Subgroups — #13131519

Order 1664.

400 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#6129	C₂	1	1	yes	maximal, characteristic
2	#11719146	C₂	1	2	yes	maximal
3	#11719125	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×#6122	C₂	1	1	yes	maximal, characteristic, complemented
5	#11719119	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×#1057	C₂	1	1	yes	maximal, characteristic, complemented
7	#11719077	C₂	1	2	yes	maximal, complemented
8	#11719068	C₂	1	2	yes	maximal, complemented
9	#11719067	C₂	1	2	yes	maximal, complemented
10	#11719051	C₂	1	1	yes	maximal, characteristic, complemented
11	#11719049	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#6129	C₂²	1	1	yes	characteristic
13	C₂×#6122	C₂²	1	2	yes	—
14	C₂²×#1057	C₂²	1	2	yes	—
15	C₄×D₄×C₁₃	C₄	1	1	yes	characteristic
16	D₄×Dic₁₃	C₄	1	1	yes	characteristic
17	C₄×C₂²⋊D₁₃	C₄	1	1	yes	characteristic
18	C₄×C₂²⋊D₁₃	C₄	1	1	yes	characteristic
19	C₂×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
20	C₂×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
21	#88146	C₂²	1	1	yes	characteristic
22	#88141	C₄	1	1	yes	characteristic
23	#88135	C₄	1	1	yes	characteristic
24	#88135	C₄	1	1	yes	characteristic
25	#88131	C₂²	1	1	yes	characteristic
26	#88131	C₂²	1	1	yes	characteristic
27	C₄²×D₁₃	C₂²	1	1	yes	characteristic
28	C₄×Dic₂₆	C₄	1	1	yes	characteristic
29	C₂²⋊#1057	C₂²	1	2	yes	—
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	2	yes	—
36	C₄⋊#1057	C₂²	1	2	yes	—
37	C₄⋊#1057	C₂²	1	2	yes	—
38	C₄×#1057	C₂²	1	1	yes	characteristic
39	C₄×#1057	C₂²	1	1	yes	characteristic
40	C₄×#1057	C₂²	1	1	yes	characteristic
41	C₄×#1057	C₂²	1	1	yes	characteristic
42	C₄×#1057	C₂²	1	2	yes	—
43	C₂×D₄×C₁₃	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₁₃	C₂×C₄	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	C₂²×Dic₁₃	C₂×C₄	1	1	yes	characteristic
49	C₂²×Dic₁₃	C₂×C₄	1	1	yes	characteristic
50	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
51	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
52	#6129	—	2	4	no	—
53	C₂×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄×D₁₃	C₂³	1	1	yes	characteristic
56	C₂×Dic₂₆	C₂×C₄	1	1	yes	characteristic
57	#6122	—	2	4	no	—
58	C₂×#1057	—	2	4	no	—
59	C₂×#1057	C₂³	1	2	yes	—
60	C₂×#1057	C₂³	1	2	yes	—
61	C₂×#1057	C₂³	1	2	yes	—
62	C₂×#1057	C₂³	1	2	yes	—
63	C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
64	C₁₃×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
65	C₁₃×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
66	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
67	C₂²⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
68	C₂²⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
69	#6103	C₂×C₄	1	1	yes	characteristic
