Subgroups — #13127783

Order 1664.

334 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11719284	C₂	1	1	yes	maximal, characteristic
2	#11719272	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#446	C₂	1	1	yes	maximal, characteristic, complemented
4	#11718512	C₂	1	1	yes	maximal, characteristic, complemented
5	#11718508	C₂	1	1	yes	maximal, characteristic
6	#11718508	C₂	1	1	yes	maximal, characteristic
7	#11718503	C₂	1	1	yes	maximal, characteristic
8	#11718490	C₂	1	1	yes	maximal, characteristic, complemented
9	#11718490	C₂	1	1	yes	maximal, characteristic, complemented
10	#11718371	C₂	1	1	yes	maximal, characteristic
11	#11718371	C₂	1	1	yes	maximal, characteristic
12	#11718363	C₂	1	1	yes	maximal, characteristic, complemented
13	#11718363	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×#6095	C₂	1	1	yes	maximal, characteristic
15	#11718344	C₂	1	1	yes	maximal, characteristic, complemented
16	C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
17	#88200	—	2	1	no	—
18	#88198	—	2	1	no	—
19	C₂×#6095	C₂²	1	1	yes	characteristic
20	#88150	—	2	1	no	—
21	#88150	—	2	1	no	—
22	C₄⋊C₄×D₁₃	—	2	1	no	—
23	#88143	—	2	1	no	—
24	#88143	—	2	1	no	—
25	#88141	—	2	1	no	—
26	C₄×D₅₂	C₂²	1	1	yes	characteristic
27	C₄×D₅₂	C₂²	1	1	yes	characteristic
28	C₄×D₅₂	C₂²	1	1	yes	characteristic
29	#88124	C₂²	1	1	yes	characteristic
30	#88124	C₂²	1	1	yes	characteristic
31	#88124	C₂²	1	1	yes	characteristic
32	#88124	C₂²	1	1	yes	characteristic
33	#88123	C₂²	1	1	yes	characteristic
34	#88123	C₂²	1	1	yes	characteristic
35	C₄×Dic₂₆	C₂²	1	1	yes	characteristic
36	C₄×Dic₂₆	C₂²	1	1	yes	characteristic
37	C₄×Dic₂₆	C₂²	1	1	yes	characteristic
38	C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
39	C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
40	C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
41	C₁₃×#97	C₂²	1	1	yes	characteristic
42	C₁₃×#97	C₂²	1	1	yes	characteristic
43	C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
44	#88061	C₂²	1	1	yes	characteristic
45	#88061	C₂²	1	1	yes	characteristic
46	#88061	C₂²	1	1	yes	characteristic
47	#88058	C₂²	1	1	yes	characteristic
48	#88058	C₂²	1	1	yes	characteristic
49	#88058	C₂²	1	1	yes	characteristic
50	#88058	C₂²	1	1	yes	characteristic
51	#88057	C₂²	1	1	yes	characteristic
52	#88057	C₂²	1	1	yes	characteristic
53	#88057	C₂²	1	1	yes	characteristic
54	#88057	C₂²	1	1	yes	characteristic
55	#88057	C₂²	1	1	yes	characteristic
56	#88056	C₂²	1	1	yes	characteristic
57	#88056	C₂²	1	1	yes	characteristic
58	#88056	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₁₃	D₄	1	1	yes	characteristic
60	C₂×D₅₂	C₂³	1	1	yes	characteristic
61	C₂×C₄×D₁₃	—	2	1	no	—
62	C₂×C₄×D₁₃	—	2	1	no	—
63	C₂×C₄×D₁₃	—	2	1	no	—
64	C₂×Dic₂₆	—	2	1	no	—
65	C₂×Dic₂₆	—	2	1	no	—
66	C₂×Dic₂₆	C₂³	1	1	yes	characteristic
67	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
68	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
69	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
70	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
71	C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
72	C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
73	C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
