Subgroups — C₁₃×#23804 #15710143

Order 3328.

1038 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#442	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×#441	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₃×#440	C₂	1	2	yes	maximal, complemented
4	C₄×C₁₃×#124	C₂	1	2	yes	maximal, complemented
5	C₁₃×#2097	C₂	1	2	yes	maximal, complemented
6	C₁₃×#2092	C₂	1	2	yes	maximal, complemented
7	C₁₃×#2015	C₂	1	2	yes	maximal, complemented
8	C₁₃×#2012	C₂	1	2	yes	maximal
9	C₁₃×#1745	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
11	C₂²×C₁₃×SD₁₆	C₂²	1	2	yes	—
12	C₂²×C₁₃×D₈	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×#124	C₂²	1	2	yes	—
14	C₂×C₄×Q₈×C₁₃	C₂²	1	2	yes	—
15	C₂×C₄×D₄×C₁₃	C₂²	1	2	yes	—
16	C₁₃×#442	—	2	4	no	—
17	C₁₃×#441	—	2	4	no	—
18	C₁₃×#440	—	2	8	no	—
19	C₂×C₁₃×#107	C₂²	1	2	yes	—
20	C₂×C₁₃×#106	C₂²	1	1	yes	characteristic
21	C₂×C₁₃×#106	C₂²	1	1	yes	characteristic
22	C₂×C₁₃×#103	C₂²	1	2	yes	—
23	C₂×C₁₃×#103	C₂²	1	2	yes	—
24	C₂×C₁₃×#102	C₂²	1	2	yes	—
25	C₂×C₁₃×#102	C₂²	1	2	yes	—
26	C₂×C₁₃×#97	C₂²	1	1	yes	characteristic
27	C₂×C₁₃×#97	C₂²	1	1	yes	characteristic
28	C₁₃×#391	C₂²	1	2	yes	—
29	C₁₃×#390	C₂²	1	2	yes	—
30	C₁₃×#388	—	2	4	no	—
31	C₁₃×#382	C₂²	1	2	yes	—
32	C₄×C₁₃×C₂²⋊C₄	—	2	4	no	—
33	C₄³×C₁₃	C₂²	1	1	yes	characteristic
34	C₁₃×#336	C₂²	1	2	yes	—
35	C₁₃×#336	C₂²	1	2	yes	—
36	C₂²×Q₈×C₁₃	C₂³	1	2	yes	—
37	C₂²×D₄×C₁₃	C₂³	1	2	yes	—
38	C₂³×C₄×C₁₃	—	2	2	no	—
39	C₂×C₁₃×Q₁₆	—	2	4	no	—
40	C₂×C₁₃×Q₁₆	—	2	2	no	—
41	C₂×C₁₃×SD₁₆	—	2	4	no	—
42	C₂×C₁₃×SD₁₆	—	2	4	no	—
43	C₂×C₁₃×SD₁₆	—	2	4	no	—
44	C₂×C₁₃×D₈	—	2	4	no	—
45	C₂×C₁₃×D₈	—	2	2	no	—
46	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
47	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
48	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
49	C₁₃×#124	—	2	4	no	—
50	C₁₃×#124	—	2	4	no	—
51	C₄×Q₈×C₁₃	—	2	4	no	—
52	C₄×Q₈×C₁₃	—	2	4	no	—
53	C₄×D₄×C₁₃	—	2	4	no	—
54	C₄×D₄×C₁₃	—	2	4	no	—
55	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
56	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
57	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
58	C₂×C₁₃×C₂²⋊C₄	—	2	4	no	—
59	C₂×C₁₃×C₂²⋊C₄	—	2	4	no	—
60	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
61	C₂×C₄²×C₁₃	—	2	2	no	—
62	C₂×C₄²×C₁₃	D₄	1	4	yes	—
