Subgroups — C₁₃×#61883 #15748222

Order 3328.

1958 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3497	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×#557	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#3392	C₂	1	8	yes	maximal, complemented
4	C₁₃×#3359	C₂	1	8	yes	maximal, complemented
5	C₁₃×#3206	C₂	1	6	yes	maximal, complemented
6	C₁₃×#3103	C₂	1	6	yes	maximal, complemented
7	C₁₃×#2982	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×2₋¹⁺⁴	—	2	1	no	—
9	C₁₃×#557	—	2	4	no	—
10	C₁₃×#557	C₂²	1	8	yes	—
11	C₁₃×#550	C₂²	1	4	yes	—
12	D₄×Q₈×C₁₃	C₂²	1	4	yes	—
13	C₁₃×#548	—	2	4	no	—
14	C₁₃×#537	—	2	3	no	—
15	C₁₃×#532	C₂²	1	6	yes	—
16	C₂×C₁₃×C₄⋊Q₈	C₂²	1	3	yes	—
17	C₂×C₁₃×#125	—	2	3	no	—
18	C₂×C₄×Q₈×C₁₃	—	2	1	no	—
19	C₂×C₄×Q₈×C₁₃	C₂²	1	3	yes	—
20	C₂×C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
21	C₁₃×#492	C₂²	1	12	yes	—
22	C₁₃×#487	C₂²	1	12	yes	—
23	C₁₃×#481	C₂²	1	4	yes	—
24	C₁₃×#475	C₂²	1	12	yes	—
25	C₁₃×#461	C₂²	1	24	yes	—
26	C₁₃×#460	C₂²	1	12	yes	—
27	C₁₃×#448	C₂²	1	12	yes	—
28	C₈×Q₈×C₁₃	C₂²	1	8	yes	—
29	C₄×C₁₃×SD₁₆	C₂²	1	24	yes	—
30	C₁₃×#431	C₂²	1	6	yes	—
31	C₁₃×2₋¹⁺⁴	—	4	8	no	—
32	C₂×C₁₃×#41	—	2	4	no	—
33	C₂×C₁₃×#41	—	2	6	no	—
34	C₂²×Q₈×C₁₃	—	2	4	no	—
35	C₂²×Q₈×C₁₃	D₄	1	1	yes	characteristic
36	C₂²×Q₈×C₁₃	C₂³	1	3	yes	—
37	C₂×C₁₃×SD₁₆	C₂³	1	24	yes	—
38	C₁₃×C₄⋊Q₈	—	2	6	no	—
39	C₁₃×C₄⋊Q₈	C₂³	1	12	yes	—
40	C₁₃×C₄⋊Q₈	C₂³	1	6	yes	—
41	C₁₃×#127	C₂³	1	6	yes	—
42	C₁₃×#125	—	2	24	no	—
43	C₁₃×#124	—	2	6	no	—
44	C₁₃×C₂²⋊Q₈	—	2	12	no	—
45	C₁₃×#121	—	2	6	no	—
46	C₁₃×#121	—	2	12	no	—
47	C₁₃×#121	C₂³	1	4	yes	—
48	C₄×Q₈×C₁₃	—	2	8	no	—
49	C₄×Q₈×C₁₃	—	2	6	no	—
50	C₄×Q₈×C₁₃	D₄	1	4	yes	—
51	C₄×Q₈×C₁₃	C₂³	1	12	yes	—
52	C₄×Q₈×C₁₃	C₂³	1	4	yes	—
53	C₄×D₄×C₁₃	—	2	6	no	—
54	C₄×D₄×C₁₃	C₂³	1	12	yes	—
55	C₂×C₁₃×C₄⋊C₄	—	2	3	no	—
56	C₂×C₁₃×C₄⋊C₄	—	2	3	no	—
57	C₂×C₁₃×C₄⋊C₄	—	2	3	no	—
58	C₂×C₁₃×C₄⋊C₄	D₄	1	3	yes	—
59	C₂×C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
60	C₂×C₄²×C₁₃	—	2	3	no	—
61	C₂×C₄²×C₁₃	C₂³	1	3	yes	—
62	C₁₃×#106	C₂³	1	8	yes	—
63	C₁₃×C₄⋊C₈	C₂³	1	12	yes	—
64	C₁₃×#103	C₂³	1	24	yes	—
65	C₁₃×#102	C₂³	1	8	yes	—
66	C₁₃×C₂²⋊C₈	C₂³	1	4	yes	—
67	C₄×C₈×C₁₃	C₂³	1	12	yes	—
68	#61883	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
69	C₁₃×#41	—	4	16	no	—
70	C₁₃×#41	—	4	24	no	—
71	C₂×Q₈×C₁₃	—	2	3	no	—
72	C₂×Q₈×C₁₃	—	2	6	no	—
73	C₂×Q₈×C₁₃	—	4	6	no	—
