Subgroups — C₁₃×#9877 #15696216

Order 3328.

518 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₁₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#1831	C₂	1	1	yes	maximal, characteristic
3	C₄×C₁₃×#97	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#1279	C₂	1	4	yes	maximal
5	C₂×C₁₃×C₄⋊Q₈	C₄	1	1	yes	characteristic
6	C₂×C₄×Q₈×C₁₃	—	2	1	no	—
7	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
8	C₂×C₁₃×#97	—	2	1	no	—
9	C₂×C₁₃×#97	—	2	1	no	—
10	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
11	C₁₃×#391	—	2	1	no	—
12	C₁₃×#389	—	2	1	no	—
13	C₁₃×#389	C₄	1	2	yes	—
14	C₁₃×#384	C₄	1	1	yes	characteristic
15	C₄×C₁₃×C₄⋊C₄	—	2	1	no	—
16	C₄×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
17	C₄³×C₁₃	C₂²	1	1	yes	characteristic
18	C₁₃×#336	—	2	1	no	—
19	C₁₃×#336	—	2	1	no	—
20	C₂²×Q₈×C₁₃	—	2	1	no	—
21	C₂²×C₈×C₁₃	—	2	2	no	—
22	C₂²×C₈×C₁₃	—	2	2	no	—
23	C₁₃×C₄⋊Q₈	—	2	4	no	—
24	C₄×Q₈×C₁₃	—	4	4	no	—
25	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
26	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
27	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
28	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
29	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
30	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₁₃	—	2	2	no	—
32	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
39	C₁₃×C₄⋊C₈	—	2	4	no	—
40	C₁₃×#97	—	2	4	no	—
41	C₁₃×#97	—	2	4	no	—
42	C₄×C₈×C₁₃	—	2	2	no	—
43	C₄×C₈×C₁₃	Q₈	1	4	yes	complemented
44	C₁₃×#95	—	2	2	no	—
45	C₁₃×#95	—	2	2	no	—
46	#9877	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
47	C₂×Q₈×C₁₃	—	2	4	no	—
48	C₂×Q₈×C₁₃	—	4	2	no	—
49	C₂×Q₈×C₁₃	—	4	2	no	—
50	C₂²×C₄×C₁₃	—	2	2	no	—
51	C₂²×C₄×C₁₃	—	2	2	no	—
52	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₁₃	—	2	4	no	—
60	C₂×C₈×C₁₃	—	2	4	no	—
61	C₂×C₈×C₁₃	—	2	4	no	—
62	C₂×C₈×C₁₃	—	4	2	no	—
63	C₂×C₈×C₁₃	—	4	2	no	—
64	C₂×C₈×C₁₃	—	4	2	no	—
65	C₁₃×C₄⋊C₄	—	2	4	no	—
66	C₁₃×C₄⋊C₄	—	2	2	no	—
67	C₁₃×C₄⋊C₄	—	2	4	no	—
68	C₁₃×C₄⋊C₄	—	4	2	no	—
69	C₁₃×C₄⋊C₄	—	4	4	no	—
70	C₁₃×C₄⋊C₄	—	4	2	no	—
71	C₁₃×C₄⋊C₄	M₁₆	1	4	yes	—
72	C₄²×C₁₃	—	2	2	no	—
73	C₄²×C₁₃	—	2	2	no	—
74	C₄²×C₁₃	—	2	2	no	—
75	C₄²×C₁₃	—	2	2	no	—
76	C₄²×C₁₃	—	2	2	no	—
77	C₄²×C₁₃	—	2	2	no	—
78	C₄²×C₁₃	—	4	4	no	—
79	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
80	C₄²×C₁₃	#41	1	2	yes	—
81	C₄×C₄⋊Q₈	C₂×C₁₃	1	1	yes	characteristic
82	#1831	C₂×C₁₃	1	1	yes	characteristic
83	C₄×#97	C₂×C₁₃	1	1	yes	characteristic
84	#1279	C₂×C₁₃	1	4	yes	—
85	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	Q₈×C₁₃	—	4	8	no	—
87	C₂×C₄×C₁₃	—	2	1	no	—
88	C₂×C₄×C₁₃	—	2	1	no	—
89	C₂×C₄×C₁₃	—	2	1	no	—
90	C₂×C₄×C₁₃	—	2	1	no	—
91	C₂×C₄×C₁₃	—	2	1	no	—
92	C₂×C₄×C₁₃	—	2	1	no	—
93	C₂×C₄×C₁₃	—	2	4	no	—
94	C₂×C₄×C₁₃	—	2	1	no	—
95	C₂×C₄×C₁₃	—	2	1	no	—
96	C₂×C₄×C₁₃	—	2	1	no	—
97	C₂×C₄×C₁₃	—	2	1	no	—
98	C₂×C₄×C₁₃	—	2	1	no	—
99	C₂×C₄×C₁₃	—	2	1	no	—
100	C₂×C₄×C₁₃	—	4	2	no	—
