Subgroups — C₁₃×#9698 #15696037

Order 3328.

622 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×#2858	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#1842	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₁₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₁₃×#332	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₃×#331	C₂	1	2	yes	maximal
6	C₂×C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
7	C₂×C₁₃×#127	C₂²	1	1	yes	characteristic
8	C₂×C₁₃×#124	—	2	2	no	—
9	C₂×C₁₃×C₄⋊C₈	—	2	2	no	—
10	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
11	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₁₃	—	2	1	no	—
13	C₁₃×#390	—	2	1	no	—
14	C₁₃×#390	—	2	1	no	—
15	C₁₃×#390	C₄	1	4	yes	—
16	C₄³×C₁₃	C₂²	1	1	yes	characteristic
17	C₁₃×#336	—	2	1	no	—
18	C₁₃×#332	—	2	4	no	—
19	C₁₃×#331	—	2	4	no	—
20	C₂²×Q₈×C₁₃	—	2	1	no	—
21	C₂²×D₄×C₁₃	—	2	1	no	—
22	C₂²×D₄×C₁₃	—	2	1	no	—
23	C₂²×D₄×C₁₃	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₁₃	—	2	2	no	—
25	C₂²×C₈×C₁₃	—	2	2	no	—
26	C₁₃×C₄⋊Q₈	—	2	2	no	—
27	C₁₃×C₄⋊Q₈	—	2	2	no	—
28	C₁₃×#127	—	2	2	no	—
29	C₁₃×#127	—	2	2	no	—
30	C₁₃×#124	—	4	4	no	—
31	C₁₃×#124	—	4	4	no	—
32	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
33	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
34	C₂×C₁₃×C₂²⋊C₄	—	2	4	no	—
35	C₂×C₁₃×C₂²⋊C₄	—	2	4	no	—
36	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₁₃	D₄	1	2	yes	—
40	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
41	C₁₃×C₄⋊C₈	—	2	2	no	—
42	C₁₃×C₄⋊C₈	—	2	2	no	—
43	C₁₃×C₄⋊C₈	—	2	2	no	—
44	C₁₃×C₄⋊C₈	—	4	4	no	—
45	C₁₃×C₄⋊C₈	D₄	1	4	yes	—
46	C₄×C₈×C₁₃	—	4	2	no	—
47	#9698	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
48	C₂⁴×C₁₃	—	2	2	no	—
49	C₂×Q₈×C₁₃	—	4	1	no	—
50	C₂×Q₈×C₁₃	—	4	1	no	—
51	C₂×Q₈×C₁₃	—	4	2	no	—
52	C₂×Q₈×C₁₃	—	4	2	no	—
53	C₂×D₄×C₁₃	—	2	4	no	—
54	C₂×D₄×C₁₃	—	4	4	no	—
55	C₂×D₄×C₁₃	—	4	2	no	—
56	C₂×D₄×C₁₃	—	4	1	no	—
57	C₂×D₄×C₁₃	—	4	1	no	—
58	C₂×D₄×C₁₃	—	4	2	no	—
59	C₂×D₄×C₁₃	—	4	2	no	—
60	C₂×D₄×C₁₃	—	4	2	no	—
61	C₂×D₄×C₁₃	—	4	1	no	—
62	C₂×D₄×C₁₃	—	4	1	no	—
63	C₂²×C₄×C₁₃	—	2	2	no	—
64	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
65	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	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	C₂×D₄	1	1	yes	characteristic
69	C₂×C₈×C₁₃	—	2	4	no	—
70	C₂×C₈×C₁₃	—	4	2	no	—
