Subgroups — C₁₃×#29427 #15715766

Order 3328.

724 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#2323	C₂	1	2	yes	maximal
2	C₁₃×#2075	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2036	C₂	1	2	yes	maximal
4	C₁₃×#2001	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#1928	C₂	1	2	yes	maximal, complemented
6	C₁₃×#1918	C₂	1	2	yes	maximal, complemented
7	C₁₃×#1878	C₂	1	2	yes	maximal, complemented
8	C₁₃×#1863	C₂	1	2	yes	maximal, complemented
9	C₁₃×#1818	C₂	1	1	yes	maximal, characteristic
10	C₂²×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×C₂²⋊Q₈	C₂²	1	2	yes	—
12	C₂×C₄×Q₈×C₁₃	C₂²	1	2	yes	—
13	C₂×C₁₃×#117	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×#107	C₂²	1	2	yes	—
15	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
16	C₂×C₁₃×#103	C₂²	1	2	yes	—
17	C₂×C₁₃×#103	C₂²	1	2	yes	—
18	C₂×C₁₃×#103	C₂²	1	2	yes	—
19	C₂×C₁₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×C₁₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
21	C₁₃×#391	C₂²	1	2	yes	—
22	C₁₃×#389	—	2	2	no	—
23	C₁₃×#389	C₂²	1	2	yes	—
24	C₁₃×#388	—	2	2	no	—
25	C₁₃×#388	—	2	2	no	—
26	C₁₃×#387	—	2	2	no	—
27	C₁₃×#380	C₂²	1	2	yes	—
28	C₄×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
29	C₁₃×#340	C₂²	1	1	yes	characteristic
30	C₁₃×#340	C₂²	1	2	yes	—
31	C₁₃×#340	C₂²	1	2	yes	—
32	C₁₃×#340	C₂²	1	1	yes	characteristic
33	C₁₃×#336	C₂²	1	2	yes	—
34	C₁₃×#336	C₂²	1	2	yes	—
35	C₂²×Q₈×C₁₃	C₂³	1	2	yes	—
36	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
37	C₂×C₁₃×Q₁₆	—	2	4	no	—
38	C₂×C₁₃×Q₁₆	—	4	1	no	—
39	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
40	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
41	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
42	C₁₃×C₂²⋊Q₈	—	2	4	no	—
43	C₁₃×C₂²⋊Q₈	—	2	4	no	—
44	C₄×Q₈×C₁₃	—	4	2	no	—
45	C₄×Q₈×C₁₃	—	4	2	no	—
46	C₁₃×#117	—	4	1	no	—
47	C₁₃×#117	—	4	1	no	—
48	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
49	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
50	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
51	C₂×C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
52	C₂×C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
53	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
54	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
55	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
56	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