101	C₂×C₄×C₁₃	—	4	2	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	#104	1	2	yes	—
106	C₂×C₄×C₁₃	#104	1	2	yes	—
107	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
108	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
109	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
110	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
111	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
112	C₂×C₄×C₁₃	#131	1	2	yes	—
113	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
114	C₂×C₄×C₁₃	#137	1	2	yes	—
115	C₈×C₁₃	—	2	8	no	—
116	C₈×C₁₃	—	8	2	no	—
117	C₂×C₄⋊Q₈	C₄×C₁₃	1	1	yes	characteristic
118	C₂×C₄×Q₈	—	2	1	no	—
119	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
120	C₂×#97	—	2	1	no	—
121	C₂×#97	—	2	1	no	—
122	C₂×C₄×C₈	C₂²×C₁₃	1	2	yes	—
123	#391	—	2	1	no	—
124	#389	—	2	1	no	—
125	#389	C₄×C₁₃	1	2	yes	—
126	#384	C₄×C₁₃	1	1	yes	characteristic
127	C₄×C₄⋊C₄	—	2	1	no	—
128	C₄×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
129	C₄³	C₂²×C₁₃	1	1	yes	characteristic
130	#336	—	2	1	no	—
131	#336	—	2	1	no	—
132	C₂²×C₁₃	#391	1	1	yes	characteristic
133	C₂²×C₁₃	#407	1	1	yes	characteristic
134	C₂²×C₁₃	#408	1	1	yes	characteristic
135	C₂²×C₁₃	#418	1	1	yes	characteristic
136	C₂²×C₁₃	#419	1	1	yes	characteristic
137	C₂²×C₁₃	C₂×#104	1	1	yes	characteristic
138	C₂²×C₁₃	C₂×#104	1	1	yes	characteristic
139	C₄×C₁₃	—	2	2	no	—
140	C₄×C₁₃	—	2	2	no	—
141	C₄×C₁₃	—	2	2	no	—
142	C₄×C₁₃	—	4	1	no	—
143	C₄×C₁₃	—	4	1	no	—
144	C₄×C₁₃	—	4	4	no	—
145	C₄×C₁₃	—	8	2	no	—
146	C₄×C₁₃	#446	1	4	yes	—
147	C₄×C₁₃	#489	1	2	yes	—
148	C₄×C₁₃	#501	1	2	yes	—
149	C₂²×Q₈	—	2	1	no	—
150	C₂²×C₈	—	2	2	no	—
151	C₂²×C₈	—	2	2	no	—
152	C₄⋊Q₈	—	2	4	no	—
153	C₄×Q₈	—	4	4	no	—
154	C₂×C₄⋊C₄	—	2	1	no	—
155	C₂×C₄⋊C₄	—	2	1	no	—
156	C₂×C₄⋊C₄	—	2	2	no	—
157	C₂×C₄⋊C₄	—	2	1	no	—
158	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	2	yes	—
159	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	2	yes	—
160	C₂×C₄²	—	2	2	no	—
161	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
162	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
163	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
164	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
165	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
166	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
167	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
168	C₄⋊C₈	—	2	4	no	—
169	#97	—	2	4	no	—
170	#97	—	2	4	no	—
171	C₄×C₈	—	2	2	no	—
172	C₄×C₈	Q₈×C₁₃	1	4	yes	complemented
173	#95	—	2	2	no	—
174	#95	—	2	2	no	—
175	C₂×C₁₃	#1484	1	1	yes	characteristic
176	C₂×C₁₃	#1485	1	1	yes	characteristic
177	C₂×C₁₃	#1490	1	1	yes	characteristic
178	C₂×C₁₃	#1491	1	1	yes	characteristic
179	C₂×C₁₃	#1985	1	1	yes	characteristic
180	C₂×C₁₃	#1993	1	1	yes	characteristic
181	C₂×C₁₃	#1994	1	1	yes	characteristic
182	C₂×Q₈	—	2	4	no	—
183	C₂×Q₈	—	4	2	no	—
184	C₂×Q₈	—	4	2	no	—
185	C₂²×C₄	—	2	2	no	—
186	C₂²×C₄	—	2	2	no	—
