Subgroups — C₁₃×#33957 #15720296

Order 3328.

444 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3383	C₂	1	2	yes	maximal
2	C₁₃×#2255	C₂	1	2	yes	maximal
3	C₁₃×#2253	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#2229	C₂	1	2	yes	maximal
5	C₁₃×#2229	C₂	1	2	yes	maximal
6	C₁₃×#2227	C₂	1	2	yes	maximal
7	C₁₃×#2183	C₂	1	1	yes	maximal, characteristic
8	C₁₃×#2176	C₂	1	2	yes	maximal
9	C₄×C₁₃×Q₃₂	C₂	1	1	yes	maximal, characteristic
10	C₁₃×#557	—	2	2	no	—
11	C₂×C₁₃×Q₃₂	C₂²	1	1	yes	characteristic
12	C₁₃×#500	C₂²	1	2	yes	—
13	C₁₃×#499	C₂²	1	2	yes	—
14	C₁₃×#499	C₂²	1	2	yes	—
15	C₁₃×#479	—	2	2	no	—
16	C₁₃×#479	—	2	2	no	—
17	C₁₃×#475	—	2	2	no	—
18	C₈×Q₈×C₁₃	C₂²	1	1	yes	characteristic
19	C₄×C₁₃×Q₁₆	C₂²	1	2	yes	—
20	C₄×C₁₃×Q₁₆	C₂²	1	2	yes	—
21	C₄×C₁₃×Q₁₆	C₂²	1	2	yes	—
22	C₁₃×#367	C₂²	1	2	yes	—
23	C₁₃×#367	C₂²	1	2	yes	—
24	C₁₃×#366	C₂²	1	1	yes	characteristic
25	C₁₃×#366	C₂²	1	1	yes	characteristic
26	C₁₃×#366	C₂²	1	1	yes	characteristic
27	C₁₃×#366	C₂²	1	2	yes	—
28	C₁₃×C₄⋊C₁₆	C₂²	1	2	yes	—
29	C₁₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
30	C₁₃×#358	C₂²	1	2	yes	—
31	C₁₃×#358	C₂²	1	2	yes	—
32	C₁₃×#358	C₂²	1	2	yes	—
33	C₁₃×#346	C₂²	1	2	yes	—
34	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
35	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
36	C₁₃×C₄⋊Q₈	—	2	2	no	—
37	C₁₃×C₄⋊Q₈	—	4	2	no	—
38	C₁₃×#125	—	2	2	no	—
39	C₁₃×#125	—	2	2	no	—
40	C₁₃×#125	—	4	2	no	—
41	C₁₃×#125	—	4	2	no	—
42	C₄×Q₈×C₁₃	—	2	2	no	—
43	C₄×Q₈×C₁₃	—	2	2	no	—
44	C₄×Q₈×C₁₃	—	2	2	no	—
45	C₄×Q₈×C₁₃	—	4	2	no	—
46	C₄×Q₈×C₁₃	D₄	1	1	yes	characteristic
47	C₁₃×Q₃₂	Q₈	1	4	yes	complemented
48	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
49	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
50	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
51	C₁₃×#107	—	2	2	no	—
52	C₁₃×#107	C₂³	1	2	yes	—
53	C₁₃×#107	C₂³	1	2	yes	—
54	C₁₃×#107	C₂³	1	2	yes	—
55	C₁₃×#106	—	2	2	no	—
56	C₁₃×#106	—	2	2	no	—
57	C₁₃×C₄⋊C₈	D₄	1	2	yes	—
58	C₁₃×C₄⋊C₈	D₄	1	1	yes	characteristic
59	C₁₃×#103	—	2	2	no	—
60	C₁₃×#103	—	2	2	no	—
61	C₁₃×#103	—	2	2	no	—
62	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
63	C₄×C₈×C₁₃	C₂³	1	2	yes	—
64	#33957	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
65	C₂×Q₈×C₁₃	—	2	2	no	—
66	C₂×Q₈×C₁₃	—	4	2	no	—
67	C₂×Q₈×C₁₃	D₈	1	1	yes	characteristic
68	C₁₃×Q₁₆	—	2	2	no	—
69	C₁₃×Q₁₆	C₂×Q₈	1	4	yes	—
70	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
72	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
73	C₁₃×C₄⋊C₄	—	2	2	no	—
74	C₁₃×C₄⋊C₄	—	2	2	no	—
75	C₁₃×C₄⋊C₄	—	2	2	no	—
76	C₁₃×C₄⋊C₄	—	4	2	no	—
77	C₁₃×C₄⋊C₄	—	4	2	no	—
78	C₁₃×C₄⋊C₄	—	4	2	no	—
79	C₁₃×C₄⋊C₄	—	4	2	no	—
80	C₁₃×C₄⋊C₄	—	4	2	no	—
81	C₁₃×C₄⋊C₄	—	4	2	no	—
82	C₁₃×C₄⋊C₄	—	4	2	no	—
83	C₁₃×C₄⋊C₄	—	4	2	no	—
84	C₁₃×C₄⋊C₄	D₈	1	2	yes	—
85	C₁₃×C₄⋊C₄	D₈	1	1	yes	characteristic
