Subgroups — C₁₃×#13396 #15699735

Order 3328.

422 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3058	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#2155	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₃×#361	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#2140	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#2117	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#1814	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#1384	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₁₃×D₈	C₂²	1	1	yes	characteristic
9	C₂×C₁₃×#131	C₂²	1	1	yes	characteristic
10	C₁₃×#535	—	2	1	no	—
11	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
13	C₁₃×#469	—	2	2	no	—
14	C₁₃×#466	—	2	2	no	—
15	C₂×C₁₃×#108	C₂²	1	1	yes	characteristic
16	C₁₃×#427	C₂²	1	1	yes	characteristic
17	C₁₃×#427	C₂²	1	1	yes	characteristic
18	C₁₃×#421	—	2	1	no	—
19	C₁₃×#418	—	2	1	no	—
20	C₁₃×#365	—	2	2	no	—
21	C₁₃×#361	—	2	2	no	—
22	C₁₃×#357	—	2	2	no	—
23	C₁₃×#350	—	2	2	no	—
24	C₁₃×#344	—	2	1	no	—
25	C₂×C₁₃×#41	D₄	1	1	yes	characteristic
26	C₂²×D₄×C₁₃	—	2	1	no	—
27	C₂×C₁₃×D₈	—	2	1	no	—
28	C₂×C₁₃×D₈	—	2	1	no	—
29	C₂×C₁₃×D₈	—	2	1	no	—
30	C₂×C₁₃×D₈	—	2	1	no	—
31	C₂×C₁₃×D₈	—	2	1	no	—
32	C₂×C₁₃×D₈	D₄	1	2	yes	—
33	C₁₃×#131	—	2	1	no	—
34	C₁₃×#131	—	2	1	no	—
35	C₁₃×#131	—	2	1	no	—
36	C₁₃×#131	—	2	1	no	—
37	C₂×C₁₃×M₁₆	—	2	1	no	—
38	C₂×C₁₃×M₁₆	—	2	1	no	—
39	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₁₃	—	2	1	no	—
41	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
42	C₁₃×#127	—	4	1	no	—
43	C₁₃×#124	—	4	1	no	—
44	C₁₃×#121	—	2	4	no	—
45	C₁₃×#120	—	4	2	no	—
46	C₁₃×#117	—	2	1	no	—
47	C₁₃×#117	—	2	1	no	—
48	C₁₃×M₃₂	—	2	1	no	—
49	C₁₃×M₃₂	—	2	1	no	—
50	C₁₃×M₃₂	—	2	1	no	—
51	C₁₃×M₃₂	—	2	1	no	—
52	C₂×C₁₃×C₁₆	—	2	1	no	—
53	C₂×C₁₃×C₁₆	D₄	1	2	yes	complemented
54	C₁₃×#108	—	2	1	no	—
55	C₁₃×#108	—	2	1	no	—
56	C₁₃×#107	—	2	1	no	—
57	C₁₃×#107	—	2	1	no	—
58	C₁₃×#106	—	2	1	no	—
59	C₁₃×#106	D₄	1	2	yes	—
60	C₁₃×#104	—	4	2	no	—
61	C₁₃×#102	—	2	4	no	—
62	#13396	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
63	C₂⁴×C₁₃	—	4	1	no	—
64	C₁₃×#41	—	2	1	no	—
65	C₁₃×#41	—	2	1	no	—
66	C₁₃×#41	—	2	1	no	—
67	C₁₃×#41	—	2	1	no	—
68	C₂×Q₈×C₁₃	SD₁₆	1	1	yes	characteristic
69	C₂×D₄×C₁₃	—	2	1	no	—
70	C₂×D₄×C₁₃	—	2	2	no	—
71	C₂×D₄×C₁₃	—	4	4	no	—
72	C₂×D₄×C₁₃	—	4	1	no	—
73	C₂×D₄×C₁₃	—	4	1	no	—
74	C₂×D₄×C₁₃	—	4	1	no	—
75	C₂×D₄×C₁₃	—	4	1	no	—
76	C₂×D₄×C₁₃	—	4	1	no	—
77	C₂×D₄×C₁₃	SD₁₆	1	1	yes	characteristic
78	C₂²×C₄×C₁₃	—	2	1	no	—
79	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
80	C₁₃×D₈	—	2	2	no	—
81	C₁₃×D₈	—	2	2	no	—
82	C₁₃×D₈	—	4	1	no	—
83	C₁₃×D₈	—	4	1	no	—
84	C₁₃×M₁₆	—	2	1	no	—
85	C₁₃×M₁₆	—	2	1	no	—
