Subgroups — #13124896

Order 1664.

434 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₁₃×#41	C₂	1	1	yes	maximal, characteristic, complemented
2	#11719182	C₂	1	1	yes	maximal, characteristic, complemented
3	#11718899	C₂	1	2	yes	maximal, complemented
4	#11718688	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×#6106	C₂	1	2	yes	maximal, complemented
6	#11718682	C₂	1	1	yes	maximal, characteristic, complemented
7	#11718676	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#6104	C₂	1	2	yes	maximal, complemented
9	#11718669	C₂	1	1	yes	maximal, characteristic, complemented
10	#11718664	C₂	1	2	yes	maximal, complemented
11	#11718586	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₁₃×#41	C₂²	1	1	yes	characteristic
13	C₂×#6127	—	2	1	no	—
14	C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
15	C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
16	C₄×D₄×C₁₃	—	2	1	no	—
17	C₄×D₄×C₁₃	—	2	1	no	—
18	C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
19	C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
20	C₁₃×#117	—	2	1	no	—
21	C₁₃×#117	C₂²	1	1	yes	characteristic
22	C₂×C₄²×C₁₃	—	2	1	no	—
23	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
24	C₂×#6106	C₂²	1	2	yes	—
25	C₂×#6104	C₂²	1	2	yes	—
26	#88184	—	2	1	no	—
27	#88184	C₂²	1	2	yes	—
28	#88129	—	2	1	no	—
29	#88128	C₂²	1	1	yes	characteristic
30	C₄×D₅₂	—	2	1	no	—
31	C₄×D₅₂	C₂²	1	2	yes	—
32	#88123	C₂²	1	1	yes	characteristic
33	#88075	C₂²	1	2	yes	—
34	#88072	C₂²	1	2	yes	—
35	C₂²⋊#1055	C₂²	1	2	yes	—
36	#88049	C₂²	1	2	yes	—
37	#88049	C₂²	1	2	yes	—
38	#88048	C₂²	1	2	yes	—
39	#88047	C₂²	1	2	yes	—
40	C₄⋊#1055	C₂²	1	1	yes	characteristic
41	C₄⋊#1055	C₂²	1	1	yes	characteristic
42	C₄×#1055	C₂²	1	2	yes	—
43	C₁₃×#41	—	2	2	no	—
44	C₁₃×#41	—	2	2	no	—
45	C₂×Q₈×C₁₃	C₂³	1	1	yes	characteristic
46	C₂×D₄×C₁₃	—	2	1	no	—
47	C₂×D₄×C₁₃	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₁₃	—	2	1	no	—
49	C₂²×C₄×C₁₃	—	2	1	no	—
50	C₂²×C₄×C₁₃	—	2	1	no	—
51	C₂²×C₄×C₁₃	D₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₃	D₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
54	C₂×C₂²⋊D₁₃	—	2	2	no	—
55	#6127	—	4	4	no	—
56	C₂×D₅₂	—	2	1	no	—
57	C₂×D₅₂	—	2	1	no	—
58	C₂×D₅₂	C₂³	1	2	yes	—
59	C₂×C₄×D₁₃	—	2	2	no	—
60	C₂×Dic₂₆	—	2	1	no	—
61	C₁₃×C₄⋊C₄	—	2	1	no	—
62	C₁₃×C₄⋊C₄	—	2	1	no	—
63	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
64	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
65	C₁₃×C₂²⋊C₄	—	2	1	no	—
66	C₁₃×C₂²⋊C₄	—	2	1	no	—
67	C₁₃×C₂²⋊C₄	—	2	1	no	—
68	C₄²×C₁₃	—	2	1	no	—
69	C₄²×C₁₃	—	2	1	no	—
70	C₄²×C₁₃	—	2	1	no	—
71	C₄²×C₁₃	D₄	1	1	yes	characteristic
72	C₄²×C₁₃	D₄	1	1	yes	characteristic
73	C₄²×C₁₃	C₂³	1	1	yes	characteristic
74	C₄²×C₁₃	C₂³	1	1	yes	characteristic
75	#6106	—	2	4	no	—
76	#6104	—	2	4	no	—
77	#6103	—	2	2	no	—
78	#6103	—	2	2	no	—
79	C₄⋊Dic₁₃	—	2	1	no	—
80	C₄⋊Dic₁₃	C₂³	1	2	yes	—
81	C₂×#1055	C₂³	1	2	yes	—
82	C₂×#1055	C₂³	1	2	yes	—
83	#3106	—	13	1	no	maximal, Sylow
84	C₂³×C₁₃	—	2	1	no	—
85	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
86	C₂²×D₁₃	—	2	2	no	—
87	Q₈×C₁₃	—	2	1	no	—
88	Q₈×C₁₃	#41	1	2	yes	complemented
89	D₄×C₁₃	—	2	4	no	—
90	D₄×C₁₃	—	2	1	no	—
91	D₄×C₁₃	#41	1	2	yes	complemented
92	C₂×C₄×C₁₃	—	2	4	no	—
