Subgroups — C₁₃×#33853 #15720192

Order 3328.

696 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#508	C₂	1	2	yes	maximal, complemented
2	C₁₃×#3287	C₂	1	2	yes	maximal, complemented
3	C₁₃×#2246	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#2215	C₂	1	2	yes	maximal, complemented
5	C₁₃×#2207	C₂	1	2	yes	maximal, complemented
6	C₁₃×#2186	C₂	1	4	yes	maximal, complemented
7	C₄×C₁₃×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
8	D₄×C₁₃×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₁₃×#135	C₂²	1	4	yes	—
10	C₁₃×#550	—	2	2	no	—
11	C₁₃×#508	—	2	8	no	—
12	C₂×C₁₃×Q₃₂	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×SD₃₂	C₂²	1	4	yes	—
14	C₂×C₁₃×D₁₆	C₂²	1	2	yes	—
15	C₂×C₁₃×D₁₆	C₂²	1	2	yes	—
16	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
17	C₁₃×#493	C₂²	1	2	yes	—
18	C₁₃×#466	C₂²	1	4	yes	—
19	C₁₃×#460	—	2	2	no	—
20	C₁₃×#449	—	2	4	no	—
21	C₄×C₁₃×Q₁₆	C₂²	1	2	yes	—
22	C₈×D₄×C₁₃	C₂²	1	1	yes	characteristic
23	C₁₃×#366	C₂²	1	1	yes	characteristic
24	C₁₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
25	C₁₃×#358	C₂²	1	2	yes	—
26	C₁₃×#357	C₂²	1	4	yes	—
27	C₁₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
28	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
29	C₂×C₁₃×#41	—	2	4	no	—
30	C₁₃×#135	—	2	8	no	—
31	C₁₃×#135	—	4	4	no	—
32	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
33	C₂×C₁₃×SD₁₆	—	2	4	no	—
34	C₂×C₁₃×D₈	—	2	2	no	—
35	C₂×C₁₃×D₈	C₂³	1	4	yes	—
36	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
37	C₁₃×#127	—	2	2	no	—
38	C₁₃×#127	—	4	2	no	—
39	C₁₃×#121	—	2	4	no	—
40	C₁₃×#121	—	4	4	no	—
41	C₄×Q₈×C₁₃	—	2	2	no	—
42	C₄×D₄×C₁₃	—	4	2	no	—
43	C₄×D₄×C₁₃	D₄	1	1	yes	characteristic
44	C₁₃×Q₃₂	D₄	1	4	yes	complemented
45	C₁₃×SD₃₂	—	2	8	no	—
46	C₁₃×D₁₆	—	2	4	no	—
47	C₁₃×D₁₆	—	4	2	no	—
48	C₂×C₁₃×C₁₆	—	2	4	no	—
49	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
50	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
51	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
52	C₁₃×#107	C₂³	1	2	yes	—
53	C₁₃×C₄⋊C₈	D₄	1	1	yes	characteristic
54	C₁₃×#103	—	2	2	no	—
55	C₁₃×#102	—	2	4	no	—
56	C₁₃×C₂²⋊C₈	D₄	1	2	yes	—
57	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
58	#33853	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
59	C₁₃×#41	—	4	8	no	—
60	C₁₃×#41	—	8	4	no	—
61	C₂×Q₈×C₁₃	—	2	2	no	—
62	C₂×D₄×C₁₃	—	2	4	no	—
63	C₂×D₄×C₁₃	—	4	4	no	—
64	C₂×D₄×C₁₃	—	4	4	no	—
65	C₂×D₄×C₁₃	—	4	2	no	—
66	C₂×D₄×C₁₃	—	4	2	no	—
67	C₂×D₄×C₁₃	D₈	1	1	yes	characteristic
68	C₂²×C₄×C₁₃	—	4	4	no	—
69	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
70	C₁₃×Q₁₆	—	2	2	no	—
71	C₁₃×Q₁₆	C₂×D₄	1	4	yes	—
72	C₁₃×SD₁₆	—	4	8	no	—
73	C₁₃×D₈	—	2	4	no	—
74	C₁₃×D₈	—	2	4	no	—
75	C₁₃×D₈	—	8	2	no	—
76	C₂×C₈×C₁₃	—	2	2	no	—
77	C₂×C₈×C₁₃	—	2	2	no	—
78	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
80	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
81	C₁₃×C₄⋊C₄	—	2	2	no	—
82	C₁₃×C₄⋊C₄	—	4	2	no	—
83	C₁₃×C₄⋊C₄	D₈	1	1	yes	characteristic
84	C₁₃×C₂²⋊C₄	—	4	4	no	—
85	C₁₃×C₂²⋊C₄	D₈	1	2	yes	—
86	C₄²×C₁₃	—	4	2	no	—
87	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
88	C₁₃×C₁₆	—	2	2	no	—
89	C₁₃×C₁₆	—	2	1	no	—
90	C₁₃×C₁₆	C₂×D₄	1	2	yes	—
91	C₂×#508	C₂×C₁₃	1	2	yes	complemented
