Subgroups — #15861807

Order 3328.

1222 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₅₂	C₂	1	2	yes	maximal
2	C₄²×D₅₂	C₂	1	1	yes	maximal, characteristic
3	C₈×C₄⋊Dic₁₃	C₂	1	2	yes	maximal, complemented
4	C₄×C₄⋊#1055	C₂	1	1	yes	maximal, characteristic, complemented
5	#13114606	C₂	1	4	yes	maximal, complemented
6	#13114593	C₂	1	2	yes	maximal, complemented
7	D₁₃×#336	C₂	1	2	yes	maximal, complemented
8	C₁₃×#1744	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₈×D₁₃	C₂²	1	4	yes	—
10	C₂×C₄×D₅₂	C₄	1	2	yes	—
11	C₂×C₄×D₅₂	C₂²	1	1	yes	characteristic
12	C₂×C₄²×D₁₃	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
14	C₄³×C₁₃	C₂²	1	1	yes	characteristic
15	C₂×#88060	C₂²	1	4	yes	—
16	C₂×C₈×Dic₁₃	C₂²	1	4	yes	—
17	C₄×#6103	C₄	1	4	yes	—
18	C₄×#6103	C₂²	1	2	yes	—
19	C₄×C₄⋊Dic₁₃	C₄	1	2	yes	—
20	C₄×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊#1055	C₂²	1	2	yes	—
22	C₂×C₄×#1055	C₂²	1	2	yes	—
23	C₈×D₅₂	C₄	1	16	yes	complemented
24	C₁₃×#336	C₂²	1	1	yes	characteristic
25	C₁₃×#336	C₂²	1	1	yes	characteristic
26	C₁₃×#336	C₂²	1	2	yes	—
27	#11718142	C₂²	1	4	yes	—
28	#11718116	C₂²	1	2	yes	—
29	C₂²×D₅₂	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄×D₁₃	C₂×C₄	1	4	yes	—
31	C₂²×C₄×D₁₃	C₂³	1	2	yes	—
32	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
33	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
34	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
35	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
37	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
38	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
39	C₂×#6103	C₂×C₄	1	2	yes	—
40	C₂×#6103	C₂×C₄	1	4	yes	—
41	C₂×C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
43	C₂×C₄×Dic₁₃	C₂×C₄	1	4	yes	—
44	C₂×C₄×Dic₁₃	C₂³	1	2	yes	—
45	C₂²×#1055	C₂³	1	4	yes	—
46	C₂×C₈×D₁₃	—	4	4	no	—
47	C₂×C₈×D₁₃	—	4	4	no	—
48	C₂×C₈×D₁₃	C₂×C₄	1	16	yes	complemented
49	C₄×D₅₂	C₈	1	16	yes	complemented
50	C₄×D₅₂	C₂×C₄	1	8	yes	—
51	C₄²×D₁₃	—	4	4	no	—
52	C₄×C₈×C₁₃	C₂×C₄	1	8	yes	complemented
53	#88071	C₂×C₄	1	4	yes	—
54	#88071	C₂×C₄	1	2	yes	—
55	#88060	C₂×C₄	1	16	yes	complemented
56	C₈×Dic₁₃	—	4	4	no	—
57	C₄⋊#1055	C₂×C₄	1	8	yes	complemented
58	C₄×#1055	—	4	2	no	—
59	#20555	—	13	1	no	maximal, Sylow
60	C₂³×D₁₃	C₂²×C₄	1	2	yes	—
61	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
66	C₂²×Dic₁₃	C₂²×C₄	1	2	yes	—
67	C₂²×Dic₁₃	C₂²×C₄	1	4	yes	—
68	C₂×D₅₂	C₄²	1	4	yes	—
69	C₂×D₅₂	C₂×C₈	1	8	yes	—
70	C₂×C₄×D₁₃	—	2	4	no	—
71	C₂×C₄×D₁₃	—	2	2	no	—
72	C₂×C₄×D₁₃	—	2	2	no	—
73	C₂×C₄×D₁₃	—	4	4	no	—
74	C₂×C₄×D₁₃	—	4	4	no	—
75	C₂×C₄×D₁₃	C₂×C₈	1	16	yes	complemented