74	C₂×Q₈×C₁₃	—	4	16	no	—
75	C₂×Q₈×C₁₃	SD₁₆	1	8	yes	complemented
76	C₂×Q₈×C₁₃	C₂×D₄	1	4	yes	—
77	C₂×Q₈×C₁₃	C₂⁴	1	6	yes	—
78	C₂×D₄×C₁₃	—	2	4	no	—
79	C₂×D₄×C₁₃	—	2	6	no	—
80	C₂×D₄×C₁₃	—	2	6	no	—
81	C₂×D₄×C₁₃	C₂⁴	1	4	yes	—
82	C₂²×C₄×C₁₃	—	2	1	no	—
83	C₂²×C₄×C₁₃	—	2	3	no	—
84	C₂²×C₄×C₁₃	—	2	12	no	—
85	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
86	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
87	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
88	C₁₃×SD₁₆	—	4	24	no	—
89	C₂×C₈×C₁₃	C₂⁴	1	8	yes	—
90	C₁₃×C₄⋊C₄	—	2	12	no	—
91	C₁₃×C₄⋊C₄	—	2	12	no	—
92	C₁₃×C₄⋊C₄	—	2	12	no	—
93	C₁₃×C₄⋊C₄	—	2	1	no	—
94	C₁₃×C₄⋊C₄	—	2	6	no	—
95	C₁₃×C₄⋊C₄	C₂×D₄	1	12	yes	—
96	C₁₃×C₄⋊C₄	C₂⁴	1	2	yes	—
97	C₁₃×C₄⋊C₄	C₂⁴	1	4	yes	—
98	C₁₃×C₂²⋊C₄	—	2	4	no	—
99	C₁₃×C₂²⋊C₄	—	2	12	no	—
100	C₄²×C₁₃	—	2	12	no	—
101	C₄²×C₁₃	C₂×D₄	1	6	yes	—
102	C₄²×C₁₃	C₂⁴	1	6	yes	—
103	#3497	C₂×C₁₃	1	1	yes	characteristic
104	C₂×#557	C₂×C₁₃	1	1	yes	characteristic, complemented
105	#3392	C₂×C₁₃	1	8	yes	complemented
106	#3359	C₂×C₁₃	1	8	yes	complemented
107	#3206	C₂×C₁₃	1	6	yes	complemented
108	#3103	C₂×C₁₃	1	6	yes	complemented
109	#2982	C₂×C₁₃	1	1	yes	characteristic, complemented
110	C₂³×C₁₃	—	2	4	no	—
111	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
112	Q₈×C₁₃	—	2	12	no	—
113	Q₈×C₁₃	—	4	6	no	—
114	Q₈×C₁₃	—	4	12	no	—
115	Q₈×C₁₃	C₂×SD₁₆	1	16	yes	—
116	D₄×C₁₃	—	2	8	no	—
117	D₄×C₁₃	—	4	8	no	—
118	D₄×C₁₃	—	4	12	no	—
119	D₄×C₁₃	—	8	6	no	—
120	C₂×C₄×C₁₃	—	2	2	no	—
121	C₂×C₄×C₁₃	—	2	2	no	—
122	C₂×C₄×C₁₃	—	2	6	no	—
123	C₂×C₄×C₁₃	—	2	4	no	—
124	C₂×C₄×C₁₃	—	4	3	no	—
125	C₂×C₄×C₁₃	—	4	2	no	—
126	C₂×C₄×C₁₃	—	4	6	no	—
127	C₂×C₄×C₁₃	—	4	24	no	—
128	C₂×C₄×C₁₃	C₂×SD₁₆	1	12	yes	—
129	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
130	C₂×C₄×C₁₃	C₂²×D₄	1	6	yes	—
131	C₂×C₄×C₁₃	C₂²×D₄	1	6	yes	—
132	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
133	C₈×C₁₃	—	2	8	no	—
134	C₂×2₋¹⁺⁴	—	2	1	no	—
135	#557	—	2	4	no	—
136	#557	C₂²×C₁₃	1	8	yes	—
137	#550	C₂²×C₁₃	1	4	yes	—
138	D₄×Q₈	C₂²×C₁₃	1	4	yes	—
139	#548	—	2	4	no	—
140	#537	—	2	3	no	—
141	#532	C₂²×C₁₃	1	6	yes	—
142	C₂×C₄⋊Q₈	C₂²×C₁₃	1	3	yes	—
143	C₂×#125	—	2	3	no	—
144	C₂×C₄×Q₈	—	2	1	no	—
145	C₂×C₄×Q₈	C₂²×C₁₃	1	3	yes	—
146	C₂×C₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic
147	#492	C₂²×C₁₃	1	12	yes	—
148	#487	C₂²×C₁₃	1	12	yes	—
149	#481	C₂²×C₁₃	1	4	yes	—
