Subgroups — #13131675

Order 1664.

406 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	#11719227	C₂	1	1	yes	maximal, characteristic
2	#11719128	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#88110	C₂	1	1	yes	maximal, characteristic, complemented
4	#11719078	C₂	1	2	yes	maximal, complemented
5	#11719075	C₂	1	2	yes	maximal, complemented
6	#11719064	C₂	1	2	yes	maximal, complemented
7	#11719063	C₂	1	2	yes	maximal, complemented
8	#11719055	C₂	1	1	yes	maximal, characteristic, complemented
9	#11719054	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×#6118	C₂	1	1	yes	maximal, characteristic, complemented
11	#11719047	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#6127	C₄	1	1	yes	characteristic
13	C₂²×#1057	C₂²	1	2	yes	—
14	C₂×#6118	C₂²	1	2	yes	—
15	C₂×D₁₃⋊C₈	C₂²	1	2	yes	—
16	C₁₃×#117	C₄	1	1	yes	characteristic
17	C₂×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
18	#88147	C₄	1	1	yes	characteristic
19	#88147	C₂²	1	1	yes	characteristic
20	#88141	C₄	1	1	yes	characteristic
21	#88141	C₂²	1	1	yes	characteristic
22	#88135	C₄	1	1	yes	characteristic
23	#88135	C₄	1	1	yes	characteristic
24	#88135	C₂²	1	1	yes	characteristic
25	#88135	C₂²	1	1	yes	characteristic
26	#88126	C₄	1	1	yes	characteristic
27	#88126	C₄	1	1	yes	characteristic
28	C₄²×D₁₃	C₂²	1	1	yes	characteristic
29	C₄²×D₁₃	C₂²	1	1	yes	characteristic
30	C₂²⋊#1057	C₂²	1	2	yes	—
31	#88112	C₂²	1	2	yes	—
32	#88112	C₂²	1	2	yes	—
33	#88111	C₂²	1	2	yes	—
34	#88111	C₂²	1	2	yes	—
35	#88110	C₂²	1	1	yes	characteristic
36	#88110	C₂²	1	1	yes	characteristic
37	#88110	C₂²	1	2	yes	—
38	#88110	C₂²	1	1	yes	characteristic
39	#88110	C₂²	1	1	yes	characteristic
40	#88110	C₂²	1	2	yes	—
41	C₄⋊#1057	C₂²	1	2	yes	—
42	C₄×#1057	C₂²	1	2	yes	—
43	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₂²⋊D₁₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₂²⋊D₁₃	C₂×C₄	1	1	yes	characteristic
46	C₂²×Dic₁₃	C₂×C₄	1	1	yes	characteristic
47	C₂²×Dic₁₃	C₂³	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×#1057	—	2	4	no	—
58	C₂×#1057	C₂³	1	2	yes	—
59	C₂×#1057	C₂³	1	2	yes	—
60	C₂×#1057	C₂³	1	2	yes	—
61	C₂×#1057	C₂³	1	2	yes	—
62	#6118	—	2	4	no	—
63	D₁₃⋊C₈	—	2	4	no	—
64	C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
65	C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
66	C₁₃×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
67	C₁₃×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
68	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
69	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
70	#6103	C₂×C₄	1	1	yes	characteristic
71	#6103	C₂×C₄	1	1	yes	characteristic
72	#6103	C₂×C₄	1	1	yes	characteristic
