Subgroups — C₁₃×#29221 #15715560

Order 3328.

638 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#2859	C₂	1	1	yes	maximal, characteristic
2	C₄×C₁₃×#125	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2100	C₂	1	2	yes	maximal
4	C₁₃×#2099	C₂	1	2	yes	maximal
5	C₁₃×#1991	C₂	1	2	yes	maximal
6	C₁₃×#1989	C₂	1	4	yes	maximal
7	C₁₃×#1835	C₂	1	2	yes	maximal
8	C₄×C₁₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
9	C₂×C₁₃×#125	C₂²	1	2	yes	—
10	C₂×C₁₃×#125	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
13	C₁₃×#389	—	2	2	no	—
14	C₁₃×#389	—	2	2	no	—
15	C₁₃×#389	C₂²	1	2	yes	—
16	C₁₃×#387	—	2	2	no	—
17	C₁₃×#384	C₂²	1	1	yes	characteristic
18	C₁₃×#384	C₂²	1	1	yes	characteristic
19	C₁₃×#382	C₂²	1	2	yes	—
20	C₁₃×#382	C₂²	1	1	yes	characteristic
21	C₄×C₁₃×C₄⋊C₄	—	2	2	no	—
22	C₄×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
23	C₄³×C₁₃	C₂²	1	1	yes	characteristic
24	C₁₃×#340	C₂²	1	4	yes	—
25	C₁₃×#340	C₂²	1	4	yes	—
26	C₁₃×#340	C₂²	1	4	yes	—
27	C₁₃×#340	C₂²	1	4	yes	—
28	C₁₃×#336	C₂²	1	2	yes	—
29	C₂²×C₈×C₁₃	C₂³	1	4	yes	—
30	C₁₃×#125	—	2	2	no	—
31	C₁₃×#125	—	2	8	no	—
32	C₁₃×#125	Q₈	1	4	yes	—
33	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
34	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
35	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
36	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
37	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
38	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
39	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
40	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
41	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
42	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
43	C₂×C₄²×C₁₃	—	2	2	no	—
44	C₂×C₄²×C₁₃	D₄	1	2	yes	—
45	C₂×C₄²×C₁₃	D₄	1	2	yes	—
46	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
49	C₁₃×C₄⋊C₈	—	2	4	no	—
50	C₄×C₈×C₁₃	—	2	4	no	—
51	C₁₃×#95	—	2	2	no	—
52	C₁₃×#95	—	2	2	no	—
53	C₁₃×#95	—	2	2	no	—
54	C₁₃×#95	—	2	4	no	—
55	#29221	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
56	C₂²×C₄×C₁₃	—	2	2	no	—
57	C₂²×C₄×C₁₃	—	2	2	no	—
58	C₂²×C₄×C₁₃	—	2	2	no	—
59	C₂²×C₄×C₁₃	—	2	2	no	—
60	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
65	C₂×C₈×C₁₃	—	2	4	no	—
66	C₂×C₈×C₁₃	—	2	4	no	—
67	C₂×C₈×C₁₃	—	2	4	no	—
68	C₁₃×C₄⋊C₄	—	2	8	no	—
69	C₁₃×C₄⋊C₄	—	2	2	no	—
70	C₁₃×C₄⋊C₄	—	2	4	no	—
71	C₁₃×C₄⋊C₄	—	2	4	no	—
72	C₁₃×C₄⋊C₄	—	2	4	no	—
73	C₁₃×C₄⋊C₄	—	4	4	no	—
74	C₁₃×C₄⋊C₄	—	4	4	no	—
75	C₁₃×C₄⋊C₄	—	4	4	no	—
76	C₁₃×C₄⋊C₄	—	4	4	no	—
77	C₁₃×C₄⋊C₄	—	4	4	no	—
78	C₁₃×C₄⋊C₄	C₂×Q₈	1	4	yes	—
79	C₄²×C₁₃	—	2	4	no	—
80	C₄²×C₁₃	—	2	2	no	—
81	C₄²×C₁₃	—	4	2	no	—
82	C₄²×C₁₃	—	4	4	no	—
83	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
84	C₄²×C₁₃	#41	1	2	yes	—
85	C₄²×C₁₃	#41	1	4	yes	—
86	#2859	C₂×C₁₃	1	1	yes	characteristic