86	C₄²×C₁₃	—	4	2	no	—
87	C₄²×C₁₃	—	4	2	no	—
88	C₄²×C₁₃	—	4	2	no	—
89	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₁₃	C₂×D₄	1	2	yes	—
91	C₁₃×C₁₆	—	2	1	no	—
92	C₁₃×C₁₆	—	2	2	no	—
93	C₁₃×C₁₆	C₂×Q₈	1	2	yes	—
94	#3383	C₂×C₁₃	1	2	yes	—
95	#2255	C₂×C₁₃	1	2	yes	—
96	#2253	C₂×C₁₃	1	1	yes	characteristic
97	#2229	C₂×C₁₃	1	2	yes	—
98	#2229	C₂×C₁₃	1	2	yes	—
99	#2227	C₂×C₁₃	1	2	yes	—
100	#2183	C₂×C₁₃	1	1	yes	characteristic
101	#2176	C₂×C₁₃	1	2	yes	—
102	C₄×Q₃₂	C₂×C₁₃	1	1	yes	characteristic
103	Q₈×C₁₃	—	2	4	no	—
104	Q₈×C₁₃	—	4	1	no	—
105	Q₈×C₁₃	—	4	2	no	—
106	Q₈×C₁₃	—	8	4	no	—
107	C₂×C₄×C₁₃	—	4	2	no	—
108	C₂×C₄×C₁₃	—	4	2	no	—
109	C₂×C₄×C₁₃	—	4	2	no	—
110	C₂×C₄×C₁₃	—	4	2	no	—
111	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
112	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
113	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
114	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
115	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
116	C₈×C₁₃	—	2	1	no	—
117	C₈×C₁₃	—	2	2	no	—
118	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
119	C₈×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
120	#557	—	2	2	no	—
121	C₂×Q₃₂	C₂²×C₁₃	1	1	yes	characteristic
122	#500	C₂²×C₁₃	1	2	yes	—
123	#499	C₂²×C₁₃	1	2	yes	—
124	#499	C₂²×C₁₃	1	2	yes	—
125	#479	—	2	2	no	—
126	#479	—	2	2	no	—
127	#475	—	2	2	no	—
128	C₈×Q₈	C₂²×C₁₃	1	1	yes	characteristic
129	C₄×Q₁₆	C₂²×C₁₃	1	2	yes	—
130	C₄×Q₁₆	C₂²×C₁₃	1	2	yes	—
131	C₄×Q₁₆	C₂²×C₁₃	1	2	yes	—
132	#367	C₂²×C₁₃	1	2	yes	—
133	#367	C₂²×C₁₃	1	2	yes	—
134	#366	C₂²×C₁₃	1	1	yes	characteristic
135	#366	C₂²×C₁₃	1	1	yes	characteristic
136	#366	C₂²×C₁₃	1	2	yes	—
137	#366	C₂²×C₁₃	1	1	yes	characteristic
138	C₄⋊C₁₆	C₂²×C₁₃	1	2	yes	—
139	C₄⋊C₁₆	C₂²×C₁₃	1	1	yes	characteristic
140	#358	C₂²×C₁₃	1	2	yes	—
141	#358	C₂²×C₁₃	1	2	yes	—
142	#358	C₂²×C₁₃	1	2	yes	—
143	#346	C₂²×C₁₃	1	2	yes	—
144	C₄×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
145	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
146	C₄×C₁₃	—	2	1	no	—
147	C₄×C₁₃	—	2	2	no	—
148	C₄×C₁₃	—	2	1	no	—
149	C₄×C₁₃	—	2	2	no	—
150	C₄×C₁₃	—	4	4	no	—
151	C₄×C₁₃	—	8	2	no	—
152	C₄×C₁₃	—	8	2	no	—
153	C₄×C₁₃	—	8	2	no	—
154	C₄×C₁₃	D₄×Q₈	1	1	yes	characteristic
155	C₄×C₁₃	#578	1	1	yes	characteristic
156	C₂×Q₁₆	C₂³×C₁₃	1	2	yes	—
157	C₄⋊Q₈	—	2	2	no	—
158	C₄⋊Q₈	—	4	2	no	—
159	#125	—	2	2	no	—
160	#125	—	2	2	no	—
161	#125	—	4	2	no	—
162	#125	—	4	2	no	—
163	C₄×Q₈	—	2	2	no	—
164	C₄×Q₈	—	2	2	no	—
165	C₄×Q₈	—	2	2	no	—
166	C₄×Q₈	—	4	2	no	—
167	C₄×Q₈	D₄×C₁₃	1	1	yes	characteristic
168	Q₃₂	Q₈×C₁₃	1	4	yes	complemented
169	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
170	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
171	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
172	#107	—	2	2	no	—
173	#107	C₂³×C₁₃	1	2	yes	—
174	#107	C₂³×C₁₃	1	2	yes	—
