Subgroups — C₁₃×#9685 #15696024

Order 3328.

640 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#2858	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#1767	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₁₃×#97	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₁₃×#329	C₂	1	4	yes	maximal, complemented
5	C₂×C₁₃×C₄⋊Q₈	C₄	1	1	yes	characteristic
6	C₂×C₁₃×#127	C₄	1	1	yes	characteristic
7	C₂×C₁₃×#124	C₄	1	2	yes	—
8	C₂×C₁₃×#124	C₂²	1	2	yes	—
9	C₂×C₁₃×C₄⋊C₈	—	2	1	no	—
10	C₂×C₁₃×#97	—	2	1	no	—
11	C₂×C₁₃×#97	C₂²	1	2	yes	—
12	C₂×C₁₃×#97	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₁₃	—	2	1	no	—
14	C₁₃×#390	—	2	1	no	—
15	C₁₃×#390	—	2	1	no	—
16	C₁₃×#390	—	2	1	no	—
17	C₁₃×#390	—	2	1	no	—
18	C₄³×C₁₃	C₂²	1	1	yes	characteristic
19	C₁₃×#336	—	2	1	no	—
20	C₁₃×#329	—	2	8	no	—
21	C₂²×Q₈×C₁₃	C₂×C₄	1	2	yes	—
22	C₂²×D₄×C₁₃	—	2	1	no	—
23	C₂²×D₄×C₁₃	—	2	1	no	—
24	C₂²×D₄×C₁₃	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₁₃	—	2	2	no	—
26	C₂²×C₈×C₁₃	—	2	2	no	—
27	C₁₃×C₄⋊Q₈	—	4	2	no	—
28	C₁₃×#127	—	4	2	no	—
29	C₁₃×#124	—	2	4	no	—
30	C₁₃×#124	—	2	4	no	—
31	C₁₃×#124	—	4	4	no	—
32	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
33	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
34	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
35	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
36	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
37	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
38	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
45	C₁₃×C₄⋊C₈	—	2	4	no	—
46	C₁₃×#97	—	2	4	no	—
47	C₁₃×#97	—	2	4	no	—
48	C₁₃×#97	—	2	2	no	—
49	C₁₃×#97	D₄	1	4	yes	complemented
50	C₄×C₈×C₁₃	—	2	4	no	—
51	#9685	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
52	C₂⁴×C₁₃	—	2	2	no	—
53	C₂×Q₈×C₁₃	—	2	4	no	—
54	C₂×Q₈×C₁₃	—	4	2	no	—
55	C₂×Q₈×C₁₃	—	4	2	no	—
56	C₂×D₄×C₁₃	—	2	4	no	—
57	C₂×D₄×C₁₃	—	4	2	no	—
58	C₂×D₄×C₁₃	—	4	2	no	—
59	C₂×D₄×C₁₃	—	4	2	no	—
60	C₂×D₄×C₁₃	—	4	2	no	—
61	C₂×D₄×C₁₃	—	4	2	no	—
62	C₂×D₄×C₁₃	—	4	2	no	—
63	C₂×D₄×C₁₃	—	4	2	no	—
64	C₂×D₄×C₁₃	—	4	2	no	—
65	C₂²×C₄×C₁₃	—	2	2	no	—
66	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
73	C₂×C₈×C₁₃	—	2	4	no	—
74	C₂×C₈×C₁₃	—	2	4	no	—
75	C₂×C₈×C₁₃	—	2	4	no	—
76	C₂×C₈×C₁₃	—	2	4	no	—
77	C₂×C₈×C₁₃	—	4	2	no	—
78	C₂×C₈×C₁₃	—	4	2	no	—
79	C₁₃×C₄⋊C₄	—	4	2	no	—
80	C₁₃×C₄⋊C₄	—	4	2	no	—
81	C₁₃×C₂²⋊C₄	—	4	4	no	—
82	C₁₃×C₂²⋊C₄	—	4	4	no	—
83	C₁₃×C₂²⋊C₄	—	4	4	no	—
84	C₁₃×C₂²⋊C₄	—	4	2	no	—
85	C₁₃×C₂²⋊C₄	—	4	2	no	—
86	C₄²×C₁₃	—	2	2	no	—
