Subgroups — C₁₃×#17991 #15704330

Order 3328.

544 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×#3170	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#2985	C₂	1	2	yes	maximal
3	C₁₃×#1587	C₂	1	4	yes	maximal, complemented
4	C₁₃×#1584	C₂	1	4	yes	maximal, complemented
5	C₁₃×#1568	C₂	1	1	yes	maximal, characteristic
6	C₁₃×#1497	C₂	1	2	yes	maximal
7	C₁₃×#1452	C₂	1	1	yes	maximal, characteristic
8	C₁₃×#533	C₂²	1	2	yes	—
9	C₁₃×#500	C₄	1	1	yes	characteristic
10	C₁₃×#494	C₄	1	1	yes	characteristic
11	C₁₃×#487	C₄	1	2	yes	—
12	C₁₃×#467	C₄	1	2	yes	—
13	C₁₃×#467	C₄	1	2	yes	—
14	C₁₃×#446	—	2	4	no	—
15	C₁₃×#446	C₂²	1	4	yes	—
16	C₈×Q₈×C₁₃	C₂²	1	4	yes	—
17	C₄×C₁₃×Q₁₆	C₂²	1	2	yes	—
18	C₄×C₁₃×Q₁₆	C₂²	1	2	yes	—
19	C₁₃×#433	—	2	4	no	—
20	C₁₃×#432	C₂²	1	2	yes	—
21	C₁₃×#431	C₂²	1	1	yes	characteristic
22	C₁₃×#431	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
24	C₁₃×#335	C₂²	1	2	yes	—
25	C₁₃×#334	C₂²	1	2	yes	—
26	C₁₃×Q₈⋊C₈	C₂²	1	8	yes	—
27	C₁₃×#322	C₂²	1	2	yes	—
28	C₁₃×#322	C₂²	1	2	yes	—
29	C₂×C₁₃×Q₁₆	C₂×C₄	1	2	yes	—
30	C₂×C₁₃×Q₁₆	C₂×C₄	1	2	yes	—
31	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
33	C₁₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
34	C₁₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
35	C₁₃×#125	—	2	2	no	—
36	C₁₃×C₂²⋊Q₈	—	2	2	no	—
37	C₁₃×C₂²⋊Q₈	C₂×C₄	1	4	yes	—
38	C₄×Q₈×C₁₃	—	2	2	no	—
39	C₄×Q₈×C₁₃	C₂³	1	4	yes	—
40	C₁₃×#117	—	2	2	no	—
41	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
42	C₁₃×#107	C₂×C₄	1	2	yes	—
43	C₁₃×#107	C₂×C₄	1	2	yes	—
44	C₁₃×C₄⋊C₈	—	2	4	no	—
45	C₁₃×C₄⋊C₈	—	2	4	no	—
46	C₁₃×C₄⋊C₈	—	2	1	no	—
47	C₁₃×C₄⋊C₈	D₄	1	2	yes	—
48	C₁₃×C₄⋊C₈	C₂³	1	4	yes	—
49	C₁₃×#103	C₂×C₄	1	4	yes	—
50	C₁₃×#103	C₂×C₄	1	4	yes	—
51	C₁₃×C₂²⋊C₈	—	2	2	no	—
52	C₁₃×C₂²⋊C₈	—	2	1	no	—
53	C₁₃×C₂²⋊C₈	D₄	1	2	yes	—
54	C₁₃×#97	—	2	4	no	—
55	C₁₃×#97	—	2	2	no	—
56	C₄×C₈×C₁₃	—	2	4	no	—
57	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
58	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
59	C₄×C₈×C₁₃	C₂³	1	2	yes	—
60	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
61	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
62	C₄×C₈×C₁₃	C₂³	1	2	yes	—
63	#17991	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
64	C₂×Q₈×C₁₃	—	2	2	no	—
65	C₂×Q₈×C₁₃	C₂²×C₄	1	4	yes	—
66	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
69	C₁₃×Q₁₆	—	4	2	no	—
70	C₁₃×Q₁₆	—	4	2	no	—
71	C₂×C₈×C₁₃	—	2	4	no	—
72	C₂×C₈×C₁₃	—	2	2	no	—
73	C₂×C₈×C₁₃	—	4	1	no	—
74	C₂×C₈×C₁₃	—	4	1	no	—
75	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₁₃	C₂×D₄	1	4	yes	—
80	C₁₃×C₄⋊C₄	—	2	2	no	—
81	C₁₃×C₄⋊C₄	—	2	2	no	—
82	C₁₃×C₄⋊C₄	—	2	2	no	—
83	C₁₃×C₄⋊C₄	—	2	4	no	—
84	C₁₃×C₄⋊C₄	—	2	4	no	—
85	C₁₃×C₄⋊C₄	C₂²×C₄	1	4	yes	—
86	C₁₃×C₂²⋊C₄	—	2	4	no	—
87	C₄²×C₁₃	—	2	4	no	—
88	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
89	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₁₃	#41	1	1	yes	characteristic
91	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
92	#3170	C₂×C₁₃	1	1	yes	characteristic
