Subgroups — C₁₃×#12738 #15699077

Order 3328.

926 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₃×#135	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×#362	C₂	1	2	yes	maximal, complemented
3	C₂×C₁₃×#361	C₂	1	2	yes	maximal, complemented
4	C₁₃×#2112	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#1810	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₁₃×#41	—	2	1	no	—
7	C₂×C₁₃×#135	—	2	4	no	—
8	C₂×C₁₃×#135	—	2	4	no	—
9	C₂×C₁₃×#135	—	2	2	no	—
10	C₂×C₁₃×#135	C₄	1	4	yes	—
11	C₂²×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
12	C₂²×C₁₃×SD₁₆	—	2	1	no	—
13	C₂²×C₁₃×D₈	C₂²	1	1	yes	characteristic
14	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
15	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
16	C₂×C₁₃×#108	C₂²	1	2	yes	—
17	C₁₃×#407	—	2	1	no	—
18	C₁₃×#362	—	2	4	no	—
19	C₁₃×#361	—	2	4	no	—
20	C₁₃×C₂²⋊C₁₆	—	2	2	no	—
21	C₁₃×#341	—	2	1	no	—
22	C₂×C₁₃×#41	—	2	8	no	—
23	C₂×C₁₃×#41	—	2	4	no	—
24	C₂×C₁₃×#41	—	4	4	no	—
25	C₂×C₁₃×#41	—	4	2	no	—
26	C₂²×Q₈×C₁₃	—	2	1	no	—
27	C₂²×D₄×C₁₃	—	2	1	no	—
28	C₂²×D₄×C₁₃	—	4	1	no	—
29	C₂³×C₄×C₁₃	—	4	1	no	—
30	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
31	C₁₃×#135	—	2	16	no	—
32	C₁₃×#135	—	2	8	no	—
33	C₁₃×#135	—	2	8	no	—
34	C₂×C₁₃×Q₁₆	—	2	2	no	—
35	C₂×C₁₃×Q₁₆	—	2	1	no	—
36	C₂×C₁₃×Q₁₆	C₂×C₄	1	2	yes	—
37	C₂×C₁₃×Q₁₆	D₄	1	4	yes	—
38	C₂×C₁₃×SD₁₆	—	2	8	no	—
39	C₂×C₁₃×SD₁₆	—	2	4	no	—
40	C₂×C₁₃×D₈	—	2	2	no	—
41	C₂×C₁₃×D₈	—	2	1	no	—
42	C₂×C₁₃×D₈	C₂×C₄	1	2	yes	—
43	C₂×C₁₃×D₈	D₄	1	4	yes	—
44	C₂×C₁₃×M₁₆	—	2	2	no	—
45	C₂²×C₈×C₁₃	—	2	2	no	—
46	C₂²×C₈×C₁₃	—	2	2	no	—
47	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
48	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
49	C₂²×C₈×C₁₃	D₄	1	2	yes	—
50	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
51	C₁₃×M₃₂	—	2	2	no	—
52	C₁₃×M₃₂	—	2	2	no	—
53	C₂×C₁₃×C₁₆	—	2	2	no	—
54	C₁₃×#108	—	2	2	no	—
55	C₁₃×#108	—	2	2	no	—
56	C₁₃×C₂²⋊C₈	—	4	2	no	—
57	#12738	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
58	C₂⁴×C₁₃	—	4	1	no	—
59	C₂⁴×C₁₃	SD₁₆	1	1	yes	characteristic
60	C₁₃×#41	—	2	16	no	—
61	C₁₃×#41	—	4	8	no	—
62	C₁₃×#41	—	4	16	no	—
63	C₂×Q₈×C₁₃	—	2	2	no	—
64	C₂×Q₈×C₁₃	—	2	4	no	—
65	C₂×Q₈×C₁₃	—	4	1	no	—
66	C₂×Q₈×C₁₃	—	4	2	no	—
67	C₂×D₄×C₁₃	—	2	2	no	—
68	C₂×D₄×C₁₃	—	2	4	no	—
69	C₂×D₄×C₁₃	—	4	8	no	—
70	C₂×D₄×C₁₃	—	4	4	no	—
71	C₂×D₄×C₁₃	—	4	1	no	—
72	C₂×D₄×C₁₃	—	4	2	no	—
73	C₂²×C₄×C₁₃	—	2	2	no	—
74	C₂²×C₄×C₁₃	—	2	2	no	—
75	C₂²×C₄×C₁₃	—	4	8	no	—
