Subgroups — #13131695

Order 1664.

464 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#88135	C₂	1	1	yes	maximal, characteristic
2	#11719169	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂²⋊#1057	C₂	1	1	yes	maximal, characteristic, complemented
4	#11719128	C₂	1	1	yes	maximal, characteristic, complemented
5	#11719127	C₂	1	1	yes	maximal, characteristic, complemented
6	D₁₃⋊(C₂²⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
7	#11719119	C₂	1	1	yes	maximal, characteristic, complemented
8	#11719066	C₂	1	2	yes	maximal, complemented
9	#11719065	C₂	1	2	yes	maximal, complemented
10	#11719064	C₂	1	2	yes	maximal, complemented
11	#11719063	C₂	1	2	yes	maximal, complemented
12	C₂²×C₂²⋊D₁₃	C₄	1	1	yes	characteristic
13	C₂³×Dic₁₃	C₂²	1	1	yes	characteristic
14	C₂²×C₄×D₁₃	C₂²	1	1	yes	characteristic
15	C₂×#6122	C₂²	1	2	yes	complemented
16	C₂²×#1057	C₂²	1	2	yes	complemented
17	C₂×#6118	C₂²	1	2	yes	complemented
18	C₂×D₁₃⋊C₈	C₂²	1	2	yes	complemented
19	C₂×C₁₃×C₂²⋊C₄	C₄	1	1	yes	characteristic
20	C₂×#6103	C₄	1	1	yes	characteristic
21	C₂×#6101	C₄	1	1	yes	characteristic
22	C₂×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
23	#88135	C₄	1	1	yes	characteristic
24	#88135	C₄	1	1	yes	characteristic
25	#88135	C₄	1	1	yes	characteristic
26	#88135	C₄	1	1	yes	characteristic
27	#88135	C₂²	1	1	yes	characteristic
28	#88135	C₂²	1	1	yes	characteristic
29	#88135	C₂²	1	1	yes	characteristic
30	#88135	C₂²	1	1	yes	characteristic
31	C₂²⋊#1057	C₂²	1	1	yes	characteristic
32	C₂²⋊#1057	C₂²	1	1	yes	characteristic
33	C₂²⋊#1057	C₂²	1	1	yes	characteristic
34	C₂²⋊#1057	C₂²	1	1	yes	characteristic
35	C₂²⋊#1057	C₂²	1	2	yes	complemented
36	C₂²⋊#1057	C₂²	1	2	yes	complemented
37	#88112	C₂²	1	2	yes	complemented
38	#88111	C₂²	1	2	yes	complemented
39	#88111	C₂²	1	2	yes	complemented
40	#88110	C₂²	1	2	yes	complemented
41	C₄⋊#1057	C₂²	1	2	yes	complemented
42	C₄×#1057	C₂²	1	2	yes	complemented
43	C₂⁴×C₁₃	C₂×C₄	1	1	yes	characteristic
44	C₂³×D₁₃	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
47	C₂×C₂²⋊D₁₃	—	2	4	no	—
48	C₂×C₂²⋊D₁₃	C₂×C₄	1	1	yes	characteristic
49	C₂×C₂²⋊D₁₃	C₂×C₄	1	1	yes	characteristic
50	C₂×C₂²⋊D₁₃	C₂×C₄	1	1	yes	characteristic
51	C₂×C₂²⋊D₁₃	C₂×C₄	1	1	yes	characteristic
