Subgroups — #13123564

Order 1664.

502 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#88161	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×M₁₆×D₁₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#11719229	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#405	C₂	1	1	yes	maximal, characteristic, complemented
5	#11718771	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#88060	C₂	1	1	yes	maximal, characteristic, complemented
7	#11718660	C₂	1	1	yes	maximal, characteristic, complemented
8	#11718367	C₂	1	2	yes	maximal, complemented
9	#11718367	C₂	1	2	yes	maximal, complemented
10	#11718349	C₂	1	2	yes	maximal, complemented
11	C₈×D₅₂	C₂	1	2	yes	maximal, complemented
12	C₂×#6127	C₂²	1	1	yes	characteristic
13	C₂²×C₄×D₁₃	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
15	C₂²×C₈×C₁₃	C₂²	1	1	yes	characteristic
16	C₁₃×#117	C₂²	1	1	yes	characteristic
17	C₂×#6099	C₂²	1	1	yes	characteristic
18	C₂²×#1055	C₂²	1	1	yes	characteristic
19	#88161	—	2	4	no	—
20	M₁₆×D₁₃	—	2	4	no	—
21	C₂×#6094	C₂²	1	2	yes	—
22	C₂×#6094	C₂²	1	2	yes	complemented
23	C₂×C₈×D₁₃	C₂²	1	2	yes	—
24	C₂×C₈×D₁₃	C₂²	1	2	yes	complemented
25	#88151	C₄	1	1	yes	characteristic
26	#88151	C₄	1	1	yes	characteristic
27	#88149	C₄	1	1	yes	characteristic
28	#88149	C₄	1	1	yes	characteristic
29	#88140	C₄	1	2	yes	—
30	#88136	C₄	1	2	yes	—
31	C₄×D₅₂	C₂²	1	2	yes	—
32	C₄×D₅₂	C₂²	1	2	yes	—
33	C₁₃×#97	C₂²	1	2	yes	complemented
34	C₄×C₈×C₁₃	C₂²	1	2	yes	complemented
35	#88060	C₂²	1	1	yes	characteristic
36	#88060	C₂²	1	1	yes	characteristic
37	#88060	C₂²	1	1	yes	characteristic
38	#88060	C₂²	1	1	yes	characteristic
39	#88060	C₂²	1	2	yes	complemented
40	#88060	C₂²	1	2	yes	complemented
41	C₄⋊#1055	C₂²	1	2	yes	complemented
42	C₄⋊#1055	C₂²	1	2	yes	complemented
43	C₂³×D₁₃	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
45	C₂×C₂²⋊D₁₃	C₂×C₄	1	2	yes	—
46	C₂²×Dic₁₃	C₂×C₄	1	1	yes	characteristic
47	#6127	—	2	2	no	—
48	#6127	—	2	2	no	—
49	C₂×D₅₂	C₂×C₄	1	2	yes	—
50	C₂×D₅₂	C₂×C₄	1	1	yes	characteristic
51	C₂×D₅₂	C₂×C₄	1	2	yes	—
52	C₂×C₄×D₁₃	—	2	2	no	—
53	C₂×C₄×D₁₃	—	2	2	no	—
54	C₂×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄×D₁₃	C₂³	1	1	yes	characteristic
57	C₂×C₄×D₁₃	C₂³	1	2	yes	—
58	C₂×C₄×D₁₃	C₂³	1	1	yes	characteristic
59	C₂×Dic₂₆	C₂×C₄	1	1	yes	characteristic
60	C₁₃×M₁₆	D₄	1	4	yes	complemented
61	C₂×C₈×C₁₃	—	2	2	no	—
62	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
63	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
64	C₂×C₈×C₁₃	C₂³	1	2	yes	—
65	C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
66	C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
67	C₁₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
68	C₄²×C₁₃	C₂³	1	2	yes	—
69	#6103	C₂×C₄	1	2	yes	—
70	#6103	C₂×C₄	1	2	yes	—
71	#6103	C₂×C₄	1	2	yes	—
72	#6103	C₂×C₄	1	2	yes	—
73	C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
74	C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
75	#6099	—	2	2	no	—
76	C₂×#1055	—	2	2	no	—
77	C₂×#1055	C₂³	1	2	yes	—
78	C₂×#1055	C₂³	1	1	yes	characteristic
