Subgroups — C₁₃×#3364 #13112422

Order 1664.

336 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#557	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₃×#556	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#488	C₂	1	2	yes	maximal
4	C₁₃×#487	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#479	C₂	1	2	yes	maximal, complemented
6	C₁₃×#474	C₂	1	1	yes	maximal, characteristic
7	C₁₃×#464	C₂	1	2	yes	maximal
8	C₁₃×#462	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈×Q₈×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×C₁₃×Q₁₆	C₂	1	2	yes	maximal, complemented
11	C₄×C₁₃×SD₁₆	C₂	1	1	yes	maximal, characteristic
12	C₂×C₁₃×Q₁₆	C₂²	1	2	yes	—
13	C₂×C₁₃×SD₁₆	C₂²	1	1	yes	characteristic
14	C₁₃×C₄⋊Q₈	—	2	1	no	—
15	C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₁₃×#126	—	2	2	no	—
18	C₁₃×#125	—	2	2	no	—
19	C₁₃×#125	C₂²	1	2	yes	—
20	C₁₃×#124	C₂²	1	2	yes	—
21	C₁₃×C₂²⋊Q₈	—	2	1	no	—
22	C₄×Q₈×C₁₃	—	2	1	no	—
23	C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
24	C₄×Q₈×C₁₃	C₂²	1	2	yes	—
25	C₄×Q₈×C₁₃	C₂²	1	2	yes	—
26	C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
27	C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
28	C₁₃×#117	—	2	1	no	—
29	C₁₃×#107	C₂²	1	2	yes	—
30	C₁₃×#106	C₂²	1	1	yes	characteristic
31	C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
32	C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
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×#102	C₂²	1	1	yes	characteristic
38	C₁₃×#102	C₂²	1	2	yes	—
39	C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
40	C₄×C₈×C₁₃	C₂²	1	2	yes	—
41	C₂×Q₈×C₁₃	—	2	1	no	—
42	C₂×Q₈×C₁₃	D₄	1	1	yes	characteristic
43	C₂×Q₈×C₁₃	C₂³	1	2	yes	—
44	C₂×Q₈×C₁₃	C₂³	1	1	yes	characteristic
45	C₂×D₄×C₁₃	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₁₃	—	2	1	no	—
47	C₁₃×Q₁₆	—	2	4	no	—
48	C₁₃×SD₁₆	—	2	2	no	—
49	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
50	C₂×C₈×C₁₃	C₂³	1	2	yes	—
51	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
52	C₁₃×C₄⋊C₄	—	2	1	no	—
53	C₁₃×C₄⋊C₄	—	2	2	no	—
54	C₁₃×C₄⋊C₄	—	2	2	no	—
55	C₁₃×C₄⋊C₄	—	2	1	no	—
56	C₁₃×C₄⋊C₄	—	2	2	no	—
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	D₄	1	1	yes	characteristic
61	C₁₃×C₄⋊C₄	D₄	1	2	yes	—
62	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
63	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
64	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
65	C₁₃×C₂²⋊C₄	—	2	2	no	—
66	C₁₃×C₂²⋊C₄	—	2	1	no	—
67	C₁₃×C₂²⋊C₄	—	2	2	no	—
68	C₄²×C₁₃	—	2	2	no	—
69	C₄²×C₁₃	—	2	2	no	—
70	C₄²×C₁₃	—	2	1	no	—
71	C₄²×C₁₃	C₂³	1	1	yes	characteristic
72	C₄²×C₁₃	C₂³	1	2	yes	—
73	#3364	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
74	C₂³×C₁₃	—	2	1	no	—
75	Q₈×C₁₃	—	2	2	no	—
76	Q₈×C₁₃	—	2	1	no	—
77	Q₈×C₁₃	—	2	2	no	—
78	Q₈×C₁₃	—	2	2	no	—
79	Q₈×C₁₃	—	4	2	no	—
80	Q₈×C₁₃	#41	1	2	yes	—