150	#475	C₂²×C₁₃	1	12	yes	—
151	#461	C₂²×C₁₃	1	24	yes	—
152	#460	C₂²×C₁₃	1	12	yes	—
153	#448	C₂²×C₁₃	1	12	yes	—
154	C₈×Q₈	C₂²×C₁₃	1	8	yes	—
155	C₄×SD₁₆	C₂²×C₁₃	1	24	yes	—
156	#431	C₂²×C₁₃	1	6	yes	—
157	C₂²×C₁₃	—	2	1	no	—
158	C₂²×C₁₃	—	2	1	no	—
159	C₂²×C₁₃	—	4	4	no	—
160	C₂²×C₁₃	—	4	4	no	—
161	C₂²×C₁₃	—	4	4	no	—
162	C₂²×C₁₃	C₂²×SD₁₆	1	2	yes	—
163	C₂²×C₁₃	C₂³×D₄	1	1	yes	characteristic
164	C₄×C₁₃	—	2	6	no	—
165	C₄×C₁₃	—	2	1	no	—
166	C₄×C₁₃	—	4	2	no	—
167	C₄×C₁₃	—	4	6	no	—
168	C₄×C₁₃	—	4	4	no	—
169	C₄×C₁₃	C₂²×SD₁₆	1	12	yes	—
170	C₄×C₁₃	#585	1	2	yes	—
171	2₋¹⁺⁴	—	4	8	no	—
172	C₂×#41	—	2	4	no	—
173	C₂×#41	—	2	6	no	—
174	C₂²×Q₈	—	2	4	no	—
175	C₂²×Q₈	D₄×C₁₃	1	1	yes	characteristic
176	C₂²×Q₈	C₂³×C₁₃	1	3	yes	—
177	C₂×SD₁₆	C₂³×C₁₃	1	24	yes	—
178	C₄⋊Q₈	—	2	6	no	—
179	C₄⋊Q₈	C₂³×C₁₃	1	12	yes	—
180	C₄⋊Q₈	C₂³×C₁₃	1	6	yes	—
181	#127	C₂³×C₁₃	1	6	yes	—
182	#125	—	2	24	no	—
183	#124	—	2	6	no	—
184	C₂²⋊Q₈	—	2	12	no	—
185	#121	—	2	6	no	—
186	#121	—	2	12	no	—
187	#121	C₂³×C₁₃	1	4	yes	—
188	C₄×Q₈	—	2	8	no	—
189	C₄×Q₈	—	2	6	no	—
190	C₄×Q₈	D₄×C₁₃	1	4	yes	—
191	C₄×Q₈	C₂³×C₁₃	1	12	yes	—
192	C₄×Q₈	C₂³×C₁₃	1	4	yes	—
193	C₄×D₄	—	2	6	no	—
194	C₄×D₄	C₂³×C₁₃	1	12	yes	—
195	C₂×C₄⋊C₄	—	2	3	no	—
196	C₂×C₄⋊C₄	—	2	3	no	—
197	C₂×C₄⋊C₄	—	2	3	no	—
198	C₂×C₄⋊C₄	D₄×C₁₃	1	3	yes	—
199	C₂×C₄⋊C₄	C₂³×C₁₃	1	1	yes	characteristic
200	C₂×C₄²	—	2	3	no	—
201	C₂×C₄²	C₂³×C₁₃	1	3	yes	—
202	#106	C₂³×C₁₃	1	8	yes	—
203	C₄⋊C₈	C₂³×C₁₃	1	12	yes	—
204	#103	C₂³×C₁₃	1	24	yes	—
205	#102	C₂³×C₁₃	1	8	yes	—
206	C₂²⋊C₈	C₂³×C₁₃	1	4	yes	—
207	C₄×C₈	C₂³×C₁₃	1	12	yes	—
208	C₂×C₁₃	—	2	2	no	—
209	C₂×C₁₃	—	8	4	no	—
210	C₂×C₁₃	#3487	1	1	yes	characteristic
211	C₂×C₁₃	C₂³×SD₁₆	1	1	yes	characteristic
212	C₂×C₁₃	#3586	1	1	yes	characteristic
213	#41	—	4	16	no	—
214	#41	—	4	24	no	—
215	C₂×Q₈	—	2	3	no	—
216	C₂×Q₈	—	2	6	no	—
217	C₂×Q₈	—	4	6	no	—
218	C₂×Q₈	—	4	16	no	—
219	C₂×Q₈	C₁₃×SD₁₆	1	8	yes	complemented
220	C₂×Q₈	C₂×D₄×C₁₃	1	4	yes	—
221	C₂×Q₈	C₂⁴×C₁₃	1	6	yes	—
222	C₂×D₄	—	2	4	no	—
223	C₂×D₄	—	2	6	no	—
224	C₂×D₄	—	2	6	no	—
225	C₂×D₄	C₂⁴×C₁₃	1	4	yes	—
226	C₂²×C₄	—	2	1	no	—
227	C₂²×C₄	—	2	3	no	—
228	C₂²×C₄	—	2	12	no	—
229	C₂²×C₄	C₂×D₄×C₁₃	1	3	yes	—
230	C₂²×C₄	C₂×D₄×C₁₃	1	3	yes	—
231	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
232	SD₁₆	—	4	24	no	—
233	C₂×C₈	C₂⁴×C₁₃	1	8	yes	—
234	C₄⋊C₄	—	2	12	no	—
235	C₄⋊C₄	—	2	12	no	—
236	C₄⋊C₄	—	2	12	no	—