57	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
58	C₂×C₁₃×C₂²⋊C₄	D₄	1	2	yes	—
59	C₂×C₄²×C₁₃	—	2	2	no	—
60	C₂×C₄²×C₁₃	—	2	2	no	—
61	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
62	C₁₃×#107	—	2	4	no	—
63	C₁₃×#107	—	4	1	no	—
64	C₁₃×#103	—	2	4	no	—
65	C₁₃×#103	—	2	4	no	—
66	C₁₃×#103	—	4	2	no	—
67	C₁₃×C₂²⋊C₈	—	4	1	no	—
68	C₁₃×C₂²⋊C₈	—	4	1	no	—
69	C₁₃×#95	—	2	2	no	—
70	C₁₃×#95	—	2	2	no	—
71	C₁₃×#95	—	2	2	no	—
72	C₁₃×#95	—	2	2	no	—
73	C₁₃×#95	—	2	2	no	—
74	#29427	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
75	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
76	C₂×Q₈×C₁₃	—	2	2	no	—
77	C₂×Q₈×C₁₃	—	2	2	no	—
78	C₂×Q₈×C₁₃	—	2	2	no	—
79	C₂×Q₈×C₁₃	—	2	2	no	—
80	C₂×Q₈×C₁₃	—	2	2	no	—
81	C₂×Q₈×C₁₃	#41	1	4	yes	—
82	C₂²×C₄×C₁₃	—	2	2	no	—
83	C₂²×C₄×C₁₃	—	2	2	no	—
84	C₂²×C₄×C₁₃	—	2	2	no	—
85	C₂²×C₄×C₁₃	—	2	2	no	—
86	C₂²×C₄×C₁₃	—	2	1	no	—
87	C₂²×C₄×C₁₃	—	2	1	no	—
88	C₂²×C₄×C₁₃	—	2	2	no	—
89	C₂²×C₄×C₁₃	—	2	2	no	—
90	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
91	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
93	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
94	C₁₃×Q₁₆	—	4	4	no	—
95	C₂×C₈×C₁₃	—	2	2	no	—
96	C₂×C₈×C₁₃	—	2	2	no	—
97	C₂×C₈×C₁₃	—	2	2	no	—
98	C₂×C₈×C₁₃	—	2	2	no	—
99	C₂×C₈×C₁₃	—	2	2	no	—
100	C₂×C₈×C₁₃	—	2	1	no	—
101	C₂×C₈×C₁₃	—	2	1	no	—
102	C₁₃×C₄⋊C₄	—	2	8	no	—
103	C₁₃×C₄⋊C₄	—	2	2	no	—
104	C₁₃×C₄⋊C₄	—	2	2	no	—
105	C₁₃×C₄⋊C₄	—	2	2	no	—
106	C₁₃×C₄⋊C₄	—	2	2	no	—
107	C₁₃×C₄⋊C₄	—	2	2	no	—
108	C₁₃×C₄⋊C₄	—	4	1	no	—
109	C₁₃×C₄⋊C₄	—	4	1	no	—
110	C₁₃×C₄⋊C₄	—	4	4	no	—
111	C₁₃×C₄⋊C₄	—	8	2	no	—
112	C₁₃×C₄⋊C₄	#41	1	4	yes	—
113	C₁₃×C₂²⋊C₄	—	2	8	no	—
114	C₁₃×C₂²⋊C₄	—	4	2	no	—
115	C₁₃×C₂²⋊C₄	—	8	2	no	—
116	C₄²×C₁₃	—	2	2	no	—
117	C₄²×C₁₃	—	2	2	no	—
118	C₄²×C₁₃	—	4	4	no	—
119	C₄²×C₁₃	—	8	2	no	—
120	#2323	C₂×C₁₃	1	2	yes	—
121	#2075	C₂×C₁₃	1	1	yes	characteristic
122	#2036	C₂×C₁₃	1	2	yes	—
123	#2001	C₂×C₁₃	1	1	yes	characteristic
124	#1928	C₂×C₁₃	1	2	yes	complemented
125	#1918	C₂×C₁₃	1	2	yes	complemented
126	#1878	C₂×C₁₃	1	2	yes	complemented
127	#1863	C₂×C₁₃	1	2	yes	complemented
128	#1818	C₂×C₁₃	1	1	yes	characteristic
129	C₂³×C₁₃	—	2	2	no	—
130	C₂³×C₁₃	—	2	1	no	—
131	C₂³×C₁₃	—	2	1	no	—
132	C₂³×C₁₃	—	2	2	no	—
133	C₂³×C₁₃	—	2	1	no	—
134	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
135	Q₈×C₁₃	—	2	4	no	—
136	Q₈×C₁₃	—	2	4	no	—
137	Q₈×C₁₃	—	4	2	no	—
138	Q₈×C₁₃	—	4	2	no	—
139	C₂×C₄×C₁₃	—	2	4	no	—