63	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
64	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
65	C₁₃×#107	—	2	4	no	—
66	C₁₃×#106	—	2	2	no	—
67	C₁₃×#106	—	2	2	no	—
68	C₁₃×#103	—	2	4	no	—
69	C₁₃×#103	—	2	4	no	—
70	C₁₃×#102	—	2	4	no	—
71	C₁₃×#102	—	2	4	no	—
72	C₁₃×#97	D₄	1	4	yes	—
73	C₁₃×#97	D₄	1	4	yes	—
74	C₁₃×#95	—	2	4	no	—
75	C₁₃×#95	—	2	4	no	—
76	#23804	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
77	C₂⁴×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	—	2	2	no	—
82	C₂×Q₈×C₁₃	—	2	2	no	—
83	C₂×Q₈×C₁₃	#41	1	4	yes	—
84	C₂×D₄×C₁₃	—	2	2	no	—
85	C₂×D₄×C₁₃	—	2	2	no	—
86	C₂×D₄×C₁₃	—	2	2	no	—
87	C₂×D₄×C₁₃	—	2	2	no	—
88	C₂×D₄×C₁₃	—	2	2	no	—
89	C₂×D₄×C₁₃	#41	1	4	yes	—
90	C₂²×C₄×C₁₃	—	2	2	no	—
91	C₂²×C₄×C₁₃	—	2	2	no	—
92	C₂²×C₄×C₁₃	—	2	2	no	—
93	C₂²×C₄×C₁₃	—	4	8	no	—
94	C₂²×C₄×C₁₃	—	4	4	no	—
95	C₂²×C₄×C₁₃	C₂×D₄	1	4	yes	—
96	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
97	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
98	C₁₃×Q₁₆	—	2	8	no	—
99	C₁₃×SD₁₆	—	2	16	no	—
100	C₁₃×D₈	—	2	8	no	—
101	C₂×C₈×C₁₃	—	2	2	no	—
102	C₂×C₈×C₁₃	—	2	2	no	—
103	C₂×C₈×C₁₃	—	2	2	no	—
104	C₂×C₈×C₁₃	—	2	2	no	—
105	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
106	C₂×C₈×C₁₃	C₂×D₄	1	4	yes	—
107	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
108	C₁₃×C₄⋊C₄	—	2	2	no	—
109	C₁₃×C₄⋊C₄	—	2	2	no	—
110	C₁₃×C₄⋊C₄	—	2	2	no	—
111	C₁₃×C₄⋊C₄	—	2	2	no	—
112	C₁₃×C₄⋊C₄	—	4	4	no	—
113	C₁₃×C₂²⋊C₄	—	2	16	no	—
114	C₁₃×C₂²⋊C₄	—	4	8	no	—
115	C₁₃×C₂²⋊C₄	—	4	4	no	—
116	C₄²×C₁₃	—	2	2	no	—
117	C₄²×C₁₃	—	2	2	no	—
118	C₄²×C₁₃	—	4	4	no	—
119	C₄²×C₁₃	—	4	4	no	—
120	C₄²×C₁₃	C₂×D₄	1	2	yes	—
121	C₄²×C₁₃	C₂×D₄	1	2	yes	—
122	C₂×#442	C₂×C₁₃	1	1	yes	characteristic
123	C₂×#441	C₂×C₁₃	1	1	yes	characteristic, complemented
124	C₂×#440	C₂×C₁₃	1	2	yes	complemented
125	C₄×#124	C₂×C₁₃	1	2	yes	complemented
126	#2097	C₂×C₁₃	1	2	yes	complemented
127	#2092	C₂×C₁₃	1	2	yes	complemented
128	#2015	C₂×C₁₃	1	2	yes	complemented
129	#2012	C₂×C₁₃	1	2	yes	—
130	#1745	C₂×C₁₃	1	1	yes	characteristic, complemented
131	C₂³×C₁₃	—	2	4	no	—
132	C₂³×C₁₃	—	4	2	no	—
133	C₂³×C₁₃	—	4	2	no	—
134	C₂³×C₁₃	—	4	4	no	—
135	C₂³×C₁₃	—	4	4	no	—
136	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
137	Q₈×C₁₃	—	2	4	no	—
138	Q₈×C₁₃	—	2	4	no	—
139	Q₈×C₁₃	—	4	2	no	—