87	C₄²×C₁₃	—	2	2	no	—
88	C₄²×C₁₃	—	2	2	no	—
89	C₄²×C₁₃	—	2	2	no	—
90	C₄²×C₁₃	—	2	2	no	—
91	C₄²×C₁₃	—	4	1	no	—
92	C₄²×C₁₃	C₂×D₄	1	2	yes	—
93	C₄²×C₁₃	#41	1	2	yes	—
94	#2858	C₂×C₁₃	1	1	yes	characteristic
95	#1767	C₂×C₁₃	1	1	yes	characteristic, complemented
96	C₄×#97	C₂×C₁₃	1	1	yes	characteristic, complemented
97	C₂×#329	C₂×C₁₃	1	4	yes	complemented
98	C₂³×C₁₃	—	4	2	no	—
99	C₂³×C₁₃	—	4	2	no	—
100	C₂³×C₁₃	—	4	2	no	—
101	C₂³×C₁₃	—	4	2	no	—
102	C₂³×C₁₃	—	4	2	no	—
103	C₂³×C₁₃	—	4	2	no	—
104	C₂³×C₁₃	—	4	2	no	—
105	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	Q₈×C₁₃	—	8	4	no	—
107	D₄×C₁₃	—	8	4	no	—
108	D₄×C₁₃	—	8	4	no	—
109	D₄×C₁₃	—	8	4	no	—
110	C₂×C₄×C₁₃	—	2	1	no	—
111	C₂×C₄×C₁₃	—	2	1	no	—
112	C₂×C₄×C₁₃	—	2	1	no	—
113	C₂×C₄×C₁₃	—	2	1	no	—
114	C₂×C₄×C₁₃	—	2	1	no	—
115	C₂×C₄×C₁₃	—	2	1	no	—
116	C₂×C₄×C₁₃	—	2	1	no	—
117	C₂×C₄×C₁₃	—	2	1	no	—
118	C₂×C₄×C₁₃	—	2	1	no	—
119	C₂×C₄×C₁₃	—	2	1	no	—
120	C₂×C₄×C₁₃	—	2	1	no	—
121	C₂×C₄×C₁₃	—	2	1	no	—
122	C₂×C₄×C₁₃	—	2	1	no	—
123	C₂×C₄×C₁₃	—	4	2	no	—
124	C₂×C₄×C₁₃	—	4	2	no	—
125	C₂×C₄×C₁₃	—	4	2	no	—
126	C₂×C₄×C₁₃	#100	1	2	yes	—
127	C₂×C₄×C₁₃	#100	1	2	yes	—
128	C₂×C₄×C₁₃	#104	1	2	yes	—
129	C₂×C₄×C₁₃	#104	1	2	yes	—
130	C₂×C₄×C₁₃	#104	1	2	yes	—
131	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
132	C₂×C₄×C₁₃	#121	1	2	yes	—
133	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
134	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
135	C₈×C₁₃	—	2	8	no	—
136	C₈×C₁₃	—	4	4	no	—
137	C₂×C₄⋊Q₈	C₄×C₁₃	1	1	yes	characteristic
138	C₂×#127	C₄×C₁₃	1	1	yes	characteristic
139	C₂×#124	C₄×C₁₃	1	2	yes	—
140	C₂×#124	C₂²×C₁₃	1	2	yes	—
141	C₂×C₄⋊C₈	—	2	1	no	—
142	C₂×#97	—	2	1	no	—
143	C₂×#97	C₂²×C₁₃	1	2	yes	—
144	C₂×#97	C₂²×C₁₃	1	2	yes	—
145	C₂×C₄×C₈	—	2	1	no	—
146	#390	—	2	1	no	—
147	#390	—	2	1	no	—
148	#390	—	2	1	no	—
149	#390	—	2	1	no	—
150	C₄³	C₂²×C₁₃	1	1	yes	characteristic
151	#336	—	2	1	no	—
152	#329	—	2	8	no	—
153	C₂²×C₁₃	—	8	2	no	—
154	C₂²×C₁₃	—	8	2	no	—
155	C₂²×C₁₃	—	8	2	no	—
156	C₂²×C₁₃	—	8	2	no	—
157	C₂²×C₁₃	—	8	2	no	—
158	C₂²×C₁₃	—	8	2	no	—
159	C₂²×C₁₃	—	8	2	no	—
160	C₂²×C₁₃	#390	1	1	yes	characteristic
161	C₂²×C₁₃	C₂×#100	1	1	yes	characteristic
162	C₂²×C₁₃	C₂×#100	1	1	yes	characteristic
163	C₂²×C₁₃	C₂×#104	1	1	yes	characteristic
164	C₂²×C₁₃	C₂×#104	1	1	yes	characteristic
165	C₂²×C₁₃	C₂×#104	1	1	yes	characteristic
166	C₂²×C₁₃	#421	1	1	yes	characteristic
167	C₄×C₁₃	—	2	2	no	—
168	C₄×C₁₃	—	2	2	no	—
169	C₄×C₁₃	—	2	1	no	—
170	C₄×C₁₃	—	2	1	no	—
171	C₄×C₁₃	—	2	2	no	—
172	C₄×C₁₃	—	2	2	no	—
