Subgroups — C₁₃×#23834 #15710173

Order 3328.

608 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#2961	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₃×#2955	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2103	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#2102	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#2080	C₂	1	2	yes	maximal, complemented
6	C₁₃×#2053	C₂	1	1	yes	maximal, characteristic
7	C₁₃×#2050	C₂	1	1	yes	maximal, characteristic
8	C₁₃×#2031	C₂	1	2	yes	maximal, complemented
9	C₁₃×#1765	C₂	1	2	yes	maximal, complemented
10	C₁₃×#1757	C₂	1	2	yes	maximal, complemented
11	C₁₃×#1747	C₂	1	1	yes	maximal, characteristic, complemented
12	C₁₃×#575	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×#131	C₂²	1	2	yes	—
14	C₁₃×#533	C₂²	1	1	yes	characteristic
15	C₁₃×#446	—	2	2	no	—
16	C₁₃×#446	—	2	2	no	—
17	C₁₃×#444	—	2	2	no	—
18	C₁₃×#443	—	2	2	no	—
19	C₁₃×#430	C₂²	1	1	yes	characteristic
20	C₁₃×#424	C₂²	1	1	yes	characteristic
21	C₁₃×#424	C₂²	1	1	yes	characteristic
22	C₁₃×#423	C₂²	1	1	yes	characteristic
23	C₁₃×#421	C₂²	1	2	yes	—
24	C₂×C₁₃×#104	C₂²	1	2	yes	—
25	C₁₃×#416	C₂²	1	2	yes	—
26	C₁₃×#413	C₂²	1	2	yes	—
27	C₁₃×#410	C₂²	1	2	yes	—
28	C₁₃×#405	C₂²	1	2	yes	—
29	C₁₃×#405	C₂²	1	1	yes	characteristic
30	C₁₃×#405	C₂²	1	1	yes	characteristic
31	C₂×C₁₃×#97	C₂²	1	1	yes	characteristic
32	C₁₃×#344	C₂²	1	2	yes	—
33	C₁₃×#344	C₂²	1	2	yes	—
34	C₁₃×#341	C₂²	1	2	yes	—
35	C₁₃×#339	C₂²	1	2	yes	—
36	C₁₃×#338	C₂²	1	1	yes	characteristic
37	C₁₃×H(3,ℤ/4)	C₂²	1	2	yes	—
38	C₁₃×H(3,ℤ/4)	C₂²	1	1	yes	characteristic
39	C₂×C₁₃×#41	C₂³	1	2	yes	—
40	C₁₃×#137	—	2	2	no	—
41	C₁₃×AGL(1,ℤ/8)	—	2	2	no	—
42	C₁₃×#135	—	2	2	no	—
43	C₁₃×#131	—	2	2	no	—
44	C₁₃×#131	—	2	2	no	—
45	C₁₃×#131	—	2	2	no	—
46	C₁₃×#131	—	2	2	no	—
47	C₂×C₁₃×M₁₆	—	2	2	no	—
48	C₂×C₁₃×M₁₆	C₂³	1	2	yes	—
49	C₂×C₁₃×M₁₆	C₂³	1	1	yes	characteristic
50	C₂×C₁₃×M₁₆	C₂³	1	2	yes	—
51	C₂×C₁₃×M₁₆	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₁₃	—	2	2	no	—
53	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
54	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
55	C₁₃×C₄⋊Q₈	—	2	1	no	—
56	C₁₃×#125	—	2	1	no	—
57	C₁₃×C₂²⋊Q₈	—	2	2	no	—
58	C₄×Q₈×C₁₃	—	2	2	no	—
59	C₁₃×#117	C₂³	1	2	yes	—
60	C₁₃×#117	C₂³	1	1	yes	characteristic
61	C₁₃×#117	C₂³	1	1	yes	characteristic
62	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