140	C₂×C₄×C₁₃	—	2	4	no	—
141	C₂×C₄×C₁₃	—	2	2	no	—
142	C₂×C₄×C₁₃	—	2	2	no	—
143	C₂×C₄×C₁₃	—	2	2	no	—
144	C₂×C₄×C₁₃	—	2	1	no	—
145	C₂×C₄×C₁₃	—	2	2	no	—
146	C₂×C₄×C₁₃	—	2	4	no	—
147	C₂×C₄×C₁₃	—	2	2	no	—
148	C₂×C₄×C₁₃	—	4	2	no	—
149	C₂×C₄×C₁₃	—	4	2	no	—
150	C₂×C₄×C₁₃	—	4	2	no	—
151	C₂×C₄×C₁₃	—	4	2	no	—
152	C₂×C₄×C₁₃	—	4	1	no	—
153	C₂×C₄×C₁₃	—	4	2	no	—
154	C₂×C₄×C₁₃	—	4	1	no	—
155	C₂×C₄×C₁₃	—	4	2	no	—
156	C₂×C₄×C₁₃	—	4	2	no	—
157	C₂×C₄×C₁₃	—	4	2	no	—
158	C₂×C₄×C₁₃	—	4	2	no	—
159	C₂×C₄×C₁₃	—	4	2	no	—
160	C₂×C₄×C₁₃	—	4	2	no	—
161	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
162	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
163	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
164	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
165	C₈×C₁₃	—	4	1	no	—
166	C₈×C₁₃	—	4	2	no	—
167	C₈×C₁₃	—	4	1	no	—
168	C₂²×Q₁₆	C₂²×C₁₃	1	1	yes	characteristic
169	C₂×C₂²⋊Q₈	C₂²×C₁₃	1	2	yes	—
170	C₂×C₄×Q₈	C₂²×C₁₃	1	2	yes	—
171	C₂×#117	C₂²×C₁₃	1	1	yes	characteristic
172	C₂×#107	C₂²×C₁₃	1	2	yes	—
173	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
174	C₂×#103	C₂²×C₁₃	1	2	yes	—
175	C₂×#103	C₂²×C₁₃	1	2	yes	—
176	C₂×#103	C₂²×C₁₃	1	2	yes	—
177	C₂×C₂²⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
178	C₂×C₂²⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
179	#391	C₂²×C₁₃	1	2	yes	—
180	#389	—	2	2	no	—
181	#389	C₂²×C₁₃	1	2	yes	—
182	#388	—	2	2	no	—
183	#388	—	2	2	no	—
184	#387	—	2	2	no	—
185	#380	C₂²×C₁₃	1	2	yes	—
186	C₄×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
187	#340	C₂²×C₁₃	1	2	yes	—
188	#340	C₂²×C₁₃	1	1	yes	characteristic
189	#340	C₂²×C₁₃	1	2	yes	—
190	#340	C₂²×C₁₃	1	1	yes	characteristic
191	#336	C₂²×C₁₃	1	2	yes	—
192	#336	C₂²×C₁₃	1	2	yes	—
193	C₂²×C₁₃	—	4	1	no	—
194	C₂²×C₁₃	—	4	2	no	—
195	C₂²×C₁₃	—	4	2	no	—
196	C₂²×C₁₃	—	4	1	no	—
197	C₂²×C₁₃	—	4	1	no	—
198	C₂²×C₁₃	#525	1	2	yes	—
199	C₂²×C₁₃	#534	1	1	yes	characteristic
200	C₂²×C₁₃	#576	1	1	yes	characteristic
201	C₂²×C₁₃	#578	1	2	yes	—
202	C₂²×C₁₃	#578	1	1	yes	characteristic
203	C₄×C₁₃	—	4	1	no	—
204	C₄×C₁₃	—	4	4	no	—
205	C₄×C₁₃	—	4	4	no	—
206	C₄×C₁₃	—	4	2	no	—
207	C₄×C₁₃	—	4	2	no	—
208	C₄×C₁₃	—	8	2	no	—
209	C₄×C₁₃	—	8	2	no	—
210	C₄×C₁₃	#551	1	1	yes	characteristic
211	C₄×C₁₃	#553	1	2	yes	—
212	C₄×C₁₃	#559	1	1	yes	characteristic
213	C₂²×Q₈	C₂³×C₁₃	1	2	yes	—
214	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
215	C₂×Q₁₆	—	2	4	no	—
216	C₂×Q₁₆	—	4	1	no	—
