Subgroups — C₁₃×#3084 #13112142

Order 1664.

376 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#544	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₃×#520	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#479	C₂	1	2	yes	maximal, complemented
4	C₁₃×#478	C₂	1	2	yes	maximal, complemented
5	C₁₃×#476	C₂	1	2	yes	maximal, complemented
6	C₁₃×#475	C₂	1	2	yes	maximal, complemented
7	C₁₃×#428	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×#106	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₃×#424	C₂	1	1	yes	maximal, characteristic, complemented
10	C₁₃×#417	C₂	1	1	yes	maximal, characteristic
11	C₁₃×#416	C₂	1	1	yes	maximal, characteristic
12	C₂×C₁₃×#41	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₁₃	C₂²	1	1	yes	characteristic
15	C₁₃×C₄⋊Q₈	—	2	1	no	—
16	C₁₃×C₄⋊Q₈	C₂²	1	2	yes	—
17	C₁₃×#125	—	2	1	no	—
18	C₁₃×#125	C₂²	1	2	yes	—
19	C₁₃×C₂²⋊Q₈	—	2	2	no	—
20	C₄×Q₈×C₁₃	C₂²	1	2	yes	—
21	C₄×D₄×C₁₃	—	2	2	no	—
22	C₄×D₄×C₁₃	C₂²	1	2	yes	—
23	C₁₃×#117	—	2	1	no	—
24	C₁₃×#117	C₂²	1	1	yes	characteristic
25	C₁₃×#117	C₂²	1	1	yes	characteristic
26	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
27	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
28	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
29	C₁₃×#107	C₂²	1	2	yes	—
30	C₁₃×#106	C₂²	1	2	yes	—
31	C₁₃×#106	C₂²	1	2	yes	—
32	C₁₃×#106	C₂²	1	2	yes	—
33	C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
34	C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
35	C₁₃×#103	C₂²	1	1	yes	characteristic
36	C₁₃×#103	C₂²	1	1	yes	characteristic
37	C₁₃×#103	C₂²	1	2	yes	—
38	C₁₃×#102	C₂²	1	1	yes	characteristic
39	C₁₃×#102	C₂²	1	1	yes	characteristic
40	C₁₃×#102	C₂²	1	2	yes	—
41	C₁₃×#41	—	2	2	no	—
42	C₁₃×#41	Q₈	1	4	yes	complemented
43	C₂×Q₈×C₁₃	—	2	2	no	—
44	C₂×Q₈×C₁₃	C₂³	1	1	yes	characteristic
45	C₂×D₄×C₁₃	—	2	1	no	—
46	C₂×D₄×C₁₃	C₂³	1	1	yes	characteristic
47	C₂²×C₄×C₁₃	—	2	1	no	—
48	C₂²×C₄×C₁₃	—	2	1	no	—
49	C₂²×C₄×C₁₃	—	2	1	no	—
50	C₂²×C₄×C₁₃	—	2	2	no	—
51	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
52	C₁₃×M₁₆	—	2	2	no	—
53	C₂×C₈×C₁₃	—	2	2	no	—
54	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
55	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
56	C₂×C₈×C₁₃	C₂³	1	2	yes	—
57	C₁₃×C₄⋊C₄	—	2	1	no	—
58	C₁₃×C₄⋊C₄	—	2	2	no	—
59	C₁₃×C₄⋊C₄	—	2	1	no	—
60	C₁₃×C₄⋊C₄	—	2	2	no	—
61	C₁₃×C₄⋊C₄	—	2	2	no	—
62	C₁₃×C₄⋊C₄	—	2	2	no	—
63	C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
64	C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
65	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
66	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
67	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
68	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
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	—	2	1	no	—
74	C₄²×C₁₃	C₂³	1	2	yes	—
75	#3084	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
76	C₂³×C₁₃	—	2	1	no	—
77	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
78	Q₈×C₁₃	—	2	1	no	—
79	Q₈×C₁₃	—	8	2	no	—
80	Q₈×C₁₃	C₂×Q₈	1	2	yes	—
81	D₄×C₁₃	—	2	4	no	—
82	D₄×C₁₃	—	2	1	no	—
83	D₄×C₁₃	C₂×Q₈	1	2	yes	—
