Subgroups — C₂²×C₁₃×#397 #15720876

Order 3328.

5230 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₁₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×C₁₃×C₂²⋊C₄	C₂	1	2	yes	maximal
3	C₂³×C₁₃×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₃×#397	C₂	1	24	yes	maximal, direct product factor
5	C₂²×C₁₃×#95	C₂	1	2	yes	maximal, complemented
6	C₂²×C₁₃×#95	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁶×C₁₃	C₂²	1	1	yes	characteristic
8	C₂⁴×C₄×C₁₃	C₂²	1	2	yes	—
9	C₂⁴×C₄×C₁₃	C₂²	1	1	yes	characteristic
10	C₂⁴×C₄×C₁₃	C₂²	1	2	yes	—
11	C₂⁴×C₄×C₁₃	C₂²	1	1	yes	characteristic
12	C₂²×C₁₃×C₄⋊C₄	—	2	8	no	—
13	C₂²×C₁₃×C₄⋊C₄	C₂²	1	12	yes	complemented
14	C₂²×C₁₃×C₂²⋊C₄	—	2	16	no	—
15	C₂²×C₁₃×C₂²⋊C₄	—	2	8	no	—
16	C₂²×C₁₃×C₂²⋊C₄	C₂²	1	24	yes	—
17	C₂²×C₁₃×C₂²⋊C₄	C₂²	1	12	yes	—
18	C₁₃×#397	C₂²	1	64	yes	direct product factor
19	C₂×C₁₃×#95	C₂²	1	24	yes	complemented
20	C₂×C₁₃×#95	C₂²	1	12	yes	complemented
21	C₂⁵×C₁₃	—	2	4	no	—
22	C₂⁵×C₁₃	—	2	4	no	—
23	C₂⁵×C₁₃	—	2	8	no	—
24	C₂⁵×C₁₃	—	2	8	no	—
25	C₂⁵×C₁₃	D₄	1	8	yes	—
26	C₂⁵×C₁₃	C₂³	1	6	yes	—
27	C₂⁵×C₁₃	C₂³	1	1	yes	characteristic
28	C₂³×C₄×C₁₃	—	2	8	no	—
29	C₂³×C₄×C₁₃	—	2	8	no	—
30	C₂³×C₄×C₁₃	—	2	8	no	—
31	C₂³×C₄×C₁₃	—	2	4	no	—
32	C₂³×C₄×C₁₃	—	2	8	no	—
33	C₂³×C₄×C₁₃	—	2	8	no	—
34	C₂³×C₄×C₁₃	—	2	8	no	—
35	C₂³×C₄×C₁₃	—	2	8	no	—
36	C₂³×C₄×C₁₃	—	2	8	no	—
37	C₂³×C₄×C₁₃	D₄	1	8	yes	—
38	C₂³×C₄×C₁₃	C₂³	1	12	yes	—
39	C₂³×C₄×C₁₃	C₂³	1	12	yes	—
40	C₂³×C₄×C₁₃	C₂³	1	6	yes	—
41	C₂³×C₄×C₁₃	C₂³	1	6	yes	—
42	C₂×C₁₃×C₄⋊C₄	—	2	48	no	—
43	C₂×C₁₃×C₄⋊C₄	C₂³	1	16	yes	complemented
44	C₂×C₁₃×C₂²⋊C₄	—	2	96	no	—
45	C₂×C₁₃×C₂²⋊C₄	—	2	48	no	—
46	C₂×C₁₃×C₂²⋊C₄	C₂³	1	32	yes	—
47	C₂×C₁₃×C₂²⋊C₄	C₂³	1	16	yes	—
48	C₁₃×#95	C₂³	1	32	yes	complemented
49	C₁₃×#95	C₂³	1	16	yes	complemented
50	C₂²×#397	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
51	C₂⁴×C₁₃	—	2	32	no	—
52	C₂⁴×C₁₃	—	2	64	no	—
53	C₂⁴×C₁₃	—	2	12	no	—
54	C₂⁴×C₁₃	—	2	12	no	—
55	C₂⁴×C₁₃	—	2	24	no	—
56	C₂⁴×C₁₃	—	2	24	no	—
57	C₂⁴×C₁₃	—	4	16	no	—
58	C₂⁴×C₁₃	—	4	16	no	—
59	C₂⁴×C₁₃	—	4	32	no	—
60	C₂⁴×C₁₃	C₂×D₄	1	24	yes	—