76	C₂²×C₄×C₁₃	—	4	1	no	—
77	C₂²×C₄×C₁₃	—	4	4	no	—
78	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
79	C₂²×C₄×C₁₃	D₈	1	2	yes	—
80	C₂²×C₄×C₁₃	SD₁₆	1	1	yes	characteristic
81	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
82	C₁₃×Q₁₆	—	2	2	no	—
83	C₁₃×Q₁₆	—	2	4	no	—
84	C₁₃×Q₁₆	C₂²⋊C₄	1	4	yes	—
85	C₁₃×SD₁₆	—	2	16	no	—
86	C₁₃×D₈	—	2	2	no	—
87	C₁₃×D₈	—	2	4	no	—
88	C₁₃×D₈	C₂²⋊C₄	1	4	yes	—
89	C₁₃×M₁₆	—	4	2	no	—
90	C₁₃×M₁₆	—	4	2	no	—
91	C₂×C₈×C₁₃	—	2	2	no	—
92	C₂×C₈×C₁₃	—	2	2	no	—
93	C₂×C₈×C₁₃	—	2	2	no	—
94	C₂×C₈×C₁₃	—	2	4	no	—
95	C₂×C₈×C₁₃	—	4	2	no	—
96	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
97	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
99	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
100	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
101	C₁₃×C₁₆	—	4	2	no	—
102	C₂²×#135	C₂×C₁₃	1	1	yes	characteristic
103	C₂×#362	C₂×C₁₃	1	2	yes	complemented
104	C₂×#361	C₂×C₁₃	1	2	yes	complemented
105	#2112	C₂×C₁₃	1	1	yes	characteristic, complemented
106	#1810	C₂×C₁₃	1	1	yes	characteristic, complemented
107	C₂³×C₁₃	—	2	2	no	—
108	C₂³×C₁₃	—	4	2	no	—
109	C₂³×C₁₃	—	4	2	no	—
110	C₂³×C₁₃	—	4	4	no	—
111	C₂³×C₁₃	—	8	4	no	—
112	C₂³×C₁₃	#102	1	2	yes	—
113	C₂³×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
114	Q₈×C₁₃	—	2	4	no	—
115	Q₈×C₁₃	—	4	2	no	—
116	Q₈×C₁₃	—	4	4	no	—
117	D₄×C₁₃	—	2	4	no	—
118	D₄×C₁₃	—	4	16	no	—
119	D₄×C₁₃	—	4	2	no	—
120	D₄×C₁₃	—	4	4	no	—
121	C₂×C₄×C₁₃	—	2	2	no	—
122	C₂×C₄×C₁₃	—	2	2	no	—
123	C₂×C₄×C₁₃	—	2	2	no	—
124	C₂×C₄×C₁₃	—	2	4	no	—
125	C₂×C₄×C₁₃	—	4	16	no	—
126	C₂×C₄×C₁₃	—	4	2	no	—
127	C₂×C₄×C₁₃	—	4	4	no	—
128	C₂×C₄×C₁₃	#102	1	2	yes	—
129	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
130	C₂×C₄×C₁₃	#120	1	2	yes	—
131	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
132	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
133	C₈×C₁₃	—	2	2	no	—
134	C₈×C₁₃	—	8	2	no	—
135	C₈×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
136	C₈×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
137	C₈×C₁₃	#120	1	2	yes	—
138	C₂²×#41	—	2	1	no	—
139	C₂×#135	—	2	4	no	—
140	C₂×#135	—	2	4	no	—
141	C₂×#135	—	2	2	no	—
142	C₂×#135	C₄×C₁₃	1	4	yes	—
143	C₂²×Q₁₆	C₂²×C₁₃	1	1	yes	characteristic
144	C₂²×SD₁₆	—	2	1	no	—
145	C₂²×D₈	C₂²×C₁₃	1	1	yes	characteristic
146	C₂³×C₈	C₂²×C₁₃	1	1	yes	characteristic
147	C₂×M₃₂	C₂²×C₁₃	1	2	yes	—
148	C₂×#108	C₂²×C₁₃	1	2	yes	—
149	#407	—	2	1	no	—
150	#362	—	2	4	no	—