217	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
218	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
219	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
220	C₂²⋊Q₈	—	2	4	no	—
221	C₂²⋊Q₈	—	2	4	no	—
222	C₄×Q₈	—	4	2	no	—
223	C₄×Q₈	—	4	2	no	—
224	#117	—	4	1	no	—
225	#117	—	4	1	no	—
226	C₂×C₄⋊C₄	—	2	2	no	—
227	C₂×C₄⋊C₄	—	2	2	no	—
228	C₂×C₄⋊C₄	—	2	2	no	—
229	C₂×C₄⋊C₄	D₄×C₁₃	1	1	yes	characteristic
230	C₂×C₄⋊C₄	D₄×C₁₃	1	1	yes	characteristic
231	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
232	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
233	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
234	C₂×C₂²⋊C₄	—	2	2	no	—
235	C₂×C₂²⋊C₄	—	2	2	no	—
236	C₂×C₂²⋊C₄	D₄×C₁₃	1	2	yes	—
237	C₂×C₄²	—	2	2	no	—
238	C₂×C₄²	—	2	2	no	—
239	C₂×C₄²	C₂³×C₁₃	1	2	yes	—
240	#107	—	2	4	no	—
241	#107	—	4	1	no	—
242	#103	—	2	4	no	—
243	#103	—	2	4	no	—
244	#103	—	4	2	no	—
245	C₂²⋊C₈	—	4	1	no	—
246	C₂²⋊C₈	—	4	1	no	—
247	#95	—	2	2	no	—
248	#95	—	2	2	no	—
249	#95	—	2	2	no	—
250	#95	—	2	2	no	—
251	#95	—	2	2	no	—
252	C₂×C₁₃	—	8	1	no	—
253	C₂×C₁₃	#2600	1	1	yes	characteristic
254	C₂×C₁₃	#3124	1	2	yes	—
255	C₂×C₁₃	#3127	1	2	yes	—
256	C₂×C₁₃	#3202	1	1	yes	characteristic
257	C₂×C₁₃	#3203	1	1	yes	characteristic
258	C₂⁴	C₂×D₄×C₁₃	1	1	yes	characteristic
259	C₂×Q₈	—	2	2	no	—
260	C₂×Q₈	—	2	2	no	—
261	C₂×Q₈	—	2	2	no	—
262	C₂×Q₈	—	2	2	no	—
263	C₂×Q₈	—	2	2	no	—
264	C₂×Q₈	C₁₃×#41	1	4	yes	—
265	C₂²×C₄	—	2	2	no	—
266	C₂²×C₄	—	2	2	no	—
267	C₂²×C₄	—	2	2	no	—
268	C₂²×C₄	—	2	2	no	—
269	C₂²×C₄	—	2	1	no	—
270	C₂²×C₄	—	2	1	no	—
271	C₂²×C₄	—	2	2	no	—
272	C₂²×C₄	—	2	2	no	—
273	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
274	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
275	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
276	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
277	Q₁₆	—	4	4	no	—
278	C₂×C₈	—	2	2	no	—
279	C₂×C₈	—	2	2	no	—
280	C₂×C₈	—	2	2	no	—
281	C₂×C₈	—	2	2	no	—
282	C₂×C₈	—	2	2	no	—
283	C₂×C₈	—	2	1	no	—
284	C₂×C₈	—	2	1	no	—
285	C₄⋊C₄	—	2	8	no	—
286	C₄⋊C₄	—	2	2	no	—
287	C₄⋊C₄	—	2	2	no	—
288	C₄⋊C₄	—	2	2	no	—
289	C₄⋊C₄	—	2	2	no	—
290	C₄⋊C₄	—	2	2	no	—
291	C₄⋊C₄	—	4	1	no	—
292	C₄⋊C₄	—	4	1	no	—
293	C₄⋊C₄	—	4	4	no	—
294	C₄⋊C₄	—	8	2	no	—
295	C₄⋊C₄	C₁₃×#41	1	4	yes	—
296	C₂²⋊C₄	—	2	8	no	—
297	C₂²⋊C₄	—	4	2	no	—
298	C₂²⋊C₄	—	8	2	no	—
299	C₄²	—	2	2	no	—
300	C₄²	—	2	2	no	—
301	C₄²	—	4	4	no	—
302	C₄²	—	8	2	no	—