93	#2985	C₂×C₁₃	1	2	yes	—
94	#1587	C₂×C₁₃	1	4	yes	complemented
95	#1584	C₂×C₁₃	1	4	yes	complemented
96	#1568	C₂×C₁₃	1	1	yes	characteristic
97	#1497	C₂×C₁₃	1	2	yes	—
98	#1452	C₂×C₁₃	1	1	yes	characteristic
99	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
100	Q₈×C₁₃	—	2	4	no	—
101	Q₈×C₁₃	—	2	4	no	—
102	Q₈×C₁₃	—	8	2	no	—
103	C₂×C₄×C₁₃	—	2	1	no	—
104	C₂×C₄×C₁₃	—	2	1	no	—
105	C₂×C₄×C₁₃	—	2	2	no	—
106	C₂×C₄×C₁₃	—	2	4	no	—
107	C₂×C₄×C₁₃	—	2	4	no	—
108	C₂×C₄×C₁₃	—	4	1	no	—
109	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
110	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
112	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
113	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
114	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
115	C₈×C₁₃	—	2	1	no	—
116	C₈×C₁₃	—	2	1	no	—
117	C₈×C₁₃	—	2	1	no	—
118	C₈×C₁₃	—	2	1	no	—
119	C₈×C₁₃	—	2	4	no	—
120	C₈×C₁₃	—	4	2	no	—
121	C₈×C₁₃	—	4	4	no	—
122	#533	C₂²×C₁₃	1	2	yes	—
123	#500	C₄×C₁₃	1	1	yes	characteristic
124	#494	C₄×C₁₃	1	1	yes	characteristic
125	#487	C₄×C₁₃	1	2	yes	—
126	#467	C₄×C₁₃	1	2	yes	—
127	#467	C₄×C₁₃	1	2	yes	—
128	#446	—	2	4	no	—
129	#446	C₂²×C₁₃	1	4	yes	—
130	C₈×Q₈	C₂²×C₁₃	1	4	yes	—
131	C₄×Q₁₆	C₂²×C₁₃	1	2	yes	—
132	C₄×Q₁₆	C₂²×C₁₃	1	2	yes	—
133	#433	—	2	4	no	—
134	#432	C₂²×C₁₃	1	2	yes	—
135	#431	C₂²×C₁₃	1	1	yes	characteristic
136	#431	C₂²×C₁₃	1	1	yes	characteristic
137	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
138	#335	C₂²×C₁₃	1	2	yes	—
139	#334	C₂²×C₁₃	1	2	yes	—
140	Q₈⋊C₈	C₂²×C₁₃	1	8	yes	—
141	#322	C₂²×C₁₃	1	2	yes	—
142	#322	C₂²×C₁₃	1	2	yes	—
143	C₂²×C₁₃	—	2	1	no	—
144	C₂²×C₁₃	—	2	1	no	—
145	C₂²×C₁₃	—	2	1	no	—
146	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
147	C₄×C₁₃	—	2	1	no	—
148	C₄×C₁₃	—	2	1	no	—
149	C₄×C₁₃	—	2	1	no	—
150	C₄×C₁₃	—	2	1	no	—
151	C₄×C₁₃	—	4	4	no	—
152	C₄×C₁₃	—	4	4	no	—
153	C₄×C₁₃	#568	1	1	yes	characteristic
154	C₄×C₁₃	#568	1	1	yes	characteristic
155	C₄×C₁₃	#578	1	2	yes	—
156	C₂×Q₁₆	C₂×C₄×C₁₃	1	2	yes	—
157	C₂×Q₁₆	C₂×C₄×C₁₃	1	2	yes	—
158	C₂²×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
159	C₂²×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
160	C₄⋊Q₈	C₂×C₄×C₁₃	1	2	yes	—
161	C₄⋊Q₈	C₂×C₄×C₁₃	1	2	yes	—
162	#125	—	2	2	no	—
163	C₂²⋊Q₈	—	2	2	no	—
164	C₂²⋊Q₈	C₂×C₄×C₁₃	1	4	yes	—
165	C₄×Q₈	—	2	2	no	—
166	C₄×Q₈	C₂³×C₁₃	1	4	yes	—
167	#117	—	2	2	no	—
168	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
169	#107	C₂×C₄×C₁₃	1	2	yes	—
170	#107	C₂×C₄×C₁₃	1	2	yes	—
171	C₄⋊C₈	—	2	4	no	—
172	C₄⋊C₈	—	2	4	no	—
173	C₄⋊C₈	—	2	1	no	—
174	C₄⋊C₈	D₄×C₁₃	1	2	yes	—
175	C₄⋊C₈	C₂³×C₁₃	1	4	yes	—
176	#103	C₂×C₄×C₁₃	1	4	yes	—
177	#103	C₂×C₄×C₁₃	1	4	yes	—
178	C₂²⋊C₈	—	2	2	no	—
179	C₂²⋊C₈	—	2	1	no	—
180	C₂²⋊C₈	D₄×C₁₃	1	2	yes	—
181	#97	—	2	4	no	—
182	#97	—	2	2	no	—
183	C₄×C₈	—	2	4	no	—
184	C₄×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
185	C₄×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
186	C₄×C₈	C₂³×C₁₃	1	2	yes	—
187	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