61	C₂⁴×C₁₃	C₂×D₄	1	4	yes	—
62	C₂⁴×C₁₃	C₂×D₄	1	4	yes	—
63	C₂⁴×C₁₃	#41	1	4	yes	—
64	C₂⁴×C₁₃	#41	1	8	yes	—
65	C₂⁴×C₁₃	#41	1	8	yes	—
66	C₂⁴×C₁₃	C₂⁴	1	4	yes	—
67	C₂⁴×C₁₃	C₂⁴	1	3	yes	—
68	C₂²×C₄×C₁₃	—	2	32	no	—
69	C₂²×C₄×C₁₃	—	2	24	no	—
70	C₂²×C₄×C₁₃	—	2	24	no	—
71	C₂²×C₄×C₁₃	—	2	24	no	—
72	C₂²×C₄×C₁₃	—	2	12	no	—
73	C₂²×C₄×C₁₃	—	2	24	no	—
74	C₂²×C₄×C₁₃	—	2	24	no	—
75	C₂²×C₄×C₁₃	—	2	24	no	—
76	C₂²×C₄×C₁₃	—	2	24	no	—
77	C₂²×C₄×C₁₃	—	2	24	no	—
78	C₂²×C₄×C₁₃	—	4	32	no	—
79	C₂²×C₄×C₁₃	—	4	32	no	—
80	C₂²×C₄×C₁₃	—	4	16	no	—
81	C₂²×C₄×C₁₃	C₂×D₄	1	24	yes	—
82	C₂²×C₄×C₁₃	C₂⁴	1	8	yes	—
83	C₂²×C₄×C₁₃	C₂⁴	1	4	yes	—
84	C₂²×C₄×C₁₃	C₂⁴	1	8	yes	—
85	C₂²×C₄×C₁₃	C₂⁴	1	4	yes	—
86	C₁₃×C₄⋊C₄	—	2	32	no	—
87	C₁₃×C₂²⋊C₄	—	2	64	no	—
88	C₁₃×C₂²⋊C₄	—	2	32	no	—
89	C₂³×C₄⋊C₄	C₂×C₁₃	1	1	yes	characteristic, complemented
90	C₂³×C₂²⋊C₄	C₂×C₁₃	1	2	yes	—
91	C₂³×C₂²⋊C₄	C₂×C₁₃	1	1	yes	characteristic
92	C₂×#397	C₂×C₁₃	1	24	yes	direct product factor
93	C₂²×#95	C₂×C₁₃	1	2	yes	complemented
94	C₂²×#95	C₂×C₁₃	1	1	yes	characteristic, complemented
95	C₂³×C₁₃	—	2	48	no	—
96	C₂³×C₁₃	—	2	96	no	—
97	C₂³×C₁₃	—	2	4	no	—
98	C₂³×C₁₃	—	2	4	no	—
99	C₂³×C₁₃	—	2	8	no	—
100	C₂³×C₁₃	—	2	8	no	—
101	C₂³×C₁₃	—	4	24	no	—
102	C₂³×C₁₃	—	4	24	no	—
103	C₂³×C₁₃	—	4	48	no	—
104	C₂³×C₁₃	—	4	128	no	—
105	C₂³×C₁₃	#121	1	16	yes	—
106	C₂³×C₁₃	#123	1	16	yes	—
107	C₂³×C₁₃	#123	1	32	yes	—
108	C₂³×C₁₃	#124	1	16	yes	—
109	C₂³×C₁₃	#126	1	32	yes	—
110	C₂³×C₁₃	C₂²×D₄	1	8	yes	—
111	C₂³×C₁₃	C₂²×D₄	1	6	yes	—
112	C₂³×C₁₃	C₂²×D₄	1	6	yes	—
113	C₂³×C₁₃	C₂×#41	1	6	yes	—
114	C₂³×C₁₃	C₂×#41	1	12	yes	—
115	C₂³×C₁₃	C₂×#41	1	12	yes	—
116	C₂³×C₁₃	C₂⁵	1	1	yes	characteristic
117	C₂×C₄×C₁₃	—	2	48	no	—
118	C₂×C₄×C₁₃	—	2	8	no	—
119	C₂×C₄×C₁₃	—	2	8	no	—
120	C₂×C₄×C₁₃	—	2	8	no	—
121	C₂×C₄×C₁₃	—	2	4	no	—
122	C₂×C₄×C₁₃	—	2	8	no	—
123	C₂×C₄×C₁₃	—	2	8	no	—
124	C₂×C₄×C₁₃	—	2	8	no	—
125	C₂×C₄×C₁₃	—	2	8	no	—
126	C₂×C₄×C₁₃	—	2	8	no	—
127	C₂×C₄×C₁₃	—	4	48	no	—
128	C₂×C₄×C₁₃	—	4	48	no	—
129	C₂×C₄×C₁₃	—	4	24	no	—
130	C₂×C₄×C₁₃	C₂²×D₄	1	8	yes	—
131	C₂⁶	C₂²×C₁₃	1	1	yes	characteristic