63	C₁₃×#108	—	2	2	no	—
64	C₁₃×C₄⋊C₈	—	2	2	no	—
65	C₁₃×C₄⋊C₈	—	2	2	no	—
66	C₁₃×C₄⋊C₈	—	2	2	no	—
67	C₁₃×#104	—	2	4	no	—
68	C₁₃×#104	—	2	2	no	—
69	C₁₃×#104	—	2	2	no	—
70	C₁₃×#103	—	2	2	no	—
71	C₁₃×#102	—	2	2	no	—
72	C₁₃×#101	—	2	2	no	—
73	C₁₃×#100	—	2	2	no	—
74	C₁₃×#99	—	2	4	no	—
75	C₁₃×#97	—	2	1	no	—
76	C₁₃×#97	—	2	1	no	—
77	C₁₃×#97	—	2	1	no	—
78	C₁₃×#97	—	2	2	no	—
79	C₁₃×#97	D₄	1	2	yes	—
80	C₄×C₈×C₁₃	—	2	1	no	—
81	C₄×C₈×C₁₃	—	2	2	no	—
82	C₄×C₈×C₁₃	D₄	1	2	yes	—
83	#23834	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
84	C₁₃×#41	—	2	2	no	—
85	C₁₃×#41	—	2	2	no	—
86	C₁₃×#41	—	2	2	no	—
87	C₁₃×#41	—	2	2	no	—
88	C₂×Q₈×C₁₃	—	2	2	no	—
89	C₂×Q₈×C₁₃	#41	1	2	yes	—
90	C₂×D₄×C₁₃	—	2	2	no	—
91	C₂×D₄×C₁₃	#41	1	2	yes	—
92	C₂²×C₄×C₁₃	—	2	1	no	—
93	C₂²×C₄×C₁₃	—	2	2	no	—
94	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
95	C₁₃×Q₁₆	—	2	2	no	—
96	C₁₃×SD₁₆	—	2	4	no	—
97	C₁₃×D₈	—	2	2	no	—
98	C₁₃×M₁₆	—	2	2	no	—
99	C₁₃×M₁₆	—	2	2	no	—
100	C₁₃×M₁₆	—	2	2	no	—
101	C₁₃×M₁₆	—	2	2	no	—
102	C₁₃×M₁₆	—	2	2	no	—
103	C₁₃×M₁₆	—	2	2	no	—
104	C₁₃×M₁₆	—	4	2	no	—
105	C₁₃×M₁₆	—	4	2	no	—
106	C₂×C₈×C₁₃	—	2	1	no	—
107	C₂×C₈×C₁₃	—	2	1	no	—
108	C₂×C₈×C₁₃	—	2	2	no	—
109	C₂×C₈×C₁₃	—	2	2	no	—
110	C₂×C₈×C₁₃	—	2	2	no	—
111	C₂×C₈×C₁₃	—	4	2	no	—
112	C₂×C₈×C₁₃	—	4	2	no	—
113	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
114	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
115	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
116	C₂×C₈×C₁₃	#41	1	2	yes	—
117	C₂×C₈×C₁₃	#41	1	2	yes	—
118	C₁₃×C₄⋊C₄	—	2	4	no	—
119	C₁₃×C₄⋊C₄	—	2	2	no	—
120	C₁₃×C₄⋊C₄	—	2	1	no	—
121	C₁₃×C₄⋊C₄	—	2	1	no	—
122	C₁₃×C₄⋊C₄	—	2	2	no	—
123	C₁₃×C₂²⋊C₄	—	2	1	no	—
124	C₁₃×C₂²⋊C₄	—	2	2	no	—
125	C₁₃×C₂²⋊C₄	#41	1	2	yes	—
126	C₄²×C₁₃	—	2	1	no	—
127	C₄²×C₁₃	—	2	1	no	—
128	C₄²×C₁₃	—	2	2	no	—
129	C₄²×C₁₃	C₂×D₄	1	2	yes	—
130	C₄²×C₁₃	#41	1	1	yes	characteristic
131	C₄²×C₁₃	#41	1	1	yes	characteristic
132	#2961	C₂×C₁₃	1	1	yes	characteristic, complemented
133	#2955	C₂×C₁₃	1	1	yes	characteristic
134	#2103	C₂×C₁₃	1	1	yes	characteristic, complemented
135	#2102	C₂×C₁₃	1	1	yes	characteristic, complemented
136	#2080	C₂×C₁₃	1	2	yes	complemented
137	#2053	C₂×C₁₃	1	1	yes	characteristic
138	#2050	C₂×C₁₃	1	1	yes	characteristic
139	#2031	C₂×C₁₃	1	2	yes	complemented
140	#1765	C₂×C₁₃	1	2	yes	complemented
141	#1757	C₂×C₁₃	1	2	yes	complemented
