Subgroups — C₁₃×#33826 #15720165

Order 3328.

488 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3177	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₃×#3176	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2229	C₂	1	2	yes	maximal, complemented
4	C₁₃×#2228	C₂	1	2	yes	maximal, complemented
5	C₁₃×#2220	C₂	1	2	yes	maximal, complemented
6	C₁₃×#2217	C₂	1	2	yes	maximal, complemented
7	C₁₃×#2208	C₂	1	2	yes	maximal, complemented
8	C₁₃×#2207	C₂	1	2	yes	maximal, complemented
9	C₁₃×#2151	C₂	1	1	yes	maximal, characteristic, complemented
10	C₁₃×#541	—	2	1	no	—
11	C₁₃×#540	—	2	1	no	—
12	C₂×C₁₃×SD₃₂	C₂²	1	2	yes	—
13	C₂×C₁₃×SD₃₂	C₂²	1	2	yes	—
14	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
15	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
16	C₁₃×#499	C₂²	1	1	yes	characteristic
17	C₁₃×#499	C₂²	1	1	yes	characteristic
18	C₁₃×#494	C₂²	1	1	yes	characteristic
19	C₁₃×#493	C₂²	1	1	yes	characteristic
20	C₁₃×#491	—	2	1	no	—
21	C₁₃×#490	—	2	1	no	—
22	C₁₃×#470	—	2	1	no	—
23	C₁₃×#469	—	2	1	no	—
24	C₁₃×#467	C₂²	1	2	yes	—
25	C₁₃×#466	C₂²	1	2	yes	—
26	C₁₃×#442	—	2	1	no	—
27	C₁₃×#441	—	2	1	no	—
28	C₄×C₁₃×Q₁₆	C₂²	1	2	yes	—
29	C₄×C₁₃×D₈	C₂²	1	2	yes	—
30	C₁₃×#405	C₂²	1	1	yes	characteristic
31	C₁₃×#367	C₂²	1	2	yes	—
32	C₁₃×#366	C₂²	1	2	yes	—
33	C₁₃×C₄⋊C₁₆	C₂²	1	2	yes	—
34	C₁₃×C₄⋊C₁₆	C₂²	1	2	yes	—
35	C₁₃×#358	C₂²	1	2	yes	—
36	C₁₃×#358	C₂²	1	2	yes	—
37	C₁₃×#357	C₂²	1	2	yes	—
38	C₁₃×#357	C₂²	1	2	yes	—
39	C₂×C₁₃×Q₁₆	—	2	1	no	—
40	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
41	C₂×C₁₃×D₈	—	2	1	no	—
42	C₂×C₁₃×D₈	C₂³	1	2	yes	—
43	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
45	C₁₃×C₄⋊Q₈	—	2	1	no	—
46	C₁₃×#127	—	2	1	no	—
47	C₁₃×#126	—	4	2	no	—
48	C₁₃×#125	—	2	1	no	—
49	C₁₃×#125	—	2	1	no	—
50	C₁₃×#125	—	4	2	no	—
51	C₁₃×#123	—	4	2	no	—
52	C₁₃×C₂²⋊Q₈	—	2	2	no	—
53	C₁₃×#121	—	2	2	no	—
54	C₁₃×#121	—	4	2	no	—
55	C₄×Q₈×C₁₃	—	2	2	no	—
56	C₄×D₄×C₁₃	—	2	2	no	—
57	C₁₃×#117	D₄	1	1	yes	characteristic
58	C₁₃×SD₃₂	—	4	2	no	—
59	C₁₃×SD₃₂	—	4	2	no	—
60	C₁₃×M₃₂	—	4	1	no	—
61	C₂×C₁₃×C₁₆	—	4	1	no	—
62	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
63	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
64	C₁₃×#107	C₂³	1	2	yes	—
65	C₁₃×#107	C₂³	1	2	yes	—
66	C₁₃×#106	—	2	1	no	—
67	C₁₃×#106	—	2	1	no	—
68	C₁₃×#103	—	2	2	no	—
69	C₁₃×#103	—	2	2	no	—
70	C₁₃×#102	—	2	2	no	—
71	C₁₃×#102	—	2	2	no	—
72	C₁₃×#97	D₄	1	1	yes	characteristic
73	C₁₃×#97	D₄	1	1	yes	characteristic
74	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
75	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
76	#33826	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