86	C₁₃×M₁₆	—	4	1	no	—
87	C₁₃×M₁₆	—	4	1	no	—
88	C₁₃×M₁₆	—	4	1	no	—
89	C₁₃×M₁₆	—	4	1	no	—
90	C₂×C₈×C₁₃	—	2	1	no	—
91	C₂×C₈×C₁₃	—	4	1	no	—
92	C₂×C₈×C₁₃	—	4	1	no	—
93	C₂×C₈×C₁₃	—	4	1	no	—
94	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
98	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
99	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
100	C₁₃×C₄⋊C₄	—	2	2	no	—
101	C₁₃×C₄⋊C₄	—	4	1	no	—
102	C₁₃×C₂²⋊C₄	—	4	4	no	—
103	C₁₃×C₂²⋊C₄	—	4	1	no	—
104	C₁₃×C₂²⋊C₄	—	4	1	no	—
105	C₄²×C₁₃	—	4	1	no	—
106	C₄²×C₁₃	—	4	1	no	—
107	C₁₃×C₁₆	—	2	2	no	—
108	C₁₃×C₁₆	—	4	1	no	—
109	#3058	C₂×C₁₃	1	1	yes	characteristic
110	#2155	C₂×C₁₃	1	1	yes	characteristic, complemented
111	C₂×#361	C₂×C₁₃	1	1	yes	characteristic, complemented
112	#2140	C₂×C₁₃	1	1	yes	characteristic, complemented
113	#2117	C₂×C₁₃	1	1	yes	characteristic, complemented
114	#1814	C₂×C₁₃	1	1	yes	characteristic, complemented
115	#1384	C₂×C₁₃	1	1	yes	characteristic, complemented
116	C₂³×C₁₃	—	2	1	no	—
117	C₂³×C₁₃	—	4	2	no	—
118	C₂³×C₁₃	—	8	1	no	—
119	C₂³×C₁₃	—	8	1	no	—
120	C₂³×C₁₃	—	16	2	no	—
121	C₂³×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
122	Q₈×C₁₃	—	2	1	no	—
123	Q₈×C₁₃	—	2	1	no	—
124	D₄×C₁₃	—	2	1	no	—
125	D₄×C₁₃	—	2	1	no	—
126	D₄×C₁₃	—	4	1	no	—
127	D₄×C₁₃	—	4	1	no	—
128	D₄×C₁₃	—	4	2	no	—
129	D₄×C₁₃	—	4	2	no	—
130	D₄×C₁₃	—	8	1	no	—
131	D₄×C₁₃	—	8	1	no	—
132	D₄×C₁₃	—	16	4	no	—
133	C₂×C₄×C₁₃	—	2	1	no	—
134	C₂×C₄×C₁₃	—	4	1	no	—
135	C₂×C₄×C₁₃	—	4	1	no	—
136	C₂×C₄×C₁₃	—	4	2	no	—
137	C₂×C₄×C₁₃	—	8	1	no	—
138	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
139	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
140	C₂×C₄×C₁₃	#123	1	1	yes	characteristic
141	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
142	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
143	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
144	C₈×C₁₃	—	4	1	no	—
145	C₈×C₁₃	—	8	1	no	—
146	C₈×C₁₃	#121	1	1	yes	characteristic
147	C₈×C₁₃	#121	1	1	yes	characteristic
148	C₈×C₁₃	#123	1	1	yes	characteristic
149	C₈×C₁₃	#124	1	1	yes	characteristic
150	C₂²×D₈	C₂²×C₁₃	1	1	yes	characteristic
151	C₂×#131	C₂²×C₁₃	1	1	yes	characteristic
152	#535	—	2	1	no	—
153	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
154	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
155	#469	—	2	2	no	—
156	#466	—	2	2	no	—
157	C₂×#108	C₂²×C₁₃	1	1	yes	characteristic
158	#427	C₂²×C₁₃	1	1	yes	characteristic
159	#427	C₂²×C₁₃	1	1	yes	characteristic
160	#421	—	2	1	no	—
161	#418	—	2	1	no	—
162	#365	—	2	2	no	—
163	#361	—	2	2	no	—
164	#357	—	2	2	no	—
165	#350	—	2	2	no	—
166	#344	—	2	1	no	—
167	C₂²×C₁₃	—	4	1	no	—
168	C₂²×C₁₃	—	4	2	no	—
169	C₂²×C₁₃	—	4	1	no	—
