Subgroups — C₁₃×#33708 #15720047

Order 3328.

590 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#499	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₃×#3147	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2245	C₂	1	2	yes	maximal, complemented
4	C₁₃×#2244	C₂	1	2	yes	maximal, complemented
5	C₁₃×#2242	C₂	1	4	yes	maximal, complemented
6	C₁₃×#2139	C₂	1	4	yes	maximal, complemented
7	C₄×C₁₃×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×#135	—	2	1	no	—
9	C₂×C₁₃×#135	C₂²	1	2	yes	—
10	C₁₃×#532	—	2	1	no	—
11	C₂×C₁₃×#125	—	2	1	no	—
12	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
13	C₁₃×#499	—	2	1	no	—
14	C₁₃×#499	—	2	1	no	—
15	C₁₃×#499	C₂²	1	2	yes	—
16	C₁₃×#499	C₂²	1	2	yes	—
17	C₁₃×#495	—	2	1	no	—
18	C₁₃×#495	C₂²	1	2	yes	—
19	C₁₃×#494	C₂²	1	1	yes	characteristic
20	C₁₃×#493	C₂²	1	1	yes	characteristic
21	C₁₃×#492	—	2	1	no	—
22	C₂×C₁₃×#107	C₂²	1	2	yes	—
23	C₂×C₁₃×#106	—	2	1	no	—
24	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
25	C₁₃×#358	C₂²	1	8	yes	—
26	C₁₃×#357	C₂²	1	8	yes	—
27	C₁₃×#346	C₂²	1	2	yes	—
28	C₄×C₁₃×C₁₆	C₂²	1	2	yes	—
29	C₂×C₁₃×#41	—	2	2	no	—
30	C₁₃×#135	—	2	8	no	—
31	C₁₃×#135	—	4	4	no	—
32	C₂×C₁₃×Q₁₆	—	2	1	no	—
33	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
34	C₂×C₁₃×SD₁₆	—	2	2	no	—
35	C₂×C₁₃×SD₁₆	—	2	2	no	—
36	C₂×C₁₃×D₈	—	2	1	no	—
37	C₂×C₁₃×D₈	C₂³	1	2	yes	—
38	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
40	C₁₃×C₄⋊Q₈	—	2	1	no	—
41	C₁₃×#127	—	2	1	no	—
42	C₁₃×#125	—	2	2	no	—
43	C₁₃×#125	—	2	2	no	—
44	C₁₃×#125	—	4	2	no	—
45	C₁₃×#124	—	2	2	no	—
46	C₁₃×#121	—	4	2	no	—
47	C₄×D₄×C₁₃	—	4	2	no	—
48	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
49	C₂×C₁₃×C₄⋊C₄	—	4	2	no	—
50	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
51	C₁₃×M₃₂	—	2	4	no	—
52	C₂×C₁₃×C₁₆	C₂³	1	4	yes	—
53	C₁₃×#107	—	2	2	no	—
54	C₁₃×#107	C₂³	1	4	yes	—
55	C₁₃×#106	—	2	2	no	—
56	C₁₃×#106	—	2	2	no	—
57	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
58	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
59	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
60	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
61	#33708	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
62	C₁₃×#41	—	4	4	no	—
63	C₁₃×#41	—	4	4	no	—
64	C₂×Q₈×C₁₃	—	2	2	no	—
65	C₂×D₄×C₁₃	—	2	2	no	—
66	C₂×D₄×C₁₃	—	4	2	no	—
67	C₂×D₄×C₁₃	—	4	2	no	—
68	C₂²×C₄×C₁₃	—	4	2	no	—
69	C₂²×C₄×C₁₃	—	4	2	no	—
70	C₂²×C₄×C₁₃	D₈	1	2	yes	—
71	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
72	C₁₃×Q₁₆	—	2	4	no	—
73	C₁₃×Q₁₆	—	4	2	no	—
74	C₁₃×SD₁₆	—	4	4	no	—
75	C₁₃×SD₁₆	—	4	4	no	—
76	C₁₃×D₈	—	2	4	no	—
77	C₁₃×D₈	—	4	2	no	—
78	C₂×C₈×C₁₃	SD₁₆	1	4	yes	—
79	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
80	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₁₃	#41	1	4	yes	—
83	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
84	C₁₃×C₄⋊C₄	—	2	4	no	—
85	C₁₃×C₄⋊C₄	—	4	2	no	—
86	C₁₃×C₄⋊C₄	—	4	4	no	—
87	C₁₃×C₄⋊C₄	—	4	4	no	—
88	C₁₃×C₄⋊C₄	—	4	2	no	—
89	C₁₃×C₂²⋊C₄	—	4	4	no	—
90	C₄²×C₁₃	D₈	1	2	yes	—
91	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
92	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
