Q₈	C₂²×C₁₃	1	6	yes	—
157	C₄×D₄	C₂²×C₁₃	1	3	yes	—
158	C₄×D₄	C₂²×C₁₃	1	3	yes	—
159	C₄×D₄	C₂²×C₁₃	1	6	yes	—
160	C₄×D₄	C₂²×C₁₃	1	3	yes	—
161	#117	C₂²×C₁₃	1	3	yes	—
162	#117	C₂²×C₁₃	1	6	yes	—
163	#117	C₂²×C₁₃	1	6	yes	—
164	#117	C₂²×C₁₃	1	6	yes	—
165	C₂×C₄²	C₂²×C₁₃	1	3	yes	—
166	C₂×C₁₃	—	4	2	no	—
167	C₂×C₁₃	—	4	3	no	—
168	C₂×C₁₃	C₂²×#41	1	1	yes	characteristic
169	C₂×C₁₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
170	C₂×C₁₃	#585	1	1	yes	characteristic
171	#41	—	2	4	no	—
172	C₂×Q₈	—	2	8	no	—
173	C₂×Q₈	C₂³×C₁₃	1	1	yes	characteristic
174	C₂×Q₈	C₂³×C₁₃	1	6	yes	—
175	C₂×Q₈	C₂³×C₁₃	1	6	yes	—
176	C₂×Q₈	C₂³×C₁₃	1	1	yes	characteristic
177	C₂×D₄	C₂³×C₁₃	1	1	yes	characteristic
178	C₂×D₄	C₂³×C₁₃	1	3	yes	—
179	C₂×D₄	C₂³×C₁₃	1	6	yes	—
180	C₂²×C₄	C₂³×C₁₃	1	3	yes	—
181	C₂²×C₄	C₂³×C₁₃	1	3	yes	—
182	C₂²×C₄	C₂³×C₁₃	1	6	yes	—
183	C₂²×C₄	C₂³×C₁₃	1	3	yes	—
184	C₄⋊C₄	C₂³×C₁₃	1	3	yes	—
185	C₄⋊C₄	C₂³×C₁₃	1	3	yes	—
186	C₄⋊C₄	C₂³×C₁₃	1	3	yes	—
187	C₄⋊C₄	C₂³×C₁₃	1	3	yes	—
188	C₄⋊C₄	C₂³×C₁₃	1	6	yes	—
189	C₄⋊C₄	C₂³×C₁₃	1	6	yes	—
190	C₄⋊C₄	C₂³×C₁₃	1	6	yes	—
191	C₄⋊C₄	C₂³×C₁₃	1	6	yes	—
192	C₄⋊C₄	C₂³×C₁₃	1	6	yes	—
193	C₄⋊C₄	C₂³×C₁₃	1	6	yes	—
194	C₂²⋊C₄	C₂³×C₁₃	1	6	yes	—
195	C₂²⋊C₄	C₂³×C₁₃	1	2	yes	—
196	C₂²⋊C₄	C₂³×C₁₃	1	6	yes	—
197	C₂²⋊C₄	C₂³×C₁₃	1	12	yes	—
198	C₂²⋊C₄	C₂³×C₁₃	1	2	yes	—
199	C₂²⋊C₄	C₂³×C₁₃	1	6	yes	—
200	C₂²⋊C₄	C₂³×C₁₃	1	6	yes	—
201	C₄²	C₂³×C₁₃	1	1	yes	characteristic
202	C₄²	C₂³×C₁₃	1	3	yes	—
203	C₄²	C₂³×C₁₃	1	3	yes	—
204	C₄²	C₂³×C₁₃	1	3	yes	—
205	C₄²	C₂³×C₁₃	1	12	yes	—
206	C₄²	C₂³×C₁₃	1	6	yes	—
207	C₁₃	#3518	1	1	yes	characteristic, direct product factor, Sylow
208	C₂³	C₂⁴×C₁₃	1	2	yes	—
209	C₂³	C₂⁴×C₁₃	1	3	yes	—
210	Q₈	—	2	12	no	—
211	Q₈	—	2	2	no	—
212	Q₈	—	4	6	no	—
213	Q₈	C₁₃×#41	1	4	yes	—
214	D₄	—	2	6	no	—
215	D₄	—	2	2	no	—
216	D₄	—	4	6	no	—
217	C₂×C₄	—	2	12	no	—
218	C₂×C₄	—	2	6	no	—
219	C₂×C₄	—	2	6	no	—
220	C₂×C₄	—	4	3	no	—