52	C₂²×Dic₁₃	—	4	2	no	—
53	C₂²×Dic₁₃	C₂×C₄	1	1	yes	characteristic
54	C₂²×Dic₁₃	C₂×C₄	1	1	yes	characteristic
55	C₂²×Dic₁₃	C₂×C₄	1	1	yes	characteristic
56	C₂²×Dic₁₃	C₂×C₄	1	1	yes	characteristic
57	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
58	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
59	C₂²×Dic₁₃	C₂³	1	1	yes	characteristic
60	C₂×C₄×D₁₃	—	4	2	no	—
61	C₂×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄×D₁₃	C₂³	1	1	yes	characteristic
64	C₂×C₄×D₁₃	C₂³	1	1	yes	characteristic
65	#6122	—	4	2	no	—
66	C₂×#1057	—	4	2	no	—
67	C₂×#1057	C₂³	1	2	yes	—
68	C₂×#1057	C₂³	1	2	yes	—
69	C₂×#1057	C₂³	1	2	yes	—
70	C₂×#1057	C₂³	1	2	yes	—
71	#6118	—	4	2	no	—
72	D₁₃⋊C₈	—	4	2	no	—
73	C₁₃×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
74	C₁₃×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
75	C₁₃×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
76	C₁₃×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
77	#6103	C₂×C₄	1	1	yes	characteristic
78	#6103	C₂×C₄	1	1	yes	characteristic
79	#6103	C₂×C₄	1	1	yes	characteristic
80	#6103	C₂×C₄	1	1	yes	characteristic
81	#6101	C₂×C₄	1	1	yes	characteristic
82	#6101	C₂×C₄	1	1	yes	characteristic
83	#6101	C₂×C₄	1	1	yes	characteristic
84	#6101	C₂×C₄	1	1	yes	characteristic
85	C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
86	C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
87	C₄×Dic₁₃	C₂³	1	1	yes	characteristic
88	C₄×Dic₁₃	C₂³	1	1	yes	characteristic
89	#2974	—	13	1	no	maximal, Sylow
90	C₂³×C₁₃	—	2	2	no	—
91	C₂³×C₁₃	—	2	2	no	—
92	C₂³×C₁₃	—	4	1	no	—
93	C₂³×C₁₃	C₂²×C₄	1	1	yes	characteristic
94	C₂³×C₁₃	C₂²×C₄	1	1	yes	characteristic
95	C₂³×C₁₃	C₂²×C₄	1	1	yes	characteristic
96	C₂²×D₁₃	—	2	2	no	—
97	C₂²×D₁₃	—	2	2	no	—
98	C₂²×D₁₃	—	4	1	no	—
99	C₂²×D₁₃	C₂²×C₄	1	1	yes	characteristic
100	C₂²×D₁₃	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₄×C₁₃	—	2	2	no	—
102	C₂×C₄×C₁₃	—	4	1	no	—
103	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
107	C₂²⋊D₁₃	—	2	2	no	—
108	C₂²⋊D₁₃	—	2	2	no	—
109	C₂²⋊D₁₃	—	2	2	no	—
110	C₂²⋊D₁₃	M₁₆	1	4	yes	complemented
111	C₂×Dic₁₃	—	2	2	no	—
112	C₂×Dic₁₃	—	2	2	no	—
113	C₂×Dic₁₃	—	2	2	no	—
114	C₂×Dic₁₃	—	2	2	no	—
115	C₂×Dic₁₃	—	4	1	no	—
116	C₂×Dic₁₃	—	4	1	no	—