71	C₂×C₈×C₁₃	—	4	2	no	—
72	C₂×C₈×C₁₃	—	4	2	no	—
73	C₂×C₈×C₁₃	—	4	4	no	—
74	C₁₃×C₄⋊C₄	—	2	4	no	—
75	C₁₃×C₄⋊C₄	—	4	2	no	—
76	C₁₃×C₂²⋊C₄	—	4	8	no	—
77	C₁₃×C₂²⋊C₄	—	4	8	no	—
78	C₄²×C₁₃	—	2	2	no	—
79	C₄²×C₁₃	—	2	1	no	—
80	C₄²×C₁₃	—	2	1	no	—
81	C₄²×C₁₃	—	4	2	no	—
82	C₄²×C₁₃	D₈	1	4	yes	—
83	C₄²×C₁₃	SD₁₆	1	4	yes	—
84	C₄²×C₁₃	C₂×D₄	1	2	yes	—
85	C₄²×C₁₃	#41	1	2	yes	—
86	#2858	C₂×C₁₃	1	1	yes	characteristic
87	#1842	C₂×C₁₃	1	1	yes	characteristic, complemented
88	C₄×C₄⋊C₈	C₂×C₁₃	1	1	yes	characteristic, complemented
89	C₂×#332	C₂×C₁₃	1	2	yes	complemented
90	C₂×#331	C₂×C₁₃	1	2	yes	—
91	C₂³×C₁₃	—	4	2	no	—
92	C₂³×C₁₃	—	4	4	no	—
93	C₂³×C₁₃	—	4	4	no	—
94	C₂³×C₁₃	—	4	2	no	—
95	C₂³×C₁₃	—	4	2	no	—
96	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	Q₈×C₁₃	—	8	2	no	—
98	Q₈×C₁₃	—	8	1	no	—
99	Q₈×C₁₃	—	8	1	no	—
100	D₄×C₁₃	—	8	4	no	—
101	D₄×C₁₃	—	8	2	no	—
102	D₄×C₁₃	—	8	2	no	—
103	D₄×C₁₃	—	8	2	no	—
104	D₄×C₁₃	—	8	1	no	—
105	D₄×C₁₃	—	8	1	no	—
106	C₂×C₄×C₁₃	—	2	2	no	—
107	C₂×C₄×C₁₃	—	2	2	no	—
108	C₂×C₄×C₁₃	—	2	2	no	—
109	C₂×C₄×C₁₃	—	2	2	no	—
110	C₂×C₄×C₁₃	—	2	2	no	—
111	C₂×C₄×C₁₃	—	4	2	no	—
112	C₂×C₄×C₁₃	—	4	4	no	—
113	C₂×C₄×C₁₃	#102	1	4	yes	—
114	C₂×C₄×C₁₃	#102	1	4	yes	—
115	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
116	C₂×C₄×C₁₃	#121	1	2	yes	—
117	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
118	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
119	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
120	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
121	C₂×C₄×C₁₃	#135	1	2	yes	—
122	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
123	C₂×C₄×C₁₃	#137	1	1	yes	characteristic
124	C₈×C₁₃	—	4	4	no	—
125	C₈×C₁₃	—	8	2	no	—
126	C₂×C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
127	C₂×#127	C₂²×C₁₃	1	1	yes	characteristic
128	C₂×#124	—	2	2	no	—
129	C₂×C₄⋊C₈	—	2	2	no	—
130	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
131	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
132	C₂×C₄×C₈	—	2	1	no	—
133	#390	—	2	1	no	—
134	#390	—	2	1	no	—
135	#390	C₄×C₁₃	1	4	yes	—
136	C₄³	C₂²×C₁₃	1	1	yes	characteristic
137	#336	—	2	1	no	—
138	#332	—	2	4	no	—
139	#331	—	2	4	no	—
140	C₂²×C₁₃	—	8	4	no	—
141	C₂²×C₁₃	—	8	4	no	—
142	C₂²×C₁₃	—	8	2	no	—
143	C₂²×C₁₃	—	8	2	no	—
144	C₂²×C₁₃	—	8	2	no	—
145	C₂²×C₁₃	#390	1	1	yes	characteristic