221	C₂×C₄	C₂⁴×C₁₃	1	3	yes	—
222	C₂×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
223	C₂×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
224	C₂×C₄	C₂⁴×C₁₃	1	2	yes	—
225	C₂×C₄	C₂⁴×C₁₃	1	6	yes	—
226	C₂×C₄	C₂⁴×C₁₃	1	6	yes	—
227	C₂×C₄	C₂⁴×C₁₃	1	3	yes	—
228	C₂×C₄	C₂⁴×C₁₃	1	3	yes	—
229	C₂×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
230	C₂²	—	2	2	no	—
231	C₂²	—	2	3	no	—
232	C₂²	—	2	3	no	—
233	C₂²	—	2	2	no	—
234	C₂²	—	2	3	no	—
235	C₂²	—	2	2	no	—
236	C₂²	C₂⁵×C₁₃	1	1	yes	characteristic
237	C₄	—	2	1	no	—
238	C₄	—	2	6	no	—
239	C₄	—	2	3	no	—
240	C₄	—	2	2	no	—
241	C₄	—	2	1	no	—
242	C₄	—	2	3	no	—
243	C₄	—	2	6	no	—
244	C₄	C₂×C₁₃×#41	1	6	yes	—
245	C₄	C₁₃×2₋¹⁺⁴	1	2	yes	—
246	C₂	—	4	2	no	—
247	C₂	—	4	3	no	—
248	C₂	C₂²×C₁₃×#41	1	1	yes	characteristic
249	C₂	C₂×C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
250	C₂	C₁₃×#585	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. Q₈²×C₁₃	C₂	semidirect
2. C₁₃×#556	C₂	semidirect
3. C₁₃×#556	C₂	semidirect
4. C₁₃×#555	C₂	semidirect
5. D₄×Q₈×C₁₃	C₂	semidirect
6. C₁₃×#548	C₂	semidirect
7. C₁₃×#542	C₂	semidirect
8. C₁₃×#533	C₂	semidirect
9. C₁₃×#529	C₂	semidirect
10. C₁₃×#519	C₂	semidirect
11. C₄×C₁₃×#41	C₂	semidirect
82. #3518	C₁₃	direct (N × H)
105. Q₈²	C₂×C₁₃	semidirect
106. #556	C₂×C₁₃	semidirect
107. #556	C₂×C₁₃	semidirect
108. #555	C₂×C₁₃	semidirect
109. D₄×Q₈	C₂×C₁₃	semidirect
110. #548	C₂×C₁₃	semidirect
111. #542	C₂×C₁₃	semidirect
112. #533	C₂×C₁₃	semidirect
113. #529	C₂×C₁₃	semidirect
114. #519	C₂×C₁₃	semidirect
115. C₄×#41	C₂×C₁₃	semidirect
207. C₁₃	#3518	direct (N × H)

Maximal subgroups

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

subgroup	index
Q₈²×C₁₃	2
C₁₃×#556	2
C₁₃×#555	2
D₄×Q₈×C₁₃	2
C₁₃×#548	2
C₁₃×#542	2
C₁₃×#533	2
C₁₃×#529	2
C₁₃×#519	2
C₄×C₁₃×#41	2
#3518	13

Subgroups — C13×#3518 #13112576

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#3518 #13112576