84	C₂×C₄×C₁₃	—	2	1	no	—
85	C₂×C₄×C₁₃	—	2	4	no	—
86	C₂×C₄×C₁₃	—	2	1	no	—
87	C₂×C₄×C₁₃	—	2	2	no	—
88	C₂×C₄×C₁₃	—	2	1	no	—
89	C₂×C₄×C₁₃	—	2	2	no	—
90	C₂×C₄×C₁₃	—	8	2	no	—
91	C₂×C₄×C₁₃	—	8	1	no	—
92	C₂×C₄×C₁₃	—	8	1	no	—
93	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
94	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
95	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
96	C₂×C₄×C₁₃	C₂×Q₈	1	2	yes	—
97	C₂×C₄×C₁₃	#41	1	2	yes	—
98	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
99	C₈×C₁₃	—	2	1	no	—
100	C₈×C₁₃	—	2	1	no	—
101	C₈×C₁₃	—	2	2	no	—
102	#544	C₂×C₁₃	1	1	yes	characteristic, complemented
103	#520	C₂×C₁₃	1	1	yes	characteristic
104	#479	C₂×C₁₃	1	2	yes	complemented
105	#478	C₂×C₁₃	1	2	yes	complemented
106	#476	C₂×C₁₃	1	2	yes	complemented
107	#475	C₂×C₁₃	1	2	yes	complemented
108	#428	C₂×C₁₃	1	1	yes	characteristic, complemented
109	C₂×#106	C₂×C₁₃	1	1	yes	characteristic, complemented
110	#424	C₂×C₁₃	1	1	yes	characteristic, complemented
111	#417	C₂×C₁₃	1	1	yes	characteristic
112	#416	C₂×C₁₃	1	1	yes	characteristic
113	C₂²×C₁₃	—	2	1	no	—
114	C₂²×C₁₃	—	2	1	no	—
115	C₂²×C₁₃	—	2	2	no	—
116	C₂²×C₁₃	—	4	1	no	—
117	C₂²×C₁₃	—	4	1	no	—
118	C₂²×C₁₃	C₂²⋊Q₈	1	2	yes	—
119	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
120	C₄×C₁₃	—	2	2	no	—
121	C₄×C₁₃	—	2	2	no	—
122	C₄×C₁₃	—	2	2	no	—
123	C₄×C₁₃	—	4	1	no	—
124	C₄×C₁₃	—	4	2	no	—
125	C₄×C₁₃	—	4	2	no	—
126	C₄×C₁₃	—	4	1	no	—
127	C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
128	C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
129	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
130	C₂×#41	C₂²×C₁₃	1	1	yes	characteristic
131	C₂×M₁₆	C₂²×C₁₃	1	1	yes	characteristic
132	C₂²×C₈	C₂²×C₁₃	1	1	yes	characteristic
133	C₄⋊Q₈	—	2	1	no	—
134	C₄⋊Q₈	C₂²×C₁₃	1	2	yes	—
135	#125	—	2	1	no	—
136	#125	C₂²×C₁₃	1	2	yes	—
137	C₂²⋊Q₈	—	2	2	no	—
138	C₄×Q₈	C₂²×C₁₃	1	2	yes	—
139	C₄×D₄	—	2	2	no	—
140	C₄×D₄	C₂²×C₁₃	1	2	yes	—
141	#117	—	2	1	no	—
142	#117	C₂²×C₁₃	1	1	yes	characteristic
143	#117	C₂²×C₁₃	1	1	yes	characteristic
144	C₂×C₄⋊C₄	—	2	1	no	—
145	C₂×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
146	C₂×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
147	#107	C₂²×C₁₃	1	2	yes	—
148	#106	C₂²×C₁₃	1	2	yes	—
149	#106	C₂²×C₁₃	1	2	yes	—
150	#106	C₂²×C₁₃	1	2	yes	—
151	C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
152	C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
153	#103	C₂²×C₁₃	1	1	yes	characteristic
154	#103	C₂²×C₁₃	1	1	yes	characteristic
155	#103	C₂²×C₁₃	1	2	yes	—
156	#102	C₂²×C₁₃	1	1	yes	characteristic
157	#102	C₂²×C₁₃	1	1	yes	characteristic
158	#102	C₂²×C₁₃	1	2	yes	—
159	C₂×C₁₃	—	2	2	no	—
160	C₂×C₁₃	—	4	2	no	—
161	C₂×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
162	C₂×C₁₃	#577	1	1	yes	characteristic
163	C₂×C₁₃	#577	1	1	yes	characteristic
164	#41	—	2	2	no	—
165	#41	Q₈×C₁₃	1	4	yes	complemented
166	C₂×Q₈	—	2	2	no	—
167	C₂×Q₈	C₂³×C₁₃	1	1	yes	characteristic
168	C₂×D₄	—	2	1	no	—
169	C₂×D₄	C₂³×C₁₃	1	1	yes	characteristic
170	C₂²×C₄	—	2	1	no	—
171	C₂²×C₄	—	2	1	no	—
172	C₂²×C₄	—	2	1	no	—
173	C₂²×C₄	—	2	2	no	—