77	C₂×Q₈×C₁₃	—	2	2	no	—
78	C₂×D₄×C₁₃	—	2	2	no	—
79	C₂×D₄×C₁₃	—	4	2	no	—
80	C₂×D₄×C₁₃	—	4	2	no	—
81	C₂²×C₄×C₁₃	—	4	2	no	—
82	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
83	C₁₃×Q₁₆	—	2	2	no	—
84	C₁₃×Q₁₆	—	2	2	no	—
85	C₁₃×Q₁₆	—	4	2	no	—
86	C₁₃×D₈	—	2	2	no	—
87	C₁₃×D₈	—	2	2	no	—
88	C₁₃×D₈	—	4	2	no	—
89	C₁₃×M₁₆	—	4	1	no	—
90	C₂×C₈×C₁₃	—	2	1	no	—
91	C₂×C₈×C₁₃	—	2	1	no	—
92	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
96	C₁₃×C₄⋊C₄	—	2	2	no	—
97	C₁₃×C₄⋊C₄	—	2	2	no	—
98	C₁₃×C₄⋊C₄	—	4	2	no	—
99	C₁₃×C₄⋊C₄	—	4	2	no	—
100	C₁₃×C₄⋊C₄	—	4	2	no	—
101	C₁₃×C₄⋊C₄	—	4	2	no	—
102	C₁₃×C₄⋊C₄	—	4	2	no	—
103	C₁₃×C₄⋊C₄	D₈	1	2	yes	—
104	C₁₃×C₂²⋊C₄	—	4	2	no	—
105	C₁₃×C₂²⋊C₄	—	4	2	no	—
106	C₁₃×C₂²⋊C₄	—	4	2	no	—
107	C₁₃×C₂²⋊C₄	D₈	1	2	yes	—
108	C₄²×C₁₃	—	4	2	no	—
109	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
110	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
111	C₁₃×C₁₆	—	2	2	no	—
112	C₁₃×C₁₆	—	2	2	no	—
113	#3177	C₂×C₁₃	1	1	yes	characteristic, complemented
114	#3176	C₂×C₁₃	1	1	yes	characteristic
115	#2229	C₂×C₁₃	1	2	yes	complemented
116	#2228	C₂×C₁₃	1	2	yes	complemented
117	#2220	C₂×C₁₃	1	2	yes	complemented
118	#2217	C₂×C₁₃	1	2	yes	complemented
119	#2208	C₂×C₁₃	1	2	yes	complemented
120	#2207	C₂×C₁₃	1	2	yes	complemented
121	#2151	C₂×C₁₃	1	1	yes	characteristic, complemented
122	C₂³×C₁₃	—	4	2	no	—
123	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
124	Q₈×C₁₃	—	4	2	no	—
125	Q₈×C₁₃	—	4	2	no	—
126	D₄×C₁₃	—	4	2	no	—
127	D₄×C₁₃	—	4	2	no	—
128	D₄×C₁₃	—	16	2	no	—
129	D₄×C₁₃	—	16	2	no	—
130	C₂×C₄×C₁₃	—	2	1	no	—
131	C₂×C₄×C₁₃	—	2	1	no	—
132	C₂×C₄×C₁₃	—	4	2	no	—
133	C₂×C₄×C₁₃	—	4	2	no	—
134	C₂×C₄×C₁₃	—	4	2	no	—
135	C₂×C₄×C₁₃	—	16	2	no	—
136	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
137	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
138	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
139	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
140	C₈×C₁₃	—	2	1	no	—
141	C₈×C₁₃	—	2	1	no	—
142	C₈×C₁₃	—	2	1	no	—
143	C₈×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
144	C₈×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
145	#541	—	2	1	no	—
146	#540	—	2	1	no	—
147	C₂×SD₃₂	C₂²×C₁₃	1	2	yes	—
148	C₂×SD₃₂	C₂²×C₁₃	1	2	yes	—
149	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
150	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
151	#499	C₂²×C₁₃	1	1	yes	characteristic
152	#499	C₂²×C₁₃	1	1	yes	characteristic
153	#494	C₂²×C₁₃	1	1	yes	characteristic
154	#493	C₂²×C₁₃	1	1	yes	characteristic
155	#491	—	2	1	no	—
156	#490	—	2	1	no	—
157	#470	—	2	1	no	—
158	#469	—	2	1	no	—
