Subgroups — #13131756

Order 1664.

388 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×#6129	C₂	1	1	yes	maximal, characteristic
2	#11719144	C₂	1	2	yes	maximal, complemented
3	#11719129	C₂	1	1	yes	maximal, characteristic, complemented
4	#11719128	C₂	1	1	yes	maximal, characteristic, complemented
5	#11719125	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#88112	C₂	1	1	yes	maximal, characteristic, complemented
7	#11719078	C₂	1	2	yes	maximal, complemented
8	#11719066	C₂	1	2	yes	maximal, complemented
9	#11719064	C₂	1	2	yes	maximal, complemented
10	#11719055	C₂	1	1	yes	maximal, characteristic, complemented
11	#11719053	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#6129	C₄	1	1	yes	characteristic
13	C₂×#6122	C₂²	1	2	yes	—
14	C₂²×#1057	C₂²	1	2	yes	—
15	C₂×#6118	C₂²	1	2	yes	—
16	C₄×D₄×C₁₃	C₄	1	1	yes	characteristic
17	D₄×Dic₁₃	C₂²	1	1	yes	characteristic
18	C₄×C₂²⋊D₁₃	C₄	1	1	yes	characteristic
19	C₄×C₂²⋊D₁₃	C₄	1	1	yes	characteristic
20	C₂×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
22	#88146	C₄	1	1	yes	characteristic
23	#88141	C₂²	1	1	yes	characteristic
24	#88135	C₂²	1	1	yes	characteristic
25	#88135	C₂²	1	1	yes	characteristic
26	#88131	C₄	1	1	yes	characteristic
27	#88131	C₄	1	1	yes	characteristic
28	C₄²×D₁₃	C₂²	1	1	yes	characteristic
29	C₄×Dic₂₆	C₄	1	1	yes	characteristic
30	C₂²⋊#1057	C₂²	1	2	yes	—
31	C₂²⋊#1057	C₂²	1	2	yes	—
32	#88112	C₂²	1	2	yes	—
33	#88112	C₂²	1	1	yes	characteristic
34	#88112	C₂²	1	1	yes	characteristic
35	#88112	C₂²	1	1	yes	characteristic
36	#88112	C₂²	1	1	yes	characteristic
37	#88112	C₂²	1	2	yes	—
38	#88111	C₂²	1	2	yes	—
39	#88110	C₂²	1	2	yes	—
40	#88110	C₂²	1	2	yes	—
41	C₄⋊#1057	C₂²	1	2	yes	—
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	—	4	2	no	—
58	C₂×#1057	—	4	2	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	#6118	—	4	2	no	—
64	C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
65	C₁₃×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
66	C₁₃×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
67	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
68	C₂²⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
69	C₂²⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
70	#6103	C₂×C₄	1	1	yes	characteristic
71	#6103	C₂×C₄	1	1	yes	characteristic
72	C₄⋊Dic₁₃	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	#6101	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	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	#2989	—	13	1	no	maximal, Sylow
87	C₂³×C₁₃	C₂²×C₄	1	1	yes	characteristic
88	C₂³×C₁₃	C₂²×C₄	1	1	yes	characteristic
89	C₂²×D₁₃	C₂²×C₄	1	1	yes	characteristic
90	D₄×C₁₃	—	2	2	no	—
91	C₂×C₄×C₁₃	—	4	1	no	—