132	C₂⁴×C₄	C₂²×C₁₃	1	2	yes	—
133	C₂⁴×C₄	C₂²×C₁₃	1	2	yes	—
134	C₂⁴×C₄	C₂²×C₁₃	1	1	yes	characteristic
135	C₂⁴×C₄	C₂²×C₁₃	1	1	yes	characteristic
136	C₂²×C₄⋊C₄	—	2	8	no	—
137	C₂²×C₄⋊C₄	C₂²×C₁₃	1	12	yes	complemented
138	C₂²×C₂²⋊C₄	—	2	16	no	—
139	C₂²×C₂²⋊C₄	—	2	8	no	—
140	C₂²×C₂²⋊C₄	C₂²×C₁₃	1	24	yes	—
141	C₂²×C₂²⋊C₄	C₂²×C₁₃	1	12	yes	—
142	#397	C₂²×C₁₃	1	64	yes	direct product factor
143	C₂×#95	C₂²×C₁₃	1	24	yes	complemented
144	C₂×#95	C₂²×C₁₃	1	12	yes	complemented
145	C₂²×C₁₃	—	2	8	no	—
146	C₂²×C₁₃	—	2	16	no	—
147	C₂²×C₁₃	—	4	96	no	—
148	C₂²×C₁₃	—	4	4	no	—
149	C₂²×C₁₃	—	4	4	no	—
150	C₂²×C₁₃	—	4	8	no	—
151	C₂²×C₁₃	#397	1	64	yes	direct product factor
152	C₂²×C₁₃	C₂×#121	1	12	yes	—
153	C₂²×C₁₃	C₂×#123	1	12	yes	—
154	C₂²×C₁₃	C₂×#123	1	24	yes	—
155	C₂²×C₁₃	C₂×#124	1	12	yes	—
156	C₂²×C₁₃	C₂×#126	1	24	yes	—
157	C₂²×C₁₃	C₂³×D₄	1	1	yes	characteristic
158	C₂²×C₁₃	C₂³×D₄	1	1	yes	characteristic
159	C₂²×C₁₃	C₂²×#41	1	1	yes	characteristic
160	C₂²×C₁₃	C₂²×#41	1	2	yes	—
161	C₂²×C₁₃	C₂²×#41	1	2	yes	—
162	C₄×C₁₃	—	2	8	no	—
163	C₄×C₁₃	—	4	8	no	—
164	C₄×C₁₃	—	4	8	no	—
165	C₄×C₁₃	—	4	4	no	—
166	C₂⁵	—	2	4	no	—
167	C₂⁵	—	2	4	no	—
168	C₂⁵	—	2	8	no	—
169	C₂⁵	—	2	8	no	—
170	C₂⁵	D₄×C₁₃	1	8	yes	—
171	C₂⁵	C₂³×C₁₃	1	6	yes	—
172	C₂⁵	C₂³×C₁₃	1	1	yes	characteristic
173	C₂³×C₄	—	2	8	no	—
174	C₂³×C₄	—	2	8	no	—
175	C₂³×C₄	—	2	8	no	—
176	C₂³×C₄	—	2	4	no	—
177	C₂³×C₄	—	2	8	no	—
178	C₂³×C₄	—	2	8	no	—
179	C₂³×C₄	—	2	8	no	—
180	C₂³×C₄	—	2	8	no	—
181	C₂³×C₄	—	2	8	no	—
182	C₂³×C₄	D₄×C₁₃	1	8	yes	—
183	C₂³×C₄	C₂³×C₁₃	1	12	yes	—
184	C₂³×C₄	C₂³×C₁₃	1	12	yes	—
185	C₂³×C₄	C₂³×C₁₃	1	6	yes	—
186	C₂³×C₄	C₂³×C₁₃	1	6	yes	—
187	C₂×C₄⋊C₄	—	2	48	no	—
188	C₂×C₄⋊C₄	C₂³×C₁₃	1	16	yes	complemented
189	C₂×C₂²⋊C₄	—	2	96	no	—
190	C₂×C₂²⋊C₄	—	2	48	no	—
191	C₂×C₂²⋊C₄	C₂³×C₁₃	1	32	yes	—
192	C₂×C₂²⋊C₄	C₂³×C₁₃	1	16	yes	—
193	#95	C₂³×C₁₃	1	32	yes	complemented
194	#95	C₂³×C₁₃	1	16	yes	complemented
195	C₂×C₁₃	—	4	8	no	—
196	C₂×C₁₃	C₂×#397	1	24	yes	direct product factor
197	C₂×C₁₃	C₂²×#121	1	1	yes	characteristic
198	C₂×C₁₃	C₂²×#123	1	1	yes	characteristic
199	C₂×C₁₃	C₂²×#123	1	2	yes	—