79	C₂×#1055	C₂³	1	1	yes	characteristic
80	#6094	—	2	4	no	—
81	#6094	—	2	4	no	—
82	C₈×D₁₃	—	2	4	no	—
83	C₈×D₁₃	—	2	4	no	—
84	#2933	—	13	1	no	maximal, Sylow
85	C₂³×C₁₃	C₂²×C₄	1	1	yes	characteristic
86	C₂²×D₁₃	—	2	2	no	—
87	C₂²×D₁₃	—	2	2	no	—
88	C₂²×D₁₃	—	4	1	no	—
89	C₂²×D₁₃	C₂²×C₄	1	1	yes	characteristic
90	C₂²×D₁₃	C₂²×C₄	1	2	yes	—
91	C₂²×D₁₃	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
93	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
94	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
96	C₂×C₄×C₁₃	#41	1	2	yes	—
97	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
98	C₂²⋊D₁₃	—	2	4	no	—
99	C₂×Dic₁₃	—	2	2	no	—
100	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
101	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
102	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
103	D₅₂	—	2	4	no	—
104	D₅₂	—	2	2	no	—
105	D₅₂	—	4	2	no	—
106	C₄×D₁₃	—	2	2	no	—
107	C₄×D₁₃	—	2	2	no	—
108	C₄×D₁₃	—	2	1	no	—
109	C₄×D₁₃	—	2	2	no	—
110	C₄×D₁₃	—	2	1	no	—
111	C₄×D₁₃	—	2	1	no	—
112	C₄×D₁₃	—	2	1	no	—
113	C₄×D₁₃	—	2	2	no	—
114	Dic₂₆	—	2	2	no	—
115	C₈×C₁₃	—	2	1	no	—
116	C₈×C₁₃	—	2	1	no	—
117	C₈×C₁₃	C₂×D₄	1	4	yes	—
118	#1055	—	2	1	no	—
119	#1055	—	2	2	no	—
120	#1055	—	2	1	no	—
121	C₂×#131	—	13	1	no	—
122	C₂²×M₁₆	—	13	1	no	—
123	#525	—	13	1	no	—
124	#436	—	13	2	no	—
125	#435	—	13	2	no	—
126	#435	—	13	2	no	—
127	C₈×D₄	—	13	2	no	—
128	#424	—	13	1	no	—
129	#408	—	13	1	no	—
130	C₂×C₂²⋊C₈	—	13	1	no	—
131	#405	D₁₃	1	1	yes	characteristic, complemented
132	C₂²×C₁₃	—	2	1	no	—
133	C₂²×C₁₃	—	2	1	no	—
134	C₂²×C₁₃	C₄×D₄	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	C₂×D₁₃	—	2	2	no	—
140	C₂×D₁₃	—	2	2	no	—
141	C₂×D₁₃	—	2	1	no	—
142	C₂×D₁₃	—	2	1	no	—
143	C₂×D₁₃	—	2	2	no	—
144	C₂×D₁₃	—	4	4	no	—
145	C₂×D₁₃	#131	1	2	yes	complemented
146	C₄×C₁₃	—	2	2	no	—
147	C₄×C₁₃	—	2	2	no	—
148	C₄×C₁₃	C₄×D₄	1	2	yes	—
149	C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
150	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
151	Dic₁₃	—	2	2	no	—
152	Dic₁₃	—	2	1	no	—
153	Dic₁₃	—	2	1	no	—
154	C₂×#41	—	13	1	no	—
155	C₂³×C₄	—	13	1	no	—
156	#131	—	26	4	no	—
157	C₂×M₁₆	—	13	1	no	—
158	C₂×M₁₆	—	13	2	no	—
159	C₂×M₁₆	—	13	2	no	—
160	C₂×M₁₆	—	26	4	no	—
161	C₂×M₁₆	C₂×D₁₃	1	1	yes	characteristic
162	C₂²×C₈	—	13	1	no	—
163	C₂²×C₈	—	13	2	no	—
164	C₂²×C₈	—	13	2	no	—
165	C₂²×C₈	C₂×D₁₃	1	1	yes	characteristic
166	#123	—	13	2	no	—
167	C₂²⋊Q₈	—	13	1	no	—
168	C₂²⋊Q₈	—	13	1	no	—
169	#121	—	13	1	no	—
170	#121	—	13	1	no	—
171	#120	—	13	2	no	—
172	C₄×D₄	—	13	2	no	—
173	C₄×D₄	—	13	2	no	—
174	#117	C₂×D₁₃	1	1	yes	characteristic
175	C₄⋊C₈	—	13	2	no	—
176	C₄⋊C₈	—	13	2	no	—
177	C₂²⋊C₈	—	13	1	no	—
178	C₂²⋊C₈	—	13	1	no	—
179	C₂²⋊C₈	—	13	1	no	—
180	C₂²⋊C₈	—	13	1	no	—
181	C₂²⋊C₈	—	13	2	no	—
182	C₂²⋊C₈	—	13	2	no	—
183	#97	C₂×D₁₃	1	2	yes	complemented
184	C₄×C₈	C₂×D₁₃	1	2	yes	complemented
185	C₂×C₁₃	—	2	2	no	—
186	C₂×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
187	C₂×C₁₃	C₂×#131	1	1	yes	characteristic
188	C₂×C₁₃	#568	1	1	yes	characteristic