117	C₂×Dic₁₃	—	4	1	no	—
118	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
119	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
120	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
121	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
122	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
123	C₂×Dic₁₃	C₂⁴	1	1	yes	characteristic
124	C₄×D₁₃	—	2	2	no	—
125	C₄×D₁₃	—	2	2	no	—
126	C₄×D₁₃	—	4	1	no	—
127	C₄×D₁₃	—	4	1	no	—
128	#1057	—	2	2	no	—
129	#1057	—	2	2	no	—
130	#1057	—	2	2	no	—
131	#1057	—	2	2	no	—
132	C₂×C₄×D₄	—	13	1	no	—
133	#436	—	13	2	no	—
134	#435	—	13	2	no	—
135	#435	—	13	2	no	—
136	C₈×D₄	—	13	2	no	—
137	#432	—	13	1	no	—
138	#431	—	13	1	no	—
139	#407	—	13	1	no	—
140	#407	—	13	1	no	—
141	C₂×C₂²⋊C₈	—	13	1	no	—
142	C₂×C₂²⋊C₈	—	13	1	no	—
143	C₂²×C₁₃	—	2	1	no	—
144	C₂²×C₁₃	—	2	1	no	—
145	C₂²×C₁₃	—	2	1	no	—
146	C₂²×C₁₃	—	2	1	no	—
147	C₂²×C₁₃	—	2	1	no	—
148	C₂²×C₁₃	—	2	1	no	—
149	C₂²×C₁₃	—	2	1	no	—
150	C₂²×C₁₃	—	4	4	no	—
151	C₂²×C₁₃	C₂×M₁₆	1	2	yes	—
152	C₂²×C₁₃	#131	1	2	yes	—
153	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
154	C₂×D₁₃	—	2	1	no	—
155	C₂×D₁₃	—	2	1	no	—
156	C₂×D₁₃	—	2	1	no	—
157	C₂×D₁₃	—	2	1	no	—
158	C₂×D₁₃	—	2	1	no	—
159	C₂×D₁₃	—	4	4	no	—
160	C₂×D₁₃	C₂×M₁₆	1	2	yes	—
161	C₄×C₁₃	—	2	1	no	—
162	C₄×C₁₃	—	2	1	no	—
163	C₄×C₁₃	—	2	1	no	—
164	C₄×C₁₃	—	2	1	no	—
165	Dic₁₃	—	2	1	no	—
166	Dic₁₃	—	2	1	no	—
167	Dic₁₃	—	2	1	no	—
168	Dic₁₃	—	2	1	no	—
169	Dic₁₃	C₂×M₁₆	1	2	yes	—
170	Dic₁₃	H(5,𝔽₂)	1	2	yes	—
171	C₂²×D₄	—	13	1	no	—
172	C₂³×C₄	—	13	1	no	—
173	C₂³×C₄	—	13	1	no	—
174	C₂×M₁₆	—	13	2	no	—
175	C₂×M₁₆	—	13	2	no	—
176	C₂²×C₈	—	13	2	no	—
177	C₂²×C₈	—	13	2	no	—
178	C₄×D₄	—	13	1	no	—
179	C₄×D₄	—	13	1	no	—
180	C₄×D₄	—	13	1	no	—
181	C₄×D₄	—	13	1	no	—
182	C₄×D₄	—	13	1	no	—
183	C₄×D₄	—	13	1	no	—
184	C₄×D₄	—	13	1	no	—
185	C₄×D₄	—	13	1	no	—
186	C₂×C₄⋊C₄	—	13	1	no	—
187	C₂×C₂²⋊C₄	—	13	1	no	—
188	C₂×C₂²⋊C₄	#261	1	1	yes	characteristic
189	C₂×C₄²	—	13	1	no	—
190	C₄⋊C₈	—	13	2	no	—
191	C₄⋊C₈	—	13	2	no	—
192	C₂²⋊C₈	—	13	2	no	—
193	C₂²⋊C₈	—	13	2	no	—
194	C₂²⋊C₈	—	13	1	no	—
195	C₂²⋊C₈	—	13	1	no	—
196	C₂²⋊C₈	—	13	1	no	—
197	C₂²⋊C₈	—	13	1	no	—
198	C₂²⋊C₈	—	13	2	no	—
199	C₂²⋊C₈	—	13	2	no	—
200	#97	—	13	2	no	—