73	#6103	C₂×C₄	1	1	yes	characteristic
74	#6101	C₂×C₄	1	1	yes	characteristic
75	#6101	C₂×C₄	1	1	yes	characteristic
76	#6101	C₂×C₄	1	1	yes	characteristic
77	#6101	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	C₂×C₄	1	1	yes	characteristic
81	C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
82	C₄×Dic₁₃	C₂³	1	1	yes	characteristic
83	C₄×Dic₁₃	C₂³	1	1	yes	characteristic
84	C₄×Dic₁₃	C₂³	1	1	yes	characteristic
85	C₄×Dic₁₃	C₂³	1	1	yes	characteristic
86	#2970	—	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	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
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	C₂²⋊D₁₃	—	2	2	no	—
98	C₂²⋊D₁₃	M₁₆	1	4	yes	complemented
99	C₂×Dic₁₃	—	2	2	no	—
100	C₂×Dic₁₃	—	4	1	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	C₂⁴	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	—	4	1	no	—
113	C₄×D₁₃	—	4	1	no	—
114	Dic₂₆	—	2	2	no	—
115	#1057	—	2	2	no	—
116	#1057	—	2	2	no	—
117	#1057	—	2	2	no	—
118	#1057	#41	1	4	yes	complemented
119	C₄×#41	—	13	1	no	—
120	#446	—	13	2	no	—
121	#435	—	13	2	no	—
122	#435	—	13	2	no	—
123	C₈×D₄	—	13	2	no	—
124	#432	—	13	1	no	—
125	#432	—	13	1	no	—
126	#432	—	13	1	no	—
127	C₄×M₁₆	—	13	1	no	—
128	C₂×#97	—	13	1	no	—
129	C₂×#97	—	13	1	no	—
130	C₂²×C₁₃	—	2	1	no	—
131	C₂²×C₁₃	—	2	1	no	—
132	C₂²×C₁₃	C₂×M₁₆	1	2	yes	—
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	—	2	1	no	—
144	C₄×C₁₃	—	2	1	no	—
145	C₄×C₁₃	—	2	1	no	—
146	Dic₁₃	—	2	1	no	—
147	Dic₁₃	—	2	1	no	—
148	Dic₁₃	—	2	1	no	—
149	Dic₁₃	—	2	1	no	—
150	Dic₁₃	C₂×M₁₆	1	2	yes	—
151	Dic₁₃	C₂×#41	1	2	yes	—
152	C₂×#41	—	13	1	no	—
153	C₂×M₁₆	—	13	2	no	—
154	C₂²×C₈	—	13	2	no	—
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	#261	1	1	yes	characteristic
167	C₂×C₄²	—	13	1	no	—
168	C₂×C₄²	—	13	1	no	—
169	C₂×C₄²	—	13	1	no	—
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	1	no	—
177	#97	—	13	1	no	—
178	#97	—	13	1	no	—
179	#97	—	13	2	no	—
180	#97	—	13	2	no	—
181	#97	—	13	1	no	—
182	C₄×C₈	—	13	2	no	—
183	C₂×C₁₃	—	2	2	no	—
184	C₂×C₁₃	C₄×#41	1	1	yes	characteristic
185	C₂×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
186	C₂×C₁₃	#568	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	—	13	1	no	—
201	C₂²×C₄	—	13	1	no	—
202	C₂²×C₄	C₂×#261	1	1	yes	characteristic
203	M₁₆	—	26	4	no	—
204	C₂×C₈	—	13	2	no	—
205	C₂×C₈	—	13	2	no	—
206	C₂×C₈	—	13	2	no	—
207	C₂×C₈	—	13	2	no	—
208	C₂×C₈	—	26	4	no	—
209	C₂×C₈	—	26	4	no	—
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	C₂×#261	1	1	yes	characteristic
215	C₄⋊C₄	C₂×#261	1	1	yes	characteristic
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	C₂×#261	1	1	yes	characteristic
