Subgroups — C₁₃×#23753 #15710092

Order 3328.

648 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#430	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#2905	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2103	C₂	1	8	yes	maximal, complemented
4	C₂×C₁₃×#344	C₂	1	2	yes	maximal
5	C₁₃×#1738	C₂	1	2	yes	maximal
6	C₂×C₁₃×#338	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₁₃×M₁₆	C₂²	1	2	yes	—
8	C₂²×C₁₃×M₁₆	C₂²	1	2	yes	—
9	C₂²×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
10	C₂²×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×#117	C₂²	1	1	yes	characteristic
12	C₁₃×#430	—	2	2	no	—
13	C₁₃×#430	C₂²	1	4	yes	—
14	C₂×C₁₃×#108	—	2	2	no	—
15	C₁₃×#424	—	2	2	no	—
16	C₁₃×#424	C₂²	1	4	yes	—
17	C₁₃×#423	—	2	2	no	—
18	C₁₃×#344	C₂²	1	8	yes	—
19	C₁₃×#341	C₂²	1	8	yes	—
20	C₁₃×#338	C₂²	1	4	yes	—
21	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
22	C₂×C₁₃×M₁₆	—	2	2	no	—
23	C₂×C₁₃×M₁₆	—	2	2	no	—
24	C₂×C₁₃×M₁₆	—	2	1	no	—
25	C₂×C₁₃×M₁₆	—	2	1	no	—
26	C₂×C₁₃×M₁₆	—	2	2	no	—
27	C₂×C₁₃×M₁₆	—	2	2	no	—
28	C₂×C₁₃×M₁₆	—	2	2	no	—
29	C₂×C₁₃×M₁₆	—	2	2	no	—
30	C₂×C₁₃×M₁₆	—	2	2	no	—
31	C₂×C₁₃×M₁₆	—	2	2	no	—
32	C₂×C₁₃×M₁₆	—	2	2	no	—
33	C₂×C₁₃×M₁₆	—	2	2	no	—
34	C₂×C₁₃×M₁₆	—	2	2	no	—
35	C₂×C₁₃×M₁₆	—	2	2	no	—
36	C₂×C₁₃×M₁₆	—	2	2	no	—
37	C₂×C₁₃×M₁₆	—	2	2	no	—
38	C₂×C₁₃×M₁₆	D₄	1	2	yes	—
39	C₂×C₁₃×M₁₆	C₂³	1	4	yes	—
40	C₂×C₁₃×M₁₆	C₂³	1	4	yes	—
41	C₂×C₁₃×M₁₆	C₂³	1	2	yes	—
42	C₂²×C₈×C₁₃	—	2	2	no	—
43	C₂²×C₈×C₁₃	—	2	2	no	—
44	C₂²×C₈×C₁₃	D₄	1	2	yes	—
45	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
46	C₁₃×#117	—	2	1	no	—
47	C₁₃×#117	—	2	1	no	—
48	C₁₃×#117	Q₈	1	2	yes	—
49	C₁₃×#117	C₂³	1	2	yes	—
50	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
51	C₂×C₁₃×C₂²⋊C₄	Q₈	1	2	yes	—
52	C₂×C₄²×C₁₃	—	2	1	no	—
53	C₁₃×#108	—	2	8	no	—
54	C₁₃×C₄⋊C₈	—	2	8	no	—
55	#23753	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
56	C₂⁴×C₁₃	C₂×Q₈	1	1	yes	characteristic
57	C₂²×C₄×C₁₃	—	2	2	no	—
58	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
61	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	#41	1	2	yes	—
64	C₂²×C₄×C₁₃	#41	1	2	yes	—
65	C₂²×C₄×C₁₃	#41	1	2	yes	—
66	C₂²×C₄×C₁₃	#41	1	2	yes	—
67	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
68	C₁₃×M₁₆	—	2	4	no	—
69	C₁₃×M₁₆	—	2	4	no	—
70	C₁₃×M₁₆	—	2	4	no	—
71	C₁₃×M₁₆	—	2	4	no	—
72	C₁₃×M₁₆	—	2	4	no	—
73	C₁₃×M₁₆	—	2	4	no	—
74	C₁₃×M₁₆	—	4	1	no	—
75	C₁₃×M₁₆	—	4	1	no	—
76	C₁₃×M₁₆	—	4	2	no	—
77	C₁₃×M₁₆	—	4	2	no	—
78	C₁₃×M₁₆	—	4	2	no	—
79	C₁₃×M₁₆	—	4	2	no	—