189	D₁₃	—	2	2	no	—
190	D₁₃	—	4	2	no	—
191	D₁₃	—	4	1	no	—
192	C₂⁴	—	13	1	no	—
193	#41	—	26	2	no	—
194	#41	—	26	2	no	—
195	C₂×Q₈	—	13	1	no	—
196	C₂×D₄	—	13	2	no	—
197	C₂×D₄	—	13	2	no	—
198	C₂×D₄	—	13	1	no	—
199	C₂×D₄	—	13	2	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	2	no	—
205	C₂²×C₄	—	13	1	no	—
206	C₂²×C₄	—	26	2	no	—
207	C₂²×C₄	—	26	2	no	—
208	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
209	M₁₆	—	26	2	no	—
210	M₁₆	—	26	4	no	—
211	M₁₆	—	26	4	no	—
212	M₁₆	D₅₂	1	4	yes	complemented
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	13	1	no	—
215	C₂×C₈	—	13	1	no	—
216	C₂×C₈	—	13	2	no	—
217	C₂×C₈	—	26	2	no	—
218	C₂×C₈	—	26	4	no	—
219	C₂×C₈	—	26	4	no	—
220	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
221	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
222	C₂×C₈	C₂²×D₁₃	1	2	yes	—
223	C₄⋊C₄	—	13	2	no	—
224	C₄⋊C₄	—	13	2	no	—
225	C₄⋊C₄	C₄×D₁₃	1	1	yes	characteristic
226	C₄⋊C₄	C₄×D₁₃	1	1	yes	characteristic
227	C₂²⋊C₄	—	13	2	no	—
228	C₂²⋊C₄	—	13	2	no	—
229	C₂²⋊C₄	—	13	2	no	—
230	C₂²⋊C₄	—	13	2	no	—
231	C₂²⋊C₄	C₄×D₁₃	1	2	yes	—
232	C₄²	C₂²×D₁₃	1	2	yes	—
233	C₁₃	#2933	1	1	yes	characteristic, complemented, Sylow
234	C₂³	—	13	1	no	—
235	C₂³	—	13	1	no	—
236	C₂³	—	13	2	no	—
237	C₂³	—	26	2	no	—
238	C₂³	—	26	2	no	—
239	C₂³	—	52	1	no	—
240	C₂³	C₂×C₄×D₁₃	1	1	yes	characteristic
241	Q₈	—	26	2	no	—
242	D₄	—	26	4	no	—
243	D₄	—	26	4	no	—
244	D₄	—	26	2	no	—
245	D₄	—	52	2	no	—
246	C₂×C₄	—	13	1	no	—
247	C₂×C₄	—	13	1	no	—
248	C₂×C₄	—	13	2	no	—
249	C₂×C₄	—	26	2	no	—
250	C₂×C₄	—	26	2	no	—
251	C₂×C₄	—	26	2	no	—
252	C₂×C₄	—	26	1	no	—
253	C₂×C₄	—	26	2	no	—
254	C₂×C₄	—	26	1	no	—
255	C₂×C₄	—	26	1	no	—
256	C₂×C₄	—	26	1	no	—
257	C₂×C₄	—	26	2	no	—
258	C₂×C₄	C₂×C₄×D₁₃	1	2	yes	—
259	C₂×C₄	C₂×C₄×D₁₃	1	2	yes	—
260	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
261	C₂×C₄	C₂×D₅₂	1	2	yes	—
262	C₂×C₄	#6127	1	2	yes	—
263	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
264	C₈	—	2	1	no	—
265	C₈	—	2	1	no	—
266	C₈	—	26	1	no	—
267	C₈	—	26	2	no	—
268	C₈	—	26	1	no	—
269	C₈	C₂×D₅₂	1	4	yes	—
270	C₂²	—	2	1	no	—
271	C₂²	—	2	1	no	—
272	C₂²	—	13	2	no	—
273	C₂²	—	26	1	no	—
274	C₂²	—	26	1	no	—
275	C₂²	—	26	1	no	—
276	C₂²	—	26	2	no	—
277	C₂²	—	26	2	no	—
278	C₂²	—	26	1	no	—
279	C₂²	—	26	1	no	—
280	C₂²	—	26	2	no	—
281	C₂²	—	52	4	no	—
282	C₂²	C₄×D₅₂	1	2	yes	—
283	C₂²	C₂²×C₄×D₁₃	1	1	yes	characteristic
284	C₄	—	2	2	no	—
285	C₄	—	2	2	no	—
286	C₄	—	26	1	no	—
287	C₄	—	26	1	no	—
288	C₄	—	26	2	no	—
289	C₄	C₄×D₅₂	1	2	yes	—
290	C₄	C₂²×D₅₂	1	1	yes	characteristic
291	C₄	C₂×#6127	1	1	yes	characteristic
292	C₂	—	2	2	no	—
293	C₂	—	26	2	no	—
294	C₂	—	52	1	no	—
295	C₂	—	52	2	no	—
296	C₂	C₂×C₄×D₅₂	1	1	yes	characteristic
297	C₂	D₁₃×#131	1	1	yes	characteristic
298	C₂	#11719455	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₃
commutator, socle	C₂²×C₁₃	C₂³×C₄
Fitting	C₁₃×#405	C₂
prime radical	#11719229	C₂
cosocle	C₂×C₄×C₁₃	C₂⁴
hypercenter	#405	D₁₃
hypocenter	C₁₃	#2933
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #405