92	#3287	C₂×C₁₃	1	2	yes	complemented
93	#2246	C₂×C₁₃	1	1	yes	characteristic, complemented
94	#2215	C₂×C₁₃	1	2	yes	complemented
95	#2207	C₂×C₁₃	1	2	yes	complemented
96	#2186	C₂×C₁₃	1	4	yes	complemented
97	C₄×Q₃₂	C₂×C₁₃	1	1	yes	characteristic, complemented
98	D₄×C₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
99	C₂³×C₁₃	—	4	4	no	—
100	C₂³×C₁₃	C₂×D₈	1	2	yes	—
101	Q₈×C₁₃	—	2	4	no	—
102	Q₈×C₁₃	—	4	2	no	—
103	D₄×C₁₃	—	2	2	no	—
104	D₄×C₁₃	—	4	4	no	—
105	D₄×C₁₃	—	4	4	no	—
106	D₄×C₁₃	—	8	8	no	—
107	D₄×C₁₃	—	8	4	no	—
108	D₄×C₁₃	—	16	2	no	—
109	D₄×C₁₃	—	16	4	no	—
110	C₂×C₄×C₁₃	—	2	2	no	—
111	C₂×C₄×C₁₃	—	2	2	no	—
112	C₂×C₄×C₁₃	—	4	2	no	—
113	C₂×C₄×C₁₃	—	4	2	no	—
114	C₂×C₄×C₁₃	—	8	8	no	—
115	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
116	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
117	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
118	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
119	C₈×C₁₃	—	2	2	no	—
120	C₈×C₁₃	—	2	1	no	—
121	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
122	C₈×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
123	C₂×#135	C₂²×C₁₃	1	4	yes	—
124	#550	—	2	2	no	—
125	#508	—	2	8	no	—
126	C₂×Q₃₂	C₂²×C₁₃	1	1	yes	characteristic
127	C₂×SD₃₂	C₂²×C₁₃	1	4	yes	—
128	C₂×D₁₆	C₂²×C₁₃	1	2	yes	—
129	C₂×D₁₆	C₂²×C₁₃	1	2	yes	—
130	C₂²×C₁₆	C₂²×C₁₃	1	2	yes	—
131	#493	C₂²×C₁₃	1	2	yes	—
132	#466	C₂²×C₁₃	1	4	yes	—
133	#460	—	2	2	no	—
134	#449	—	2	4	no	—
135	C₄×Q₁₆	C₂²×C₁₃	1	2	yes	—
136	C₈×D₄	C₂²×C₁₃	1	1	yes	characteristic
137	#366	C₂²×C₁₃	1	1	yes	characteristic
138	C₄⋊C₁₆	C₂²×C₁₃	1	1	yes	characteristic
139	#358	C₂²×C₁₃	1	2	yes	—
140	#357	C₂²×C₁₃	1	4	yes	—
141	C₂²⋊C₁₆	C₂²×C₁₃	1	2	yes	—
142	C₄×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
143	C₂²×C₁₃	—	2	2	no	—
144	C₂²×C₁₃	—	2	2	no	—
145	C₂²×C₁₃	—	2	2	no	—
146	C₂²×C₁₃	—	8	4	no	—
147	C₂²×C₁₃	—	8	4	no	—
148	C₂²×C₁₃	—	8	4	no	—
149	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
150	C₄×C₁₃	—	2	1	no	—
151	C₄×C₁₃	—	2	2	no	—
152	C₄×C₁₃	—	4	4	no	—
153	C₄×C₁₃	—	8	2	no	—
154	C₄×C₁₃	#508	1	2	yes	complemented
155	C₄×C₁₃	D₄²	1	1	yes	characteristic
156	C₄×C₁₃	#576	1	1	yes	characteristic
157	C₂×#41	—	2	4	no	—
158	#135	—	2	8	no	—
159	#135	—	4	4	no	—
160	C₂×Q₁₆	C₂³×C₁₃	1	2	yes	—
161	C₂×SD₁₆	—	2	4	no	—
162	C₂×D₈	—	2	2	no	—
163	C₂×D₈	C₂³×C₁₃	1	4	yes	—
164	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
165	#127	—	2	2	no	—
166	#127	—	4	2	no	—
167	#121	—	2	4	no	—
168	#121	—	4	4	no	—
169	C₄×Q₈	—	2	2	no	—
170	C₄×D₄	—	4	2	no	—
171	C₄×D₄	D₄×C₁₃	1	1	yes	characteristic
172	Q₃₂	D₄×C₁₃	1	4	yes	complemented
173	SD₃₂	—	2	8	no	—
174	D₁₆	—	2	4	no	—
175	D₁₆	—	4	2	no	—
176	C₂×C₁₆	—	2	4	no	—
177	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
178	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
179	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
180	#107	C₂³×C₁₃	1	2	yes	—
181	C₄⋊C₈	D₄×C₁₃	1	1	yes	characteristic
182	#103	—	2	2	no	—
183	#102	—	2	4	no	—
184	C₂²⋊C₈	D₄×C₁₃	1	2	yes	—
185	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
186	C₂×C₁₃	—	4	2	no	—
187	C₂×C₁₃	—	16	4	no	—
188	C₂×C₁₃	D₄×D₈	1	1	yes	characteristic
189	C₂×C₁₃	C₂×#508	1	1	yes	characteristic
