Subgroups — #13123958

Order 1664.

398 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#6127	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#11718772	C₂	1	2	yes	maximal, complemented
4	C₈×C₂²⋊D₁₃	C₂	1	2	yes	maximal, complemented
5	#11718587	C₂	1	1	yes	maximal, characteristic, complemented
6	#11718498	C₂	1	1	yes	maximal, characteristic, complemented
7	#11718497	C₂	1	2	yes	maximal, complemented
8	#11718496	C₂	1	1	yes	maximal, characteristic, complemented
9	#11718494	C₂	1	1	yes	maximal, characteristic, complemented
10	D₁₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
11	#11718489	C₂	1	2	yes	maximal, complemented
12	C₂×#6127	C₄	1	1	yes	characteristic
13	C₂²×C₈×C₁₃	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
15	C₄×C₂²⋊D₁₃	C₄	1	1	yes	characteristic
16	C₄×C₂²⋊D₁₃	C₄	1	1	yes	characteristic
17	C₄×C₂²⋊D₁₃	C₂²	1	1	yes	characteristic
18	C₄×C₂²⋊D₁₃	C₂²	1	1	yes	characteristic
19	#88180	C₄	1	1	yes	characteristic
20	C₂×#6094	C₂²	1	2	yes	—
21	C₂×C₈×D₁₃	C₂²	1	2	yes	—
22	C₄×D₅₂	C₄	1	1	yes	characteristic
23	C₄×D₅₂	C₂²	1	1	yes	characteristic
24	#88126	C₄	1	1	yes	characteristic
25	#88126	C₄	1	1	yes	characteristic
26	C₄²×D₁₃	C₂²	1	1	yes	characteristic
27	C₄²×D₁₃	C₂²	1	1	yes	characteristic
28	C₄×Dic₂₆	C₄	1	1	yes	characteristic
29	C₄×Dic₂₆	C₂²	1	1	yes	characteristic
30	C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
31	C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
32	C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
33	C₂²⋊#1055	C₂²	1	2	yes	—
34	#88060	C₂²	1	2	yes	—
35	#88060	C₂²	1	2	yes	—
36	#88055	C₂²	1	2	yes	—
37	#88054	C₂²	1	2	yes	—
38	#88054	C₂²	1	2	yes	—
39	C₈×Dic₁₃	C₂²	1	2	yes	—
40	C₄⋊#1055	C₂²	1	1	yes	characteristic
41	C₄⋊#1055	C₂²	1	1	yes	characteristic
42	#88043	C₂²	1	2	yes	—
43	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
46	C₂×C₂²⋊D₁₃	C₂×C₄	1	1	yes	characteristic
47	C₂×C₂²⋊D₁₃	C₂×C₄	1	1	yes	characteristic
48	#6127	—	2	4	no	—
49	C₂×D₅₂	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄×D₁₃	C₂³	1	1	yes	characteristic
55	C₂×C₄×D₁₃	C₂³	1	1	yes	characteristic
56	C₂×Dic₂₆	C₂×C₄	1	1	yes	characteristic
57	C₂×C₈×C₁₃	—	4	2	no	—
58	C₂×C₈×C₁₃	C₂³	1	2	yes	—
59	C₂×C₈×C₁₃	C₂³	1	2	yes	—
60	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
61	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
62	C₄²×C₁₃	C₂³	1	1	yes	characteristic
63	C₄²×C₁₃	C₂³	1	1	yes	characteristic
64	C₂²⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
65	C₂²⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