142	#1747	C₂×C₁₃	1	1	yes	characteristic, complemented
143	C₂³×C₁₃	—	2	2	no	—
144	C₂³×C₁₃	C₂×#41	1	1	yes	characteristic
145	Q₈×C₁₃	—	2	2	no	—
146	Q₈×C₁₃	—	2	2	no	—
147	Q₈×C₁₃	—	8	2	no	—
148	D₄×C₁₃	—	2	2	no	—
149	D₄×C₁₃	—	2	2	no	—
150	D₄×C₁₃	—	4	2	no	—
151	D₄×C₁₃	—	4	2	no	—
152	C₂×C₄×C₁₃	—	2	1	no	—
153	C₂×C₄×C₁₃	—	2	1	no	—
154	C₂×C₄×C₁₃	—	2	2	no	—
155	C₂×C₄×C₁₃	—	2	2	no	—
156	C₂×C₄×C₁₃	—	2	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	—	8	1	no	—
161	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
162	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
163	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
164	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
165	C₈×C₁₃	—	2	2	no	—
166	C₈×C₁₃	—	4	1	no	—
167	C₈×C₁₃	—	4	1	no	—
168	C₈×C₁₃	—	4	2	no	—
169	C₈×C₁₃	—	4	2	no	—
170	C₈×C₁₃	#124	1	1	yes	characteristic
171	C₈×C₁₃	#124	1	2	yes	—
172	C₈×C₁₃	#124	1	1	yes	characteristic
173	#575	C₂²×C₁₃	1	1	yes	characteristic
174	C₂×#131	C₂²×C₁₃	1	2	yes	—
175	#533	C₂²×C₁₃	1	1	yes	characteristic
176	#446	—	2	2	no	—
177	#446	—	2	2	no	—
178	#444	—	2	2	no	—
179	#443	—	2	2	no	—
180	#430	C₂²×C₁₃	1	1	yes	characteristic
181	#424	C₂²×C₁₃	1	1	yes	characteristic
182	#424	C₂²×C₁₃	1	1	yes	characteristic
183	#423	C₂²×C₁₃	1	1	yes	characteristic
184	#421	C₂²×C₁₃	1	2	yes	—
185	C₂×#104	C₂²×C₁₃	1	2	yes	—
186	#416	C₂²×C₁₃	1	2	yes	—
187	#413	C₂²×C₁₃	1	2	yes	—
188	#410	C₂²×C₁₃	1	2	yes	—
189	#405	C₂²×C₁₃	1	1	yes	characteristic
190	#405	C₂²×C₁₃	1	2	yes	—
191	#405	C₂²×C₁₃	1	1	yes	characteristic
192	C₂×#97	C₂²×C₁₃	1	1	yes	characteristic
193	#344	C₂²×C₁₃	1	2	yes	—
194	#344	C₂²×C₁₃	1	2	yes	—
195	#341	C₂²×C₁₃	1	2	yes	—
196	#339	C₂²×C₁₃	1	2	yes	—
197	#338	C₂²×C₁₃	1	1	yes	characteristic
198	H(3,ℤ/4)	C₂²×C₁₃	1	1	yes	characteristic
199	H(3,ℤ/4)	C₂²×C₁₃	1	2	yes	—
200	C₂²×C₁₃	—	4	1	no	—
201	C₂²×C₁₃	—	4	2	no	—
202	C₂²×C₁₃	—	8	2	no	—
203	C₂²×C₁₃	#525	1	1	yes	characteristic
204	C₂²×C₁₃	#529	1	2	yes	—
205	C₄×C₁₃	—	4	1	no	—
206	C₄×C₁₃	—	4	1	no	—
207	C₄×C₁₃	—	4	2	no	—
208	C₄×C₁₃	—	4	2	no	—
209	C₄×C₁₃	—	4	2	no	—
210	C₄×C₁₃	—	8	2	no	—
211	C₄×C₁₃	#525	1	1	yes	characteristic
212	C₄×C₁₃	C₂×#124	1	1	yes	characteristic
213	C₄×C₁₃	#529	1	2	yes	—
214	C₂×#41	C₂³×C₁₃	1	2	yes	—
215	#137	—	2	2	no	—
216	AGL(1,ℤ/8)	—	2	2	no	—
217	#135	—	2	2	no	—
218	#131	—	2	2	no	—
219	#131	—	2	2	no	—
220	#131	—	2	2	no	—