70	#6103	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	C₂×C₄	1	1	yes	characteristic
81	C₄×Dic₁₃	C₂³	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	#2922	—	13	1	no	maximal, Sylow
86	C₂³×C₁₃	C₂²×C₄	1	1	yes	characteristic
87	C₂³×C₁₃	C₂²×C₄	1	1	yes	characteristic
88	C₂²×D₁₃	C₂²×C₄	1	1	yes	characteristic
89	D₄×C₁₃	—	2	2	no	—
90	C₂×C₄×C₁₃	—	2	2	no	—
91	C₂×C₄×C₁₃	—	2	2	no	—
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	—	2	2	no	—
99	C₂×Dic₁₃	—	2	2	no	—
100	C₂×Dic₁₃	—	2	2	no	—
101	C₂×Dic₁₃	—	2	2	no	—
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	C₂×Dic₁₃	#41	1	4	yes	—
109	C₂×Dic₁₃	C₂⁴	1	1	yes	characteristic
110	C₄×D₁₃	—	2	2	no	—
111	C₄×D₁₃	—	2	2	no	—
112	C₄×D₁₃	—	2	2	no	—
113	Dic₂₆	—	2	2	no	—
114	#1057	—	2	2	no	—
115	#1057	—	2	2	no	—
116	#1057	—	2	2	no	—
117	#1057	—	2	2	no	—
118	C₄×#41	—	13	1	no	—
119	#433	—	13	2	no	—
120	#433	—	13	2	no	—
121	#431	—	13	1	no	—
122	#431	—	13	1	no	—
123	#431	—	13	1	no	—
124	#431	—	13	1	no	—
125	#424	—	13	2	no	—
126	#408	—	13	2	no	—
127	C₄×M₁₆	—	13	1	no	—
128	C₂×C₄×C₈	—	13	1	no	—
129	C₂²×C₁₃	—	2	1	no	—
130	C₂²×C₁₃	—	2	1	no	—
131	C₂²×C₁₃	—	2	1	no	—
132	C₂²×C₁₃	—	2	1	no	—
133	C₂²×C₁₃	—	2	1	no	—
134	C₂²×C₁₃	#117	1	2	yes	—
135	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
136	C₂×D₁₃	—	2	1	no	—
137	C₂×D₁₃	—	2	1	no	—
138	C₂×D₁₃	—	2	1	no	—
139	C₄×C₁₃	—	2	1	no	—
140	C₄×C₁₃	—	2	1	no	—
141	C₄×C₁₃	—	2	1	no	—
142	C₄×C₁₃	—	2	1	no	—
143	C₄×C₁₃	#131	1	2	yes	—
144	Dic₁₃	—	2	1	no	—
145	Dic₁₃	—	2	1	no	—
146	Dic₁₃	—	2	1	no	—
147	Dic₁₃	—	2	1	no	—
148	Dic₁₃	#117	1	2	yes	—
149	Dic₁₃	#131	1	2	yes	—
150	Dic₁₃	C₂×#41	1	2	yes	—
151	C₂×#41	—	13	1	no	—
152	C₂×M₁₆	—	13	2	no	—
153	C₂²×C₈	—	13	2	no	—
154	C₄×Q₈	—	13	1	no	—
155	C₄×Q₈	—	13	1	no	—
156	C₄×D₄	—	13	1	no	—
157	C₄×D₄	—	13	1	no	—
158	C₄×D₄	—	13	1	no	—
159	C₄×D₄	—	13	1	no	—
160	C₄×D₄	—	13	1	no	—
161	C₄×D₄	#261	1	1	yes	characteristic
162	#117	—	13	1	no	—
163	#117	—	13	1	no	—
164	#117	—	13	1	no	—
165	C₂×C₄²	—	13	1	no	—
166	C₂×C₄²	—	13	1	no	—
167	C₂×C₄²	—	13	1	no	—
168	C₄⋊C₈	—	13	2	no	—
169	C₄⋊C₈	—	13	2	no	—
170	C₄⋊C₈	—	13	2	no	—
171	C₄⋊C₈	—	13	2	no	—
172	C₂²⋊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	C₄×C₈	—	13	1	no	—
178	C₄×C₈	—	13	1	no	—
179	C₄×C₈	—	13	2	no	—
180	C₄×C₈	—	13	1	no	—
181	C₄×C₈	—	13	1	no	—
182	C₂×C₁₃	—	2	2	no	—
183	C₂×C₁₃	—	4	1	no	—
184	C₂×C₁₃	C₂×#117	1	1	yes	characteristic
185	C₂×C₁₃	C₂×#131	1	1	yes	characteristic
186	C₂×C₁₃	C₂×#131	1	1	yes	characteristic
187	D₁₃	—	4	1	no	—
188	#41	—	26	4	no	—
189	C₂×Q₈	—	13	1	no	—
190	C₂×D₄	—	13	1	no	—
191	C₂×D₄	—	13	1	no	—
192	C₂×D₄	C₂×#261	1	1	yes	characteristic
193	C₂²×C₄	—	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	C₂×#261	1	1	yes	characteristic
201	C₂²×C₄	C₂×#261	1	1	yes	characteristic