66	#6103	C₂×C₄	1	1	yes	characteristic
67	#6103	C₂×C₄	1	1	yes	characteristic
68	#6103	C₂×C₄	1	1	yes	characteristic
69	#6103	C₂×C₄	1	1	yes	characteristic
70	C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
71	C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
72	#6101	C₂×C₄	1	1	yes	characteristic
73	#6101	C₂×C₄	1	1	yes	characteristic
74	#6101	C₂×C₄	1	1	yes	characteristic
75	#6101	C₂×C₄	1	1	yes	characteristic
76	C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
77	C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
78	C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
79	C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
80	C₄×Dic₁₃	C₂³	1	1	yes	characteristic
81	C₄×Dic₁₃	C₂³	1	1	yes	characteristic
82	C₂×#1055	C₂³	1	2	yes	—
83	C₂×#1055	C₂³	1	2	yes	—
84	#6094	—	4	2	no	—
85	C₈×D₁₃	—	4	2	no	—
86	#2990	—	13	1	no	maximal, Sylow
87	C₂³×C₁₃	C₂²×C₄	1	1	yes	characteristic
88	C₂²×D₁₃	C₂²×C₄	1	1	yes	characteristic
89	C₂²×D₁₃	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₄×C₁₃	—	2	2	no	—
91	C₂×C₄×C₁₃	—	2	2	no	—
92	C₂×C₄×C₁₃	—	2	1	no	—
93	C₂×C₄×C₁₃	—	2	1	no	—
94	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
100	C₂²⋊D₁₃	—	2	2	no	—
101	C₂²⋊D₁₃	M₁₆	1	4	yes	complemented
102	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
103	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
104	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
105	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
106	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
107	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
108	D₅₂	—	2	2	no	—
109	C₄×D₁₃	—	2	2	no	—
110	C₄×D₁₃	—	2	2	no	—
111	C₄×D₁₃	—	2	2	no	—
112	C₄×D₁₃	—	2	1	no	—
113	C₄×D₁₃	—	2	1	no	—
114	C₄×D₁₃	—	2	1	no	—
115	C₄×D₁₃	—	2	1	no	—
116	C₄×D₁₃	—	2	2	no	—
117	Dic₂₆	—	2	2	no	—
118	C₈×C₁₃	—	2	2	no	—
119	C₈×C₁₃	—	2	2	no	—
120	#1055	—	2	2	no	—
121	#1055	—	2	2	no	—
122	C₄×#41	—	13	1	no	—
123	#446	—	13	2	no	—
124	#435	—	13	2	no	—
125	#435	—	13	2	no	—
126	C₈×D₄	—	13	2	no	—
127	#432	—	13	1	no	—
128	#432	—	13	1	no	—
129	#431	—	13	1	no	—
130	#423	—	13	1	no	—
131	C₂×C₄⋊C₈	—	13	1	no	—
132	C₂×C₄⋊C₈	D₁₃	1	1	yes	characteristic, complemented
133	C₂²×C₁₃	—	2	1	no	—
134	C₂²×C₁₃	—	2	1	no	—
135	C₂²×C₁₃	C₂×M₁₆	1	2	yes	—
136	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
137	C₂×D₁₃	—	2	1	no	—
138	C₂×D₁₃	—	2	1	no	—
139	C₂×D₁₃	—	2	1	no	—
140	C₂×D₁₃	—	2	1	no	—
141	C₂×D₁₃	—	2	1	no	—
142	C₂×D₁₃	C₂×M₁₆	1	2	yes	—
143	C₄×C₁₃	—	2	1	no	—
144	C₄×C₁₃	—	2	1	no	—