200	C₂×C₁₃	C₂²×#124	1	1	yes	characteristic
201	C₂×C₁₃	C₂²×#126	1	2	yes	—
202	C₂⁴	—	2	32	no	—
203	C₂⁴	—	2	64	no	—
204	C₂⁴	—	2	12	no	—
205	C₂⁴	—	2	12	no	—
206	C₂⁴	—	2	24	no	—
207	C₂⁴	—	2	24	no	—
208	C₂⁴	—	4	16	no	—
209	C₂⁴	—	4	16	no	—
210	C₂⁴	—	4	32	no	—
211	C₂⁴	C₂×D₄×C₁₃	1	24	yes	—
212	C₂⁴	C₂×D₄×C₁₃	1	4	yes	—
213	C₂⁴	C₂×D₄×C₁₃	1	4	yes	—
214	C₂⁴	C₁₃×#41	1	4	yes	—
215	C₂⁴	C₁₃×#41	1	8	yes	—
216	C₂⁴	C₁₃×#41	1	8	yes	—
217	C₂⁴	C₂⁴×C₁₃	1	4	yes	—
218	C₂⁴	C₂⁴×C₁₃	1	3	yes	—
219	C₂²×C₄	—	2	32	no	—
220	C₂²×C₄	—	2	24	no	—
221	C₂²×C₄	—	2	24	no	—
222	C₂²×C₄	—	2	24	no	—
223	C₂²×C₄	—	2	12	no	—
224	C₂²×C₄	—	2	24	no	—
225	C₂²×C₄	—	2	24	no	—
226	C₂²×C₄	—	2	24	no	—
227	C₂²×C₄	—	2	24	no	—
228	C₂²×C₄	—	2	24	no	—
229	C₂²×C₄	—	4	32	no	—
230	C₂²×C₄	—	4	32	no	—
231	C₂²×C₄	—	4	16	no	—
232	C₂²×C₄	C₂×D₄×C₁₃	1	24	yes	—
233	C₂²×C₄	C₂⁴×C₁₃	1	8	yes	—
234	C₂²×C₄	C₂⁴×C₁₃	1	4	yes	—
235	C₂²×C₄	C₂⁴×C₁₃	1	8	yes	—
236	C₂²×C₄	C₂⁴×C₁₃	1	4	yes	—
237	C₄⋊C₄	—	2	32	no	—
238	C₂²⋊C₄	—	2	64	no	—
239	C₂²⋊C₄	—	2	32	no	—
240	C₁₃	C₂²×#397	1	1	yes	characteristic, direct product factor, Sylow
241	C₂³	—	2	48	no	—
242	C₂³	—	2	96	no	—
243	C₂³	—	2	4	no	—
244	C₂³	—	2	4	no	—
245	C₂³	—	2	8	no	—
246	C₂³	—	2	8	no	—
247	C₂³	—	4	128	no	—
248	C₂³	—	4	24	no	—
249	C₂³	—	4	24	no	—
250	C₂³	—	4	48	no	—
251	C₂³	C₁₃×#121	1	16	yes	—
252	C₂³	C₁₃×#123	1	16	yes	—
253	C₂³	C₁₃×#123	1	32	yes	—
254	C₂³	C₁₃×#124	1	16	yes	—
255	C₂³	C₁₃×#126	1	32	yes	—
256	C₂³	C₂²×D₄×C₁₃	1	8	yes	—
257	C₂³	C₂²×D₄×C₁₃	1	6	yes	—
258	C₂³	C₂²×D₄×C₁₃	1	6	yes	—
259	C₂³	C₂×C₁₃×#41	1	6	yes	—
260	C₂³	C₂×C₁₃×#41	1	12	yes	—
261	C₂³	C₂×C₁₃×#41	1	12	yes	—
262	C₂³	C₂⁵×C₁₃	1	1	yes	characteristic
263	C₂×C₄	—	2	48	no	—
264	C₂×C₄	—	2	8	no	—
265	C₂×C₄	—	2	8	no	—
266	C₂×C₄	—	2	8	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	8	no	—
269	C₂×C₄	—	2	8	no	—
270	C₂×C₄	—	2	8	no	—
271	C₂×C₄	—	2	8	no	—
272	C₂×C₄	—	2	8	no	—
273	C₂×C₄	—	4	48	no	—
274	C₂×C₄	—	4	48	no	—
275	C₂×C₄	—	4	24	no	—
276	C₂×C₄	C₂²×D₄×C₁₃	1	8	yes	—
277	C₂²	—	2	8	no	—
278	C₂²	—	2	16	no	—
279	C₂²	—	4	96	no	—
280	C₂²	—	4	4	no	—