140	Q₈×C₁₃	—	4	2	no	—
141	D₄×C₁₃	—	2	4	no	—
142	D₄×C₁₃	—	2	4	no	—
143	D₄×C₁₃	—	4	2	no	—
144	D₄×C₁₃	—	4	2	no	—
145	C₂×C₄×C₁₃	—	2	4	no	—
146	C₂×C₄×C₁₃	—	2	4	no	—
147	C₂×C₄×C₁₃	—	2	4	no	—
148	C₂×C₄×C₁₃	—	2	4	no	—
149	C₂×C₄×C₁₃	—	2	4	no	—
150	C₂×C₄×C₁₃	—	4	2	no	—
151	C₂×C₄×C₁₃	—	4	2	no	—
152	C₂×C₄×C₁₃	—	4	2	no	—
153	C₂×C₄×C₁₃	—	4	2	no	—
154	C₂×C₄×C₁₃	—	4	4	no	—
155	C₂×C₄×C₁₃	—	4	4	no	—
156	C₂×C₄×C₁₃	—	8	8	no	—
157	C₂×C₄×C₁₃	#127	1	4	yes	—
158	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
159	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
160	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
161	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
162	C₈×C₁₃	—	2	2	no	—
163	C₈×C₁₃	—	2	4	no	—
164	C₈×C₁₃	—	2	2	no	—
165	C₂²×Q₁₆	C₂²×C₁₃	1	1	yes	characteristic
166	C₂²×SD₁₆	C₂²×C₁₃	1	2	yes	—
167	C₂²×D₈	C₂²×C₁₃	1	1	yes	characteristic
168	C₂×#124	C₂²×C₁₃	1	2	yes	—
169	C₂×C₄×Q₈	C₂²×C₁₃	1	2	yes	—
170	C₂×C₄×D₄	C₂²×C₁₃	1	2	yes	—
171	#442	—	2	4	no	—
172	#441	—	2	4	no	—
173	#440	—	2	8	no	—
174	C₂×#107	C₂²×C₁₃	1	2	yes	—
175	C₂×#106	C₂²×C₁₃	1	1	yes	characteristic
176	C₂×#106	C₂²×C₁₃	1	1	yes	characteristic
177	C₂×#103	C₂²×C₁₃	1	2	yes	—
178	C₂×#103	C₂²×C₁₃	1	2	yes	—
179	C₂×#102	C₂²×C₁₃	1	2	yes	—
180	C₂×#102	C₂²×C₁₃	1	2	yes	—
181	C₂×#97	C₂²×C₁₃	1	1	yes	characteristic
182	C₂×#97	C₂²×C₁₃	1	1	yes	characteristic
183	#391	C₂²×C₁₃	1	2	yes	—
184	#390	C₂²×C₁₃	1	2	yes	—
185	#388	—	2	4	no	—
186	#382	C₂²×C₁₃	1	2	yes	—
187	C₄×C₂²⋊C₄	—	2	4	no	—
188	C₄³	C₂²×C₁₃	1	1	yes	characteristic
189	#336	C₂²×C₁₃	1	2	yes	—
190	#336	C₂²×C₁₃	1	2	yes	—
191	C₂²×C₁₃	—	4	8	no	—
192	C₂²×C₁₃	—	4	4	no	—
193	C₂²×C₁₃	—	8	4	no	—
194	C₂²×C₁₃	—	8	2	no	—
195	C₂²×C₁₃	—	8	2	no	—
196	C₂²×C₁₃	#525	1	2	yes	—
197	C₂²×C₁₃	C₂×#127	1	1	yes	characteristic
198	C₂²×C₁₃	#575	1	4	yes	—
199	C₄×C₁₃	—	2	4	no	—
200	C₄×C₁₃	—	4	4	no	—
201	C₄×C₁₃	—	4	4	no	—
202	C₄×C₁₃	—	8	2	no	—
203	C₄×C₁₃	—	8	2	no	—
204	C₄×C₁₃	#525	1	1	yes	characteristic
205	C₄×C₁₃	#525	1	2	yes	—
206	C₄×C₁₃	#525	1	1	yes	characteristic
207	C₂²×Q₈	C₂³×C₁₃	1	2	yes	—
208	C₂²×D₄	C₂³×C₁₃	1	2	yes	—
209	C₂³×C₄	—	2	2	no	—
210	C₂×Q₁₆	—	2	4	no	—
211	C₂×Q₁₆	—	2	2	no	—
212	C₂×SD₁₆	—	2	4	no	—
213	C₂×SD₁₆	—	2	4	no	—
214	C₂×SD₁₆	—	2	4	no	—
215	C₂×D₈	—	2	4	no	—
216	C₂×D₈	—	2	2	no	—