173	C₄×C₁₃	—	4	1	no	—
174	C₄×C₁₃	—	8	2	no	—
175	C₄×C₁₃	#329	1	4	yes	complemented
176	C₂²×Q₈	C₂×C₄×C₁₃	1	2	yes	—
177	C₂²×D₄	—	2	1	no	—
178	C₂²×D₄	—	2	1	no	—
179	C₂²×D₄	C₂×C₄×C₁₃	1	2	yes	—
180	C₂²×C₈	—	2	2	no	—
181	C₂²×C₈	—	2	2	no	—
182	C₄⋊Q₈	—	4	2	no	—
183	#127	—	4	2	no	—
184	#124	—	2	4	no	—
185	#124	—	2	4	no	—
186	#124	—	4	4	no	—
187	C₂×C₄⋊C₄	—	2	1	no	—
188	C₂×C₄⋊C₄	—	2	1	no	—
189	C₂×C₂²⋊C₄	—	2	2	no	—
190	C₂×C₂²⋊C₄	—	2	2	no	—
191	C₂×C₂²⋊C₄	—	2	2	no	—
192	C₂×C₂²⋊C₄	—	2	2	no	—
193	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
194	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
195	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
196	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
197	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
198	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
199	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
200	C₄⋊C₈	—	2	4	no	—
201	#97	—	2	4	no	—
202	#97	—	2	4	no	—
203	#97	—	2	2	no	—
204	#97	D₄×C₁₃	1	4	yes	complemented
205	C₄×C₈	—	2	4	no	—
206	C₂×C₁₃	—	16	2	no	—
207	C₂×C₁₃	C₂×#329	1	1	yes	characteristic
208	C₂×C₁₃	#1525	1	1	yes	characteristic
209	C₂×C₁₃	#1527	1	1	yes	characteristic
210	C₂×C₁₃	#1530	1	1	yes	characteristic
211	C₂×C₁₃	#1962	1	1	yes	characteristic
212	C₂×C₁₃	#1965	1	1	yes	characteristic
213	C₂×C₁₃	#1980	1	1	yes	characteristic
214	C₂⁴	—	2	2	no	—
215	C₂×Q₈	—	2	4	no	—
216	C₂×Q₈	—	4	2	no	—
217	C₂×Q₈	—	4	2	no	—
218	C₂×D₄	—	2	4	no	—
219	C₂×D₄	—	4	2	no	—
220	C₂×D₄	—	4	2	no	—
221	C₂×D₄	—	4	2	no	—
222	C₂×D₄	—	4	2	no	—
223	C₂×D₄	—	4	2	no	—
224	C₂×D₄	—	4	2	no	—
225	C₂×D₄	—	4	2	no	—
226	C₂×D₄	—	4	2	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
229	C₂²×C₄	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
230	C₂²×C₄	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
231	C₂²×C₄	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
232	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
233	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
234	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
235	C₂×C₈	—	2	4	no	—
236	C₂×C₈	—	2	4	no	—
237	C₂×C₈	—	2	4	no	—
238	C₂×C₈	—	2	4	no	—
239	C₂×C₈	—	4	2	no	—
240	C₂×C₈	—	4	2	no	—
241	C₄⋊C₄	—	4	2	no	—
242	C₄⋊C₄	—	4	2	no	—
243	C₂²⋊C₄	—	4	4	no	—
244	C₂²⋊C₄	—	4	4	no	—
245	C₂²⋊C₄	—	4	4	no	—
246	C₂²⋊C₄	—	4	2	no	—
247	C₂²⋊C₄	—	4	2	no	—
248	C₄²	—	2	2	no	—
249	C₄²	—	2	2	no	—
250	C₄²	—	2	2	no	—
251	C₄²	—	2	2	no	—
252	C₄²	—	2	2	no	—
253	C₄²	—	4	1	no	—
254	C₄²	C₂×D₄×C₁₃	1	2	yes	—