174	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
175	M₁₆	—	2	2	no	—
176	C₂×C₈	—	2	2	no	—
177	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
178	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
179	C₂×C₈	C₂³×C₁₃	1	2	yes	—
180	C₄⋊C₄	—	2	1	no	—
181	C₄⋊C₄	—	2	2	no	—
182	C₄⋊C₄	—	2	1	no	—
183	C₄⋊C₄	—	2	2	no	—
184	C₄⋊C₄	—	2	2	no	—
185	C₄⋊C₄	—	2	2	no	—
186	C₄⋊C₄	D₄×C₁₃	1	1	yes	characteristic
187	C₄⋊C₄	D₄×C₁₃	1	1	yes	characteristic
188	C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
189	C₄⋊C₄	C₂³×C₁₃	1	1	yes	characteristic
190	C₄⋊C₄	C₂³×C₁₃	1	1	yes	characteristic
191	C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
192	C₂²⋊C₄	—	2	1	no	—
193	C₂²⋊C₄	—	2	2	no	—
194	C₂²⋊C₄	D₄×C₁₃	1	2	yes	—
195	C₄²	—	2	2	no	—
196	C₄²	—	2	1	no	—
197	C₄²	C₂³×C₁₃	1	2	yes	—
198	C₁₃	#3084	1	1	yes	characteristic, direct product factor, Sylow
199	C₂³	—	2	1	no	—
200	C₂³	C₂×D₄×C₁₃	1	1	yes	characteristic
201	Q₈	—	2	1	no	—
202	Q₈	—	8	2	no	—
203	Q₈	C₂×Q₈×C₁₃	1	2	yes	—
204	D₄	—	2	4	no	—
205	D₄	—	2	1	no	—
206	D₄	C₂×Q₈×C₁₃	1	2	yes	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	8	2	no	—
214	C₂×C₄	—	8	1	no	—
215	C₂×C₄	—	8	1	no	—
216	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
217	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
218	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
219	C₂×C₄	C₂×Q₈×C₁₃	1	2	yes	—
220	C₂×C₄	C₁₃×#41	1	2	yes	—
221	C₂×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
222	C₈	—	2	1	no	—
223	C₈	—	2	1	no	—
224	C₈	—	2	2	no	—
225	C₂²	—	2	1	no	—
226	C₂²	—	2	1	no	—
227	C₂²	—	2	2	no	—
228	C₂²	—	4	1	no	—
229	C₂²	—	4	1	no	—
230	C₂²	C₁₃×C₂²⋊Q₈	1	2	yes	—
231	C₂²	C₂²×D₄×C₁₃	1	1	yes	characteristic
232	C₄	—	2	2	no	—
233	C₄	—	2	2	no	—
234	C₄	—	2	2	no	—
235	C₄	—	4	1	no	—
236	C₄	—	4	2	no	—
237	C₄	—	4	2	no	—
238	C₄	—	4	1	no	—
239	C₄	C₁₃×C₂²⋊Q₈	1	2	yes	—
240	C₄	C₂²×Q₈×C₁₃	1	1	yes	characteristic
241	C₄	C₂×C₁₃×#41	1	1	yes	characteristic
242	C₂	—	2	2	no	—
243	C₂	—	4	2	no	—
244	C₂	C₂×C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
245	C₂	C₁₃×#577	1	1	yes	characteristic
246	C₂	C₁₃×#577	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₃
prime radical	C₂×C₁₃×#41	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₁₃×#117 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₃×#544	C₂	semidirect
3. C₁₃×#479	C₂	semidirect
4. C₁₃×#478	C₂	semidirect
5. C₁₃×#476	C₂	semidirect
6. C₁₃×#475	C₂	semidirect
7. C₁₃×#428	C₂	semidirect
8. C₂×C₁₃×#106	C₂	semidirect
9. C₁₃×#424	C₂	semidirect
42. C₁₃×#41	Q₈	semidirect
75. #3084	C₁₃	direct (N × H)
102. #544	C₂×C₁₃	semidirect
104. #479	C₂×C₁₃	semidirect
105. #478	C₂×C₁₃	semidirect
106. #476	C₂×C₁₃	semidirect
107. #475	C₂×C₁₃	semidirect
108. #428	C₂×C₁₃	semidirect
109. C₂×#106	C₂×C₁₃	semidirect
110. #424	C₂×C₁₃	semidirect
165. #41	Q₈×C₁₃	semidirect
198. C₁₃	#3084	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₃×#544	2
C₁₃×#520	2
C₁₃×#479	2
C₁₃×#478	2
C₁₃×#476	2
C₁₃×#475	2
C₁₃×#428	2
C₂×C₁₃×#106	2
C₁₃×#424	2
C₁₃×#417	2
C₁₃×#416	2
#3084	13

Subgroups — C13×#3084 #13112142

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#3084 #13112142