Subgroups — #13123798

Order 1664.

412 conjugacy classes of subgroups.

The subgroup lattice has 294 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	C₄×C₁₃×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#11718789	C₂	1	2	yes	maximal, complemented
4	#11718788	C₂	1	2	yes	maximal, complemented
5	#11718586	C₂	1	1	yes	maximal, characteristic, complemented
6	#11718367	C₂	1	2	yes	maximal, complemented
7	#11718362	C₂	1	2	yes	maximal, complemented
8	C₄×#6094	C₂	1	1	yes	maximal, characteristic, complemented
9	#11718346	C₂	1	1	yes	maximal, characteristic, complemented
10	#11718346	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₈×D₁₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#6127	C₄	1	1	yes	characteristic
13	C₂×C₁₃×M₁₆	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×#97	C₂²	1	2	yes	—
31	C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
32	C₄×C₈×C₁₃	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	2	yes	—
41	C₄×#1055	C₂²	1	1	yes	characteristic
42	C₄×#1055	C₂²	1	1	yes	characteristic
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	C₁₃×M₁₆	—	2	4	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	—	2	4	no	—
85	C₈×D₁₃	—	2	4	no	—
86	#2965	—	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	2	no	—
93	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
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	C₂⁴	1	1	yes	characteristic
99	C₂²⋊D₁₃	—	2	2	no	—
100	C₂²⋊D₁₃	—	2	2	no	—
101	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
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	D₅₂	—	2	2	no	—
108	C₄×D₁₃	—	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	2	no	—
113	C₄×D₁₃	M₁₆	1	4	yes	complemented
114	Dic₂₆	—	2	2	no	—
115	C₈×C₁₃	—	2	2	no	—
116	C₈×C₁₃	#41	1	4	yes	complemented
117	#1055	—	2	2	no	—
118	#1055	—	2	2	no	—
119	C₄×#41	—	13	1	no	—
120	#446	—	13	2	no	—
121	#436	—	13	2	no	—
122	#435	—	13	2	no	—
123	#435	—	13	2	no	—
124	#431	—	13	1	no	—
125	#431	—	13	1	no	—
126	#431	—	13	1	no	—
127	C₄×M₁₆	—	13	1	no	—
128	C₄×M₁₆	D₁₃	1	1	yes	characteristic, complemented
129	C₂×C₄×C₈	—	13	1	no	—
130	C₂²×C₁₃	—	2	1	no	—
131	C₂²×C₁₃	—	2	1	no	—
132	C₂²×C₁₃	—	2	1	no	—
133	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
134	C₂×D₁₃	—	2	1	no	—
135	C₂×D₁₃	—	2	1	no	—
136	C₂×D₁₃	—	2	1	no	—
137	C₂×D₁₃	—	2	1	no	—
138	C₂×D₁₃	—	2	1	no	—
139	C₂×D₁₃	C₂×M₁₆	1	2	yes	—
140	C₄×C₁₃	—	2	1	no	—
141	C₄×C₁₃	—	2	1	no	—
142	C₄×C₁₃	—	2	1	no	—
143	C₄×C₁₃	C₂×M₁₆	1	2	yes	—
144	C₄×C₁₃	#131	1	2	yes	—
145	C₄×C₁₃	C₂×#41	1	2	yes	—
146	Dic₁₃	—	2	1	no	—
147	Dic₁₃	—	2	1	no	—
148	Dic₁₃	—	2	1	no	—
149	Dic₁₃	—	2	1	no	—
150	Dic₁₃	—	2	1	no	—
151	Dic₁₃	C₂×M₁₆	1	2	yes	—
152	C₂×#41	—	13	1	no	—
153	C₂×M₁₆	—	13	2	no	—
154	C₂×M₁₆	C₂×D₁₃	1	2	yes	—
155	C₂²×C₈	—	13	2	no	—
156	C₄×Q₈	—	13	1	no	—
157	C₄×Q₈	—	13	1	no	—
158	C₄×D₄	—	13	1	no	—
159	C₄×D₄	—	13	1	no	—
160	C₄×D₄	—	13	1	no	—
161	C₄×D₄	—	13	1	no	—
162	C₄×D₄	—	13	1	no	—
163	C₄×D₄	—	13	1	no	—
164	#117	—	13	1	no	—
165	#117	—	13	1	no	—
166	#117	—	13	1	no	—
167	C₂×C₄²	—	13	1	no	—
168	C₂×C₄²	—	13	1	no	—
169	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
170	C₄⋊C₈	—	13	2	no	—
171	C₄⋊C₈	—	13	2	no	—
172	C₄⋊C₈	—	13	2	no	—
173	C₂²⋊C₈	—	13	2	no	—
174	C₂²⋊C₈	—	13	2	no	—
175	C₂²⋊C₈	—	13	2	no	—
176	#97	—	13	2	no	—
177	#97	C₂×D₁₃	1	2	yes	—
178	C₄×C₈	—	13	2	no	—
179	C₄×C₈	—	13	1	no	—
180	C₄×C₈	—	13	1	no	—
181	C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
182	C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
183	C₂×C₁₃	—	4	1	no	—
184	C₂×C₁₃	C₄×#41	1	1	yes	characteristic
185	C₂×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
186	C₂×C₁₃	C₂×#131	1	1	yes	characteristic
187	D₁₃	—	2	2	no	—
188	D₁₃	—	4	1	no	—