255	C₄²	C₁₃×#41	1	2	yes	—
256	C₁₃	#9685	1	1	yes	characteristic, direct product factor, Sylow
257	C₂³	—	4	2	no	—
258	C₂³	—	4	2	no	—
259	C₂³	—	4	2	no	—
260	C₂³	—	4	2	no	—
261	C₂³	—	4	2	no	—
262	C₂³	—	4	2	no	—
263	C₂³	—	4	2	no	—
264	C₂³	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
265	Q₈	—	8	4	no	—
266	D₄	—	8	4	no	—
267	D₄	—	8	4	no	—
268	D₄	—	8	4	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	C₁₃×#100	1	2	yes	—
286	C₂×C₄	C₁₃×#100	1	2	yes	—
287	C₂×C₄	C₁₃×#104	1	2	yes	—
288	C₂×C₄	C₁₃×#104	1	2	yes	—
289	C₂×C₄	C₁₃×#104	1	2	yes	—
290	C₂×C₄	C₄×D₄×C₁₃	1	2	yes	—
291	C₂×C₄	C₁₃×#121	1	2	yes	—
292	C₂×C₄	C₁₃×#124	1	1	yes	characteristic
293	C₂×C₄	C₁₃×#127	1	1	yes	characteristic
294	C₈	—	2	8	no	—
295	C₈	—	4	4	no	—
296	C₂²	—	8	2	no	—
297	C₂²	—	8	2	no	—
298	C₂²	—	8	2	no	—
299	C₂²	—	8	2	no	—
300	C₂²	—	8	2	no	—
301	C₂²	—	8	2	no	—
302	C₂²	—	8	2	no	—
303	C₂²	C₁₃×#390	1	1	yes	characteristic
304	C₂²	C₂×C₁₃×#100	1	1	yes	characteristic
305	C₂²	C₂×C₁₃×#100	1	1	yes	characteristic
306	C₂²	C₂×C₁₃×#104	1	1	yes	characteristic
307	C₂²	C₂×C₁₃×#104	1	1	yes	characteristic
308	C₂²	C₂×C₁₃×#104	1	1	yes	characteristic
309	C₂²	C₁₃×#421	1	1	yes	characteristic
310	C₄	—	2	2	no	—
311	C₄	—	2	2	no	—
312	C₄	—	2	2	no	—
313	C₄	—	2	2	no	—
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	4	1	no	—
317	C₄	—	8	2	no	—
318	C₄	C₁₃×#329	1	4	yes	complemented
319	C₂	—	16	2	no	—
320	C₂	C₂×C₁₃×#329	1	1	yes	characteristic
321	C₂	C₁₃×#1525	1	1	yes	characteristic
322	C₂	C₁₃×#1527	1	1	yes	characteristic
323	C₂	C₁₃×#1530	1	1	yes	characteristic
324	C₂	C₁₃×#1962	1	1	yes	characteristic
325	C₂	C₁₃×#1965	1	1	yes	characteristic
326	C₂	C₁₃×#1980	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₁₃
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃
prime radical	C₂×C₁₃×#127	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ C₄³×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₃×#1767	C₂	semidirect
3. C₄×C₁₃×#97	C₂	semidirect
4. C₂×C₁₃×#329	C₂	semidirect
49. C₁₃×#97	D₄	semidirect
51. #9685	C₁₃	direct (N × H)
95. #1767	C₂×C₁₃	semidirect
96. C₄×#97	C₂×C₁₃	semidirect
97. C₂×#329	C₂×C₁₃	semidirect
175. C₄×C₁₃	#329	semidirect
204. #97	D₄×C₁₃	semidirect
256. C₁₃	#9685	direct (N × H)
318. C₄	C₁₃×#329	semidirect

Maximal subgroups

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

subgroup	index
C₁₃×#2858	2
C₁₃×#1767	2
C₄×C₁₃×#97	2
C₂×C₁₃×#329	2
#9685	13

Subgroups — C13×#9685 #15696024

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#9685 #15696024