81	D₄×C₁₃	—	2	1	no	—
82	D₄×C₁₃	—	2	1	no	—
83	C₂×C₄×C₁₃	—	2	1	no	—
84	C₂×C₄×C₁₃	—	2	1	no	—
85	C₂×C₄×C₁₃	—	2	2	no	—
86	C₂×C₄×C₁₃	—	2	1	no	—
87	C₂×C₄×C₁₃	—	2	2	no	—
88	C₂×C₄×C₁₃	—	4	2	no	—
89	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
90	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
91	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
92	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
93	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
94	C₈×C₁₃	—	2	1	no	—
95	C₈×C₁₃	—	2	2	no	—
96	C₈×C₁₃	—	2	1	no	—
97	#557	C₂×C₁₃	1	1	yes	characteristic, complemented
98	#556	C₂×C₁₃	1	1	yes	characteristic
99	#488	C₂×C₁₃	1	2	yes	—
100	#487	C₂×C₁₃	1	1	yes	characteristic, complemented
101	#479	C₂×C₁₃	1	2	yes	complemented
102	#474	C₂×C₁₃	1	1	yes	characteristic
103	#464	C₂×C₁₃	1	2	yes	—
104	#462	C₂×C₁₃	1	1	yes	characteristic, complemented
105	C₈×Q₈	C₂×C₁₃	1	1	yes	characteristic, complemented
106	C₄×Q₁₆	C₂×C₁₃	1	2	yes	complemented
107	C₄×SD₁₆	C₂×C₁₃	1	1	yes	characteristic
108	C₂²×C₁₃	—	4	1	no	—
109	C₂²×C₁₃	—	4	1	no	—
110	C₂²×C₁₃	—	4	1	no	—
111	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
112	C₄×C₁₃	—	2	2	no	—
113	C₄×C₁₃	—	2	2	no	—
114	C₄×C₁₃	—	2	2	no	—
115	C₄×C₁₃	—	2	1	no	—
116	C₄×C₁₃	—	4	2	no	—
117	C₄×C₁₃	—	4	1	no	—
118	C₄×C₁₃	—	4	2	no	—
119	C₄×C₁₃	—	4	1	no	—
120	C₄×C₁₃	#135	1	2	yes	—
121	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
122	C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
123	C₂×Q₁₆	C₂²×C₁₃	1	2	yes	—
124	C₂×SD₁₆	C₂²×C₁₃	1	1	yes	characteristic
125	C₄⋊Q₈	—	2	1	no	—
126	C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
127	C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
128	#126	—	2	2	no	—
129	#125	—	2	2	no	—
130	#125	C₂²×C₁₃	1	2	yes	—
131	#124	C₂²×C₁₃	1	2	yes	—
132	C₂²⋊Q₈	—	2	1	no	—
133	C₄×Q₈	—	2	1	no	—
134	C₄×Q₈	C₂²×C₁₃	1	2	yes	—
135	C₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic
136	C₄×Q₈	C₂²×C₁₃	1	2	yes	—
137	C₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic
138	C₄×D₄	C₂²×C₁₃	1	1	yes	characteristic
139	#117	—	2	1	no	—
140	#107	C₂²×C₁₃	1	2	yes	—
141	#106	C₂²×C₁₃	1	1	yes	characteristic
142	C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
143	C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
144	#103	C₂²×C₁₃	1	1	yes	characteristic
145	#103	C₂²×C₁₃	1	2	yes	—
146	#103	C₂²×C₁₃	1	2	yes	—
147	#103	C₂²×C₁₃	1	2	yes	—
148	#102	C₂²×C₁₃	1	1	yes	characteristic
149	#102	C₂²×C₁₃	1	2	yes	—
150	C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
151	C₄×C₈	C₂²×C₁₃	1	2	yes	—
152	C₂×C₁₃	—	8	1	no	—
153	C₂×C₁₃	#548	1	1	yes	characteristic
154	C₂×C₁₃	C₂×#135	1	1	yes	characteristic
155	C₂×C₁₃	#578	1	1	yes	characteristic
156	C₂×Q₈	—	2	1	no	—
157	C₂×Q₈	D₄×C₁₃	1	1	yes	characteristic
158	C₂×Q₈	C₂³×C₁₃	1	2	yes	—
159	C₂×Q₈	C₂³×C₁₃	1	1	yes	characteristic