189	#41	—	26	4	no	—
190	C₂×Q₈	—	13	1	no	—
191	C₂×D₄	—	13	1	no	—
192	C₂×D₄	—	13	1	no	—
193	C₂×D₄	—	13	1	no	—
194	C₂²×C₄	—	13	1	no	—
195	C₂²×C₄	—	13	1	no	—
196	C₂²×C₄	—	13	1	no	—
197	C₂²×C₄	—	13	1	no	—
198	C₂²×C₄	—	13	1	no	—
199	C₂²×C₄	—	13	1	no	—
200	C₂²×C₄	C₄×D₁₃	1	1	yes	characteristic
201	C₂²×C₄	C₄×D₁₃	1	1	yes	characteristic
202	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
203	M₁₆	—	2	4	no	—
204	M₁₆	—	26	4	no	—
205	C₂×C₈	—	13	2	no	—
206	C₂×C₈	—	13	2	no	—
207	C₂×C₈	—	26	4	no	—
208	C₂×C₈	C₂²×D₁₃	1	2	yes	—
209	C₂×C₈	C₂²×D₁₃	1	2	yes	—
210	C₄⋊C₄	—	13	1	no	—
211	C₄⋊C₄	—	13	1	no	—
212	C₄⋊C₄	—	13	1	no	—
213	C₄⋊C₄	—	13	1	no	—
214	C₄⋊C₄	—	13	1	no	—
215	C₄⋊C₄	—	13	1	no	—
216	C₂²⋊C₄	—	13	1	no	—
217	C₂²⋊C₄	—	13	1	no	—
218	C₂²⋊C₄	—	13	1	no	—
219	C₂²⋊C₄	—	13	1	no	—
220	C₂²⋊C₄	—	13	1	no	—
221	C₂²⋊C₄	—	13	1	no	—
222	C₄²	—	13	1	no	—
223	C₄²	—	13	1	no	—
224	C₄²	—	13	1	no	—
225	C₄²	—	13	1	no	—
226	C₄²	—	13	1	no	—
227	C₄²	—	13	1	no	—
228	C₄²	C₄×D₁₃	1	1	yes	characteristic
229	C₄²	C₄×D₁₃	1	1	yes	characteristic
230	C₄²	C₂²×D₁₃	1	1	yes	characteristic
231	C₄²	C₂²×D₁₃	1	1	yes	characteristic
232	C₁₃	#2965	1	1	yes	characteristic, complemented, Sylow
233	C₂³	—	13	1	no	—
234	C₂³	—	13	1	no	—
235	C₂³	C₂×C₄×D₁₃	1	1	yes	characteristic
236	Q₈	—	26	2	no	—
237	D₄	—	26	2	no	—
238	D₄	—	26	2	no	—
239	D₄	—	26	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	13	1	no	—
244	C₂×C₄	—	13	4	no	—
245	C₂×C₄	—	13	1	no	—
246	C₂×C₄	—	13	1	no	—
247	C₂×C₄	—	13	1	no	—
248	C₂×C₄	—	13	1	no	—
249	C₂×C₄	—	13	1	no	—
250	C₂×C₄	—	26	2	no	—
251	C₂×C₄	—	26	2	no	—
252	C₂×C₄	—	26	2	no	—
253	C₂×C₄	—	26	2	no	—
254	C₂×C₄	—	26	2	no	—
255	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
256	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
257	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
258	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
259	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
260	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
261	C₈	—	2	2	no	—
262	C₈	—	26	2	no	—
263	C₈	—	26	2	no	—
264	C₈	#6127	1	4	yes	complemented
265	C₂²	—	2	1	no	—
266	C₂²	—	2	1	no	—
267	C₂²	—	2	1	no	—
268	C₂²	—	13	2	no	—
269	C₂²	—	26	1	no	—
270	C₂²	—	26	1	no	—
271	C₂²	—	26	1	no	—
272	C₂²	—	26	1	no	—
273	C₂²	—	26	1	no	—
274	C₂²	C₂²×C₄×D₁₃	1	1	yes	characteristic
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	2	1	no	—
278	C₄	—	13	2	no	—
279	C₄	—	26	1	no	—
280	C₄	—	26	1	no	—
281	C₄	—	26	1	no	—
282	C₄	—	26	1	no	—
283	C₄	—	26	1	no	—
284	C₄	M₁₆×D₁₃	1	2	yes	—
285	C₄	#88161	1	2	yes	—
286	C₄	C₂×#6127	1	2	yes	—
287	C₂	—	4	1	no	—
288	C₂	—	26	2	no	—
289	C₂	—	52	1	no	—
290	C₂	C₄×#6127	1	1	yes	characteristic
291	C₂	C₂×M₁₆×D₁₃	1	1	yes	characteristic
292	C₂	C₂×#88161	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₂²×D₁₃
commutator, socle	C₂²×C₁₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₃
Fitting	C₄×C₁₃×M₁₆	C₂
prime radical	C₂×#6127	C₄
cosocle	C₂×C₄×C₁₃	C₂⁴
hypercenter	C₄×M₁₆	D₁₃
hypocenter	C₁₃	#2965
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄² ◁ C₄×M₁₆

Complements

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

N	H ≅ G/N	type
2. C₄×C₁₃×M₁₆	C₂	semidirect
3. #11718789	C₂	semidirect
4. #11718788	C₂	semidirect
5. #11718586	C₂	semidirect
6. #11718367	C₂	semidirect
7. #11718362	C₂	semidirect
8. C₄×#6094	C₂	semidirect
9. #11718346	C₂	semidirect
10. #11718346	C₂	semidirect
11. C₄×C₈×D₁₃	C₂	semidirect
113. C₄×D₁₃	M₁₆	semidirect
116. C₈×C₁₃	#41	semidirect
128. C₄×M₁₆	D₁₃	semidirect
232. C₁₃	#2965	semidirect
264. C₈	#6127	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
C₄×C₁₃×M₁₆	2
#11718789	2
#11718788	2
#11718586	2
#11718367	2
#11718362	2
C₄×#6094	2
#11718346	2
C₄×C₈×D₁₃	2
#2965	13