188	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
189	C₄×C₈	C₂³×C₁₃	1	2	yes	—
190	C₂×C₁₃	—	4	1	no	—
191	C₂×C₁₃	#2932	1	1	yes	characteristic
192	C₂×C₁₃	#2950	1	1	yes	characteristic
193	C₂×C₁₃	#2957	1	1	yes	characteristic
194	C₂×Q₈	—	2	2	no	—
195	C₂×Q₈	C₂²×C₄×C₁₃	1	4	yes	—
196	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
197	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
198	C₂²×C₄	C₁₃×#41	1	1	yes	characteristic
199	Q₁₆	—	4	2	no	—
200	Q₁₆	—	4	2	no	—
201	C₂×C₈	—	2	4	no	—
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	—	4	1	no	—
204	C₂×C₈	—	4	1	no	—
205	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
206	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
207	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
208	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
209	C₂×C₈	C₂×D₄×C₁₃	1	4	yes	—
210	C₄⋊C₄	—	2	2	no	—
211	C₄⋊C₄	—	2	2	no	—
212	C₄⋊C₄	—	2	2	no	—
213	C₄⋊C₄	—	2	4	no	—
214	C₄⋊C₄	—	2	4	no	—
215	C₄⋊C₄	C₂²×C₄×C₁₃	1	4	yes	—
216	C₂²⋊C₄	—	2	4	no	—
217	C₄²	—	2	4	no	—
218	C₄²	C₂²×C₄×C₁₃	1	1	yes	characteristic
219	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
220	C₄²	C₁₃×#41	1	1	yes	characteristic
221	C₄²	C₂⁴×C₁₃	1	1	yes	characteristic
222	C₁₃	#17991	1	1	yes	characteristic, direct product factor, Sylow
223	C₂³	C₄×D₄×C₁₃	1	1	yes	characteristic
224	Q₈	—	2	4	no	—
225	Q₈	—	2	4	no	—
226	Q₈	—	8	2	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	4	1	no	—
233	C₂×C₄	C₄×D₄×C₁₃	1	1	yes	characteristic
234	C₂×C₄	C₄×D₄×C₁₃	1	1	yes	characteristic
235	C₂×C₄	C₄×D₄×C₁₃	1	1	yes	characteristic
236	C₂×C₄	C₂³×C₄×C₁₃	1	1	yes	characteristic
237	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
238	C₂×C₄	C₂×C₁₃×#41	1	1	yes	characteristic
239	C₈	—	2	1	no	—
240	C₈	—	2	1	no	—
241	C₈	—	2	1	no	—
242	C₈	—	2	1	no	—
243	C₈	—	2	4	no	—
244	C₈	—	4	2	no	—
245	C₈	—	4	4	no	—
246	C₂²	—	2	1	no	—
247	C₂²	—	2	1	no	—
248	C₂²	—	2	1	no	—
249	C₂²	C₂×C₄×D₄×C₁₃	1	1	yes	characteristic
250	C₄	—	2	1	no	—
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	2	1	no	—
254	C₄	—	4	4	no	—
255	C₄	—	4	4	no	—
256	C₄	C₁₃×#568	1	1	yes	characteristic
257	C₄	C₁₃×#568	1	1	yes	characteristic
258	C₄	C₁₃×#578	1	2	yes	—
259	C₂	—	4	1	no	—
260	C₂	C₁₃×#2932	1	1	yes	characteristic
261	C₂	C₁₃×#2950	1	1	yes	characteristic
262	C₂	C₁₃×#2957	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₃
Frattini, cosocle	C₄²	C₂⁴×C₁₃
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	C₂³×C₁₃	C₄×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃ ◁ C₁₃×#431 ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₃×#1587	C₂	semidirect
4. C₁₃×#1584	C₂	semidirect
63. #17991	C₁₃	direct (N × H)
94. #1587	C₂×C₁₃	semidirect
95. #1584	C₂×C₁₃	semidirect
222. C₁₃	#17991	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×#3170	2
C₁₃×#2985	2
C₁₃×#1587	2
C₁₃×#1584	2
C₁₃×#1568	2
C₁₃×#1497	2
C₁₃×#1452	2
#17991	13

Subgroups — C13×#17991 #15704330

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#17991 #15704330