Subgroups — C₂×C₁₃×#2096 #15710067

Order 3328.

1238 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×#106	C₂	1	2	yes	maximal
2	C₂²×C₁₃×#103	C₂	1	2	yes	maximal
3	C₂×C₁₃×#391	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₃×#384	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#2096	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₁₃×#336	C₂	1	1	yes	maximal, characteristic
7	C₂³×Q₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂³×C₈×C₁₃	C₂²	1	2	yes	—
9	C₂²×C₁₃×C₄⋊C₄	—	2	2	no	—
10	C₂²×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
11	C₂²×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
12	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×#106	—	2	8	no	—
14	C₂×C₁₃×#106	C₂²	1	8	yes	—
15	C₂×C₁₃×#103	—	2	8	no	—
16	C₂×C₁₃×#103	C₂²	1	8	yes	—
17	C₁₃×#391	—	2	2	no	—
18	C₁₃×#391	C₂²	1	4	yes	—
19	C₁₃×#384	—	2	2	no	—
20	C₁₃×#384	C₂²	1	4	yes	—
21	C₂×C₁₃×#95	—	2	2	no	—
22	C₁₃×#336	C₂²	1	4	yes	—
23	C₂²×Q₈×C₁₃	—	2	1	no	—
24	C₂²×Q₈×C₁₃	—	2	2	no	—
25	C₂²×Q₈×C₁₃	—	2	4	no	—
26	C₂²×Q₈×C₁₃	—	2	4	no	—
27	C₂²×Q₈×C₁₃	Q₈	1	4	yes	—
28	C₂²×Q₈×C₁₃	C₂³	1	2	yes	—
29	C₂³×C₄×C₁₃	—	2	1	no	—
30	C₂³×C₄×C₁₃	—	2	2	no	—
31	C₂³×C₄×C₁₃	—	2	1	no	—
32	C₂³×C₄×C₁₃	D₄	1	2	yes	—
33	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
34	C₂²×C₈×C₁₃	—	2	4	no	—
35	C₂²×C₈×C₁₃	—	2	4	no	—
36	C₂²×C₈×C₁₃	D₄	1	8	yes	—
37	C₂²×C₈×C₁₃	C₂³	1	4	yes	—
38	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
39	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
40	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
41	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
42	C₂×C₁₃×C₄⋊C₄	—	2	8	no	—
43	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
44	C₂×C₁₃×C₄⋊C₄	—	4	8	no	—
45	C₂×C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
46	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
47	C₂×C₄²×C₁₃	—	2	4	no	—
48	C₂×C₄²×C₁₃	D₄	1	2	yes	—
49	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
50	C₁₃×#106	—	2	16	no	—
51	C₁₃×#103	—	2	16	no	—
52	C₁₃×#95	—	2	8	no	—
53	C₂×#2096	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
54	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
55	C₂×Q₈×C₁₃	—	2	4	no	—
56	C₂×Q₈×C₁₃	—	2	2	no	—
57	C₂×Q₈×C₁₃	—	2	16	no	—
58	C₂×Q₈×C₁₃	—	2	4	no	—
59	C₂×Q₈×C₁₃	—	2	4	no	—
60	C₂×Q₈×C₁₃	—	4	2	no	—
61	C₂×Q₈×C₁₃	—	4	8	no	—
62	C₂×Q₈×C₁₃	—	4	2	no	—
63	C₂×Q₈×C₁₃	C₂×Q₈	1	4	yes	—
64	C₂²×C₄×C₁₃	—	2	4	no	—
65	C₂²×C₄×C₁₃	—	2	4	no	—
66	C₂²×C₄×C₁₃	—	2	4	no	—
67	C₂²×C₄×C₁₃	—	2	2	no	—