175	#107	C₂³×C₁₃	1	2	yes	—
176	#106	—	2	2	no	—
177	#106	—	2	2	no	—
178	C₄⋊C₈	D₄×C₁₃	1	2	yes	—
179	C₄⋊C₈	D₄×C₁₃	1	1	yes	characteristic
180	#103	—	2	2	no	—
181	#103	—	2	2	no	—
182	#103	—	2	2	no	—
183	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
184	C₄×C₈	C₂³×C₁₃	1	2	yes	—
185	C₂×C₁₃	Q₈×D₈	1	1	yes	characteristic
186	C₂×C₁₃	#3414	1	1	yes	characteristic
187	C₂×C₁₃	#3417	1	1	yes	characteristic
188	C₂×Q₈	—	2	2	no	—
189	C₂×Q₈	—	4	2	no	—
190	C₂×Q₈	C₁₃×D₈	1	1	yes	characteristic
191	Q₁₆	—	2	2	no	—
192	Q₁₆	C₂×Q₈×C₁₃	1	4	yes	—
193	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
194	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
195	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
196	C₄⋊C₄	—	2	2	no	—
197	C₄⋊C₄	—	2	2	no	—
198	C₄⋊C₄	—	2	2	no	—
199	C₄⋊C₄	—	4	2	no	—
200	C₄⋊C₄	—	4	2	no	—
201	C₄⋊C₄	—	4	2	no	—
202	C₄⋊C₄	—	4	2	no	—
203	C₄⋊C₄	—	4	2	no	—
204	C₄⋊C₄	—	4	2	no	—
205	C₄⋊C₄	—	4	2	no	—
206	C₄⋊C₄	—	4	2	no	—
207	C₄⋊C₄	C₁₃×D₈	1	2	yes	—
208	C₄⋊C₄	C₁₃×D₈	1	1	yes	characteristic
209	C₄²	—	4	2	no	—
210	C₄²	—	4	2	no	—
211	C₄²	—	4	2	no	—
212	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
213	C₄²	C₂×D₄×C₁₃	1	2	yes	—
214	C₁₆	—	2	1	no	—
215	C₁₆	—	2	2	no	—
216	C₁₆	C₂×Q₈×C₁₃	1	2	yes	—
217	C₁₃	#33957	1	1	yes	characteristic, direct product factor, Sylow
218	Q₈	—	2	4	no	—
219	Q₈	—	4	1	no	—
220	Q₈	—	4	2	no	—
221	Q₈	—	8	4	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	—	4	2	no	—
224	C₂×C₄	—	4	2	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
227	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
228	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
229	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
230	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
231	C₈	—	2	1	no	—
232	C₈	—	2	2	no	—
233	C₈	C₂²×Q₈×C₁₃	1	1	yes	characteristic
234	C₈	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
235	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
236	C₄	—	2	1	no	—
237	C₄	—	2	2	no	—
238	C₄	—	2	1	no	—
239	C₄	—	2	2	no	—
240	C₄	—	4	4	no	—
241	C₄	—	8	2	no	—
242	C₄	—	8	2	no	—
243	C₄	—	8	2	no	—
244	C₄	D₄×Q₈×C₁₃	1	1	yes	characteristic
245	C₄	C₁₃×#578	1	1	yes	characteristic
246	C₂	Q₈×C₁₃×D₈	1	1	yes	characteristic
247	C₂	C₁₃×#3414	1	1	yes	characteristic
248	C₂	C₁₃×#3417	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₁₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₄×Q₈×C₁₃ ◁ C₈×Q₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
47. C₁₃×Q₃₂	Q₈	semidirect
64. #33957	C₁₃	direct (N × H)
168. Q₃₂	Q₈×C₁₃	semidirect
217. C₁₃	#33957	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×#3383	2
C₁₃×#2255	2
C₁₃×#2253	2
C₁₃×#2229	2
C₁₃×#2227	2
C₁₃×#2183	2
C₁₃×#2176	2
C₄×C₁₃×Q₃₂	2
#33957	13

Subgroups — C13×#33957 #15720296

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33957 #15720296