Complements

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

N	H ≅ G/N	type
1. C₂×#88161	C₂	semidirect
2. C₂×M₁₆×D₁₃	C₂	semidirect
4. C₁₃×#405	C₂	semidirect
5. #11718771	C₂	semidirect
6. C₂×#88060	C₂	semidirect
7. #11718660	C₂	semidirect
8. #11718367	C₂	semidirect
9. #11718367	C₂	semidirect
10. #11718349	C₂	semidirect
11. C₈×D₅₂	C₂	semidirect
22. C₂×#6094	C₂²	semidirect
24. C₂×C₈×D₁₃	C₂²	semidirect
33. C₁₃×#97	C₂²	semidirect
34. C₄×C₈×C₁₃	C₂²	semidirect
39. #88060	C₂²	semidirect
40. #88060	C₂²	semidirect
41. C₄⋊#1055	C₂²	semidirect
42. C₄⋊#1055	C₂²	semidirect
60. C₁₃×M₁₆	D₄	semidirect
131. #405	D₁₃	semidirect
145. C₂×D₁₃	#131	semidirect
183. #97	C₂×D₁₃	semidirect
184. C₄×C₈	C₂×D₁₃	semidirect
212. M₁₆	D₅₂	semidirect
233. C₁₃	#2933	semidirect

Maximal subgroups

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

subgroup	index
C₂×#88161	2
C₂×M₁₆×D₁₃	2
#11719229	2
C₁₃×#405	2
#11718771	2
C₂×#88060	2
#11718660	2
#11718367	2
#11718349	2
C₈×D₅₂	2
#2933	13