68	C₂²×C₄×C₁₃	—	2	2	no	—
69	C₂²×C₄×C₁₃	—	2	4	no	—
70	C₂²×C₄×C₁₃	—	4	2	no	—
71	C₂²×C₄×C₁₃	—	4	2	no	—
72	C₂²×C₄×C₁₃	—	4	4	no	—
73	C₂²×C₄×C₁₃	—	4	2	no	—
74	C₂²×C₄×C₁₃	—	4	8	no	—
75	C₂²×C₄×C₁₃	—	4	4	no	—
76	C₂²×C₄×C₁₃	SD₁₆	1	8	yes	—
77	C₂²×C₄×C₁₃	C₂×D₄	1	4	yes	—
78	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₁₃	C₂×D₄	1	4	yes	—
80	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
81	C₂²×C₄×C₁₃	#41	1	4	yes	—
82	C₂²×C₄×C₁₃	#41	1	2	yes	—
83	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
84	C₂×C₈×C₁₃	—	2	16	no	—
85	C₂×C₈×C₁₃	—	2	4	no	—
86	C₂×C₈×C₁₃	—	2	4	no	—
87	C₂×C₈×C₁₃	C₂×D₄	1	8	yes	—
88	C₁₃×C₄⋊C₄	—	2	4	no	—
89	C₁₃×C₄⋊C₄	—	2	4	no	—
90	C₁₃×C₄⋊C₄	—	2	4	no	—
91	C₁₃×C₄⋊C₄	—	4	4	no	—
92	C₁₃×C₄⋊C₄	—	4	2	no	—
93	C₁₃×C₄⋊C₄	—	4	16	no	—
94	C₄²×C₁₃	—	2	4	no	—
95	C₄²×C₁₃	—	2	4	no	—
96	C₂²×#106	C₂×C₁₃	1	2	yes	—
97	C₂²×#103	C₂×C₁₃	1	2	yes	—
98	C₂×#391	C₂×C₁₃	1	1	yes	characteristic
99	C₂×#384	C₂×C₁₃	1	1	yes	characteristic
100	#2096	C₂×C₁₃	1	8	yes	direct product factor
101	C₂×#336	C₂×C₁₃	1	1	yes	characteristic
102	C₂³×C₁₃	C₂²⋊Q₈	1	2	yes	—
103	C₂³×C₁₃	#123	1	2	yes	—
104	C₂³×C₁₃	#127	1	2	yes	—
105	C₂³×C₁₃	C₂×SD₁₆	1	4	yes	—
106	C₂³×C₁₃	#137	1	4	yes	—
107	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
108	Q₈×C₁₃	—	2	8	no	—
109	Q₈×C₁₃	—	4	4	no	—
110	Q₈×C₁₃	—	4	8	no	—
111	C₂×C₄×C₁₃	—	2	4	no	—
112	C₂×C₄×C₁₃	—	2	4	no	—
113	C₂×C₄×C₁₃	—	2	16	no	—
114	C₂×C₄×C₁₃	—	2	4	no	—
115	C₂×C₄×C₁₃	—	4	2	no	—
116	C₂×C₄×C₁₃	—	4	2	no	—
117	C₂×C₄×C₁₃	—	4	8	no	—
118	C₂×C₄×C₁₃	—	4	4	no	—
119	C₂×C₄×C₁₃	—	4	2	no	—
120	C₂×C₄×C₁₃	—	4	8	no	—
121	C₂×C₄×C₁₃	—	4	4	no	—
122	C₂×C₄×C₁₃	—	8	8	no	—
123	C₂×C₄×C₁₃	—	8	4	no	—
124	C₂×C₄×C₁₃	C₂²⋊Q₈	1	8	yes	—
125	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
126	C₂×C₄×C₁₃	#123	1	8	yes	—
127	C₂×C₄×C₁₃	#123	1	2	yes	—
128	C₂×C₄×C₁₃	#127	1	2	yes	—
129	C₂×C₄×C₁₃	C₂×SD₁₆	1	8	yes	—
130	C₂×C₄×C₁₃	C₂²×D₄	1	2	yes	—
131	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
132	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
133	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
134	C₈×C₁₃	—	2	8	no	—
135	C₂³×Q₈	C₂²×C₁₃	1	1	yes	characteristic
136	C₂³×C₈	C₂²×C₁₃	1	2	yes	—
137	C₂²×C₄⋊C₄	—	2	2	no	—
138	C₂²×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
139	C₂²×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
140	C₂²×C₄²	C₂²×C₁₃	1	1	yes	characteristic
141	C₂×#106	—	2	8	no	—
142	C₂×#106	C₂²×C₁₃	1	8	yes	—
143	C₂×#103	—	2	8	no	—
144	C₂×#103	C₂²×C₁₃	1	8	yes	—
145	#391	—	2	2	no	—
