Subgroups — C₁₃×#32668 #15719007

Order 3328.

920 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#2548	C₂	1	2	yes	maximal, complemented
2	C₁₃×#2054	C₂	1	2	yes	maximal, complemented
3	C₁₃×#2011	C₂	1	4	yes	maximal, complemented
4	C₁₃×#1984	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#1970	C₂	1	2	yes	maximal, complemented
6	C₁₃×#1925	C₂	1	4	yes	maximal, complemented
7	C₂²×C₁₃×D₈	C₂²	1	2	yes	—
8	C₂²×C₁₃×D₈	C₂²	1	2	yes	—
9	C₂×C₁₃×#127	C₂²	1	2	yes	—
10	C₂×C₁₃×#124	—	2	2	no	—
11	C₂×C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×#107	C₂²	1	2	yes	—
13	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×#102	C₂²	1	4	yes	—
15	C₂×C₁₃×#102	C₂²	1	4	yes	—
16	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
17	C₁₃×#394	—	2	2	no	—
18	C₁₃×#394	—	2	2	no	—
19	C₁₃×#390	—	2	2	no	—
20	C₁₃×#390	C₂²	1	4	yes	—
21	C₁₃×#390	C₂²	1	4	yes	—
22	C₄×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
23	C₁₃×#340	C₂²	1	2	yes	—
24	C₁₃×#336	C₂²	1	4	yes	—
25	C₂²×Q₈×C₁₃	D₄	1	1	yes	characteristic
26	C₂²×D₄×C₁₃	—	2	2	no	—
27	C₂²×D₄×C₁₃	—	2	2	no	—
28	C₂²×D₄×C₁₃	C₂³	1	4	yes	—
29	C₂×C₁₃×D₈	—	2	4	no	—
30	C₂×C₁₃×D₈	—	4	2	no	—
31	C₂×C₁₃×D₈	—	4	2	no	—
32	C₂×C₁₃×D₈	—	4	4	no	—
33	C₂×C₁₃×D₈	—	4	2	no	—
34	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
35	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
36	C₁₃×#127	—	4	2	no	—
37	C₁₃×#127	—	4	2	no	—
38	C₁₃×#124	—	4	8	no	—
39	C₄×Q₈×C₁₃	—	4	1	no	—
40	C₄×Q₈×C₁₃	—	4	1	no	—
41	C₂×C₁₃×C₄⋊C₄	D₄	1	2	yes	—
42	C₂×C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
43	C₂×C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
44	C₂×C₁₃×C₂²⋊C₄	—	2	4	no	—
45	C₂×C₁₃×C₂²⋊C₄	—	2	4	no	—
46	C₂×C₁₃×C₂²⋊C₄	—	2	4	no	—
47	C₂×C₁₃×C₂²⋊C₄	—	2	4	no	—
48	C₂×C₄²×C₁₃	—	2	2	no	—
49	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
50	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
51	C₁₃×#107	—	4	2	no	—
52	C₁₃×C₄⋊C₈	—	4	1	no	—
53	C₁₃×#102	—	2	8	no	—
54	C₁₃×#102	—	4	4	no	—
55	C₄×C₈×C₁₃	—	4	1	no	—
56	C₁₃×#95	—	2	2	no	—
57	#32668	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
58	C₂⁴×C₁₃	—	2	4	no	—
59	C₂×Q₈×C₁₃	—	4	1	no	—
60	C₂×Q₈×C₁₃	—	4	1	no	—
61	C₂×Q₈×C₁₃	D₈	1	4	yes	complemented
62	C₂×D₄×C₁₃	—	2	4	no	—
63	C₂×D₄×C₁₃	—	2	4	no	—
64	C₂×D₄×C₁₃	—	2	4	no	—
65	C₂×D₄×C₁₃	—	2	4	no	—
66	C₂×D₄×C₁₃	—	2	4	no	—
67	C₂×D₄×C₁₃	—	2	4	no	—
68	C₂×D₄×C₁₃	—	4	4	no	—
69	C₂×D₄×C₁₃	—	4	4	no	—
70	C₂×D₄×C₁₃	—	4	4	no	—
71	C₂×D₄×C₁₃	—	4	4	no	—
72	C₂×D₄×C₁₃	—	8	4	no	—
73	C₂²×C₄×C₁₃	—	2	4	no	—
74	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