303	C₁₃	#29427	1	1	yes	characteristic, direct product factor, Sylow
304	C₂³	—	2	2	no	—
305	C₂³	—	2	1	no	—
306	C₂³	—	2	1	no	—
307	C₂³	—	2	2	no	—
308	C₂³	—	2	1	no	—
309	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
310	Q₈	—	2	4	no	—
311	Q₈	—	2	4	no	—
312	Q₈	—	4	2	no	—
313	Q₈	—	4	2	no	—
314	C₂×C₄	—	2	4	no	—
315	C₂×C₄	—	2	4	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	2	4	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	—	4	2	no	—
327	C₂×C₄	—	4	1	no	—
328	C₂×C₄	—	4	2	no	—
329	C₂×C₄	—	4	1	no	—
330	C₂×C₄	—	4	2	no	—
331	C₂×C₄	—	4	2	no	—
332	C₂×C₄	—	4	2	no	—
333	C₂×C₄	—	4	2	no	—
334	C₂×C₄	—	4	2	no	—
335	C₂×C₄	—	4	2	no	—
336	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
337	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
338	C₂×C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
339	C₂×C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
340	C₈	—	4	1	no	—
341	C₈	—	4	2	no	—
342	C₈	—	4	1	no	—
343	C₂²	—	4	1	no	—
344	C₂²	—	4	2	no	—
345	C₂²	—	4	2	no	—
346	C₂²	—	4	1	no	—
347	C₂²	—	4	1	no	—
348	C₂²	C₁₃×#525	1	2	yes	—
349	C₂²	C₁₃×#534	1	1	yes	characteristic
350	C₂²	C₁₃×#576	1	1	yes	characteristic
351	C₂²	C₁₃×#578	1	2	yes	—
352	C₂²	C₁₃×#578	1	1	yes	characteristic
353	C₄	—	4	1	no	—
354	C₄	—	4	4	no	—
355	C₄	—	4	4	no	—
356	C₄	—	4	2	no	—
357	C₄	—	4	2	no	—
358	C₄	—	8	2	no	—
359	C₄	—	8	2	no	—
360	C₄	C₁₃×#551	1	1	yes	characteristic
361	C₄	C₁₃×#553	1	2	yes	—
362	C₄	C₁₃×#559	1	1	yes	characteristic
363	C₂	—	8	1	no	—
364	C₂	C₁₃×#2600	1	1	yes	characteristic
365	C₂	C₁₃×#3124	1	2	yes	—
366	C₂	C₁₃×#3127	1	2	yes	—
367	C₂	C₁₃×#3202	1	1	yes	characteristic
368	C₂	C₁₃×#3203	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃
prime radical	C₂⁴×C₁₃	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ C₂×C₁₃×#117 ◁ G

Complements

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

N	H ≅ G/N	type
5. C₁₃×#1928	C₂	semidirect
6. C₁₃×#1918	C₂	semidirect
7. C₁₃×#1878	C₂	semidirect
8. C₁₃×#1863	C₂	semidirect
74. #29427	C₁₃	direct (N × H)
124. #1928	C₂×C₁₃	semidirect
125. #1918	C₂×C₁₃	semidirect
126. #1878	C₂×C₁₃	semidirect
127. #1863	C₂×C₁₃	semidirect
303. C₁₃	#29427	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×#2323	2
C₁₃×#2075	2
C₁₃×#2036	2
C₁₃×#2001	2
C₁₃×#1928	2
C₁₃×#1918	2
C₁₃×#1878	2
C₁₃×#1863	2
C₁₃×#1818	2
#29427	13

Subgroups — C13×#29427 #15715766

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#29427 #15715766