146	#391	C₂²×C₁₃	1	4	yes	—
147	#384	—	2	2	no	—
148	#384	C₂²×C₁₃	1	4	yes	—
149	C₂×#95	—	2	2	no	—
150	#336	C₂²×C₁₃	1	4	yes	—
151	C₂²×C₁₃	#399	1	4	yes	—
152	C₂²×C₁₃	#475	1	8	yes	—
153	C₂²×C₁₃	#483	1	8	yes	—
154	C₂²×C₁₃	#492	1	4	yes	—
155	C₂²×C₁₃	#497	1	4	yes	—
156	C₂²×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
157	C₂²×C₁₃	C₂×#123	1	1	yes	characteristic
158	C₂²×C₁₃	C₂×#127	1	1	yes	characteristic
159	C₂²×C₁₃	C₂²×SD₁₆	1	2	yes	—
160	C₂²×C₁₃	C₂×#137	1	2	yes	—
161	C₄×C₁₃	—	2	8	no	—
162	C₄×C₁₃	—	4	4	no	—
163	C₄×C₁₃	—	8	4	no	—
164	C₄×C₁₃	—	8	2	no	—
165	C₄×C₁₃	#399	1	4	yes	—
166	C₄×C₁₃	C₂×C₂²⋊Q₈	1	2	yes	—
167	C₄×C₁₃	C₂×#123	1	2	yes	—
168	C₂²×Q₈	—	2	1	no	—
169	C₂²×Q₈	—	2	2	no	—
170	C₂²×Q₈	—	2	4	no	—
171	C₂²×Q₈	—	2	4	no	—
172	C₂²×Q₈	Q₈×C₁₃	1	4	yes	—
173	C₂²×Q₈	C₂³×C₁₃	1	2	yes	—
174	C₂³×C₄	—	2	1	no	—
175	C₂³×C₄	—	2	2	no	—
176	C₂³×C₄	—	2	1	no	—
177	C₂³×C₄	D₄×C₁₃	1	2	yes	—
178	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
179	C₂²×C₈	—	2	4	no	—
180	C₂²×C₈	—	2	4	no	—
181	C₂²×C₈	D₄×C₁₃	1	8	yes	—
182	C₂²×C₈	C₂³×C₁₃	1	4	yes	—
183	C₂×C₄⋊C₄	—	2	4	no	—
184	C₂×C₄⋊C₄	—	2	4	no	—
185	C₂×C₄⋊C₄	—	2	4	no	—
186	C₂×C₄⋊C₄	—	2	8	no	—
187	C₂×C₄⋊C₄	—	2	2	no	—
188	C₂×C₄⋊C₄	—	2	1	no	—
189	C₂×C₄⋊C₄	—	4	8	no	—
190	C₂×C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
191	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
192	C₂×C₄²	—	2	4	no	—
193	C₂×C₄²	D₄×C₁₃	1	2	yes	—
194	C₂×C₄²	C₂³×C₁₃	1	2	yes	—
195	#106	—	2	16	no	—
196	#103	—	2	16	no	—
197	#95	—	2	8	no	—
198	C₂×C₁₃	#2096	1	8	yes	direct product factor
199	C₂×C₁₃	C₂×#399	1	1	yes	characteristic
200	C₂×C₁₃	C₂×#475	1	2	yes	—
201	C₂×C₁₃	C₂×#483	1	2	yes	—
202	C₂×C₁₃	C₂×#492	1	1	yes	characteristic
203	C₂×C₁₃	C₂×#497	1	1	yes	characteristic
204	C₂⁴	C₂×D₄×C₁₃	1	1	yes	characteristic
205	C₂×Q₈	—	2	4	no	—
206	C₂×Q₈	—	2	2	no	—
207	C₂×Q₈	—	2	16	no	—
208	C₂×Q₈	—	2	4	no	—
209	C₂×Q₈	—	2	4	no	—
210	C₂×Q₈	—	4	8	no	—
211	C₂×Q₈	—	4	2	no	—
212	C₂×Q₈	—	4	2	no	—
213	C₂×Q₈	C₂×Q₈×C₁₃	1	4	yes	—
214	C₂²×C₄	—	2	4	no	—
215	C₂²×C₄	—	2	4	no	—
216	C₂²×C₄	—	2	4	no	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	2	4	no	—
220	C₂²×C₄	—	4	2	no	—
221	C₂²×C₄	—	4	2	no	—
222	C₂²×C₄	—	4	4	no	—
223	C₂²×C₄	—	4	2	no	—
224	C₂²×C₄	—	4	8	no	—
225	C₂²×C₄	—	4	4	no	—
226	C₂²×C₄	C₁₃×SD₁₆	1	8	yes	—
227	C₂²×C₄	C₂×D₄×C₁₃	1	4	yes	—
228	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
229	C₂²×C₄	C₂×D₄×C₁₃	1	4	yes	—
230	C₂²×C₄	C₂×Q₈×C₁₃	1	2	yes	—
231	C₂²×C₄	C₁₃×#41	1	4	yes	—
232	C₂²×C₄	C₁₃×#41	1	2	yes	—