170	C₂²×C₁₃	—	8	4	no	—
171	C₂²×C₁₃	—	8	2	no	—
172	C₂²×C₁₃	—	16	2	no	—
173	C₂²×C₁₃	—	16	2	no	—
174	C₂²×C₁₃	#450	1	1	yes	characteristic
175	C₂²×C₁₃	#469	1	1	yes	characteristic
176	C₂²×C₁₃	#482	1	1	yes	characteristic
177	C₄×C₁₃	—	4	1	no	—
178	C₄×C₁₃	—	8	2	no	—
179	C₄×C₁₃	—	16	1	no	—
180	C₄×C₁₃	#394	1	1	yes	characteristic
181	C₄×C₁₃	#449	1	1	yes	characteristic
182	C₄×C₁₃	#465	1	1	yes	characteristic
183	C₄×C₁₃	#481	1	1	yes	characteristic
184	C₂×#41	D₄×C₁₃	1	1	yes	characteristic
185	C₂²×D₄	—	2	1	no	—
186	C₂×D₈	—	2	1	no	—
187	C₂×D₈	—	2	1	no	—
188	C₂×D₈	—	2	1	no	—
189	C₂×D₈	—	2	1	no	—
190	C₂×D₈	—	2	1	no	—
191	C₂×D₈	D₄×C₁₃	1	2	yes	—
192	#131	—	2	1	no	—
193	#131	—	2	1	no	—
194	#131	—	2	1	no	—
195	#131	—	2	1	no	—
196	C₂×M₁₆	—	2	1	no	—
197	C₂×M₁₆	—	2	1	no	—
198	C₂×M₁₆	D₄×C₁₃	1	1	yes	characteristic
199	C₂²×C₈	—	2	1	no	—
200	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
201	#127	—	4	1	no	—
202	#124	—	4	1	no	—
203	#121	—	2	4	no	—
204	#120	—	4	2	no	—
205	#117	—	2	1	no	—
206	#117	—	2	1	no	—
207	M₃₂	—	2	1	no	—
208	M₃₂	—	2	1	no	—
209	M₃₂	—	2	1	no	—
210	M₃₂	—	2	1	no	—
211	C₂×C₁₆	—	2	1	no	—
212	C₂×C₁₆	D₄×C₁₃	1	2	yes	complemented
213	#108	—	2	1	no	—
214	#108	—	2	1	no	—
215	#107	—	2	1	no	—
216	#107	—	2	1	no	—
217	#106	—	2	1	no	—
218	#106	D₄×C₁₃	1	2	yes	—
219	#104	—	4	2	no	—
220	#102	—	2	4	no	—
221	C₂×C₁₃	—	2	1	no	—
222	C₂×C₁₃	—	2	1	no	—
223	C₂×C₁₃	—	2	1	no	—
224	C₂×C₁₃	—	8	1	no	—
225	C₂×C₁₃	—	16	2	no	—
226	C₂×C₁₃	#2034	1	1	yes	characteristic
227	C₂⁴	—	4	1	no	—
228	#41	—	2	1	no	—
229	#41	—	2	1	no	—
230	#41	—	2	1	no	—
231	#41	—	2	1	no	—
232	C₂×Q₈	C₁₃×SD₁₆	1	1	yes	characteristic
233	C₂×D₄	—	2	1	no	—
234	C₂×D₄	—	2	2	no	—
235	C₂×D₄	—	4	4	no	—
236	C₂×D₄	—	4	1	no	—
237	C₂×D₄	—	4	1	no	—
238	C₂×D₄	—	4	1	no	—
239	C₂×D₄	—	4	1	no	—
240	C₂×D₄	—	4	1	no	—
241	C₂×D₄	C₁₃×SD₁₆	1	1	yes	characteristic
242	C₂²×C₄	—	2	1	no	—
243	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
244	D₈	—	2	2	no	—
245	D₈	—	2	2	no	—
246	D₈	—	4	1	no	—
247	D₈	—	4	1	no	—
248	M₁₆	—	2	1	no	—
249	M₁₆	—	2	1	no	—
250	M₁₆	—	4	1	no	—
251	M₁₆	—	4	1	no	—
252	M₁₆	—	4	1	no	—
253	M₁₆	—	4	1	no	—
254	C₂×C₈	—	2	1	no	—
255	C₂×C₈	—	4	1	no	—
256	C₂×C₈	—	4	1	no	—
257	C₂×C₈	—	4	1	no	—
258	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
259	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
260	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
261	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
262	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
263	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	4	1	no	—
266	C₂²⋊C₄	—	4	4	no	—
267	C₂²⋊C₄	—	4	1	no	—
268	C₂²⋊C₄	—	4	1	no	—
269	C₄²	—	4	1	no	—
270	C₄²	—	4	1	no	—