159	#467	C₂²×C₁₃	1	2	yes	—
160	#466	C₂²×C₁₃	1	2	yes	—
161	#442	—	2	1	no	—
162	#441	—	2	1	no	—
163	C₄×Q₁₆	C₂²×C₁₃	1	2	yes	—
164	C₄×D₈	C₂²×C₁₃	1	2	yes	—
165	#405	C₂²×C₁₃	1	1	yes	characteristic
166	#367	C₂²×C₁₃	1	2	yes	—
167	#366	C₂²×C₁₃	1	2	yes	—
168	C₄⋊C₁₆	C₂²×C₁₃	1	2	yes	—
169	C₄⋊C₁₆	C₂²×C₁₃	1	2	yes	—
170	#358	C₂²×C₁₃	1	2	yes	—
171	#358	C₂²×C₁₃	1	2	yes	—
172	#357	C₂²×C₁₃	1	2	yes	—
173	#357	C₂²×C₁₃	1	2	yes	—
174	C₂²×C₁₃	—	2	2	no	—
175	C₂²×C₁₃	—	2	1	no	—
176	C₂²×C₁₃	—	8	2	no	—
177	C₂²×C₁₃	—	8	2	no	—
178	C₂²×C₁₃	—	8	2	no	—
179	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
180	C₄×C₁₃	—	2	2	no	—
181	C₄×C₁₃	—	2	2	no	—
182	C₄×C₁₃	—	2	1	no	—
183	C₄×C₁₃	—	8	2	no	—
184	C₄×C₁₃	—	8	2	no	—
185	C₄×C₁₃	—	8	2	no	—
186	C₄×C₁₃	#537	1	1	yes	characteristic
187	C₄×C₁₃	#575	1	1	yes	characteristic
188	C₂×Q₁₆	—	2	1	no	—
189	C₂×Q₁₆	C₂³×C₁₃	1	2	yes	—
190	C₂×D₈	—	2	1	no	—
191	C₂×D₈	C₂³×C₁₃	1	2	yes	—
192	C₂×M₁₆	D₄×C₁₃	1	1	yes	characteristic
193	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
194	C₄⋊Q₈	—	2	1	no	—
195	#127	—	2	1	no	—
196	#126	—	4	2	no	—
197	#125	—	2	1	no	—
198	#125	—	2	1	no	—
199	#125	—	4	2	no	—
200	#123	—	4	2	no	—
201	C₂²⋊Q₈	—	2	2	no	—
202	#121	—	2	2	no	—
203	#121	—	4	2	no	—
204	C₄×Q₈	—	2	2	no	—
205	C₄×D₄	—	2	2	no	—
206	#117	D₄×C₁₃	1	1	yes	characteristic
207	SD₃₂	—	4	2	no	—
208	SD₃₂	—	4	2	no	—
209	M₃₂	—	4	1	no	—
210	C₂×C₁₆	—	4	1	no	—
211	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
212	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
213	#107	C₂³×C₁₃	1	2	yes	—
214	#107	C₂³×C₁₃	1	2	yes	—
215	#106	—	2	1	no	—
216	#106	—	2	1	no	—
217	#103	—	2	2	no	—
218	#103	—	2	2	no	—
219	#102	—	2	2	no	—
220	#102	—	2	2	no	—
221	#97	D₄×C₁₃	1	1	yes	characteristic
222	#97	D₄×C₁₃	1	1	yes	characteristic
223	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
224	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
225	C₂×C₁₃	—	4	1	no	—
226	C₂×C₁₃	—	16	2	no	—
227	C₂×C₁₃	#3245	1	1	yes	characteristic
228	C₂×C₁₃	#3416	1	2	yes	—
229	C₂×Q₈	—	2	2	no	—
230	C₂×D₄	—	2	2	no	—
231	C₂×D₄	—	4	2	no	—
232	C₂×D₄	—	4	2	no	—
233	C₂²×C₄	—	4	2	no	—
234	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
235	Q₁₆	—	2	2	no	—
236	Q₁₆	—	2	2	no	—
237	Q₁₆	—	4	2	no	—
238	D₈	—	2	2	no	—
239	D₈	—	2	2	no	—
240	D₈	—	4	2	no	—
241	M₁₆	—	4	1	no	—
242	C₂×C₈	—	2	1	no	—
243	C₂×C₈	—	2	1	no	—
244	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
245	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
246	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
247	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
248	C₄⋊C₄	—	2	2	no	—