221	#131	—	2	2	no	—
222	C₂×M₁₆	—	2	2	no	—
223	C₂×M₁₆	C₂³×C₁₃	1	2	yes	—
224	C₂×M₁₆	C₂³×C₁₃	1	2	yes	—
225	C₂×M₁₆	C₂³×C₁₃	1	1	yes	characteristic
226	C₂×M₁₆	C₂³×C₁₃	1	1	yes	characteristic
227	C₂²×C₈	—	2	2	no	—
228	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
229	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
230	C₄⋊Q₈	—	2	1	no	—
231	#125	—	2	1	no	—
232	C₂²⋊Q₈	—	2	2	no	—
233	C₄×Q₈	—	2	2	no	—
234	#117	C₂³×C₁₃	1	2	yes	—
235	#117	C₂³×C₁₃	1	1	yes	characteristic
236	#117	C₂³×C₁₃	1	1	yes	characteristic
237	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
238	#108	—	2	2	no	—
239	C₄⋊C₈	—	2	2	no	—
240	C₄⋊C₈	—	2	2	no	—
241	C₄⋊C₈	—	2	2	no	—
242	#104	—	2	4	no	—
243	#104	—	2	2	no	—
244	#104	—	2	2	no	—
245	#103	—	2	2	no	—
246	#102	—	2	2	no	—
247	#101	—	2	2	no	—
248	#100	—	2	2	no	—
249	#99	—	2	4	no	—
250	#97	—	2	1	no	—
251	#97	—	2	1	no	—
252	#97	—	2	1	no	—
253	#97	—	2	2	no	—
254	#97	D₄×C₁₃	1	2	yes	—
255	C₄×C₈	—	2	1	no	—
256	C₄×C₈	—	2	2	no	—
257	C₄×C₈	D₄×C₁₃	1	2	yes	—
258	C₂×C₁₃	—	2	1	no	—
259	C₂×C₁₃	—	2	2	no	—
260	C₂×C₁₃	—	8	2	no	—
261	C₂×C₁₃	#2398	1	1	yes	characteristic
262	#41	—	2	2	no	—
263	#41	—	2	2	no	—
264	#41	—	2	2	no	—
265	#41	—	2	2	no	—
266	C₂×Q₈	—	2	2	no	—
267	C₂×Q₈	C₁₃×#41	1	2	yes	—
268	C₂×D₄	—	2	2	no	—
269	C₂×D₄	C₁₃×#41	1	2	yes	—
270	C₂²×C₄	—	2	1	no	—
271	C₂²×C₄	—	2	2	no	—
272	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
273	Q₁₆	—	2	2	no	—
274	SD₁₆	—	2	4	no	—
275	D₈	—	2	2	no	—
276	M₁₆	—	2	2	no	—
277	M₁₆	—	2	2	no	—
278	M₁₆	—	2	2	no	—
279	M₁₆	—	2	2	no	—
280	M₁₆	—	2	2	no	—
281	M₁₆	—	2	2	no	—
282	M₁₆	—	4	2	no	—
283	M₁₆	—	4	2	no	—
284	C₂×C₈	—	2	1	no	—
285	C₂×C₈	—	2	1	no	—
286	C₂×C₈	—	2	2	no	—
287	C₂×C₈	—	2	2	no	—
288	C₂×C₈	—	2	2	no	—
289	C₂×C₈	—	4	2	no	—
290	C₂×C₈	—	4	2	no	—
291	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
292	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
293	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
294	C₂×C₈	C₁₃×#41	1	2	yes	—
295	C₂×C₈	C₁₃×#41	1	2	yes	—
296	C₄⋊C₄	—	2	4	no	—
297	C₄⋊C₄	—	2	2	no	—
298	C₄⋊C₄	—	2	1	no	—
299	C₄⋊C₄	—	2	1	no	—
300	C₄⋊C₄	—	2	2	no	—
301	C₂²⋊C₄	—	2	1	no	—
302	C₂²⋊C₄	—	2	2	no	—
303	C₂²⋊C₄	C₁₃×#41	1	2	yes	—
304	C₄²	—	2	1	no	—
305	C₄²	—	2	1	no	—
306	C₄²	—	2	2	no	—
307	C₄²	C₂×D₄×C₁₃	1	2	yes	—
308	C₄²	C₁₃×#41	1	1	yes	characteristic
309	C₄²	C₁₃×#41	1	1	yes	characteristic