281	C₂²	—	4	4	no	—
282	C₂²	—	4	8	no	—
283	C₂²	C₁₃×#397	1	64	yes	direct product factor
284	C₂²	C₂×C₁₃×#121	1	12	yes	—
285	C₂²	C₂×C₁₃×#123	1	12	yes	—
286	C₂²	C₂×C₁₃×#123	1	24	yes	—
287	C₂²	C₂×C₁₃×#124	1	12	yes	—
288	C₂²	C₂×C₁₃×#126	1	24	yes	—
289	C₂²	C₂³×D₄×C₁₃	1	1	yes	characteristic
290	C₂²	C₂³×D₄×C₁₃	1	1	yes	characteristic
291	C₂²	C₂²×C₁₃×#41	1	1	yes	characteristic
292	C₂²	C₂²×C₁₃×#41	1	2	yes	—
293	C₂²	C₂²×C₁₃×#41	1	2	yes	—
294	C₄	—	2	8	no	—
295	C₄	—	4	8	no	—
296	C₄	—	4	8	no	—
297	C₄	—	4	4	no	—
298	C₂	—	4	8	no	—
299	C₂	C₂×C₁₃×#397	1	24	yes	direct product factor
300	C₂	C₂²×C₁₃×#121	1	1	yes	characteristic
301	C₂	C₂²×C₁₃×#123	1	1	yes	characteristic
302	C₂	C₂²×C₁₃×#123	1	2	yes	—
303	C₂	C₂²×C₁₃×#124	1	1	yes	characteristic
304	C₂	C₂²×C₁₃×#126	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁵×C₁₃	C₂³
commutator, Frattini, cosocle	C₂³	C₂⁵×C₁₃
prime radical	C₂⁶×C₁₃	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂³×C₁₃×C₄⋊C₄	C₂	semidirect
4. C₂×C₁₃×#397	C₂	direct (N × H)
5. C₂²×C₁₃×#95	C₂	semidirect
6. C₂²×C₁₃×#95	C₂	semidirect
13. C₂²×C₁₃×C₄⋊C₄	C₂²	semidirect
18. C₁₃×#397	C₂²	direct (N × H)
19. C₂×C₁₃×#95	C₂²	semidirect
20. C₂×C₁₃×#95	C₂²	semidirect
43. C₂×C₁₃×C₄⋊C₄	C₂³	semidirect
48. C₁₃×#95	C₂³	semidirect
49. C₁₃×#95	C₂³	semidirect
50. C₂²×#397	C₁₃	direct (N × H)
89. C₂³×C₄⋊C₄	C₂×C₁₃	semidirect
92. C₂×#397	C₂×C₁₃	direct (N × H)
93. C₂²×#95	C₂×C₁₃	semidirect
94. C₂²×#95	C₂×C₁₃	semidirect
137. C₂²×C₄⋊C₄	C₂²×C₁₃	semidirect
142. #397	C₂²×C₁₃	direct (N × H)
143. C₂×#95	C₂²×C₁₃	semidirect
144. C₂×#95	C₂²×C₁₃	semidirect
151. C₂²×C₁₃	#397	direct (N × H)
188. C₂×C₄⋊C₄	C₂³×C₁₃	semidirect
193. #95	C₂³×C₁₃	semidirect
194. #95	C₂³×C₁₃	semidirect
196. C₂×C₁₃	C₂×#397	direct (N × H)
240. C₁₃	C₂²×#397	direct (N × H)
283. C₂²	C₁₃×#397	direct (N × H)
299. C₂	C₂×C₁₃×#397	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₁₃×C₄⋊C₄	2
C₂³×C₁₃×C₂²⋊C₄	2
C₂×C₁₃×#397	2
C₂²×C₁₃×#95	2
C₂²×#397	13

Subgroups — C22×C13×#397 #15720876

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₁₃×#397 #15720876