87	C₄×#125	C₂×C₁₃	1	1	yes	characteristic
88	#2100	C₂×C₁₃	1	2	yes	—
89	#2099	C₂×C₁₃	1	2	yes	—
90	#1991	C₂×C₁₃	1	2	yes	—
91	#1989	C₂×C₁₃	1	4	yes	—
92	#1835	C₂×C₁₃	1	2	yes	—
93	C₄×C₄⋊C₈	C₂×C₁₃	1	1	yes	characteristic
94	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
95	C₂×C₄×C₁₃	—	2	2	no	—
96	C₂×C₄×C₁₃	—	2	2	no	—
97	C₂×C₄×C₁₃	—	2	2	no	—
98	C₂×C₄×C₁₃	—	2	2	no	—
99	C₂×C₄×C₁₃	—	2	2	no	—
100	C₂×C₄×C₁₃	—	2	2	no	—
101	C₂×C₄×C₁₃	—	2	4	no	—
102	C₂×C₄×C₁₃	—	4	2	no	—
103	C₂×C₄×C₁₃	—	4	2	no	—
104	C₂×C₄×C₁₃	—	4	2	no	—
105	C₂×C₄×C₁₃	—	4	2	no	—
106	C₂×C₄×C₁₃	—	4	2	no	—
107	C₂×C₄×C₁₃	—	4	4	no	—
108	C₂×C₄×C₁₃	—	4	2	no	—
109	C₂×C₄×C₁₃	—	4	2	no	—
110	C₂×C₄×C₁₃	—	4	4	no	—
111	C₂×C₄×C₁₃	C₂²⋊Q₈	1	4	yes	—
112	C₂×C₄×C₁₃	#124	1	2	yes	—
113	C₂×C₄×C₁₃	#124	1	2	yes	—
114	C₂×C₄×C₁₃	#135	1	4	yes	—
115	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
116	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
117	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
118	C₂×C₄×C₁₃	2₋¹⁺⁴	1	2	yes	—
119	C₈×C₁₃	—	4	4	no	—
120	C₂×#125	C₂²×C₁₃	1	2	yes	—
121	C₂×#125	C₂²×C₁₃	1	1	yes	characteristic
122	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
123	C₂×C₄×C₈	C₂²×C₁₃	1	2	yes	—
124	#389	—	2	2	no	—
125	#389	—	2	2	no	—
126	#389	C₂²×C₁₃	1	2	yes	—
127	#387	—	2	2	no	—
128	#384	C₂²×C₁₃	1	1	yes	characteristic
129	#384	C₂²×C₁₃	1	1	yes	characteristic
130	#382	C₂²×C₁₃	1	1	yes	characteristic
131	#382	C₂²×C₁₃	1	2	yes	—
132	C₄×C₄⋊C₄	—	2	2	no	—
133	C₄×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
134	C₄³	C₂²×C₁₃	1	1	yes	characteristic
135	#340	C₂²×C₁₃	1	4	yes	—
136	#340	C₂²×C₁₃	1	4	yes	—
137	#340	C₂²×C₁₃	1	4	yes	—
138	#340	C₂²×C₁₃	1	4	yes	—
139	#336	C₂²×C₁₃	1	2	yes	—
140	C₂²×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
141	C₂²×C₁₃	C₂×#124	1	1	yes	characteristic
142	C₂²×C₁₃	#536	1	1	yes	characteristic
143	C₂²×C₁₃	C₂×#135	1	2	yes	—
144	C₂²×C₁₃	#575	1	2	yes	—
145	C₄×C₁₃	—	2	4	no	—
146	C₄×C₁₃	—	2	2	no	—
147	C₄×C₁₃	—	2	2	no	—
148	C₄×C₁₃	—	2	4	no	—
149	C₄×C₁₃	—	4	4	no	—
150	C₄×C₁₃	—	8	2	no	—
151	C₄×C₁₃	—	8	2	no	—
152	C₄×C₁₃	—	8	2	no	—
153	C₄×C₁₃	#556	1	2	yes	—
154	C₄×C₁₃	#557	1	2	yes	—
155	C₂²×C₈	C₂³×C₁₃	1	4	yes	—
156	#125	—	2	2	no	—
157	#125	—	2	8	no	—
158	#125	Q₈×C₁₃	1	4	yes	—
159	C₂×C₄⋊C₄	—	2	2	no	—
160	C₂×C₄⋊C₄	—	2	2	no	—
161	C₂×C₄⋊C₄	—	2	2	no	—
162	C₂×C₄⋊C₄	—	2	2	no	—
163	C₂×C₄⋊C₄	—	2	2	no	—
164	C₂×C₄⋊C₄	—	2	2	no	—
165	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
166	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
167	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
168	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
169	C₂×C₄²	—	2	2	no	—
170	C₂×C₄²	D₄×C₁₃	1	2	yes	—
171	C₂×C₄²	D₄×C₁₃	1	2	yes	—
172	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
173	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