76	C₂×C₄×D₁₃	C₂²×C₄	1	8	yes	—
77	C₂×C₈×C₁₃	—	2	2	no	—
78	C₂×C₈×C₁₃	—	2	2	no	—
79	C₂×C₈×C₁₃	—	2	2	no	—
80	C₂×C₈×C₁₃	—	2	2	no	—
81	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
82	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
83	C₄²×C₁₃	—	2	2	no	—
84	C₄²×C₁₃	—	2	1	no	—
85	C₄²×C₁₃	—	2	1	no	—
86	C₄²×C₁₃	—	4	2	no	—
87	C₄²×C₁₃	C₂×C₈	1	8	yes	complemented
88	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
89	#6103	C₄²	1	8	yes	—
90	#6103	C₂×C₈	1	16	yes	complemented
91	C₄⋊Dic₁₃	C₄²	1	4	yes	—
92	C₄⋊Dic₁₃	C₂×C₈	1	8	yes	complemented
93	C₄×Dic₁₃	—	2	4	no	—
94	C₄×Dic₁₃	—	4	4	no	—
95	C₂×#1055	—	2	4	no	—
96	C₂×#1055	—	2	4	no	—
97	C₂×#1055	C₂²×C₄	1	8	yes	—
98	C₈×D₁₃	—	4	16	no	—
99	C₂×C₈×D₄	—	13	2	no	—
100	C₄²×D₄	—	13	1	no	—
101	C₈×C₄⋊C₄	—	13	2	no	—
102	C₄×C₄⋊C₈	—	13	1	no	—
103	C₈×C₂²⋊C₄	—	13	4	no	—
104	C₄×C₂²⋊C₈	—	13	2	no	—
105	#1744	D₁₃	1	1	yes	characteristic, complemented
106	C₂×#336	—	13	2	no	—
107	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
108	C₂²×D₁₃	—	2	2	no	—
109	C₂²×D₁₃	—	2	2	no	—
110	C₂²×D₁₃	—	2	4	no	—
111	C₂²×D₁₃	C₂×C₄²	1	4	yes	—
112	C₂²×D₁₃	C₂²×C₈	1	8	yes	—
113	C₂×C₄×C₁₃	—	2	2	no	—
114	C₂×C₄×C₁₃	—	2	2	no	—
115	C₂×C₄×C₁₃	—	2	2	no	—
116	C₂×C₄×C₁₃	—	2	2	no	—
117	C₂×C₄×C₁₃	—	2	2	no	—
118	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
119	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
120	C₂×C₄×C₁₃	C₂²×C₈	1	4	yes	—
121	C₂×C₄×C₁₃	C₂²×C₈	1	4	yes	—
122	C₂×C₄×C₁₃	C₂²×C₈	1	4	yes	—
123	C₂×C₄×C₁₃	C₂³×C₄	1	2	yes	—
124	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
125	C₂×C₄×C₁₃	H(5,𝔽₂)	1	2	yes	—
126	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
127	C₂×Dic₁₃	—	2	4	no	—
128	C₂×Dic₁₃	—	2	2	no	—
129	C₂×Dic₁₃	—	2	2	no	—
130	C₂×Dic₁₃	—	2	4	no	—
131	C₂×Dic₁₃	C₂×C₄²	1	4	yes	—
132	C₂×Dic₁₃	C₂²×C₈	1	8	yes	—
133	D₅₂	C₄×C₈	1	16	yes	complemented
134	C₄×D₁₃	—	2	8	no	—
135	C₄×D₁₃	—	2	8	no	—
136	C₄×D₁₃	—	4	16	no	—
137	C₈×C₁₃	—	2	4	no	—
138	C₈×C₁₃	—	2	4	no	—
139	#1055	—	2	8	no	—
140	C₂³×C₈	—	13	4	no	—
141	C₂×C₄×D₄	—	13	2	no	—
142	C₂×C₄×D₄	—	13	1	no	—
143	C₂²×C₄²	—	13	2	no	—
144	C₈×D₄	—	13	16	no	—
145	C₂×C₄⋊C₈	—	13	2	no	—
146	C₂×C₂²⋊C₈	—	13	4	no	—
147	C₂×C₄×C₈	—	13	4	no	—
148	C₂×C₄×C₈	—	13	2	no	—
149	C₂×C₄×C₈	C₂×D₁₃	1	2	yes	—
150	C₄×C₄⋊C₄	—	13	2	no	—
151	C₄×C₄⋊C₄	—	13	1	no	—
152	C₄×C₂²⋊C₄	—	13	4	no	—
153	C₄×C₂²⋊C₄	—	13	2	no	—
154	C₄³	C₂×D₁₃	1	1	yes	characteristic
155	#336	—	13	4	no	—
156	#336	—	13	2	no	—
157	#336	C₂×D₁₃	1	1	yes	characteristic
158	#336	C₂×D₁₃	1	1	yes	characteristic
159	#336	C₂×D₁₃	1	2	yes	—
160	C₂²×C₁₃	C₂²×C₄²	1	1	yes	characteristic