93	C₂×C₄×C₁₃	—	2	1	no	—
94	C₂×C₄×C₁₃	—	2	2	no	—
95	C₂×C₄×C₁₃	—	2	2	no	—
96	C₂×C₄×C₁₃	—	2	1	no	—
97	C₂×C₄×C₁₃	—	2	1	no	—
98	C₂×C₄×C₁₃	—	2	1	no	—
99	C₂×C₄×C₁₃	—	2	1	no	—
100	C₂×C₄×C₁₃	—	4	2	no	—
101	C₂×C₄×C₁₃	—	4	2	no	—
102	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
104	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
105	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
106	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
107	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
108	C₂²⋊D₁₃	—	4	4	no	—
109	C₂×Dic₁₃	—	2	2	no	—
110	D₅₂	—	2	2	no	—
111	D₅₂	—	2	2	no	—
112	D₅₂	—	4	2	no	—
113	D₅₂	—	4	2	no	—
114	C₄×D₁₃	—	4	4	no	—
115	Dic₂₆	—	4	2	no	—
116	#1055	—	2	2	no	—
117	#1055	—	2	2	no	—
118	#532	—	13	1	no	—
119	C₄×#41	D₁₃	1	1	yes	characteristic, complemented
120	#482	—	13	2	no	—
121	#477	—	13	1	no	—
122	#476	—	13	1	no	—
123	#460	—	13	1	no	—
124	#459	—	13	1	no	—
125	#449	—	13	2	no	—
126	C₄×SD₁₆	—	13	2	no	—
127	C₄×D₈	—	13	2	no	—
128	#431	—	13	1	no	—
129	C₂²×C₁₃	—	2	1	no	—
130	C₂²×C₁₃	—	2	2	no	—
131	C₂²×C₁₃	—	2	1	no	—
132	C₂²×C₁₃	—	2	1	no	—
133	C₂²×C₁₃	—	4	1	no	—
134	C₂²×C₁₃	—	4	1	no	—
135	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
136	C₂×D₁₃	—	4	2	no	—
137	C₂×D₁₃	—	4	2	no	—
138	C₂×D₁₃	—	4	2	no	—
139	C₄×C₁₃	—	2	1	no	—
140	C₄×C₁₃	—	2	2	no	—
141	C₄×C₁₃	—	2	1	no	—
142	C₄×C₁₃	—	2	1	no	—
143	C₄×C₁₃	—	4	1	no	—
144	C₄×C₁₃	—	4	1	no	—
145	C₄×C₁₃	#135	1	2	yes	—
146	C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
147	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
148	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
149	Dic₁₃	—	4	2	no	—
150	C₂×#41	—	26	1	no	—
151	C₂×#41	C₂×D₁₃	1	1	yes	characteristic
152	C₂×SD₁₆	—	13	2	no	—
153	C₂×D₈	—	13	2	no	—
154	C₄⋊Q₈	—	13	1	no	—
155	#127	—	13	1	no	—
156	#124	—	26	1	no	—
157	#121	—	13	2	no	—
158	#121	—	26	1	no	—
159	C₄×Q₈	C₂×D₁₃	1	1	yes	characteristic
160	C₄×Q₈	C₂×D₁₃	1	1	yes	characteristic
161	C₄×D₄	—	2	1	no	—
162	C₄×D₄	—	2	1	no	—
163	C₄×D₄	—	13	2	no	—
164	C₄×D₄	—	26	1	no	—
165	C₄×D₄	C₂×D₁₃	1	1	yes	characteristic
166	C₄×D₄	C₂×D₁₃	1	1	yes	characteristic
167	#117	—	2	1	no	—
168	#117	C₂×D₁₃	1	1	yes	characteristic
169	C₂×C₄²	—	2	1	no	—
170	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
171	#107	—	13	2	no	—
172	#106	—	13	2	no	—
173	C₄⋊C₈	—	13	1	no	—
174	C₄⋊C₈	—	13	1	no	—
175	#103	—	13	2	no	—
176	#102	—	13	2	no	—
177	#102	—	13	2	no	—
178	#102	—	13	2	no	—
179	C₂²⋊C₈	—	13	2	no	—
180	C₄×C₈	—	13	2	no	—
181	C₂×C₁₃	—	4	1	no	—
182	C₂×C₁₃	—	4	2	no	—
183	C₂×C₁₃	#525	1	1	yes	characteristic
184	C₂×C₁₃	C₂×#135	1	1	yes	characteristic
185	C₂×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
186	D₁₃	—	8	2	no	—
187	#41	—	2	2	no	—
188	#41	—	2	2	no	—
189	#41	—	52	4	no	—
190	C₂×Q₈	—	26	1	no	—
191	C₂×Q₈	C₂²×D₁₃	1	1	yes	characteristic
192	C₂×D₄	—	2	1	no	—
193	C₂×D₄	—	13	2	no	—
194	C₂×D₄	—	26	2	no	—
195	C₂×D₄	—	26	1	no	—
196	C₂×D₄	—	26	1	no	—
197	C₂×D₄	C₂²×D₁₃	1	1	yes	characteristic
198	C₂²×C₄	—	2	1	no	—
199	C₂²×C₄	—	2	1	no	—
200	C₂²×C₄	—	2	1	no	—
201	C₂²×C₄	—	26	2	no	—
202	C₂²×C₄	C₂²⋊D₁₃	1	1	yes	characteristic
203	C₂²×C₄	C₂²⋊D₁₃	1	1	yes	characteristic
204	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
205	SD₁₆	—	26	4	no	—
206	D₈	—	26	4	no	—