93	C₁₃×C₁₆	—	2	4	no	—
94	C₂×#499	C₂×C₁₃	1	1	yes	characteristic, complemented
95	#3147	C₂×C₁₃	1	1	yes	characteristic
96	#2245	C₂×C₁₃	1	2	yes	complemented
97	#2244	C₂×C₁₃	1	2	yes	complemented
98	#2242	C₂×C₁₃	1	4	yes	complemented
99	#2139	C₂×C₁₃	1	4	yes	complemented
100	C₄×M₃₂	C₂×C₁₃	1	1	yes	characteristic, complemented
101	C₂³×C₁₃	—	4	2	no	—
102	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
103	Q₈×C₁₃	—	4	2	no	—
104	Q₈×C₁₃	—	4	2	no	—
105	D₄×C₁₃	—	4	2	no	—
106	D₄×C₁₃	—	4	2	no	—
107	D₄×C₁₃	—	8	4	no	—
108	D₄×C₁₃	—	8	4	no	—
109	C₂×C₄×C₁₃	—	4	2	no	—
110	C₂×C₄×C₁₃	—	4	2	no	—
111	C₂×C₄×C₁₃	—	4	4	no	—
112	C₂×C₄×C₁₃	—	8	4	no	—
113	C₂×C₄×C₁₃	—	8	4	no	—
114	C₂×C₄×C₁₃	#124	1	2	yes	—
115	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
116	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
117	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
118	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
119	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
120	C₈×C₁₃	#124	1	2	yes	—
121	C₈×C₁₃	C₂×SD₁₆	1	2	yes	—
122	C₈×C₁₃	C₂×SD₁₆	1	2	yes	—
123	C₈×C₁₃	C₂×#41	1	2	yes	—
124	C₂×#135	—	2	1	no	—
125	C₂×#135	C₂²×C₁₃	1	2	yes	—
126	#532	—	2	1	no	—
127	C₂×#125	—	2	1	no	—
128	C₂×M₃₂	C₂²×C₁₃	1	2	yes	—
129	#499	—	2	1	no	—
130	#499	—	2	1	no	—
131	#499	C₂²×C₁₃	1	2	yes	—
132	#499	C₂²×C₁₃	1	2	yes	—
133	#495	—	2	1	no	—
134	#495	C₂²×C₁₃	1	2	yes	—
135	#494	C₂²×C₁₃	1	1	yes	characteristic
136	#493	C₂²×C₁₃	1	1	yes	characteristic
137	#492	—	2	1	no	—
138	C₂×#107	C₂²×C₁₃	1	2	yes	—
139	C₂×#106	—	2	1	no	—
140	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
141	#358	C₂²×C₁₃	1	8	yes	—
142	#357	C₂²×C₁₃	1	8	yes	—
143	#346	C₂²×C₁₃	1	2	yes	—
144	C₄×C₁₆	C₂²×C₁₃	1	2	yes	—
145	C₂²×C₁₃	—	2	2	no	—
146	C₂²×C₁₃	—	8	2	no	—
147	C₂²×C₁₃	—	8	4	no	—
148	C₂²×C₁₃	#486	1	2	yes	—
149	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
150	C₄×C₁₃	—	2	2	no	—
151	C₄×C₁₃	—	2	2	no	—
152	C₄×C₁₃	—	8	2	no	—
153	C₄×C₁₃	—	8	4	no	—
154	C₄×C₁₃	#486	1	2	yes	—
155	C₄×C₁₃	C₂×#124	1	1	yes	characteristic
156	C₄×C₁₃	C₂²×SD₁₆	1	1	yes	characteristic
157	C₂×#41	—	2	2	no	—
158	#135	—	2	8	no	—
159	#135	—	4	4	no	—
160	C₂×Q₁₆	—	2	1	no	—
161	C₂×Q₁₆	C₂³×C₁₃	1	2	yes	—
162	C₂×SD₁₆	—	2	2	no	—
163	C₂×SD₁₆	—	2	2	no	—
164	C₂×D₈	—	2	1	no	—
165	C₂×D₈	C₂³×C₁₃	1	2	yes	—
166	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
167	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
168	C₄⋊Q₈	—	2	1	no	—
169	#127	—	2	1	no	—
170	#125	—	2	2	no	—
171	#125	—	2	2	no	—
172	#125	—	4	2	no	—
173	#124	—	2	2	no	—
174	#121	—	4	2	no	—
175	C₄×D₄	—	4	2	no	—
176	C₂×C₄⋊C₄	—	2	2	no	—
177	C₂×C₄⋊C₄	—	4	2	no	—
178	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
179	M₃₂	—	2	4	no	—
180	C₂×C₁₆	C₂³×C₁₃	1	4	yes	—
181	#107	—	2	2	no	—
182	#107	C₂³×C₁₃	1	4	yes	—
183	#106	—	2	2	no	—
184	#106	—	2	2	no	—
185	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
186	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
187	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
188	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
189	C₂×C₁₃	—	2	2	no	—