80	C₁₃×M₁₆	—	4	2	no	—
81	C₂×C₈×C₁₃	—	2	2	no	—
82	C₂×C₈×C₁₃	—	2	2	no	—
83	C₂×C₈×C₁₃	—	2	2	no	—
84	C₂×C₈×C₁₃	—	2	4	no	—
85	C₂×C₈×C₁₃	—	2	4	no	—
86	C₂×C₈×C₁₃	—	2	4	no	—
87	C₂×C₈×C₁₃	—	4	2	no	—
88	C₂×C₈×C₁₃	—	4	2	no	—
89	C₂×C₈×C₁₃	—	4	2	no	—
90	C₂×C₈×C₁₃	—	4	2	no	—
91	C₂×C₈×C₁₃	C₂×D₄	1	4	yes	—
92	C₁₃×C₄⋊C₄	—	2	2	no	—
93	C₁₃×C₄⋊C₄	—	2	2	no	—
94	C₁₃×C₂²⋊C₄	—	2	2	no	—
95	C₁₃×C₂²⋊C₄	C₂×Q₈	1	4	yes	—
96	C₄²×C₁₃	—	2	2	no	—
97	C₄²×C₁₃	—	2	2	no	—
98	C₂×#430	C₂×C₁₃	1	1	yes	characteristic
99	#2905	C₂×C₁₃	1	1	yes	characteristic
100	#2103	C₂×C₁₃	1	8	yes	complemented
101	C₂×#344	C₂×C₁₃	1	2	yes	—
102	#1738	C₂×C₁₃	1	2	yes	—
103	C₂×#338	C₂×C₁₃	1	1	yes	characteristic
104	C₂³×C₁₃	—	8	1	no	—
105	C₂³×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
106	C₂³×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
107	C₂³×C₁₃	#125	1	2	yes	—
108	C₂³×C₁₃	#125	1	2	yes	—
109	C₂³×C₁₃	C₂²×Q₈	1	1	yes	characteristic
110	C₂×C₄×C₁₃	—	2	4	no	—
111	C₂×C₄×C₁₃	—	8	2	no	—
112	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
113	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
114	C₂×C₄×C₁₃	#123	1	1	yes	characteristic
115	C₂×C₄×C₁₃	#123	1	1	yes	characteristic
116	C₂×C₄×C₁₃	#123	1	2	yes	—
117	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
118	C₂×C₄×C₁₃	#124	1	2	yes	—
119	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
120	C₂×C₄×C₁₃	#125	1	2	yes	—
121	C₂×C₄×C₁₃	#125	1	2	yes	—
122	C₂×C₄×C₁₃	#126	1	2	yes	—
123	C₂×C₄×C₁₃	#126	1	2	yes	—
124	C₂×C₄×C₁₃	#126	1	2	yes	—
125	C₂×C₄×C₁₃	#126	1	2	yes	—
126	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
127	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
128	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
129	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
130	C₈×C₁₃	—	2	4	no	—
131	C₈×C₁₃	—	4	2	no	—
132	C₈×C₁₃	—	4	4	no	—
133	C₈×C₁₃	—	4	4	no	—
134	C₂²×M₁₆	C₂²×C₁₃	1	2	yes	—
135	C₂²×M₁₆	C₂²×C₁₃	1	2	yes	—
136	C₂²×M₁₆	C₂²×C₁₃	1	1	yes	characteristic
137	C₂²×M₁₆	C₂²×C₁₃	1	1	yes	characteristic
138	C₂×#117	C₂²×C₁₃	1	1	yes	characteristic
139	#430	—	2	2	no	—
140	#430	C₂²×C₁₃	1	4	yes	—
141	C₂×#108	—	2	2	no	—
142	#424	—	2	2	no	—
143	#424	C₂²×C₁₃	1	4	yes	—
144	#423	—	2	2	no	—
145	#344	C₂²×C₁₃	1	8	yes	—
146	#341	C₂²×C₁₃	1	8	yes	—
147	#338	C₂²×C₁₃	1	4	yes	—
148	C₂²×C₁₃	—	4	1	no	—
149	C₂²×C₁₃	—	4	1	no	—
150	C₂²×C₁₃	—	4	2	no	—
151	C₂²×C₁₃	—	4	2	no	—
152	C₂²×C₁₃	—	4	1	no	—
153	C₂²×C₁₃	#400	1	1	yes	characteristic
154	C₂²×C₁₃	#400	1	1	yes	characteristic
155	C₂²×C₁₃	#400	1	2	yes	—