271	C₁₆	—	2	2	no	—
272	C₁₆	—	4	1	no	—
273	C₁₃	#13396	1	1	yes	characteristic, direct product factor, Sylow
274	C₂³	—	2	1	no	—
275	C₂³	—	4	2	no	—
276	C₂³	—	8	1	no	—
277	C₂³	—	8	1	no	—
278	C₂³	—	16	2	no	—
279	C₂³	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
280	Q₈	—	2	1	no	—
281	Q₈	—	2	1	no	—
282	D₄	—	2	1	no	—
283	D₄	—	2	1	no	—
284	D₄	—	4	1	no	—
285	D₄	—	4	1	no	—
286	D₄	—	4	2	no	—
287	D₄	—	4	2	no	—
288	D₄	—	8	1	no	—
289	D₄	—	8	1	no	—
290	D₄	—	16	4	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	4	1	no	—
293	C₂×C₄	—	4	1	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	8	1	no	—
296	C₂×C₄	C₁₃×#120	1	1	yes	characteristic
297	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
298	C₂×C₄	C₁₃×#123	1	1	yes	characteristic
299	C₂×C₄	C₂×C₁₃×SD₁₆	1	1	yes	characteristic
300	C₂×C₄	C₁₃×#135	1	1	yes	characteristic
301	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
302	C₈	—	4	1	no	—
303	C₈	—	8	1	no	—
304	C₈	C₁₃×#121	1	1	yes	characteristic
305	C₈	C₁₃×#121	1	1	yes	characteristic
306	C₈	C₁₃×#123	1	1	yes	characteristic
307	C₈	C₁₃×#124	1	1	yes	characteristic
308	C₂²	—	4	1	no	—
309	C₂²	—	4	2	no	—
310	C₂²	—	4	1	no	—
311	C₂²	—	8	4	no	—
312	C₂²	—	8	2	no	—
313	C₂²	—	16	2	no	—
314	C₂²	—	16	2	no	—
315	C₂²	C₁₃×#450	1	1	yes	characteristic
316	C₂²	C₁₃×#469	1	1	yes	characteristic
317	C₂²	C₁₃×#482	1	1	yes	characteristic
318	C₄	—	4	1	no	—
319	C₄	—	8	2	no	—
320	C₄	—	16	1	no	—
321	C₄	C₁₃×#394	1	1	yes	characteristic
322	C₄	C₁₃×#449	1	1	yes	characteristic
323	C₄	C₁₃×#465	1	1	yes	characteristic
324	C₄	C₁₃×#481	1	1	yes	characteristic
325	C₂	—	2	1	no	—
326	C₂	—	2	1	no	—
327	C₂	—	2	1	no	—
328	C₂	—	8	1	no	—
329	C₂	—	16	2	no	—
330	C₂	C₁₃×#2034	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₁₃	#2034
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₁₃
prime radical	C₁₃×#3058	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₃ ◁ C₂²×C₄×C₁₃ ◁ C₂×C₁₃×#131 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₃×#2155	C₂	semidirect
3. C₂×C₁₃×#361	C₂	semidirect
4. C₁₃×#2140	C₂	semidirect
5. C₁₃×#2117	C₂	semidirect
6. C₁₃×#1814	C₂	semidirect
7. C₁₃×#1384	C₂	semidirect
53. C₂×C₁₃×C₁₆	D₄	semidirect
62. #13396	C₁₃	direct (N × H)
110. #2155	C₂×C₁₃	semidirect
111. C₂×#361	C₂×C₁₃	semidirect
112. #2140	C₂×C₁₃	semidirect
113. #2117	C₂×C₁₃	semidirect
114. #1814	C₂×C₁₃	semidirect
115. #1384	C₂×C₁₃	semidirect
212. C₂×C₁₆	D₄×C₁₃	semidirect
273. C₁₃	#13396	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×#3058	2
C₁₃×#2155	2
C₂×C₁₃×#361	2
C₁₃×#2140	2
C₁₃×#2117	2
C₁₃×#1814	2
C₁₃×#1384	2
#13396	13

Subgroups — C13×#13396 #15699735

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#13396 #15699735