75	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
79	C₁₃×D₈	—	4	8	no	—
80	C₁₃×D₈	—	4	8	no	—
81	C₂×C₈×C₁₃	—	2	2	no	—
82	C₂×C₈×C₁₃	—	2	2	no	—
83	C₂×C₈×C₁₃	—	2	4	no	—
84	C₂×C₈×C₁₃	—	2	2	no	—
85	C₂×C₈×C₁₃	#41	1	4	yes	—
86	C₁₃×C₄⋊C₄	—	2	4	no	—
87	C₁₃×C₄⋊C₄	—	2	4	no	—
88	C₁₃×C₄⋊C₄	—	4	2	no	—
89	C₁₃×C₄⋊C₄	—	4	1	no	—
90	C₁₃×C₂²⋊C₄	—	4	8	no	—
91	C₁₃×C₂²⋊C₄	—	4	8	no	—
92	C₁₃×C₂²⋊C₄	—	8	4	no	—
93	C₁₃×C₂²⋊C₄	—	8	4	no	—
94	C₄²×C₁₃	—	2	1	no	—
95	C₄²×C₁₃	—	2	1	no	—
96	C₄²×C₁₃	—	2	2	no	—
97	C₄²×C₁₃	—	2	2	no	—
98	C₄²×C₁₃	—	4	4	no	—
99	#2548	C₂×C₁₃	1	2	yes	complemented
100	#2054	C₂×C₁₃	1	2	yes	complemented
101	#2011	C₂×C₁₃	1	4	yes	complemented
102	#1984	C₂×C₁₃	1	1	yes	characteristic, complemented
103	#1970	C₂×C₁₃	1	2	yes	complemented
104	#1925	C₂×C₁₃	1	4	yes	complemented
105	C₂³×C₁₃	—	4	4	no	—
106	C₂³×C₁₃	—	4	4	no	—
107	C₂³×C₁₃	—	4	4	no	—
108	C₂³×C₁₃	—	4	4	no	—
109	C₂³×C₁₃	—	4	4	no	—
110	C₂³×C₁₃	—	4	4	no	—
111	C₂³×C₁₃	—	4	4	no	—
112	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
113	Q₈×C₁₃	—	4	4	no	—
114	D₄×C₁₃	—	4	4	no	—
115	D₄×C₁₃	—	4	4	no	—
116	D₄×C₁₃	—	4	4	no	—
117	D₄×C₁₃	—	4	4	no	—
118	D₄×C₁₃	—	8	8	no	—
119	D₄×C₁₃	—	8	8	no	—
120	C₂×C₄×C₁₃	—	2	2	no	—
121	C₂×C₄×C₁₃	—	2	1	no	—
122	C₂×C₄×C₁₃	—	2	2	no	—
123	C₂×C₄×C₁₃	—	2	1	no	—
124	C₂×C₄×C₁₃	—	2	1	no	—
125	C₂×C₄×C₁₃	—	2	2	no	—
126	C₂×C₄×C₁₃	—	2	2	no	—
127	C₂×C₄×C₁₃	—	2	4	no	—
128	C₂×C₄×C₁₃	—	4	4	no	—
129	C₂×C₄×C₁₃	—	4	4	no	—
130	C₂×C₄×C₁₃	—	4	4	no	—
131	C₂×C₄×C₁₃	C₂×D₈	1	4	yes	—
132	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
133	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
134	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
135	C₂×C₄×C₁₃	H(5,𝔽₂)	1	2	yes	—
136	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
137	C₈×C₁₃	—	2	4	no	—
138	C₈×C₁₃	—	4	2	no	—
139	C₂²×D₈	C₂²×C₁₃	1	2	yes	—
140	C₂²×D₈	C₂²×C₁₃	1	2	yes	—
141	C₂×#127	C₂²×C₁₃	1	2	yes	—
142	C₂×#124	—	2	2	no	—
143	C₂×C₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic
144	C₂×#107	C₂²×C₁₃	1	2	yes	—
145	C₂×C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
146	C₂×#102	C₂²×C₁₃	1	4	yes	—
147	C₂×#102	C₂²×C₁₃	1	4	yes	—
148	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
149	#394	—	2	2	no	—
150	#394	—	2	2	no	—
151	#390	—	2	2	no	—
152	#390	C₂²×C₁₃	1	4	yes	—
153	#390	C₂²×C₁₃	1	4	yes	—
154	C₄×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
155	#340	C₂²×C₁₃	1	2	yes	—
156	#336	C₂²×C₁₃	1	4	yes	—
157	C₂²×C₁₃	—	8	4	no	—
158	C₂²×C₁₃	—	8	4	no	—
159	C₂²×C₁₃	—	8	4	no	—
160	C₂²×C₁₃	—	8	4	no	—