233	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
234	C₂×C₈	—	2	16	no	—
235	C₂×C₈	—	2	4	no	—
236	C₂×C₈	—	2	4	no	—
237	C₂×C₈	C₂×D₄×C₁₃	1	8	yes	—
238	C₄⋊C₄	—	2	4	no	—
239	C₄⋊C₄	—	2	4	no	—
240	C₄⋊C₄	—	2	4	no	—
241	C₄⋊C₄	—	4	4	no	—
242	C₄⋊C₄	—	4	2	no	—
243	C₄⋊C₄	—	4	16	no	—
244	C₄²	—	2	4	no	—
245	C₄²	—	2	4	no	—
246	C₁₃	C₂×#2096	1	1	yes	characteristic, direct product factor, Sylow
247	C₂³	C₁₃×C₂²⋊Q₈	1	2	yes	—
248	C₂³	C₁₃×#123	1	2	yes	—
249	C₂³	C₁₃×#127	1	2	yes	—
250	C₂³	C₂×C₁₃×SD₁₆	1	4	yes	—
251	C₂³	C₁₃×#137	1	4	yes	—
252	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
253	Q₈	—	2	8	no	—
254	Q₈	—	4	8	no	—
255	Q₈	—	4	4	no	—
256	C₂×C₄	—	2	16	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	4	8	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	4	4	no	—
264	C₂×C₄	—	4	2	no	—
265	C₂×C₄	—	4	8	no	—
266	C₂×C₄	—	4	4	no	—
267	C₂×C₄	—	8	8	no	—
268	C₂×C₄	—	8	4	no	—
269	C₂×C₄	C₁₃×C₂²⋊Q₈	1	2	yes	—
270	C₂×C₄	C₁₃×C₂²⋊Q₈	1	8	yes	—
271	C₂×C₄	C₁₃×#123	1	2	yes	—
272	C₂×C₄	C₁₃×#123	1	8	yes	—
273	C₂×C₄	C₁₃×#127	1	2	yes	—
274	C₂×C₄	C₂×C₁₃×SD₁₆	1	8	yes	—
275	C₂×C₄	C₂²×D₄×C₁₃	1	2	yes	—
276	C₂×C₄	C₂²×Q₈×C₁₃	1	1	yes	characteristic
277	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
278	C₂×C₄	C₂×C₁₃×#41	1	1	yes	characteristic
279	C₈	—	2	8	no	—
280	C₂²	C₁₃×#399	1	4	yes	—
281	C₂²	C₁₃×#475	1	8	yes	—
282	C₂²	C₁₃×#483	1	8	yes	—
283	C₂²	C₁₃×#492	1	4	yes	—
284	C₂²	C₁₃×#497	1	4	yes	—
285	C₂²	C₂×C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
286	C₂²	C₂×C₁₃×#123	1	1	yes	characteristic
287	C₂²	C₂×C₁₃×#127	1	1	yes	characteristic
288	C₂²	C₂²×C₁₃×SD₁₆	1	2	yes	—
289	C₂²	C₂×C₁₃×#137	1	2	yes	—
290	C₄	—	2	8	no	—
291	C₄	—	4	4	no	—
292	C₄	—	8	4	no	—
293	C₄	—	8	2	no	—
294	C₄	C₁₃×#399	1	4	yes	—
295	C₄	C₂×C₁₃×C₂²⋊Q₈	1	2	yes	—
296	C₄	C₂×C₁₃×#123	1	2	yes	—
297	C₂	C₁₃×#2096	1	8	yes	direct product factor
298	C₂	C₂×C₁₃×#399	1	1	yes	characteristic
299	C₂	C₂×C₁₃×#475	1	2	yes	—
300	C₂	C₂×C₁₃×#483	1	2	yes	—
301	C₂	C₂×C₁₃×#492	1	1	yes	characteristic
302	C₂	C₂×C₁₃×#497	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
5. C₁₃×#2096	C₂	direct (N × H)
53. C₂×#2096	C₁₃	direct (N × H)
100. #2096	C₂×C₁₃	direct (N × H)
198. C₂×C₁₃	#2096	direct (N × H)
246. C₁₃	C₂×#2096	direct (N × H)
297. C₂	C₁₃×#2096	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×#106	2
C₂²×C₁₃×#103	2
C₂×C₁₃×#391	2
C₂×C₁₃×#384	2
C₁₃×#2096	2
C₂×C₁₃×#336	2
C₂×#2096	13

Subgroups — C2×C13×#2096 #15710067

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₃×#2096 #15710067