187	C₂²×C₄	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
188	C₂²×C₄	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
189	C₂²×C₄	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
190	C₂²×C₄	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
191	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
192	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
193	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
194	C₂×C₈	—	2	4	no	—
195	C₂×C₈	—	2	4	no	—
196	C₂×C₈	—	2	4	no	—
197	C₂×C₈	—	4	2	no	—
198	C₂×C₈	—	4	2	no	—
199	C₂×C₈	—	4	2	no	—
200	C₄⋊C₄	—	2	4	no	—
201	C₄⋊C₄	—	2	2	no	—
202	C₄⋊C₄	—	2	4	no	—
203	C₄⋊C₄	—	4	2	no	—
204	C₄⋊C₄	—	4	4	no	—
205	C₄⋊C₄	—	4	2	no	—
206	C₄⋊C₄	C₁₃×M₁₆	1	4	yes	—
207	C₄²	—	2	2	no	—
208	C₄²	—	2	2	no	—
209	C₄²	—	2	2	no	—
210	C₄²	—	2	2	no	—
211	C₄²	—	2	2	no	—
212	C₄²	—	2	2	no	—
213	C₄²	—	4	4	no	—
214	C₄²	C₂×Q₈×C₁₃	1	2	yes	—
215	C₄²	C₁₃×#41	1	2	yes	—
216	C₁₃	#9877	1	1	yes	characteristic, direct product factor, Sylow
217	C₂³	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
218	Q₈	—	4	8	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	4	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	C₁₃×#104	1	2	yes	—
238	C₂×C₄	C₁₃×#104	1	2	yes	—
239	C₂×C₄	C₄×Q₈×C₁₃	1	2	yes	—
240	C₂×C₄	C₁₃×C₂²⋊Q₈	1	2	yes	—
241	C₂×C₄	C₁₃×#124	1	1	yes	characteristic
242	C₂×C₄	C₁₃×C₄⋊Q₈	1	1	yes	characteristic
243	C₂×C₄	C₂×C₁₃×M₁₆	1	2	yes	—
244	C₂×C₄	C₁₃×#131	1	2	yes	—
245	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	2	yes	—
246	C₂×C₄	C₁₃×#137	1	2	yes	—
247	C₈	—	2	8	no	—
248	C₈	—	8	2	no	—
249	C₂²	C₁₃×#391	1	1	yes	characteristic
250	C₂²	C₁₃×#407	1	1	yes	characteristic
251	C₂²	C₁₃×#408	1	1	yes	characteristic
252	C₂²	C₁₃×#418	1	1	yes	characteristic
253	C₂²	C₁₃×#419	1	1	yes	characteristic
254	C₂²	C₂×C₁₃×#104	1	1	yes	characteristic
255	C₂²	C₂×C₁₃×#104	1	1	yes	characteristic
256	C₄	—	2	2	no	—
257	C₄	—	2	2	no	—
258	C₄	—	2	2	no	—
259	C₄	—	4	1	no	—
260	C₄	—	4	1	no	—
261	C₄	—	4	4	no	—
262	C₄	—	8	2	no	—
263	C₄	C₁₃×#446	1	4	yes	—
264	C₄	C₁₃×#489	1	2	yes	—
265	C₄	C₁₃×#501	1	2	yes	—
266	C₂	C₁₃×#1484	1	1	yes	characteristic
267	C₂	C₁₃×#1485	1	1	yes	characteristic
268	C₂	C₁₃×#1490	1	1	yes	characteristic
269	C₂	C₁₃×#1491	1	1	yes	characteristic
270	C₂	C₁₃×#1985	1	1	yes	characteristic
271	C₂	C₁₃×#1993	1	1	yes	characteristic
272	C₂	C₁₃×#1994	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₁₃	C₂×D₄
commutator	C₂²×C₄	C₂²×C₄×C₁₃
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃
socle, prime radical	C₂³×C₁₃	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₃ ◁ C₄³×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
43. C₄×C₈×C₁₃	Q₈	semidirect
46. #9877	C₁₃	direct (N × H)
172. C₄×C₈	Q₈×C₁₃	semidirect
216. C₁₃	#9877	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₄×C₁₃×C₄⋊Q₈	2
C₁₃×#1831	2
C₄×C₁₃×#97	2
C₁₃×#1279	2
#9877	13

Subgroups — C13×#9877 #15696216

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#9877 #15696216