92	C₂×C₄×C₁₃	—	4	1	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	C₂²⋊D₁₃	—	2	2	no	—
99	C₂²⋊D₁₃	—	2	2	no	—
100	C₂×Dic₁₃	—	2	2	no	—
101	C₂×Dic₁₃	—	2	1	no	—
102	C₂×Dic₁₃	—	2	1	no	—
103	C₂×Dic₁₃	—	2	1	no	—
104	C₂×Dic₁₃	—	2	1	no	—
105	C₂×Dic₁₃	M₁₆	1	4	yes	complemented
106	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
107	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
108	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
109	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
110	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
111	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
112	C₂×Dic₁₃	C₂⁴	1	1	yes	characteristic
113	C₄×D₁₃	—	2	2	no	—
114	C₄×D₁₃	—	2	1	no	—
115	C₄×D₁₃	—	2	1	no	—
116	C₄×D₁₃	—	4	1	no	—
117	Dic₂₆	—	2	2	no	—
118	#1057	—	2	2	no	—
119	#1057	—	2	2	no	—
120	#1057	—	2	2	no	—
121	#1057	—	2	2	no	—
122	C₄×#41	—	13	1	no	—
123	#446	—	13	2	no	—
124	#436	—	13	2	no	—
125	#435	—	13	2	no	—
126	#435	—	13	2	no	—
127	#432	—	13	1	no	—
128	#432	—	13	1	no	—
129	#431	—	13	1	no	—
130	#423	—	13	1	no	—
131	#423	—	13	1	no	—
132	C₂×C₄⋊C₈	—	13	1	no	—
133	C₂²×C₁₃	—	2	1	no	—
134	C₂²×C₁₃	—	2	1	no	—
135	C₂²×C₁₃	—	2	1	no	—
136	C₂²×C₁₃	—	2	1	no	—
137	C₂²×C₁₃	—	2	1	no	—
138	C₂²×C₁₃	C₂×M₁₆	1	2	yes	—
139	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
140	C₂×D₁₃	—	2	1	no	—
141	C₂×D₁₃	—	2	1	no	—
142	C₂×D₁₃	—	2	1	no	—
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	—	2	1	no	—
148	Dic₁₃	—	2	1	no	—
149	Dic₁₃	—	2	1	no	—
150	Dic₁₃	—	2	1	no	—
151	Dic₁₃	—	2	1	no	—
152	Dic₁₃	C₂×M₁₆	1	2	yes	—
153	Dic₁₃	C₂×M₁₆	1	2	yes	—
154	Dic₁₃	H(5,𝔽₂)	1	1	yes	characteristic
155	Dic₁₃	2₋¹⁺⁴	1	1	yes	characteristic
156	C₂×#41	—	13	1	no	—
157	C₂×M₁₆	—	13	2	no	—
158	C₂×M₁₆	—	13	2	no	—
159	C₂²×C₈	—	13	2	no	—
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	#261	1	1	yes	characteristic
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	—	13	1	no	—
174	C₄⋊C₈	—	13	2	no	—
175	C₄⋊C₈	—	13	1	no	—
176	C₄⋊C₈	—	13	1	no	—
177	C₄⋊C₈	—	13	2	no	—
178	C₄⋊C₈	—	13	1	no	—
179	C₄⋊C₈	—	13	2	no	—
180	C₄⋊C₈	—	13	1	no	—
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	—	4	1	no	—
189	C₂×C₁₃	#520	1	1	yes	characteristic
190	C₂×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
191	C₂×C₁₃	#568	1	1	yes	characteristic
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	C₂×#261	1	1	yes	characteristic
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	—	13	1	no	—
205	C₂²×C₄	C₂×#261	1	1	yes	characteristic
206	C₂²×C₄	C₂×#261	1	1	yes	characteristic
207	M₁₆	—	52	2	no	—
208	M₁₆	—	52	2	no	—
209	C₂×C₈	—	13	2	no	—
210	C₂×C₈	—	13	2	no	—
211	C₂×C₈	—	13	2	no	—
212	C₂×C₈	—	13	2	no	—