161	C₂²×C₁₃	#518	1	1	yes	characteristic
162	C₂²×C₁₃	#520	1	1	yes	characteristic
163	C₂²×C₁₃	C₂³×C₈	1	2	yes	—
164	C₂²×C₁₃	#568	1	2	yes	—
165	C₂×D₁₃	—	2	4	no	—
166	C₂×D₁₃	—	2	8	no	—
167	C₂×D₁₃	—	2	8	no	—
168	C₂×D₁₃	—	2	4	no	—
169	C₂×D₁₃	C₂×C₄×C₈	1	8	yes	—
170	C₄×C₁₃	—	2	2	no	—
171	C₄×C₁₃	—	2	4	no	—
172	C₄×C₁₃	—	2	4	no	—
173	C₄×C₁₃	C₂×C₄×C₈	1	4	yes	—
174	C₄×C₁₃	#518	1	2	yes	—
175	C₄×C₁₃	#520	1	2	yes	—
176	Dic₁₃	—	2	8	no	—
177	Dic₁₃	—	2	4	no	—
178	C₂²×D₄	—	13	1	no	—
179	C₂³×C₄	—	13	2	no	—
180	C₂³×C₄	—	13	4	no	—
181	C₂²×C₈	—	13	4	no	—
182	C₂²×C₈	—	13	16	no	—
183	C₂²×C₈	—	52	4	no	—
184	C₂²×C₈	—	52	4	no	—
185	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
186	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
187	C₄×D₄	—	13	16	no	—
188	C₄×D₄	—	13	8	no	—
189	C₂×C₄⋊C₄	—	13	1	no	—
190	C₂×C₄⋊C₄	—	13	2	no	—
191	C₂×C₂²⋊C₄	—	13	2	no	—
192	C₂×C₂²⋊C₄	—	13	4	no	—
193	C₂×C₄²	—	13	2	no	—
194	C₂×C₄²	—	13	4	no	—
195	C₂×C₄²	—	52	4	no	—
196	C₂×C₄²	C₄×D₁₃	1	2	yes	—
197	C₂×C₄²	C₄×D₁₃	1	2	yes	—
198	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
199	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
200	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
201	C₄⋊C₈	—	13	8	no	—
202	C₂²⋊C₈	—	13	16	no	—
203	C₄×C₈	—	52	2	no	—
204	C₄×C₈	—	52	4	no	—
205	C₄×C₈	C₄×D₁₃	1	8	yes	complemented
206	#95	—	13	2	no	—
207	#95	—	13	4	no	—
208	C₂×C₁₃	#2275	1	1	yes	characteristic
209	C₂×C₁₃	C₂²×C₄×C₈	1	1	yes	characteristic
210	C₂×C₁₃	#2875	1	1	yes	characteristic
211	C₂×C₁₃	#2972	1	2	yes	—
212	C₂×C₁₃	#2988	1	2	yes	—
213	D₁₃	—	2	8	no	—
214	C₂⁴	—	13	2	no	—
215	C₂×D₄	—	13	4	no	—
216	C₂×D₄	—	13	8	no	—
217	C₂²×C₄	—	13	2	no	—
218	C₂²×C₄	—	13	4	no	—
219	C₂²×C₄	—	13	8	no	—
220	C₂²×C₄	—	13	16	no	—
221	C₂²×C₄	—	26	4	no	—
222	C₂²×C₄	—	26	2	no	—
223	C₂²×C₄	—	26	2	no	—
224	C₂²×C₄	—	52	4	no	—
225	C₂²×C₄	—	52	4	no	—
226	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
227	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
228	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
229	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
230	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	13	8	no	—
236	C₂×C₈	—	26	4	no	—
237	C₂×C₈	—	26	4	no	—
238	C₂×C₈	—	52	16	no	—
239	C₂×C₈	C₂×C₄×D₁₃	1	4	yes	—
240	C₂×C₈	C₂×C₄×D₁₃	1	4	yes	—
241	C₄⋊C₄	—	13	4	no	—
242	C₄⋊C₄	—	13	8	no	—
243	C₂²⋊C₄	—	13	8	no	—
244	C₂²⋊C₄	—	13	16	no	—
245	C₄²	—	2	1	no	—
246	C₄²	—	2	2	no	—
247	C₄²	—	2	1	no	—
248	C₄²	—	4	2	no	—
249	C₄²	—	26	4	no	—
250	C₄²	—	52	4	no	—
251	C₄²	C₈×D₁₃	1	8	yes	complemented
252	C₄²	C₂×C₄×D₁₃	1	4	yes	—