202	M₁₆	—	26	4	no	—
203	C₂×C₈	—	13	2	no	—
204	C₂×C₈	—	13	2	no	—
205	C₂×C₈	—	13	2	no	—
206	C₂×C₈	—	13	2	no	—
207	C₂×C₈	—	26	4	no	—
208	C₄⋊C₄	—	13	1	no	—
209	C₄⋊C₄	—	13	1	no	—
210	C₄⋊C₄	—	13	1	no	—
211	C₄⋊C₄	—	13	1	no	—
212	C₄⋊C₄	—	13	1	no	—
213	C₄⋊C₄	C₂×#261	1	1	yes	characteristic
214	C₂²⋊C₄	—	13	1	no	—
215	C₂²⋊C₄	—	13	1	no	—
216	C₂²⋊C₄	—	13	1	no	—
217	C₂²⋊C₄	—	13	1	no	—
218	C₂²⋊C₄	C₂×#261	1	1	yes	characteristic
219	C₂²⋊C₄	C₂×#261	1	1	yes	characteristic
220	C₄²	—	13	1	no	—
221	C₄²	—	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	—	13	1	no	—
229	C₄²	C₂×#261	1	1	yes	characteristic
230	C₁₃	#2922	1	1	yes	characteristic, complemented, Sylow
231	C₂³	—	13	1	no	—
232	C₂³	C₂²×#261	1	1	yes	characteristic
233	C₂³	C₂²×#261	1	1	yes	characteristic
234	Q₈	—	26	2	no	—
235	D₄	—	2	2	no	—
236	D₄	—	26	2	no	—
237	D₄	—	26	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	13	4	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	—	13	1	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	—	26	2	no	—
253	C₂×C₄	—	26	2	no	—
254	C₂×C₄	C₂²×#261	1	1	yes	characteristic
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	C₈	—	26	2	no	—
260	C₈	—	26	2	no	—
261	C₈	—	26	2	no	—
262	C₈	—	26	2	no	—
263	C₂²	—	2	1	no	—
264	C₂²	—	2	1	no	—
265	C₂²	—	2	1	no	—
266	C₂²	—	2	1	no	—
267	C₂²	—	2	1	no	—
268	C₂²	—	26	1	no	—
269	C₂²	—	26	1	no	—
270	C₂²	—	26	1	no	—
271	C₂²	#88237	1	2	yes	—
272	C₂²	C₂³×#261	1	1	yes	characteristic
273	C₄	—	2	1	no	—
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	13	2	no	—
278	C₄	—	13	2	no	—
279	C₄	—	13	2	no	—
280	C₄	—	26	1	no	—
281	C₄	—	26	1	no	—
282	C₄	—	26	1	no	—
283	C₄	—	26	1	no	—
284	C₄	#88238	1	2	yes	—
285	C₂	—	2	2	no	—
286	C₂	—	4	1	no	—
287	C₂	—	52	1	no	—
288	C₂	C₂×#88237	1	1	yes	characteristic
289	C₂	C₂×#88238	1	1	yes	characteristic
290	C₂	D₁₃⋊#131	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×#261
commutator, socle	C₂²×C₁₃	C₂³×C₄
Frattini	C₂²	C₂³×#261
Fitting	C₄×D₄×C₁₃	C₄
prime radical	C₂×#6129	C₂²
cosocle	C₂×Dic₁₃	C₂⁴
hypercenter	C₄×D₄	#261
hypocenter	C₁₃	#2922
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×D₄

Complements

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

N	H ≅ G/N	type
3. #11719125	C₂	semidirect
4. C₄×#6122	C₂	semidirect
5. #11719119	C₂	semidirect
6. C₂×C₄×#1057	C₂	semidirect
7. #11719077	C₂	semidirect
8. #11719068	C₂	semidirect
9. #11719067	C₂	semidirect
10. #11719051	C₂	semidirect
11. #11719049	C₂	semidirect
230. C₁₃	#2922	semidirect

Maximal subgroups

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

subgroup	index
C₄×#6129	2
#11719146	2
#11719125	2
C₄×#6122	2
#11719119	2
C₂×C₄×#1057	2
#11719077	2
#11719068	2
#11719067	2
#11719051	2
#11719049	2
#2922	13