213	C₂×C₈	—	52	2	no	—
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	C₂×#261	1	1	yes	characteristic
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	C₂×#261	1	1	yes	characteristic
225	C₂²⋊C₄	C₂×#261	1	1	yes	characteristic
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	—	13	1	no	—
233	C₄²	—	13	1	no	—
234	C₄²	—	13	1	no	—
235	C₄²	C₂×#261	1	1	yes	characteristic
236	C₁₃	#2989	1	1	yes	characteristic, complemented, Sylow
237	C₂³	—	13	1	no	—
238	C₂³	C₂²×#261	1	1	yes	characteristic
239	C₂³	C₂²×#261	1	1	yes	characteristic
240	Q₈	—	26	2	no	—
241	D₄	—	2	2	no	—
242	D₄	—	26	2	no	—
243	D₄	—	26	2	no	—
244	C₂×C₄	—	4	1	no	—
245	C₂×C₄	—	4	1	no	—
246	C₂×C₄	—	13	4	no	—
247	C₂×C₄	—	13	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	1	no	—
255	C₂×C₄	—	26	1	no	—
256	C₂×C₄	—	26	2	no	—
257	C₂×C₄	—	26	1	no	—
258	C₂×C₄	—	26	1	no	—
259	C₂×C₄	—	26	1	no	—
260	C₂×C₄	—	26	1	no	—
261	C₂×C₄	—	26	2	no	—
262	C₂×C₄	—	52	1	no	—
263	C₂×C₄	C₂²×#261	1	1	yes	characteristic
264	C₂×C₄	C₂²×#261	1	1	yes	characteristic
265	C₂×C₄	C₂²×#261	1	1	yes	characteristic
266	C₂×C₄	C₂²×#261	1	1	yes	characteristic
267	C₂×C₄	C₂²×#261	1	1	yes	characteristic
268	C₈	—	26	2	no	—
269	C₈	—	26	2	no	—
270	C₈	—	26	2	no	—
271	C₈	—	26	2	no	—
272	C₂²	—	2	1	no	—
273	C₂²	—	2	1	no	—
274	C₂²	—	2	1	no	—
275	C₂²	—	2	1	no	—
276	C₂²	—	2	1	no	—
277	C₂²	—	26	1	no	—
278	C₂²	—	26	1	no	—
279	C₂²	—	26	1	no	—
280	C₂²	D₁₃⋊M₁₆	1	2	yes	—
281	C₂²	C₂³×#261	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	—	2	1	no	—
287	C₄	—	13	2	no	—
288	C₄	—	13	2	no	—
289	C₄	—	13	1	no	—
290	C₄	—	13	1	no	—
291	C₄	—	26	1	no	—
292	C₄	—	26	1	no	—
293	C₄	—	26	1	no	—
294	C₄	—	26	1	no	—
295	C₂	—	2	2	no	—
296	C₂	—	4	1	no	—
297	C₂	—	52	1	no	—
298	C₂	C₂×D₁₃⋊M₁₆	1	1	yes	characteristic
299	C₂	#11719621	1	1	yes	characteristic
300	C₂	#11719631	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#261
commutator, socle	C₂²×C₁₃	C₂³×C₄
Fitting	C₄×D₄×C₁₃	C₄
prime radical	C₂×#6129	C₄
cosocle	C₂×Dic₁₃	C₂⁴
hypercenter	C₄×D₄	#261
hypocenter	C₁₃	#2989
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄×D₄

Complements

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

N	H ≅ G/N	type
2. #11719144	C₂	semidirect
3. #11719129	C₂	semidirect
4. #11719128	C₂	semidirect
5. #11719125	C₂	semidirect
6. C₂×#88112	C₂	semidirect
7. #11719078	C₂	semidirect
8. #11719066	C₂	semidirect
9. #11719064	C₂	semidirect
10. #11719055	C₂	semidirect
11. #11719053	C₂	semidirect
105. C₂×Dic₁₃	M₁₆	semidirect
236. C₁₃	#2989	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
#11719144	2
#11719129	2
#11719128	2
#11719125	2
C₂×#88112	2
#11719078	2
#11719066	2
#11719064	2
#11719055	2
#11719053	2
#2989	13