145	C₄×C₁₃	—	2	1	no	—
146	C₄×C₁₃	—	2	1	no	—
147	C₄×C₁₃	#131	1	2	yes	—
148	C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
149	C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
150	Dic₁₃	—	2	1	no	—
151	Dic₁₃	—	2	1	no	—
152	Dic₁₃	—	2	1	no	—
153	Dic₁₃	—	2	1	no	—
154	Dic₁₃	—	2	1	no	—
155	Dic₁₃	C₂×M₁₆	1	2	yes	—
156	C₂×#41	—	13	1	no	—
157	C₂×M₁₆	—	13	2	no	—
158	C₂²×C₈	—	13	2	no	—
159	C₂²×C₈	C₂×D₁₃	1	2	yes	—
160	C₄×Q₈	—	13	1	no	—
161	C₄×Q₈	—	13	1	no	—
162	C₄×D₄	—	13	1	no	—
163	C₄×D₄	—	13	1	no	—
164	C₄×D₄	—	13	1	no	—
165	C₄×D₄	—	13	1	no	—
166	C₄×D₄	—	13	1	no	—
167	C₄×D₄	—	13	1	no	—
168	#117	—	13	1	no	—
169	#117	—	13	1	no	—
170	#117	—	13	1	no	—
171	C₂×C₄²	—	13	1	no	—
172	C₂×C₄²	—	13	1	no	—
173	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
174	C₄⋊C₈	—	13	2	no	—
175	C₄⋊C₈	—	13	2	no	—
176	C₄⋊C₈	—	13	1	no	—
177	C₄⋊C₈	—	13	1	no	—
178	C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
179	C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
180	C₄⋊C₈	C₂×D₁₃	1	2	yes	—
181	C₂²⋊C₈	—	13	2	no	—
182	C₂²⋊C₈	—	13	2	no	—
183	C₂²⋊C₈	—	13	2	no	—
184	#97	—	13	2	no	—
185	#97	—	13	2	no	—
186	C₄×C₈	—	13	2	no	—
187	C₂×C₁₃	—	2	2	no	—
188	C₂×C₁₃	#520	1	1	yes	characteristic
189	C₂×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
190	C₂×C₁₃	C₂×#131	1	1	yes	characteristic
191	D₁₃	—	2	2	no	—
192	D₁₃	—	4	1	no	—
193	#41	—	26	4	no	—
194	C₂×Q₈	—	13	1	no	—
195	C₂×D₄	—	13	1	no	—
196	C₂×D₄	—	13	1	no	—
197	C₂×D₄	—	13	1	no	—
198	C₂²×C₄	—	13	1	no	—
199	C₂²×C₄	—	13	1	no	—
200	C₂²×C₄	—	13	1	no	—
201	C₂²×C₄	—	13	1	no	—
202	C₂²×C₄	—	13	1	no	—
203	C₂²×C₄	—	13	1	no	—
204	C₂²×C₄	C₄×D₁₃	1	1	yes	characteristic
205	C₂²×C₄	C₄×D₁₃	1	1	yes	characteristic
206	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
207	M₁₆	—	52	2	no	—
208	C₂×C₈	—	4	2	no	—
209	C₂×C₈	—	13	2	no	—
210	C₂×C₈	—	13	2	no	—
211	C₂×C₈	—	52	2	no	—
212	C₂×C₈	C₂²×D₁₃	1	2	yes	—
213	C₂×C₈	C₂²×D₁₃	1	2	yes	—
214	C₄⋊C₄	—	13	1	no	—
215	C₄⋊C₄	—	13	1	no	—
216	C₄⋊C₄	—	13	1	no	—
217	C₄⋊C₄	—	13	1	no	—
218	C₄⋊C₄	—	13	1	no	—
219	C₄⋊C₄	—	13	1	no	—
220	C₂²⋊C₄	—	13	1	no	—
221	C₂²⋊C₄	—	13	1	no	—
222	C₂²⋊C₄	—	13	1	no	—
223	C₂²⋊C₄	—	13	1	no	—
224	C₂²⋊C₄	—	13	1	no	—
225	C₂²⋊C₄	—	13	1	no	—
226	C₄²	—	13	1	no	—
227	C₄²	—	13	1	no	—
228	C₄²	—	13	1	no	—
229	C₄²	—	13	1	no	—
230	C₄²	—	13	1	no	—
231	C₄²	—	13	1	no	—
232	C₄²	C₄×D₁₃	1	1	yes	characteristic
233	C₄²	C₄×D₁₃	1	1	yes	characteristic
234	C₄²	C₂²×D₁₃	1	1	yes	characteristic
235	C₄²	C₂²×D₁₃	1	1	yes	characteristic
236	C₁₃	#2990	1	1	yes	characteristic, complemented, Sylow