217	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
218	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
219	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
220	#124	—	2	4	no	—
221	#124	—	2	4	no	—
222	C₄×Q₈	—	2	4	no	—
223	C₄×Q₈	—	2	4	no	—
224	C₄×D₄	—	2	4	no	—
225	C₄×D₄	—	2	4	no	—
226	C₂×C₄⋊C₄	—	2	2	no	—
227	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
228	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
229	C₂×C₂²⋊C₄	—	2	4	no	—
230	C₂×C₂²⋊C₄	—	2	2	no	—
231	C₂×C₂²⋊C₄	—	2	4	no	—
232	C₂×C₄²	—	2	2	no	—
233	C₂×C₄²	D₄×C₁₃	1	4	yes	—
234	C₂×C₄²	C₂³×C₁₃	1	2	yes	—
235	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
236	#107	—	2	4	no	—
237	#106	—	2	2	no	—
238	#106	—	2	2	no	—
239	#103	—	2	4	no	—
240	#103	—	2	4	no	—
241	#102	—	2	4	no	—
242	#102	—	2	4	no	—
243	#97	D₄×C₁₃	1	4	yes	—
244	#97	D₄×C₁₃	1	4	yes	—
245	#95	—	2	4	no	—
246	#95	—	2	4	no	—
247	C₂×C₁₃	—	8	4	no	—
248	C₂×C₁₃	#2397	1	1	yes	characteristic
249	C₂×C₁₃	#3111	1	4	yes	—
250	C₂×C₁₃	#3153	1	2	yes	—
251	C₂⁴	—	2	2	no	—
252	C₂×Q₈	—	2	2	no	—
253	C₂×Q₈	—	2	2	no	—
254	C₂×Q₈	—	2	2	no	—
255	C₂×Q₈	—	2	2	no	—
256	C₂×Q₈	—	2	2	no	—
257	C₂×Q₈	C₁₃×#41	1	4	yes	—
258	C₂×D₄	—	2	2	no	—
259	C₂×D₄	—	2	2	no	—
260	C₂×D₄	—	2	2	no	—
261	C₂×D₄	—	2	2	no	—
262	C₂×D₄	—	2	2	no	—
263	C₂×D₄	C₁₃×#41	1	4	yes	—
264	C₂²×C₄	—	2	2	no	—
265	C₂²×C₄	—	2	2	no	—
266	C₂²×C₄	—	2	2	no	—
267	C₂²×C₄	—	4	8	no	—
268	C₂²×C₄	—	4	4	no	—
269	C₂²×C₄	C₂×D₄×C₁₃	1	4	yes	—
270	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
271	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
272	Q₁₆	—	2	8	no	—
273	SD₁₆	—	2	16	no	—
274	D₈	—	2	8	no	—
275	C₂×C₈	—	2	2	no	—
276	C₂×C₈	—	2	2	no	—
277	C₂×C₈	—	2	2	no	—
278	C₂×C₈	—	2	2	no	—
279	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
280	C₂×C₈	C₂×D₄×C₁₃	1	4	yes	—
281	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
282	C₄⋊C₄	—	2	2	no	—
283	C₄⋊C₄	—	2	2	no	—
284	C₄⋊C₄	—	2	2	no	—
285	C₄⋊C₄	—	2	2	no	—
286	C₄⋊C₄	—	4	4	no	—
287	C₂²⋊C₄	—	2	16	no	—
288	C₂²⋊C₄	—	4	8	no	—
289	C₂²⋊C₄	—	4	4	no	—
290	C₄²	—	2	2	no	—
291	C₄²	—	2	2	no	—
292	C₄²	—	4	4	no	—
293	C₄²	—	4	4	no	—
294	C₄²	C₂×D₄×C₁₃	1	2	yes	—
295	C₄²	C₂×D₄×C₁₃	1	2	yes	—
296	C₁₃	#23804	1	1	yes	characteristic, direct product factor, Sylow
297	C₂³	—	2	4	no	—
298	C₂³	—	4	2	no	—
299	C₂³	—	4	2	no	—
300	C₂³	—	4	4	no	—