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	4	2	no	—
251	C₄⋊C₄	—	4	2	no	—
252	C₄⋊C₄	—	4	2	no	—
253	C₄⋊C₄	—	4	2	no	—
254	C₄⋊C₄	—	4	2	no	—
255	C₄⋊C₄	C₁₃×D₈	1	2	yes	—
256	C₂²⋊C₄	—	4	2	no	—
257	C₂²⋊C₄	—	4	2	no	—
258	C₂²⋊C₄	—	4	2	no	—
259	C₂²⋊C₄	C₁₃×D₈	1	2	yes	—
260	C₄²	—	4	2	no	—
261	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
262	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
263	C₁₆	—	2	2	no	—
264	C₁₆	—	2	2	no	—
265	C₁₃	#33826	1	1	yes	characteristic, direct product factor, Sylow
266	C₂³	—	4	2	no	—
267	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
268	Q₈	—	4	2	no	—
269	Q₈	—	4	2	no	—
270	D₄	—	4	2	no	—
271	D₄	—	4	2	no	—
272	D₄	—	16	2	no	—
273	D₄	—	16	2	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	16	2	no	—
280	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
281	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
282	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
283	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
284	C₈	—	2	1	no	—
285	C₈	—	2	1	no	—
286	C₈	—	2	1	no	—
287	C₈	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
288	C₈	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
289	C₂²	—	2	2	no	—
290	C₂²	—	2	1	no	—
291	C₂²	—	8	2	no	—
292	C₂²	—	8	2	no	—
293	C₂²	—	8	2	no	—
294	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
295	C₄	—	2	2	no	—
296	C₄	—	2	2	no	—
297	C₄	—	2	1	no	—
298	C₄	—	8	2	no	—
299	C₄	—	8	2	no	—
300	C₄	—	8	2	no	—
301	C₄	C₁₃×#537	1	1	yes	characteristic
302	C₄	C₁₃×#575	1	1	yes	characteristic
303	C₂	—	4	1	no	—
304	C₂	—	16	2	no	—
305	C₂	C₁₃×#3245	1	1	yes	characteristic
306	C₂	C₁₃×#3416	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
prime radical	C₁₃×#3176	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₁₃×#117 ◁ C₁₃×#405 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₃×#3177	C₂	semidirect
3. C₁₃×#2229	C₂	semidirect
4. C₁₃×#2228	C₂	semidirect
5. C₁₃×#2220	C₂	semidirect
6. C₁₃×#2217	C₂	semidirect
7. C₁₃×#2208	C₂	semidirect
8. C₁₃×#2207	C₂	semidirect
9. C₁₃×#2151	C₂	semidirect
76. #33826	C₁₃	direct (N × H)
113. #3177	C₂×C₁₃	semidirect
115. #2229	C₂×C₁₃	semidirect
116. #2228	C₂×C₁₃	semidirect
117. #2220	C₂×C₁₃	semidirect
118. #2217	C₂×C₁₃	semidirect
119. #2208	C₂×C₁₃	semidirect
120. #2207	C₂×C₁₃	semidirect
121. #2151	C₂×C₁₃	semidirect
265. C₁₃	#33826	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×#3177	2
C₁₃×#3176	2
C₁₃×#2229	2
C₁₃×#2228	2
C₁₃×#2220	2
C₁₃×#2217	2
C₁₃×#2208	2
C₁₃×#2207	2
C₁₃×#2151	2
#33826	13

Subgroups — C13×#33826 #15720165

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33826 #15720165