237	C₂³	—	13	1	no	—
238	C₂³	—	13	1	no	—
239	C₂³	C₂×C₄×D₁₃	1	1	yes	characteristic
240	Q₈	—	26	2	no	—
241	D₄	—	13	4	no	—
242	D₄	—	26	2	no	—
243	D₄	—	26	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	13	1	no	—
249	C₂×C₄	—	13	1	no	—
250	C₂×C₄	—	13	1	no	—
251	C₂×C₄	—	13	1	no	—
252	C₂×C₄	—	13	1	no	—
253	C₂×C₄	—	13	1	no	—
254	C₂×C₄	—	26	2	no	—
255	C₂×C₄	—	26	2	no	—
256	C₂×C₄	—	26	2	no	—
257	C₂×C₄	—	26	2	no	—
258	C₂×C₄	—	26	1	no	—
259	C₂×C₄	—	26	1	no	—
260	C₂×C₄	—	26	1	no	—
261	C₂×C₄	—	26	1	no	—
262	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
263	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
264	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
265	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
266	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
267	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
268	C₈	—	2	2	no	—
269	C₈	—	2	2	no	—
270	C₈	—	26	2	no	—
271	C₈	—	26	2	no	—
272	C₂²	—	2	1	no	—
273	C₂²	—	2	1	no	—
274	C₂²	—	13	2	no	—
275	C₂²	—	26	1	no	—
276	C₂²	—	26	1	no	—
277	C₂²	—	26	1	no	—
278	C₂²	—	26	1	no	—
279	C₂²	—	26	1	no	—
280	C₂²	M₁₆×D₁₃	1	2	yes	—
281	C₂²	C₂²×C₄×D₁₃	1	1	yes	characteristic
282	C₄	—	2	1	no	—
283	C₄	—	2	1	no	—
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	13	2	no	—
287	C₄	—	26	1	no	—
288	C₄	—	26	1	no	—
289	C₄	—	26	1	no	—
290	C₄	—	26	1	no	—
291	C₄	—	26	1	no	—
292	C₄	#88155	1	2	yes	—
293	C₄	#88251	1	1	yes	characteristic
294	C₄	#88254	1	1	yes	characteristic
295	C₂	—	2	2	no	—
296	C₂	—	26	2	no	—
297	C₂	—	52	1	no	—
298	C₂	#11719209	1	1	yes	characteristic
299	C₂	C₂×#88155	1	1	yes	characteristic
300	C₂	C₂×M₁₆×D₁₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₃
commutator, socle	C₂²×C₁₃	C₂³×C₄
Fitting	C₂×C₁₃×C₄⋊C₈	C₂
prime radical	C₂×#6127	C₄
cosocle	C₂×C₄×C₁₃	C₂⁴
hypercenter	C₂×C₄⋊C₈	D₁₃
hypocenter	C₁₃	#2990
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
2. C₂×C₁₃×C₄⋊C₈	C₂	semidirect
3. #11718772	C₂	semidirect
4. C₈×C₂²⋊D₁₃	C₂	semidirect
5. #11718587	C₂	semidirect
6. #11718498	C₂	semidirect
7. #11718497	C₂	semidirect
8. #11718496	C₂	semidirect
9. #11718494	C₂	semidirect
10. D₁₃×C₄⋊C₈	C₂	semidirect
11. #11718489	C₂	semidirect
101. C₂²⋊D₁₃	M₁₆	semidirect
132. C₂×C₄⋊C₈	D₁₃	semidirect
236. C₁₃	#2990	semidirect

Maximal subgroups

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

subgroup	index
C₄×#6127	2
C₂×C₁₃×C₄⋊C₈	2
#11718772	2
C₈×C₂²⋊D₁₃	2
#11718587	2
#11718498	2
#11718497	2
#11718496	2
#11718494	2
D₁₃×C₄⋊C₈	2
#11718489	2
#2990	13