207	C₂×C₈	—	13	2	no	—
208	C₂×C₈	—	13	2	no	—
209	C₄⋊C₄	—	2	1	no	—
210	C₄⋊C₄	—	2	1	no	—
211	C₄⋊C₄	—	13	2	no	—
212	C₄⋊C₄	—	26	1	no	—
213	C₄⋊C₄	C₂²×D₁₃	1	1	yes	characteristic
214	C₄⋊C₄	C₂²×D₁₃	1	1	yes	characteristic
215	C₂²⋊C₄	—	2	1	no	—
216	C₂²⋊C₄	—	2	1	no	—
217	C₂²⋊C₄	—	2	1	no	—
218	C₂²⋊C₄	—	26	2	no	—
219	C₂²⋊C₄	—	26	2	no	—
220	C₄²	—	2	1	no	—
221	C₄²	—	2	1	no	—
222	C₄²	—	2	1	no	—
223	C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
224	C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
225	C₄²	C₂²×D₁₃	1	1	yes	characteristic
226	C₄²	C₂²×D₁₃	1	1	yes	characteristic
227	C₁₃	#3106	1	1	yes	characteristic, complemented, Sylow
228	C₂³	—	2	1	no	—
229	C₂³	—	26	2	no	—
230	C₂³	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
231	Q₈	—	2	1	no	—
232	Q₈	—	52	2	no	—
233	Q₈	#6127	1	2	yes	complemented
234	D₄	—	2	4	no	—
235	D₄	—	2	1	no	—
236	D₄	—	26	2	no	—
237	D₄	—	26	2	no	—
238	D₄	—	52	2	no	—
239	D₄	—	52	4	no	—
240	D₄	—	52	2	no	—
241	D₄	#6127	1	2	yes	complemented
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	2	1	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	—	2	1	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	—	26	2	no	—
253	C₂×C₄	—	52	4	no	—
254	C₂×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
255	C₂×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
256	C₂×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
257	C₂×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
258	C₂×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
259	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
260	C₈	—	26	2	no	—
261	C₈	—	26	2	no	—
262	C₂²	—	2	1	no	—
263	C₂²	—	2	2	no	—
264	C₂²	—	2	1	no	—
265	C₂²	—	2	1	no	—
266	C₂²	—	4	1	no	—
267	C₂²	—	4	1	no	—
268	C₂²	—	52	2	no	—
269	C₂²	—	52	2	no	—
270	C₂²	—	52	2	no	—
271	C₂²	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
272	C₄	—	2	1	no	—
273	C₄	—	2	2	no	—
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	4	1	no	—
277	C₄	—	4	1	no	—
278	C₄	—	52	2	no	—
279	C₄	#88203	1	2	yes	—
280	C₄	#88204	1	2	yes	—
281	C₄	C₂×#6127	1	1	yes	characteristic
282	C₄	C₂×#6127	1	1	yes	characteristic
283	C₂	—	4	1	no	—
284	C₂	—	4	2	no	—
285	C₂	—	104	2	no	—
286	C₂	#11719496	1	1	yes	characteristic
287	C₂	C₂×#88203	1	1	yes	characteristic
288	C₂	C₂×#88204	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₃
commutator, cosocle	C₂×C₄×C₁₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃
Fitting	C₄×C₁₃×#41	C₂
socle	C₂²×C₁₃	C₂²×D₄
hypercenter	C₄×#41	D₁₃
hypocenter	C₁₃	#3106
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
1. C₄×C₁₃×#41	C₂	semidirect
2. #11719182	C₂	semidirect
3. #11718899	C₂	semidirect
4. #11718688	C₂	semidirect
5. C₄×#6106	C₂	semidirect
6. #11718682	C₂	semidirect
7. #11718676	C₂	semidirect
8. C₄×#6104	C₂	semidirect
9. #11718669	C₂	semidirect
10. #11718664	C₂	semidirect
11. #11718586	C₂	semidirect
88. Q₈×C₁₃	#41	semidirect
91. D₄×C₁₃	#41	semidirect
119. C₄×#41	D₁₃	semidirect
227. C₁₃	#3106	semidirect
233. Q₈	#6127	semidirect
241. D₄	#6127	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₁₃×#41	2
#11719182	2
#11718899	2
#11718688	2
C₄×#6106	2
#11718682	2
#11718676	2
C₄×#6104	2
#11718669	2
#11718664	2
#11718586	2
#3106	13