156	C₂²×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
157	C₂²×C₁₃	C₂×#125	1	2	yes	—
158	C₄×C₁₃	—	4	4	no	—
159	C₄×C₁₃	#400	1	2	yes	—
160	C₄×C₁₃	#400	1	1	yes	characteristic
161	C₄×C₁₃	#400	1	1	yes	characteristic
162	C₄×C₁₃	C₂×#123	1	1	yes	characteristic
163	C₄×C₁₃	C₂×#124	1	1	yes	characteristic
164	C₄×C₁₃	C₂×#126	1	2	yes	—
165	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
166	C₂×M₁₆	—	2	2	no	—
167	C₂×M₁₆	—	2	2	no	—
168	C₂×M₁₆	—	2	1	no	—
169	C₂×M₁₆	—	2	1	no	—
170	C₂×M₁₆	—	2	2	no	—
171	C₂×M₁₆	—	2	2	no	—
172	C₂×M₁₆	—	2	2	no	—
173	C₂×M₁₆	—	2	2	no	—
174	C₂×M₁₆	—	2	2	no	—
175	C₂×M₁₆	—	2	2	no	—
176	C₂×M₁₆	—	2	2	no	—
177	C₂×M₁₆	—	2	2	no	—
178	C₂×M₁₆	—	2	2	no	—
179	C₂×M₁₆	—	2	2	no	—
180	C₂×M₁₆	—	2	2	no	—
181	C₂×M₁₆	—	2	2	no	—
182	C₂×M₁₆	D₄×C₁₃	1	2	yes	—
183	C₂×M₁₆	C₂³×C₁₃	1	4	yes	—
184	C₂×M₁₆	C₂³×C₁₃	1	4	yes	—
185	C₂×M₁₆	C₂³×C₁₃	1	2	yes	—
186	C₂²×C₈	—	2	2	no	—
187	C₂²×C₈	—	2	2	no	—
188	C₂²×C₈	D₄×C₁₃	1	2	yes	—
189	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
190	#117	—	2	1	no	—
191	#117	—	2	1	no	—
192	#117	Q₈×C₁₃	1	2	yes	—
193	#117	C₂³×C₁₃	1	2	yes	—
194	C₂×C₄⋊C₄	—	2	1	no	—
195	C₂×C₂²⋊C₄	Q₈×C₁₃	1	2	yes	—
196	C₂×C₄²	—	2	1	no	—
197	#108	—	2	8	no	—
198	C₄⋊C₈	—	2	8	no	—
199	C₂×C₁₃	—	2	1	no	—
200	C₂×C₁₃	—	2	2	no	—
201	C₂×C₁₃	—	2	2	no	—
202	C₂×C₁₃	—	2	1	no	—
203	C₂×C₁₃	—	2	1	no	—
204	C₂×C₁₃	C₂×#400	1	1	yes	characteristic
205	C₂⁴	C₂×Q₈×C₁₃	1	1	yes	characteristic
206	C₂²×C₄	—	2	2	no	—
207	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
208	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
209	C₂²×C₄	C₂×Q₈×C₁₃	1	1	yes	characteristic
210	C₂²×C₄	C₁₃×#41	1	1	yes	characteristic
211	C₂²×C₄	C₁₃×#41	1	1	yes	characteristic
212	C₂²×C₄	C₁₃×#41	1	2	yes	—
213	C₂²×C₄	C₁₃×#41	1	2	yes	—
214	C₂²×C₄	C₁₃×#41	1	2	yes	—
215	C₂²×C₄	C₁₃×#41	1	2	yes	—
216	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
217	M₁₆	—	2	4	no	—
218	M₁₆	—	2	4	no	—
219	M₁₆	—	2	4	no	—
220	M₁₆	—	2	4	no	—
221	M₁₆	—	2	4	no	—
222	M₁₆	—	2	4	no	—
223	M₁₆	—	4	1	no	—
224	M₁₆	—	4	1	no	—
225	M₁₆	—	4	2	no	—
226	M₁₆	—	4	2	no	—
227	M₁₆	—	4	2	no	—
228	M₁₆	—	4	2	no	—
229	M₁₆	—	4	2	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	4	no	—
234	C₂×C₈	—	2	4	no	—
235	C₂×C₈	—	2	4	no	—
236	C₂×C₈	—	4	2	no	—
237	C₂×C₈	—	4	2	no	—
238	C₂×C₈	—	4	2	no	—
239	C₂×C₈	—	4	2	no	—
240	C₂×C₈	C₂×D₄×C₁₃	1	4	yes	—
241	C₄⋊C₄	—	2	2	no	—
242	C₄⋊C₄	—	2	2	no	—
243	C₂²⋊C₄	—	2	2	no	—
244	C₂²⋊C₄	C₂×Q₈×C₁₃	1	4	yes	—
245	C₄²	—	2	2	no	—
246	C₄²	—	2	2	no	—