151	#361	—	2	4	no	—
152	C₂²⋊C₁₆	—	2	2	no	—
153	#341	—	2	1	no	—
154	C₂²×C₁₃	—	2	2	no	—
155	C₂²×C₁₃	—	2	4	no	—
156	C₂²×C₁₃	—	2	2	no	—
157	C₂²×C₁₃	—	4	4	no	—
158	C₂²×C₁₃	—	4	4	no	—
159	C₂²×C₁₃	—	8	8	no	—
160	C₂²×C₁₃	—	8	4	no	—
161	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
162	C₂²×C₁₃	#450	1	2	yes	—
163	C₄×C₁₃	—	2	2	no	—
164	C₄×C₁₃	—	4	4	no	—
165	C₄×C₁₃	#379	1	1	yes	characteristic
166	C₄×C₁₃	#418	1	1	yes	characteristic
167	C₄×C₁₃	#447	1	2	yes	—
168	C₂×#41	—	2	8	no	—
169	C₂×#41	—	2	4	no	—
170	C₂×#41	—	4	4	no	—
171	C₂×#41	—	4	2	no	—
172	C₂²×Q₈	—	2	1	no	—
173	C₂²×D₄	—	2	1	no	—
174	C₂²×D₄	—	4	1	no	—
175	C₂³×C₄	—	4	1	no	—
176	C₂³×C₄	D₄×C₁₃	1	1	yes	characteristic
177	#135	—	2	16	no	—
178	#135	—	2	8	no	—
179	#135	—	2	8	no	—
180	C₂×Q₁₆	—	2	2	no	—
181	C₂×Q₁₆	—	2	1	no	—
182	C₂×Q₁₆	C₂×C₄×C₁₃	1	2	yes	—
183	C₂×Q₁₆	D₄×C₁₃	1	4	yes	—
184	C₂×SD₁₆	—	2	8	no	—
185	C₂×SD₁₆	—	2	4	no	—
186	C₂×D₈	—	2	2	no	—
187	C₂×D₈	—	2	1	no	—
188	C₂×D₈	C₂×C₄×C₁₃	1	2	yes	—
189	C₂×D₈	D₄×C₁₃	1	4	yes	—
190	C₂×M₁₆	—	2	2	no	—
191	C₂²×C₈	—	2	2	no	—
192	C₂²×C₈	—	2	2	no	—
193	C₂²×C₈	C₂×C₄×C₁₃	1	2	yes	—
194	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
195	C₂²×C₈	D₄×C₁₃	1	2	yes	—
196	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
197	M₃₂	—	2	2	no	—
198	M₃₂	—	2	2	no	—
199	C₂×C₁₆	—	2	2	no	—
200	#108	—	2	2	no	—
201	#108	—	2	2	no	—
202	C₂²⋊C₈	—	4	2	no	—
203	C₂×C₁₃	—	2	2	no	—
204	C₂×C₁₃	—	4	2	no	—
205	C₂×C₁₃	—	8	4	no	—
206	C₂×C₁₃	#1786	1	1	yes	characteristic
207	C₂×C₁₃	#2148	1	2	yes	—
208	C₂⁴	—	4	1	no	—
209	C₂⁴	C₁₃×SD₁₆	1	1	yes	characteristic
210	#41	—	2	16	no	—
211	#41	—	4	8	no	—
212	#41	—	4	16	no	—
213	C₂×Q₈	—	2	2	no	—
214	C₂×Q₈	—	2	4	no	—
215	C₂×Q₈	—	4	1	no	—
216	C₂×Q₈	—	4	2	no	—
217	C₂×D₄	—	2	2	no	—
218	C₂×D₄	—	2	4	no	—
219	C₂×D₄	—	4	8	no	—
220	C₂×D₄	—	4	4	no	—
221	C₂×D₄	—	4	1	no	—
222	C₂×D₄	—	4	2	no	—
223	C₂²×C₄	—	2	2	no	—
224	C₂²×C₄	—	2	2	no	—
225	C₂²×C₄	—	4	8	no	—
226	C₂²×C₄	—	4	1	no	—
227	C₂²×C₄	—	4	4	no	—
228	C₂²×C₄	C₁₃×C₂²⋊C₄	1	2	yes	—
229	C₂²×C₄	C₁₃×D₈	1	2	yes	—
230	C₂²×C₄	C₁₃×SD₁₆	1	1	yes	characteristic
231	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
232	Q₁₆	—	2	2	no	—
233	Q₁₆	—	2	4	no	—
234	Q₁₆	C₁₃×C₂²⋊C₄	1	4	yes	—
235	SD₁₆	—	2	16	no	—
236	D₈	—	2	2	no	—
237	D₈	—	2	4	no	—
238	D₈	C₁₃×C₂²⋊C₄	1	4	yes	—
239	M₁₆	—	4	2	no	—
240	M₁₆	—	4	2	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	2	4	no	—
245	C₂×C₈	—	4	2	no	—
246	C₂×C₈	C₁₃×C₂²⋊C₄	1	2	yes	—