190	C₂×C₁₃	—	16	2	no	—
191	C₂×C₁₃	C₂×#486	1	1	yes	characteristic
192	C₂×C₁₃	#3414	1	2	yes	—
193	#41	—	4	4	no	—
194	#41	—	4	4	no	—
195	C₂×Q₈	—	2	2	no	—
196	C₂×D₄	—	2	2	no	—
197	C₂×D₄	—	4	2	no	—
198	C₂×D₄	—	4	2	no	—
199	C₂²×C₄	—	4	2	no	—
200	C₂²×C₄	—	4	2	no	—
201	C₂²×C₄	C₁₃×D₈	1	2	yes	—
202	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
203	Q₁₆	—	2	4	no	—
204	Q₁₆	—	4	2	no	—
205	SD₁₆	—	4	4	no	—
206	SD₁₆	—	4	4	no	—
207	D₈	—	2	4	no	—
208	D₈	—	4	2	no	—
209	C₂×C₈	C₁₃×SD₁₆	1	4	yes	—
210	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
211	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
212	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
213	C₂×C₈	C₁₃×#41	1	4	yes	—
214	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
215	C₄⋊C₄	—	2	4	no	—
216	C₄⋊C₄	—	4	2	no	—
217	C₄⋊C₄	—	4	4	no	—
218	C₄⋊C₄	—	4	4	no	—
219	C₄⋊C₄	—	4	2	no	—
220	C₂²⋊C₄	—	4	4	no	—
221	C₄²	C₁₃×D₈	1	2	yes	—
222	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
223	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
224	C₁₆	—	2	4	no	—
225	C₁₃	#33708	1	1	yes	characteristic, direct product factor, Sylow
226	C₂³	—	4	2	no	—
227	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
228	Q₈	—	4	2	no	—
229	Q₈	—	4	2	no	—
230	D₄	—	4	2	no	—
231	D₄	—	4	2	no	—
232	D₄	—	8	4	no	—
233	D₄	—	8	4	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	4	4	no	—
237	C₂×C₄	—	8	4	no	—
238	C₂×C₄	—	8	4	no	—
239	C₂×C₄	C₁₃×#124	1	2	yes	—
240	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
241	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
242	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
243	C₂×C₄	C₂×C₁₃×SD₁₆	1	2	yes	—
244	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
245	C₈	C₁₃×#124	1	2	yes	—
246	C₈	C₂×C₁₃×SD₁₆	1	2	yes	—
247	C₈	C₂×C₁₃×SD₁₆	1	2	yes	—
248	C₈	C₂×C₁₃×#41	1	2	yes	—
249	C₂²	—	2	2	no	—
250	C₂²	—	8	2	no	—
251	C₂²	—	8	4	no	—
252	C₂²	C₁₃×#486	1	2	yes	—
253	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
254	C₄	—	2	2	no	—
255	C₄	—	2	2	no	—
256	C₄	—	8	2	no	—
257	C₄	—	8	4	no	—
258	C₄	C₁₃×#486	1	2	yes	—
259	C₄	C₂×C₁₃×#124	1	1	yes	characteristic
260	C₄	C₂²×C₁₃×SD₁₆	1	1	yes	characteristic
261	C₂	—	2	2	no	—
262	C₂	—	16	2	no	—
263	C₂	C₂×C₁₃×#486	1	1	yes	characteristic
264	C₂	C₁₃×#3414	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
prime radical	C₁₃×#3147	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂×C₄²×C₁₃ ◁ C₂×C₄×C₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₃×#499	C₂	semidirect
3. C₁₃×#2245	C₂	semidirect
4. C₁₃×#2244	C₂	semidirect
5. C₁₃×#2242	C₂	semidirect
6. C₁₃×#2139	C₂	semidirect
7. C₄×C₁₃×M₃₂	C₂	semidirect
61. #33708	C₁₃	direct (N × H)
94. C₂×#499	C₂×C₁₃	semidirect
96. #2245	C₂×C₁₃	semidirect
97. #2244	C₂×C₁₃	semidirect
98. #2242	C₂×C₁₃	semidirect
99. #2139	C₂×C₁₃	semidirect
100. C₄×M₃₂	C₂×C₁₃	semidirect
225. C₁₃	#33708	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₁₃×#499	2
C₁₃×#3147	2
C₁₃×#2245	2
C₁₃×#2244	2
C₁₃×#2242	2
C₁₃×#2139	2
C₄×C₁₃×M₃₂	2
#33708	13

Subgroups — C13×#33708 #15720047

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33708 #15720047