160	C₂×D₄	C₂³×C₁₃	1	1	yes	characteristic
161	C₂²×C₄	—	2	1	no	—
162	Q₁₆	—	2	4	no	—
163	SD₁₆	—	2	2	no	—
164	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
165	C₂×C₈	C₂³×C₁₃	1	2	yes	—
166	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
167	C₄⋊C₄	—	2	1	no	—
168	C₄⋊C₄	—	2	2	no	—
169	C₄⋊C₄	—	2	2	no	—
170	C₄⋊C₄	—	2	1	no	—
171	C₄⋊C₄	—	2	2	no	—
172	C₄⋊C₄	—	2	1	no	—
173	C₄⋊C₄	—	2	2	no	—
174	C₄⋊C₄	—	2	1	no	—
175	C₄⋊C₄	D₄×C₁₃	1	1	yes	characteristic
176	C₄⋊C₄	D₄×C₁₃	1	2	yes	—
177	C₄⋊C₄	C₂³×C₁₃	1	1	yes	characteristic
178	C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
179	C₄⋊C₄	C₂³×C₁₃	1	1	yes	characteristic
180	C₂²⋊C₄	—	2	2	no	—
181	C₂²⋊C₄	—	2	1	no	—
182	C₂²⋊C₄	—	2	2	no	—
183	C₄²	—	2	2	no	—
184	C₄²	—	2	2	no	—
185	C₄²	—	2	1	no	—
186	C₄²	C₂³×C₁₃	1	1	yes	characteristic
187	C₄²	C₂³×C₁₃	1	2	yes	—
188	C₁₃	#3364	1	1	yes	characteristic, direct product factor, Sylow
189	C₂³	—	2	1	no	—
190	Q₈	—	2	2	no	—
191	Q₈	—	2	1	no	—
192	Q₈	—	2	2	no	—
193	Q₈	—	2	2	no	—
194	Q₈	—	4	2	no	—
195	Q₈	C₁₃×#41	1	2	yes	—
196	D₄	—	2	1	no	—
197	D₄	—	2	1	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	4	2	no	—
204	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
205	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
206	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
207	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
208	C₂×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
209	C₈	—	2	1	no	—
210	C₈	—	2	2	no	—
211	C₈	—	2	1	no	—
212	C₂²	—	4	1	no	—
213	C₂²	—	4	1	no	—
214	C₂²	—	4	1	no	—
215	C₂²	C₂²×D₄×C₁₃	1	1	yes	characteristic
216	C₄	—	2	2	no	—
217	C₄	—	2	2	no	—
218	C₄	—	2	2	no	—
219	C₄	—	2	1	no	—
220	C₄	—	4	2	no	—
221	C₄	—	4	1	no	—
222	C₄	—	4	2	no	—
223	C₄	—	4	1	no	—
224	C₄	C₁₃×#135	1	2	yes	—
225	C₄	C₂×C₁₃×#41	1	1	yes	characteristic
226	C₄	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
227	C₂	—	8	1	no	—
228	C₂	C₁₃×#548	1	1	yes	characteristic
229	C₂	C₂×C₁₃×#135	1	1	yes	characteristic
230	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	C₂×D₄×C₁₃	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₄×Q₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₃×#557	C₂	semidirect
4. C₁₃×#487	C₂	semidirect
5. C₁₃×#479	C₂	semidirect
8. C₁₃×#462	C₂	semidirect
9. C₈×Q₈×C₁₃	C₂	semidirect
10. C₄×C₁₃×Q₁₆	C₂	semidirect
73. #3364	C₁₃	direct (N × H)
97. #557	C₂×C₁₃	semidirect
100. #487	C₂×C₁₃	semidirect
101. #479	C₂×C₁₃	semidirect
104. #462	C₂×C₁₃	semidirect
105. C₈×Q₈	C₂×C₁₃	semidirect
106. C₄×Q₁₆	C₂×C₁₃	semidirect
188. C₁₃	#3364	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×#557	2
C₁₃×#556	2
C₁₃×#488	2
C₁₃×#487	2
C₁₃×#479	2
C₁₃×#474	2
C₁₃×#464	2
C₁₃×#462	2
C₈×Q₈×C₁₃	2
C₄×C₁₃×Q₁₆	2
C₄×C₁₃×SD₁₆	2
#3364	13

Subgroups — C13×#3364 #13112422

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#3364 #13112422