247	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
248	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
249	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
250	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
251	C₁₆	—	4	2	no	—
252	C₁₃	#12738	1	1	yes	characteristic, direct product factor, Sylow
253	C₂³	—	2	2	no	—
254	C₂³	—	4	2	no	—
255	C₂³	—	4	2	no	—
256	C₂³	—	4	4	no	—
257	C₂³	—	8	4	no	—
258	C₂³	C₁₃×#102	1	2	yes	—
259	C₂³	C₂×C₁₃×SD₁₆	1	1	yes	characteristic
260	Q₈	—	2	4	no	—
261	Q₈	—	4	2	no	—
262	Q₈	—	4	4	no	—
263	D₄	—	2	4	no	—
264	D₄	—	4	16	no	—
265	D₄	—	4	2	no	—
266	D₄	—	4	4	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	4	16	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	4	4	no	—
274	C₂×C₄	C₁₃×#102	1	2	yes	—
275	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
276	C₂×C₄	C₁₃×#120	1	2	yes	—
277	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
278	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	2	yes	—
279	C₈	—	2	2	no	—
280	C₈	—	8	2	no	—
281	C₈	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
282	C₈	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
283	C₈	C₁₃×#120	1	2	yes	—
284	C₂²	—	2	2	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	2	4	no	—
287	C₂²	—	4	4	no	—
288	C₂²	—	4	4	no	—
289	C₂²	—	8	8	no	—
290	C₂²	—	8	4	no	—
291	C₂²	C₂×C₁₃×#102	1	1	yes	characteristic
292	C₂²	C₁₃×#450	1	2	yes	—
293	C₄	—	2	2	no	—
294	C₄	—	4	4	no	—
295	C₄	C₁₃×#379	1	1	yes	characteristic
296	C₄	C₁₃×#418	1	1	yes	characteristic
297	C₄	C₁₃×#447	1	2	yes	—
298	C₂	—	2	2	no	—
299	C₂	—	4	2	no	—
300	C₂	—	8	4	no	—
301	C₂	C₁₃×#1786	1	1	yes	characteristic
302	C₂	C₁₃×#2148	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₁₃
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃
prime radical	C₂²×C₁₃×#135	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂³×C₄×C₁₃ ◁ C₂³×C₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₁₃×#362	C₂	semidirect
3. C₂×C₁₃×#361	C₂	semidirect
4. C₁₃×#2112	C₂	semidirect
5. C₁₃×#1810	C₂	semidirect
57. #12738	C₁₃	direct (N × H)
103. C₂×#362	C₂×C₁₃	semidirect
104. C₂×#361	C₂×C₁₃	semidirect
105. #2112	C₂×C₁₃	semidirect
106. #1810	C₂×C₁₃	semidirect
252. C₁₃	#12738	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₁₃×#135	2
C₂×C₁₃×#362	2
C₂×C₁₃×#361	2
C₁₃×#2112	2
C₁₃×#1810	2
#12738	13

Subgroups — C13×#12738 #15699077

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#12738 #15699077