253	C₁₃	#20555	1	1	yes	characteristic, complemented, Sylow
254	C₂³	—	13	4	no	—
255	C₂³	—	13	8	no	—
256	C₂³	—	26	4	no	—
257	C₂³	—	26	2	no	—
258	C₂³	—	26	2	no	—
259	C₂³	C₂²×C₄×D₁₃	1	1	yes	characteristic
260	D₄	—	13	16	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	13	4	no	—
267	C₂×C₄	—	13	8	no	—
268	C₂×C₄	—	26	4	no	—
269	C₂×C₄	—	26	4	no	—
270	C₂×C₄	—	26	2	no	—
271	C₂×C₄	—	26	2	no	—
272	C₂×C₄	—	26	8	no	—
273	C₂×C₄	—	26	8	no	—
274	C₂×C₄	—	52	16	no	—
275	C₂×C₄	C₄²×D₁₃	1	2	yes	—
276	C₂×C₄	C₄²×D₁₃	1	2	yes	—
277	C₂×C₄	C₂×C₈×D₁₃	1	4	yes	—
278	C₂×C₄	C₂×C₈×D₁₃	1	4	yes	—
279	C₂×C₄	C₂×C₈×D₁₃	1	4	yes	—
280	C₂×C₄	C₂²×C₄×D₁₃	1	2	yes	—
281	C₂×C₄	#88256	1	1	yes	characteristic
282	C₂×C₄	#88256	1	2	yes	—
283	C₂×C₄	#88257	1	1	yes	characteristic
284	C₈	—	2	4	no	—
285	C₈	—	2	4	no	—
286	C₈	—	26	8	no	—
287	C₂²	—	13	8	no	—
288	C₂²	—	26	4	no	—
289	C₂²	—	26	8	no	—
290	C₂²	—	26	8	no	—
291	C₂²	—	26	4	no	—
292	C₂²	C₂×C₄²×D₁₃	1	1	yes	characteristic
293	C₂²	#11719226	1	1	yes	characteristic
294	C₂²	#11719228	1	1	yes	characteristic
295	C₂²	C₂²×C₈×D₁₃	1	2	yes	—
296	C₂²	#11719450	1	2	yes	—
297	C₄	—	2	4	no	—
298	C₄	—	2	4	no	—
299	C₄	—	2	2	no	—
300	C₄	—	26	4	no	—
301	C₄	—	26	8	no	—
302	C₄	C₄×C₈×D₁₃	1	4	yes	—
303	C₄	#11719226	1	2	yes	—
304	C₄	#11719228	1	2	yes	—
305	C₂	—	26	8	no	—
306	C₂	#13117198	1	1	yes	characteristic
307	C₂	C₂×C₄×C₈×D₁₃	1	1	yes	characteristic
308	C₂	#13123261	1	1	yes	characteristic
309	C₂	#13123859	1	2	yes	—
310	C₂	#13123944	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₁₃
commutator	C₂×C₁₃	C₂²×C₄×C₈
Fitting	C₁₃×#1744	C₂
socle	C₂³×C₁₃	C₂³×C₄
prime radical	C₂²×D₅₂	C₂×C₄
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	#1744	D₁₃
hypocenter	C₁₃	#20555
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ #1744

Complements

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

N	H ≅ G/N	type
3. C₈×C₄⋊Dic₁₃	C₂	semidirect
4. C₄×C₄⋊#1055	C₂	semidirect
5. #13114606	C₂	semidirect
6. #13114593	C₂	semidirect
7. D₁₃×#336	C₂	semidirect
8. C₁₃×#1744	C₂	semidirect
23. C₈×D₅₂	C₄	semidirect
48. C₂×C₈×D₁₃	C₂×C₄	semidirect
49. C₄×D₅₂	C₈	semidirect
52. C₄×C₈×C₁₃	C₂×C₄	semidirect
55. #88060	C₂×C₄	semidirect
57. C₄⋊#1055	C₂×C₄	semidirect
75. C₂×C₄×D₁₃	C₂×C₈	semidirect
87. C₄²×C₁₃	C₂×C₈	semidirect
90. #6103	C₂×C₈	semidirect
92. C₄⋊Dic₁₃	C₂×C₈	semidirect
105. #1744	D₁₃	semidirect
133. D₅₂	C₄×C₈	semidirect
205. C₄×C₈	C₄×D₁₃	semidirect
251. C₄²	C₈×D₁₃	semidirect
253. C₁₃	#20555	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×D₅₂	2
C₄²×D₅₂	2
C₈×C₄⋊Dic₁₃	2
C₄×C₄⋊#1055	2
#13114606	2
#13114593	2
D₁₃×#336	2
C₁₃×#1744	2
#20555	13