190	C₂×C₁₃	#3415	1	1	yes	characteristic
191	#41	—	4	8	no	—
192	#41	—	8	4	no	—
193	C₂×Q₈	—	2	2	no	—
194	C₂×D₄	—	2	4	no	—
195	C₂×D₄	—	4	4	no	—
196	C₂×D₄	—	4	4	no	—
197	C₂×D₄	—	4	2	no	—
198	C₂×D₄	—	4	2	no	—
199	C₂×D₄	C₁₃×D₈	1	1	yes	characteristic
200	C₂²×C₄	—	4	4	no	—
201	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
202	Q₁₆	—	2	2	no	—
203	Q₁₆	C₂×D₄×C₁₃	1	4	yes	—
204	SD₁₆	—	4	8	no	—
205	D₈	—	2	4	no	—
206	D₈	—	2	4	no	—
207	D₈	—	8	2	no	—
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	—	2	2	no	—
210	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
211	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
212	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
213	C₄⋊C₄	—	2	2	no	—
214	C₄⋊C₄	—	4	2	no	—
215	C₄⋊C₄	C₁₃×D₈	1	1	yes	characteristic
216	C₂²⋊C₄	—	4	4	no	—
217	C₂²⋊C₄	C₁₃×D₈	1	2	yes	—
218	C₄²	—	4	2	no	—
219	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
220	C₁₆	—	2	2	no	—
221	C₁₆	—	2	1	no	—
222	C₁₆	C₂×D₄×C₁₃	1	2	yes	—
223	C₁₃	#33853	1	1	yes	characteristic, direct product factor, Sylow
224	C₂³	—	4	4	no	—
225	C₂³	C₂×C₁₃×D₈	1	2	yes	—
226	Q₈	—	2	4	no	—
227	Q₈	—	4	2	no	—
228	D₄	—	2	2	no	—
229	D₄	—	4	4	no	—
230	D₄	—	4	4	no	—
231	D₄	—	8	8	no	—
232	D₄	—	8	4	no	—
233	D₄	—	16	2	no	—
234	D₄	—	16	4	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	8	8	no	—
240	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
241	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
242	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
243	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
244	C₈	—	2	2	no	—
245	C₈	—	2	1	no	—
246	C₈	C₂²×D₄×C₁₃	1	1	yes	characteristic
247	C₈	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
248	C₂²	—	2	2	no	—
249	C₂²	—	2	2	no	—
250	C₂²	—	2	2	no	—
251	C₂²	—	8	4	no	—
252	C₂²	—	8	4	no	—
253	C₂²	—	8	4	no	—
254	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
255	C₄	—	2	2	no	—
256	C₄	—	2	1	no	—
257	C₄	—	4	4	no	—
258	C₄	—	8	2	no	—
259	C₄	C₁₃×#508	1	2	yes	complemented
260	C₄	D₄²×C₁₃	1	1	yes	characteristic
261	C₄	C₁₃×#576	1	1	yes	characteristic
262	C₂	—	4	2	no	—
263	C₂	—	16	4	no	—
264	C₂	D₄×C₁₃×D₈	1	1	yes	characteristic
265	C₂	C₂×C₁₃×#508	1	1	yes	characteristic
266	C₂	C₁₃×#3415	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₄×D₄×C₁₃ ◁ C₈×D₄×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₃×#508	C₂	semidirect
2. C₁₃×#3287	C₂	semidirect
3. C₁₃×#2246	C₂	semidirect
4. C₁₃×#2215	C₂	semidirect
5. C₁₃×#2207	C₂	semidirect
6. C₁₃×#2186	C₂	semidirect
7. C₄×C₁₃×Q₃₂	C₂	semidirect
8. D₄×C₁₃×C₁₆	C₂	semidirect
44. C₁₃×Q₃₂	D₄	semidirect
58. #33853	C₁₃	direct (N × H)
91. C₂×#508	C₂×C₁₃	semidirect
92. #3287	C₂×C₁₃	semidirect
93. #2246	C₂×C₁₃	semidirect
94. #2215	C₂×C₁₃	semidirect
95. #2207	C₂×C₁₃	semidirect
96. #2186	C₂×C₁₃	semidirect
97. C₄×Q₃₂	C₂×C₁₃	semidirect
98. D₄×C₁₆	C₂×C₁₃	semidirect
154. C₄×C₁₃	#508	semidirect
172. Q₃₂	D₄×C₁₃	semidirect
223. C₁₃	#33853	direct (N × H)
259. C₄	C₁₃×#508	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₁₃×#508	2
C₁₃×#3287	2
C₁₃×#2246	2
C₁₃×#2215	2
C₁₃×#2207	2
C₁₃×#2186	2
C₄×C₁₃×Q₃₂	2
D₄×C₁₃×C₁₆	2
#33853	13

Subgroups — C13×#33853 #15720192

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33853 #15720192