221	C₂²⋊C₄	C₂×#261	1	1	yes	characteristic
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	—	13	1	no	—
229	C₄²	—	13	1	no	—
230	C₄²	C₂×#261	1	1	yes	characteristic
231	C₄²	C₂×#261	1	1	yes	characteristic
232	C₁₃	#2970	1	1	yes	characteristic, complemented, Sylow
233	C₂³	—	13	1	no	—
234	C₂³	—	13	1	no	—
235	C₂³	C₂²×#261	1	1	yes	characteristic
236	Q₈	—	26	2	no	—
237	D₄	—	13	4	no	—
238	D₄	—	26	2	no	—
239	D₄	—	26	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	13	1	no	—
242	C₂×C₄	—	13	1	no	—
243	C₂×C₄	—	13	1	no	—
244	C₂×C₄	—	13	1	no	—
245	C₂×C₄	—	13	1	no	—
246	C₂×C₄	—	13	1	no	—
247	C₂×C₄	—	26	2	no	—
248	C₂×C₄	—	26	2	no	—
249	C₂×C₄	—	26	2	no	—
250	C₂×C₄	—	26	2	no	—
251	C₂×C₄	—	26	2	no	—
252	C₂×C₄	—	52	1	no	—
253	C₂×C₄	—	52	1	no	—
254	C₂×C₄	—	52	1	no	—
255	C₂×C₄	C₂²×#261	1	1	yes	characteristic
256	C₂×C₄	C₂²×#261	1	1	yes	characteristic
257	C₂×C₄	C₂²×#261	1	1	yes	characteristic
258	C₂×C₄	C₂²×#261	1	1	yes	characteristic
259	C₂×C₄	C₂²×#261	1	1	yes	characteristic
260	C₂×C₄	C₂²×#261	1	1	yes	characteristic
261	C₈	—	13	4	no	—
262	C₈	—	26	2	no	—
263	C₈	—	26	2	no	—
264	C₈	—	26	2	no	—
265	C₂²	—	2	1	no	—
266	C₂²	—	2	1	no	—
267	C₂²	—	13	2	no	—
268	C₂²	—	26	1	no	—
269	C₂²	—	26	1	no	—
270	C₂²	—	26	1	no	—
271	C₂²	—	26	1	no	—
272	C₂²	—	26	1	no	—
273	C₂²	D₁₃⋊M₁₆	1	2	yes	—
274	C₂²	C₂³×#261	1	1	yes	characteristic
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	2	1	no	—
278	C₄	—	2	1	no	—
279	C₄	—	2	1	no	—
280	C₄	—	2	1	no	—
281	C₄	—	13	2	no	—
282	C₄	—	13	2	no	—
283	C₄	—	26	1	no	—
284	C₄	—	26	1	no	—
285	C₄	—	26	1	no	—
286	C₄	—	26	1	no	—
287	C₂	—	2	2	no	—
288	C₂	—	26	2	no	—
289	C₂	—	52	1	no	—
290	C₂	C₂×D₁₃⋊M₁₆	1	1	yes	characteristic
291	C₂	#11719629	1	1	yes	characteristic
292	C₂	#41×#261	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#261
commutator, socle	C₂²×C₁₃	C₂³×C₄
Fitting	C₁₃×#117	C₄
prime radical	C₂×#6127	C₄
cosocle	C₂×Dic₁₃	C₂⁴
hypercenter	#117	#261
hypocenter	C₁₃	#2970
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ #117

Complements

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

N	H ≅ G/N	type
2. #11719128	C₂	semidirect
3. C₂×#88110	C₂	semidirect
4. #11719078	C₂	semidirect
5. #11719075	C₂	semidirect
6. #11719064	C₂	semidirect
7. #11719063	C₂	semidirect
8. #11719055	C₂	semidirect
9. #11719054	C₂	semidirect
10. C₄×#6118	C₂	semidirect
11. #11719047	C₂	semidirect
98. C₂²⋊D₁₃	M₁₆	semidirect
118. #1057	#41	semidirect
232. C₁₃	#2970	semidirect

Maximal subgroups

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

subgroup	index
#11719227	2
#11719128	2
C₂×#88110	2
#11719078	2
#11719075	2
#11719064	2
#11719063	2
#11719055	2
#11719054	2
C₄×#6118	2
#11719047	2
#2970	13