201	C₄×C₈	—	13	2	no	—
202	C₂×C₁₃	—	2	2	no	—
203	C₂×C₁₃	—	2	2	no	—
204	C₂×C₁₃	—	4	1	no	—
205	C₂×C₁₃	#518	1	1	yes	characteristic
206	C₂×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
207	C₂×C₁₃	C₂×#131	1	1	yes	characteristic
208	D₁₃	—	2	2	no	—
209	D₁₃	—	4	1	no	—
210	C₂⁴	—	13	1	no	—
211	C₂⁴	C₂×#261	1	1	yes	characteristic
212	C₂×D₄	—	13	1	no	—
213	C₂×D₄	—	13	1	no	—
214	C₂×D₄	—	13	1	no	—
215	C₂×D₄	—	13	1	no	—
216	C₂×D₄	—	26	4	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	C₂²×C₄	—	13	1	no	—
223	C₂²×C₄	—	13	1	no	—
224	C₂²×C₄	—	13	1	no	—
225	C₂²×C₄	—	13	1	no	—
226	C₂²×C₄	—	13	1	no	—
227	C₂²×C₄	—	13	1	no	—
228	C₂²×C₄	—	52	2	no	—
229	C₂²×C₄	—	52	2	no	—
230	C₂²×C₄	C₂×#261	1	1	yes	characteristic
231	C₂²×C₄	C₂×#261	1	1	yes	characteristic
232	M₁₆	—	52	2	no	—
233	M₁₆	—	52	2	no	—
234	C₂×C₈	—	13	2	no	—
235	C₂×C₈	—	13	2	no	—
236	C₂×C₈	—	13	2	no	—
237	C₂×C₈	—	13	2	no	—
238	C₂×C₈	—	52	2	no	—
239	C₂×C₈	—	52	2	no	—
240	C₄⋊C₄	—	13	1	no	—
241	C₄⋊C₄	—	13	1	no	—
242	C₄⋊C₄	—	13	1	no	—
243	C₄⋊C₄	—	13	1	no	—
244	C₂²⋊C₄	—	13	1	no	—
245	C₂²⋊C₄	—	13	1	no	—
246	C₂²⋊C₄	—	13	1	no	—
247	C₂²⋊C₄	—	13	1	no	—
248	C₂²⋊C₄	C₂×#261	1	1	yes	characteristic
249	C₂²⋊C₄	C₂×#261	1	1	yes	characteristic
250	C₂²⋊C₄	C₂×#261	1	1	yes	characteristic
251	C₂²⋊C₄	C₂×#261	1	1	yes	characteristic
252	C₄²	—	13	1	no	—
253	C₄²	—	13	1	no	—
254	C₄²	—	13	1	no	—
255	C₄²	—	13	1	no	—
256	C₁₃	#2974	1	1	yes	characteristic, complemented, Sylow
257	C₂³	—	2	2	no	—
258	C₂³	—	2	2	no	—
259	C₂³	—	4	1	no	—
260	C₂³	—	13	1	no	—
261	C₂³	—	13	1	no	—
262	C₂³	—	26	2	no	—
263	C₂³	—	26	2	no	—
264	C₂³	—	52	1	no	—
265	C₂³	C₂²×#261	1	1	yes	characteristic
266	C₂³	C₂²×#261	1	1	yes	characteristic
267	C₂³	C₂²×#261	1	1	yes	characteristic
268	D₄	—	13	4	no	—
269	D₄	—	26	2	no	—
270	D₄	—	26	2	no	—
271	D₄	—	26	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	4	1	no	—
274	C₂×C₄	—	13	1	no	—
275	C₂×C₄	—	13	1	no	—
276	C₂×C₄	—	13	1	no	—
277	C₂×C₄	—	13	1	no	—
278	C₂×C₄	—	13	1	no	—
279	C₂×C₄	—	13	1	no	—
280	C₂×C₄	—	26	2	no	—
281	C₂×C₄	—	26	2	no	—
282	C₂×C₄	—	26	2	no	—
283	C₂×C₄	—	26	2	no	—
284	C₂×C₄	—	26	2	no	—
285	C₂×C₄	—	26	2	no	—
286	C₂×C₄	—	52	1	no	—
287	C₂×C₄	—	52	1	no	—
288	C₂×C₄	—	52	1	no	—
289	C₂×C₄	—	52	1	no	—
290	C₂×C₄	—	52	1	no	—