174	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
175	C₄⋊C₈	—	2	4	no	—
176	C₄×C₈	—	2	4	no	—
177	#95	—	2	2	no	—
178	#95	—	2	2	no	—
179	#95	—	2	2	no	—
180	#95	—	2	4	no	—
181	C₂×C₁₃	#2568	1	1	yes	characteristic
182	C₂×C₁₃	#3076	1	2	yes	—
183	C₂×C₁₃	#3131	1	1	yes	characteristic
184	C₂×C₁₃	#3142	1	1	yes	characteristic
185	C₂×C₁₃	#3232	1	2	yes	—
186	C₂²×C₄	—	2	2	no	—
187	C₂²×C₄	—	2	2	no	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	2	2	no	—
190	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
191	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
192	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
193	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
194	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
195	C₂×C₈	—	2	4	no	—
196	C₂×C₈	—	2	4	no	—
197	C₂×C₈	—	2	4	no	—
198	C₄⋊C₄	—	2	8	no	—
199	C₄⋊C₄	—	2	2	no	—
200	C₄⋊C₄	—	2	4	no	—
201	C₄⋊C₄	—	2	4	no	—
202	C₄⋊C₄	—	2	4	no	—
203	C₄⋊C₄	—	4	4	no	—
204	C₄⋊C₄	—	4	4	no	—
205	C₄⋊C₄	—	4	4	no	—
206	C₄⋊C₄	—	4	4	no	—
207	C₄⋊C₄	—	4	4	no	—
208	C₄⋊C₄	C₂×Q₈×C₁₃	1	4	yes	—
209	C₄²	—	2	4	no	—
210	C₄²	—	2	2	no	—
211	C₄²	—	4	2	no	—
212	C₄²	—	4	4	no	—
213	C₄²	C₂×Q₈×C₁₃	1	2	yes	—
214	C₄²	C₁₃×#41	1	2	yes	—
215	C₄²	C₁₃×#41	1	4	yes	—
216	C₁₃	#29221	1	1	yes	characteristic, direct product factor, Sylow
217	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	4	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	4	4	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	—	4	4	no	—
234	C₂×C₄	C₁₃×C₂²⋊Q₈	1	4	yes	—
235	C₂×C₄	C₁₃×#124	1	2	yes	—
236	C₂×C₄	C₁₃×#124	1	2	yes	—
237	C₂×C₄	C₁₃×#135	1	4	yes	—
238	C₂×C₄	C₂²×Q₈×C₁₃	1	1	yes	characteristic
239	C₂×C₄	C₂×C₁₃×#41	1	1	yes	characteristic
240	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
241	C₂×C₄	C₁₃×2₋¹⁺⁴	1	2	yes	—
242	C₈	—	4	4	no	—
243	C₂²	C₂×C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
244	C₂²	C₂×C₁₃×#124	1	1	yes	characteristic
245	C₂²	C₁₃×#536	1	1	yes	characteristic
246	C₂²	C₂×C₁₃×#135	1	2	yes	—
247	C₂²	C₁₃×#575	1	2	yes	—
248	C₄	—	2	4	no	—
249	C₄	—	2	2	no	—
250	C₄	—	2	2	no	—
251	C₄	—	2	4	no	—
252	C₄	—	4	4	no	—
253	C₄	—	8	2	no	—
254	C₄	—	8	2	no	—
255	C₄	—	8	2	no	—
256	C₄	C₁₃×#556	1	2	yes	—
257	C₄	C₁₃×#557	1	2	yes	—
258	C₂	C₁₃×#2568	1	1	yes	characteristic
259	C₂	C₁₃×#3076	1	2	yes	—
260	C₂	C₁₃×#3131	1	1	yes	characteristic
261	C₂	C₁₃×#3142	1	1	yes	characteristic
262	C₂	C₁₃×#3232	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃
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
55. #29221	C₁₃	direct (N × H)
216. C₁₃	#29221	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×#2859	2
C₄×C₁₃×#125	2
C₁₃×#2100	2
C₁₃×#2099	2
C₁₃×#1991	2
C₁₃×#1989	2
C₁₃×#1835	2
C₄×C₁₃×C₄⋊C₈	2
#29221	13

Subgroups — C13×#29221 #15715560

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#29221 #15715560