146	C₂²×C₁₃	#413	1	2	yes	—
147	C₂²×C₁₃	C₂×#102	1	2	yes	—
148	C₂²×C₁₃	#416	1	1	yes	characteristic
149	C₂²×C₁₃	#417	1	1	yes	characteristic
150	C₄×C₁₃	—	2	4	no	—
151	C₄×C₁₃	—	2	4	no	—
152	C₄×C₁₃	—	2	2	no	—
153	C₄×C₁₃	—	8	2	no	—
154	C₄×C₁₃	#463	1	2	yes	—
155	C₄×C₁₃	#464	1	2	yes	—
156	C₄×C₁₃	#486	1	2	yes	—
157	C₄×C₁₃	#492	1	1	yes	characteristic
158	C₄×C₁₃	#493	1	1	yes	characteristic
159	C₂²×Q₈	—	2	1	no	—
160	C₂²×D₄	—	2	1	no	—
161	C₂²×D₄	—	2	1	no	—
162	C₂²×D₄	C₂×C₄×C₁₃	1	2	yes	—
163	C₂²×C₈	—	2	2	no	—
164	C₂²×C₈	—	2	2	no	—
165	C₄⋊Q₈	—	2	2	no	—
166	C₄⋊Q₈	—	2	2	no	—
167	#127	—	2	2	no	—
168	#127	—	2	2	no	—
169	#124	—	4	4	no	—
170	#124	—	4	4	no	—
171	C₂×C₄⋊C₄	—	2	1	no	—
172	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	2	yes	—
173	C₂×C₂²⋊C₄	—	2	4	no	—
174	C₂×C₂²⋊C₄	—	2	4	no	—
175	C₂×C₄²	C₂×C₄×C₁₃	1	2	yes	—
176	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
177	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
178	C₂×C₄²	D₄×C₁₃	1	2	yes	—
179	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
180	C₄⋊C₈	—	2	2	no	—
181	C₄⋊C₈	—	2	2	no	—
182	C₄⋊C₈	—	2	2	no	—
183	C₄⋊C₈	—	4	4	no	—
184	C₄⋊C₈	D₄×C₁₃	1	4	yes	—
185	C₄×C₈	—	4	2	no	—
186	C₂×C₁₃	—	16	2	no	—
187	C₂×C₁₃	#1540	1	2	yes	—
188	C₂×C₁₃	#1547	1	2	yes	—
189	C₂×C₁₃	#1957	1	1	yes	characteristic
190	C₂×C₁₃	#1967	1	1	yes	characteristic
191	C₂×C₁₃	#1983	1	1	yes	characteristic
192	C₂⁴	—	2	2	no	—
193	C₂×Q₈	—	4	1	no	—
194	C₂×Q₈	—	4	1	no	—
195	C₂×Q₈	—	4	2	no	—
196	C₂×Q₈	—	4	2	no	—
197	C₂×D₄	—	2	4	no	—
198	C₂×D₄	—	4	4	no	—
199	C₂×D₄	—	4	2	no	—
200	C₂×D₄	—	4	1	no	—
201	C₂×D₄	—	4	1	no	—
202	C₂×D₄	—	4	2	no	—
203	C₂×D₄	—	4	2	no	—
204	C₂×D₄	—	4	2	no	—
205	C₂×D₄	—	4	1	no	—
206	C₂×D₄	—	4	1	no	—
207	C₂²×C₄	—	2	2	no	—
208	C₂²×C₄	C₁₃×C₂²⋊C₄	1	2	yes	—
209	C₂²×C₄	C₁₃×C₂²⋊C₄	1	2	yes	—
210	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
211	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
212	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
213	C₂×C₈	—	2	4	no	—
214	C₂×C₈	—	4	2	no	—
215	C₂×C₈	—	4	2	no	—
216	C₂×C₈	—	4	2	no	—
217	C₂×C₈	—	4	4	no	—
218	C₄⋊C₄	—	2	4	no	—
219	C₄⋊C₄	—	4	2	no	—
220	C₂²⋊C₄	—	4	8	no	—
221	C₂²⋊C₄	—	4	8	no	—
222	C₄²	—	2	2	no	—
223	C₄²	—	2	1	no	—
224	C₄²	—	2	1	no	—
225	C₄²	—	4	2	no	—
226	C₄²	C₁₃×D₈	1	4	yes	—
227	C₄²	C₁₃×SD₁₆	1	4	yes	—
228	C₄²	C₂×D₄×C₁₃	1	2	yes	—