247	C₁₃	#23753	1	1	yes	characteristic, direct product factor, Sylow
248	C₂³	—	8	1	no	—
249	C₂³	C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
250	C₂³	C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
251	C₂³	C₁₃×#125	1	2	yes	—
252	C₂³	C₁₃×#125	1	2	yes	—
253	C₂³	C₂²×Q₈×C₁₃	1	1	yes	characteristic
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	8	2	no	—
256	C₂×C₄	C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
257	C₂×C₄	C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
258	C₂×C₄	C₁₃×#123	1	1	yes	characteristic
259	C₂×C₄	C₁₃×#123	1	1	yes	characteristic
260	C₂×C₄	C₁₃×#123	1	2	yes	—
261	C₂×C₄	C₁₃×#124	1	1	yes	characteristic
262	C₂×C₄	C₁₃×#124	1	2	yes	—
263	C₂×C₄	C₁₃×#124	1	1	yes	characteristic
264	C₂×C₄	C₁₃×#125	1	2	yes	—
265	C₂×C₄	C₁₃×#125	1	2	yes	—
266	C₂×C₄	C₁₃×#126	1	2	yes	—
267	C₂×C₄	C₁₃×#126	1	2	yes	—
268	C₂×C₄	C₁₃×#126	1	2	yes	—
269	C₂×C₄	C₁₃×#126	1	2	yes	—
270	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
271	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
272	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
273	C₂×C₄	C₂×C₁₃×#41	1	1	yes	characteristic
274	C₈	—	2	4	no	—
275	C₈	—	4	2	no	—
276	C₈	—	4	4	no	—
277	C₈	—	4	4	no	—
278	C₂²	—	4	1	no	—
279	C₂²	—	4	1	no	—
280	C₂²	—	4	2	no	—
281	C₂²	—	4	2	no	—
282	C₂²	—	4	1	no	—
283	C₂²	C₁₃×#400	1	1	yes	characteristic
284	C₂²	C₁₃×#400	1	1	yes	characteristic
285	C₂²	C₁₃×#400	1	2	yes	—
286	C₂²	C₂×C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
287	C₂²	C₂×C₁₃×#125	1	2	yes	—
288	C₄	—	4	4	no	—
289	C₄	C₁₃×#400	1	2	yes	—
290	C₄	C₁₃×#400	1	1	yes	characteristic
291	C₄	C₁₃×#400	1	1	yes	characteristic
292	C₄	C₂×C₁₃×#123	1	1	yes	characteristic
293	C₄	C₂×C₁₃×#124	1	1	yes	characteristic
294	C₄	C₂×C₁₃×#126	1	2	yes	—
295	C₂	—	2	1	no	—
296	C₂	—	2	2	no	—
297	C₂	—	2	1	no	—
298	C₂	—	2	1	no	—
299	C₂	—	2	2	no	—
300	C₂	C₂×C₁₃×#400	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₁₃	C₂×#124
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃
socle	C₂×C₁₃	C₂×#400
prime radical	C₂⁴×C₁₃	C₂×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₃ ◁ C₂³×C₄×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₃×#2103	C₂	semidirect
55. #23753	C₁₃	direct (N × H)
100. #2103	C₂×C₁₃	semidirect
247. C₁₃	#23753	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×#430	2
C₁₃×#2905	2
C₁₃×#2103	2
C₂×C₁₃×#344	2
C₁₃×#1738	2
C₂×C₁₃×#338	2
#23753	13

Subgroups — C13×#23753 #15710092

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#23753 #15710092