301	C₂³	—	4	4	no	—
302	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
303	Q₈	—	2	4	no	—
304	Q₈	—	2	4	no	—
305	Q₈	—	4	2	no	—
306	Q₈	—	4	2	no	—
307	D₄	—	2	4	no	—
308	D₄	—	2	4	no	—
309	D₄	—	4	2	no	—
310	D₄	—	4	2	no	—
311	C₂×C₄	—	2	4	no	—
312	C₂×C₄	—	2	4	no	—
313	C₂×C₄	—	2	4	no	—
314	C₂×C₄	—	2	4	no	—
315	C₂×C₄	—	2	4	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	—	4	2	no	—
318	C₂×C₄	—	4	2	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	—	4	4	no	—
321	C₂×C₄	—	4	4	no	—
322	C₂×C₄	—	8	8	no	—
323	C₂×C₄	C₁₃×#127	1	4	yes	—
324	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
325	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
326	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
327	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
328	C₈	—	2	2	no	—
329	C₈	—	2	4	no	—
330	C₈	—	2	2	no	—
331	C₂²	—	4	8	no	—
332	C₂²	—	4	4	no	—
333	C₂²	—	8	4	no	—
334	C₂²	—	8	2	no	—
335	C₂²	—	8	2	no	—
336	C₂²	C₁₃×#525	1	2	yes	—
337	C₂²	C₂×C₁₃×#127	1	1	yes	characteristic
338	C₂²	C₁₃×#575	1	4	yes	—
339	C₄	—	2	4	no	—
340	C₄	—	4	4	no	—
341	C₄	—	4	4	no	—
342	C₄	—	8	2	no	—
343	C₄	—	8	2	no	—
344	C₄	C₁₃×#525	1	1	yes	characteristic
345	C₄	C₁₃×#525	1	2	yes	—
346	C₄	C₁₃×#525	1	1	yes	characteristic
347	C₂	—	8	4	no	—
348	C₂	C₁₃×#2397	1	1	yes	characteristic
349	C₂	C₁₃×#3111	1	4	yes	—
350	C₂	C₁₃×#3153	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃
prime radical	C₂²×C₁₃×D₈	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ 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. C₂×C₁₃×#441	C₂	semidirect
3. C₂×C₁₃×#440	C₂	semidirect
4. C₄×C₁₃×#124	C₂	semidirect
5. C₁₃×#2097	C₂	semidirect
6. C₁₃×#2092	C₂	semidirect
7. C₁₃×#2015	C₂	semidirect
9. C₁₃×#1745	C₂	semidirect
76. #23804	C₁₃	direct (N × H)
123. C₂×#441	C₂×C₁₃	semidirect
124. C₂×#440	C₂×C₁₃	semidirect
125. C₄×#124	C₂×C₁₃	semidirect
126. #2097	C₂×C₁₃	semidirect
127. #2092	C₂×C₁₃	semidirect
128. #2015	C₂×C₁₃	semidirect
130. #1745	C₂×C₁₃	semidirect
296. C₁₃	#23804	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₁₃×#442	2
C₂×C₁₃×#441	2
C₂×C₁₃×#440	2
C₄×C₁₃×#124	2
C₁₃×#2097	2
C₁₃×#2092	2
C₁₃×#2015	2
C₁₃×#2012	2
C₁₃×#1745	2
#23804	13

Subgroups — C13×#23804 #15710143

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#23804 #15710143