229	C₄²	C₁₃×#41	1	2	yes	—
230	C₁₃	#9698	1	1	yes	characteristic, direct product factor, Sylow
231	C₂³	—	4	2	no	—
232	C₂³	—	4	4	no	—
233	C₂³	—	4	4	no	—
234	C₂³	—	4	2	no	—
235	C₂³	—	4	2	no	—
236	C₂³	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
237	Q₈	—	8	2	no	—
238	Q₈	—	8	1	no	—
239	Q₈	—	8	1	no	—
240	D₄	—	8	4	no	—
241	D₄	—	8	2	no	—
242	D₄	—	8	2	no	—
243	D₄	—	8	2	no	—
244	D₄	—	8	1	no	—
245	D₄	—	8	1	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	—	4	4	no	—
253	C₂×C₄	C₁₃×#102	1	4	yes	—
254	C₂×C₄	C₁₃×#102	1	4	yes	—
255	C₂×C₄	C₄×D₄×C₁₃	1	2	yes	—
256	C₂×C₄	C₁₃×#121	1	2	yes	—
257	C₂×C₄	C₁₃×#124	1	1	yes	characteristic
258	C₂×C₄	C₁₃×#127	1	1	yes	characteristic
259	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
260	C₂×C₄	C₂×C₁₃×SD₁₆	1	2	yes	—
261	C₂×C₄	C₁₃×#135	1	2	yes	—
262	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
263	C₂×C₄	C₁₃×#137	1	1	yes	characteristic
264	C₈	—	4	4	no	—
265	C₈	—	8	2	no	—
266	C₂²	—	8	2	no	—
267	C₂²	—	8	4	no	—
268	C₂²	—	8	4	no	—
269	C₂²	—	8	2	no	—
270	C₂²	—	8	2	no	—
271	C₂²	C₁₃×#390	1	1	yes	characteristic
272	C₂²	C₁₃×#413	1	2	yes	—
273	C₂²	C₂×C₁₃×#102	1	2	yes	—
274	C₂²	C₁₃×#416	1	1	yes	characteristic
275	C₂²	C₁₃×#417	1	1	yes	characteristic
276	C₄	—	2	4	no	—
277	C₄	—	2	4	no	—
278	C₄	—	2	2	no	—
279	C₄	—	8	2	no	—
280	C₄	C₁₃×#463	1	2	yes	—
281	C₄	C₁₃×#464	1	2	yes	—
282	C₄	C₁₃×#486	1	2	yes	—
283	C₄	C₁₃×#492	1	1	yes	characteristic
284	C₄	C₁₃×#493	1	1	yes	characteristic
285	C₂	—	16	2	no	—
286	C₂	C₁₃×#1540	1	2	yes	—
287	C₂	C₁₃×#1547	1	2	yes	—
288	C₂	C₁₃×#1957	1	1	yes	characteristic
289	C₂	C₁₃×#1967	1	1	yes	characteristic
290	C₂	C₁₃×#1983	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₁₃
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃
prime radical	C₂×C₁₃×#127	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ 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. C₁₃×#1842	C₂	semidirect
3. C₄×C₁₃×C₄⋊C₈	C₂	semidirect
4. C₂×C₁₃×#332	C₂	semidirect
47. #9698	C₁₃	direct (N × H)
87. #1842	C₂×C₁₃	semidirect
88. C₄×C₄⋊C₈	C₂×C₁₃	semidirect
89. C₂×#332	C₂×C₁₃	semidirect
230. C₁₃	#9698	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×#2858	2
C₁₃×#1842	2
C₄×C₁₃×C₄⋊C₈	2
C₂×C₁₃×#332	2
C₂×C₁₃×#331	2
#9698	13

Subgroups — C13×#9698 #15696037

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#9698 #15696037