161	C₂²×C₁₃	—	8	4	no	—
162	C₂²×C₁₃	—	8	4	no	—
163	C₂²×C₁₃	—	8	4	no	—
164	C₂²×C₁₃	#534	1	1	yes	characteristic
165	C₂²×C₁₃	#550	1	2	yes	—
166	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
167	C₂²×C₁₃	#575	1	1	yes	characteristic
168	C₂²×C₁₃	#576	1	2	yes	—
169	C₄×C₁₃	—	2	4	no	—
170	C₄×C₁₃	—	2	2	no	—
171	C₄×C₁₃	—	4	1	no	—
172	C₄×C₁₃	—	4	2	no	—
173	C₄×C₁₃	—	8	4	no	—
174	C₄×C₁₃	#538	1	2	yes	—
175	C₄×C₁₃	#555	1	1	yes	characteristic
176	C₄×C₁₃	H(3,𝔽₄)	1	1	yes	characteristic
177	C₂²×Q₈	D₄×C₁₃	1	1	yes	characteristic
178	C₂²×D₄	—	2	2	no	—
179	C₂²×D₄	—	2	2	no	—
180	C₂²×D₄	C₂³×C₁₃	1	4	yes	—
181	C₂×D₈	—	2	4	no	—
182	C₂×D₈	—	4	2	no	—
183	C₂×D₈	—	4	2	no	—
184	C₂×D₈	—	4	4	no	—
185	C₂×D₈	—	4	2	no	—
186	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
187	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
188	#127	—	4	2	no	—
189	#127	—	4	2	no	—
190	#124	—	4	8	no	—
191	C₄×Q₈	—	4	1	no	—
192	C₄×Q₈	—	4	1	no	—
193	C₂×C₄⋊C₄	D₄×C₁₃	1	2	yes	—
194	C₂×C₄⋊C₄	D₄×C₁₃	1	1	yes	characteristic
195	C₂×C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
196	C₂×C₂²⋊C₄	—	2	4	no	—
197	C₂×C₂²⋊C₄	—	2	4	no	—
198	C₂×C₂²⋊C₄	—	2	4	no	—
199	C₂×C₂²⋊C₄	—	2	4	no	—
200	C₂×C₄²	—	2	2	no	—
201	C₂×C₄²	C₂³×C₁₃	1	2	yes	—
202	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
203	#107	—	4	2	no	—
204	C₄⋊C₈	—	4	1	no	—
205	#102	—	2	8	no	—
206	#102	—	4	4	no	—
207	C₄×C₈	—	4	1	no	—
208	#95	—	2	2	no	—
209	C₂×C₁₃	—	16	4	no	—
210	C₂×C₁₃	#2733	1	1	yes	characteristic
211	C₂×C₁₃	#3186	1	1	yes	characteristic
212	C₂×C₁₃	#3199	1	1	yes	characteristic
213	C₂×C₁₃	#3391	1	2	yes	—
214	C₂×C₁₃	#3403	1	2	yes	—
215	C₂⁴	—	2	4	no	—
216	C₂×Q₈	—	4	1	no	—
217	C₂×Q₈	—	4	1	no	—
218	C₂×Q₈	C₁₃×D₈	1	4	yes	complemented
219	C₂×D₄	—	2	4	no	—
220	C₂×D₄	—	2	4	no	—
221	C₂×D₄	—	2	4	no	—
222	C₂×D₄	—	2	4	no	—
223	C₂×D₄	—	2	4	no	—
224	C₂×D₄	—	2	4	no	—
225	C₂×D₄	—	4	4	no	—
226	C₂×D₄	—	4	4	no	—
227	C₂×D₄	—	4	4	no	—
228	C₂×D₄	—	4	4	no	—
229	C₂×D₄	—	8	4	no	—
230	C₂²×C₄	—	2	4	no	—
231	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
232	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
233	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
234	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
235	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
236	D₈	—	4	8	no	—
237	D₈	—	4	8	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	2	4	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	C₁₃×#41	1	4	yes	—
243	C₄⋊C₄	—	2	4	no	—
244	C₄⋊C₄	—	2	4	no	—
245	C₄⋊C₄	—	4	2	no	—
246	C₄⋊C₄	—	4	1	no	—
247	C₂²⋊C₄	—	4	8	no	—