237	C₄⋊C₄	—	2	1	no	—
238	C₄⋊C₄	—	2	6	no	—
239	C₄⋊C₄	C₂×D₄×C₁₃	1	12	yes	—
240	C₄⋊C₄	C₂⁴×C₁₃	1	2	yes	—
241	C₄⋊C₄	C₂⁴×C₁₃	1	4	yes	—
242	C₂²⋊C₄	—	2	4	no	—
243	C₂²⋊C₄	—	2	12	no	—
244	C₄²	—	2	12	no	—
245	C₄²	C₂×D₄×C₁₃	1	6	yes	—
246	C₄²	C₂⁴×C₁₃	1	6	yes	—
247	C₁₃	#61883	1	1	yes	characteristic, direct product factor, Sylow
248	C₂³	—	2	4	no	—
249	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
250	Q₈	—	2	12	no	—
251	Q₈	—	4	6	no	—
252	Q₈	—	4	12	no	—
253	Q₈	C₂×C₁₃×SD₁₆	1	16	yes	—
254	D₄	—	2	8	no	—
255	D₄	—	4	8	no	—
256	D₄	—	4	12	no	—
257	D₄	—	8	6	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	6	no	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	4	3	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	4	6	no	—
265	C₂×C₄	—	4	24	no	—
266	C₂×C₄	C₂×C₁₃×SD₁₆	1	12	yes	—
267	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
268	C₂×C₄	C₂²×D₄×C₁₃	1	6	yes	—
269	C₂×C₄	C₂²×D₄×C₁₃	1	6	yes	—
270	C₂×C₄	C₂⁵×C₁₃	1	1	yes	characteristic
271	C₈	—	2	8	no	—
272	C₂²	—	2	1	no	—
273	C₂²	—	2	1	no	—
274	C₂²	—	4	4	no	—
275	C₂²	—	4	4	no	—
276	C₂²	—	4	4	no	—
277	C₂²	C₂²×C₁₃×SD₁₆	1	2	yes	—
278	C₂²	C₂³×D₄×C₁₃	1	1	yes	characteristic
279	C₄	—	2	6	no	—
280	C₄	—	2	1	no	—
281	C₄	—	4	4	no	—
282	C₄	—	4	2	no	—
283	C₄	—	4	6	no	—
284	C₄	C₂²×C₁₃×SD₁₆	1	12	yes	—
285	C₄	C₁₃×#585	1	2	yes	—
286	C₂	—	2	2	no	—
287	C₂	—	8	4	no	—
288	C₂	C₁₃×#3487	1	1	yes	characteristic
289	C₂	C₂³×C₁₃×SD₁₆	1	1	yes	characteristic
290	C₂	C₁₃×#3586	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₁₃
prime radical	C₁₃×#3497	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂×C₄×Q₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₁₃×#557	C₂	semidirect
3. C₁₃×#3392	C₂	semidirect
4. C₁₃×#3359	C₂	semidirect
5. C₁₃×#3206	C₂	semidirect
6. C₁₃×#3103	C₂	semidirect
7. C₁₃×#2982	C₂	semidirect
68. #61883	C₁₃	direct (N × H)
75. C₂×Q₈×C₁₃	SD₁₆	semidirect
104. C₂×#557	C₂×C₁₃	semidirect
105. #3392	C₂×C₁₃	semidirect
106. #3359	C₂×C₁₃	semidirect
107. #3206	C₂×C₁₃	semidirect
108. #3103	C₂×C₁₃	semidirect
109. #2982	C₂×C₁₃	semidirect
219. C₂×Q₈	C₁₃×SD₁₆	semidirect
247. C₁₃	#61883	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×#3497	2
C₂×C₁₃×#557	2
C₁₃×#3392	2
C₁₃×#3359	2
C₁₃×#3206	2
C₁₃×#3103	2
C₁₃×#2982	2
#61883	13

Subgroups — C13×#61883 #15748222

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#61883 #15748222