310	C₁₃	#23834	1	1	yes	characteristic, direct product factor, Sylow
311	C₂³	—	2	2	no	—
312	C₂³	C₂×C₁₃×#41	1	1	yes	characteristic
313	Q₈	—	2	2	no	—
314	Q₈	—	2	2	no	—
315	Q₈	—	8	2	no	—
316	D₄	—	2	2	no	—
317	D₄	—	2	2	no	—
318	D₄	—	4	2	no	—
319	D₄	—	4	2	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	2	2	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	—	4	2	no	—
327	C₂×C₄	—	4	2	no	—
328	C₂×C₄	—	8	1	no	—
329	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
330	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
331	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
332	C₂×C₄	C₂×C₁₃×#41	1	1	yes	characteristic
333	C₈	—	2	2	no	—
334	C₈	—	4	1	no	—
335	C₈	—	4	1	no	—
336	C₈	—	4	2	no	—
337	C₈	—	4	2	no	—
338	C₈	C₁₃×#124	1	1	yes	characteristic
339	C₈	C₁₃×#124	1	2	yes	—
340	C₈	C₁₃×#124	1	1	yes	characteristic
341	C₂²	—	4	1	no	—
342	C₂²	—	4	2	no	—
343	C₂²	—	8	2	no	—
344	C₂²	C₁₃×#525	1	1	yes	characteristic
345	C₂²	C₁₃×#529	1	2	yes	—
346	C₄	—	4	1	no	—
347	C₄	—	4	1	no	—
348	C₄	—	4	2	no	—
349	C₄	—	4	2	no	—
350	C₄	—	4	2	no	—
351	C₄	—	8	2	no	—
352	C₄	C₁₃×#525	1	1	yes	characteristic
353	C₄	C₂×C₁₃×#124	1	1	yes	characteristic
354	C₄	C₁₃×#529	1	2	yes	—
355	C₂	—	2	1	no	—
356	C₂	—	2	2	no	—
357	C₂	—	8	2	no	—
358	C₂	C₁₃×#2398	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₁₃	C₂×#124
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃
socle	C₂×C₁₃	#2398
prime radical	C₁₃×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₃ ◁ C₂²×C₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₃×#2961	C₂	semidirect
3. C₁₃×#2103	C₂	semidirect
4. C₁₃×#2102	C₂	semidirect
5. C₁₃×#2080	C₂	semidirect
8. C₁₃×#2031	C₂	semidirect
9. C₁₃×#1765	C₂	semidirect
10. C₁₃×#1757	C₂	semidirect
11. C₁₃×#1747	C₂	semidirect
83. #23834	C₁₃	direct (N × H)
132. #2961	C₂×C₁₃	semidirect
134. #2103	C₂×C₁₃	semidirect
135. #2102	C₂×C₁₃	semidirect
136. #2080	C₂×C₁₃	semidirect
139. #2031	C₂×C₁₃	semidirect
140. #1765	C₂×C₁₃	semidirect
141. #1757	C₂×C₁₃	semidirect
142. #1747	C₂×C₁₃	semidirect
310. C₁₃	#23834	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×#2961	2
C₁₃×#2955	2
C₁₃×#2103	2
C₁₃×#2102	2
C₁₃×#2080	2
C₁₃×#2053	2
C₁₃×#2050	2
C₁₃×#2031	2
C₁₃×#1765	2
C₁₃×#1757	2
C₁₃×#1747	2
#23834	13

Subgroups — C13×#23834 #15710173

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#23834 #15710173