248	C₂²⋊C₄	—	4	8	no	—
249	C₂²⋊C₄	—	8	4	no	—
250	C₂²⋊C₄	—	8	4	no	—
251	C₄²	—	2	1	no	—
252	C₄²	—	2	1	no	—
253	C₄²	—	2	2	no	—
254	C₄²	—	2	2	no	—
255	C₄²	—	4	4	no	—
256	C₁₃	#32668	1	1	yes	characteristic, direct product factor, Sylow
257	C₂³	—	4	4	no	—
258	C₂³	—	4	4	no	—
259	C₂³	—	4	4	no	—
260	C₂³	—	4	4	no	—
261	C₂³	—	4	4	no	—
262	C₂³	—	4	4	no	—
263	C₂³	—	4	4	no	—
264	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
265	Q₈	—	4	4	no	—
266	D₄	—	4	4	no	—
267	D₄	—	4	4	no	—
268	D₄	—	4	4	no	—
269	D₄	—	4	4	no	—
270	D₄	—	8	8	no	—
271	D₄	—	8	8	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	4	no	—
280	C₂×C₄	—	4	4	no	—
281	C₂×C₄	—	4	4	no	—
282	C₂×C₄	—	4	4	no	—
283	C₂×C₄	C₂×C₁₃×D₈	1	4	yes	—
284	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
285	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
286	C₂×C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
287	C₂×C₄	C₁₃×H(5,𝔽₂)	1	2	yes	—
288	C₂×C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
289	C₈	—	2	4	no	—
290	C₈	—	4	2	no	—
291	C₂²	—	8	4	no	—
292	C₂²	—	8	4	no	—
293	C₂²	—	8	4	no	—
294	C₂²	—	8	4	no	—
295	C₂²	—	8	4	no	—
296	C₂²	—	8	4	no	—
297	C₂²	—	8	4	no	—
298	C₂²	C₁₃×#534	1	1	yes	characteristic
299	C₂²	C₁₃×#550	1	2	yes	—
300	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
301	C₂²	C₁₃×#575	1	1	yes	characteristic
302	C₂²	C₁₃×#576	1	2	yes	—
303	C₄	—	2	4	no	—
304	C₄	—	2	2	no	—
305	C₄	—	4	1	no	—
306	C₄	—	4	2	no	—
307	C₄	—	8	4	no	—
308	C₄	C₁₃×#538	1	2	yes	—
309	C₄	C₁₃×#555	1	1	yes	characteristic
310	C₄	C₁₃×H(3,𝔽₄)	1	1	yes	characteristic
311	C₂	—	16	4	no	—
312	C₂	C₁₃×#2733	1	1	yes	characteristic
313	C₂	C₁₃×#3186	1	1	yes	characteristic
314	C₂	C₁₃×#3199	1	1	yes	characteristic
315	C₂	C₁₃×#3391	1	2	yes	—
316	C₂	C₁₃×#3403	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ C₂×C₄×Q₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₃×#2548	C₂	semidirect
2. C₁₃×#2054	C₂	semidirect
3. C₁₃×#2011	C₂	semidirect
4. C₁₃×#1984	C₂	semidirect
5. C₁₃×#1970	C₂	semidirect
6. C₁₃×#1925	C₂	semidirect
57. #32668	C₁₃	direct (N × H)
61. C₂×Q₈×C₁₃	D₈	semidirect
99. #2548	C₂×C₁₃	semidirect
100. #2054	C₂×C₁₃	semidirect
101. #2011	C₂×C₁₃	semidirect
102. #1984	C₂×C₁₃	semidirect
103. #1970	C₂×C₁₃	semidirect
104. #1925	C₂×C₁₃	semidirect
218. C₂×Q₈	C₁₃×D₈	semidirect
256. C₁₃	#32668	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×#2548	2
C₁₃×#2054	2
C₁₃×#2011	2
C₁₃×#1984	2
C₁₃×#1970	2
C₁₃×#1925	2
#32668	13

Subgroups — C13×#32668 #15719007

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#32668 #15719007