291	C₂×C₄	C₂²×#261	1	1	yes	characteristic
292	C₂×C₄	C₂²×#261	1	1	yes	characteristic
293	C₂×C₄	C₂²×#261	1	1	yes	characteristic
294	C₂×C₄	C₂²×#261	1	1	yes	characteristic
295	C₈	—	26	2	no	—
296	C₈	—	26	2	no	—
297	C₈	—	26	2	no	—
298	C₈	—	26	2	no	—
299	C₂²	—	2	1	no	—
300	C₂²	—	2	1	no	—
301	C₂²	—	2	1	no	—
302	C₂²	—	2	1	no	—
303	C₂²	—	2	1	no	—
304	C₂²	—	2	1	no	—
305	C₂²	—	2	1	no	—
306	C₂²	—	4	4	no	—
307	C₂²	—	13	2	no	—
308	C₂²	—	26	1	no	—
309	C₂²	—	26	1	no	—
310	C₂²	—	26	1	no	—
311	C₂²	—	26	1	no	—
312	C₂²	—	26	1	no	—
313	C₂²	—	52	4	no	—
314	C₂²	D₁₃⋊M₁₆	1	2	yes	—
315	C₂²	#88238	1	2	yes	—
316	C₂²	C₂³×#261	1	1	yes	characteristic
317	C₄	—	2	1	no	—
318	C₄	—	2	1	no	—
319	C₄	—	2	1	no	—
320	C₄	—	2	1	no	—
321	C₄	—	13	2	no	—
322	C₄	—	13	2	no	—
323	C₄	—	26	1	no	—
324	C₄	—	26	1	no	—
325	C₄	—	26	1	no	—
326	C₄	—	26	1	no	—
327	C₂	—	2	2	no	—
328	C₂	—	2	2	no	—
329	C₂	—	4	1	no	—
330	C₂	—	26	2	no	—
331	C₂	—	52	1	no	—
332	C₂	C₂×D₁₃⋊M₁₆	1	1	yes	characteristic
333	C₂	C₂×#88238	1	1	yes	characteristic
334	C₂	#11719623	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#261
commutator, socle	C₂²×C₁₃	C₂³×C₄
Fitting	C₂×C₁₃×C₂²⋊C₄	C₄
prime radical	C₂²×C₂²⋊D₁₃	C₄
cosocle	C₂×Dic₁₃	C₂⁴
hypercenter	C₂×C₂²⋊C₄	#261
hypocenter	C₁₃	#2974
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
2. #11719169	C₂	semidirect
3. C₂×C₂²⋊#1057	C₂	semidirect
4. #11719128	C₂	semidirect
5. #11719127	C₂	semidirect
6. D₁₃⋊(C₂²⋊C₈)	C₂	semidirect
7. #11719119	C₂	semidirect
8. #11719066	C₂	semidirect
9. #11719065	C₂	semidirect
10. #11719064	C₂	semidirect
11. #11719063	C₂	semidirect
15. C₂×#6122	C₂²	semidirect
16. C₂²×#1057	C₂²	semidirect
17. C₂×#6118	C₂²	semidirect
18. C₂×D₁₃⋊C₈	C₂²	semidirect
35. C₂²⋊#1057	C₂²	semidirect
36. C₂²⋊#1057	C₂²	semidirect
37. #88112	C₂²	semidirect
38. #88111	C₂²	semidirect
39. #88111	C₂²	semidirect
40. #88110	C₂²	semidirect
41. C₄⋊#1057	C₂²	semidirect
42. C₄×#1057	C₂²	semidirect
110. C₂²⋊D₁₃	M₁₆	semidirect
256. C₁₃	#2974	semidirect

Maximal subgroups

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

subgroup	index
C₂×#88135	2
#11719169	2
C₂×C₂²⋊#1057	2
#11719128	2
#11719127	2
D₁₃⋊(C₂²⋊C₈)	2
#11719119	2
#11719066	2
#11719065	2
#11719064	2
#11719063	2
#2974	13