74	C₄²×C₁₃	C₂³	1	1	yes	characteristic
75	C₄²×C₁₃	C₂³	1	1	yes	characteristic
76	C₄²×C₁₃	C₂³	1	1	yes	characteristic
77	#6103	—	2	1	no	—
78	#6103	—	2	1	no	—
79	#6103	—	2	1	no	—
80	C₄⋊Dic₁₃	—	2	1	no	—
81	C₄⋊Dic₁₃	—	2	1	no	—
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	C₄⋊Dic₁₃	C₂³	1	1	yes	characteristic
87	C₄⋊Dic₁₃	C₂³	1	1	yes	characteristic
88	#6101	—	2	1	no	—
89	#6101	—	2	1	no	—
90	#6101	—	2	1	no	—
91	#6101	—	2	1	no	—
92	#6101	—	2	1	no	—
93	#6101	—	2	1	no	—
94	#6101	—	2	1	no	—
95	#6101	—	2	1	no	—
96	#6101	—	2	1	no	—
97	#6101	—	2	1	no	—
98	#6101	—	2	1	no	—
99	C₄×Dic₁₃	—	2	1	no	—
100	C₄×Dic₁₃	—	2	1	no	—
101	C₄×Dic₁₃	—	2	1	no	—
102	#6095	Q₈	1	4	yes	complemented
103	#3388	—	13	1	no	maximal, Sylow
104	C₂²×D₁₃	—	2	1	no	—
105	Q₈×C₁₃	—	4	1	no	—
106	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
107	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
108	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
109	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
110	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
111	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
112	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
113	C₂×Dic₁₃	—	2	1	no	—
114	C₂×Dic₁₃	—	2	1	no	—
115	C₂×Dic₁₃	—	2	1	no	—
116	C₂×Dic₁₃	—	2	1	no	—
117	C₂×Dic₁₃	—	2	1	no	—
118	C₂×Dic₁₃	—	2	1	no	—
119	C₂×Dic₁₃	—	2	1	no	—
120	D₅₂	—	2	1	no	—
121	D₅₂	C₂×Q₈	1	2	yes	—
122	C₄×D₁₃	—	4	2	no	—
123	C₄×D₁₃	—	8	1	no	—
124	C₄×D₁₃	—	8	1	no	—
125	Dic₂₆	—	2	1	no	—
126	Dic₂₆	—	4	2	no	—
127	Dic₂₆	—	8	1	no	—
128	Dic₂₆	C₂×Q₈	1	2	yes	—
129	C₈×C₁₃	—	2	1	no	—
130	C₈×C₁₃	—	2	1	no	—
131	C₈×C₁₃	—	2	1	no	—
132	C₈×C₁₃	C₂×Q₈	1	2	yes	—
133	#557	—	13	1	no	—
134	#554	—	13	1	no	—
135	#501	—	13	1	no	—
136	#501	—	13	1	no	—
137	#499	—	13	1	no	—
138	#479	—	13	1	no	—
139	#479	—	13	1	no	—
140	#478	—	13	1	no	—
141	#478	—	13	1	no	—
142	#476	—	13	1	no	—
143	#475	—	13	1	no	—
144	#446	D₁₃	1	1	yes	characteristic, complemented
145	#440	—	13	1	no	—
146	#440	—	13	1	no	—
147	C₄×SD₁₆	—	13	1	no	—
148	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
149	C₂×D₁₃	—	2	2	no	—
150	C₂×D₁₃	—	4	1	no	—
151	C₂×D₁₃	—	4	1	no	—
152	C₄×C₁₃	—	2	1	no	—
153	C₄×C₁₃	—	2	1	no	—
154	C₄×C₁₃	—	2	1	no	—
155	C₄×C₁₃	—	2	1	no	—
156	C₄×C₁₃	—	2	1	no	—
157	C₄×C₁₃	—	2	1	no	—
158	C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
159	C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
160	Dic₁₃	—	2	2	no	—
161	Dic₁₃	—	4	1	no	—
162	Dic₁₃	—	4	1	no	—
163	Dic₁₃	—	4	1	no	—
164	Dic₁₃	—	4	1	no	—
165	Dic₁₃	—	4	1	no	—
166	Dic₁₃	—	4	1	no	—
167	C₂×SD₁₆	—	13	1	no	—
168	C₄⋊Q₈	—	13	1	no	—
169	C₄⋊Q₈	—	13	1	no	—
170	C₄⋊Q₈	—	26	1	no	—
171	#125	—	13	1	no	—
172	#125	—	13	1	no	—
173	#125	—	13	1	no	—
174	#125	—	13	1	no	—
175	#125	—	26	1	no	—
176	#125	—	26	1	no	—
177	C₂²⋊Q₈	—	26	1	no	—
178	C₂²⋊Q₈	—	26	1	no	—
179	C₂²⋊Q₈	—	26	1	no	—
180	C₄×Q₈	—	13	1	no	—
181	C₄×Q₈	—	13	1	no	—
182	C₄×Q₈	—	13	1	no	—
183	C₄×Q₈	—	26	1	no	—
184	C₄×Q₈	C₂×D₁₃	1	1	yes	characteristic
185	C₄×D₄	—	13	1	no	—
186	C₄×D₄	—	13	1	no	—
187	C₄×D₄	—	13	1	no	—
188	C₂×C₄⋊C₄	—	26	1	no	—
189	#107	—	13	1	no	—
190	#107	—	13	1	no	—
191	#107	—	13	1	no	—
192	#107	—	13	1	no	—
193	#106	—	13	1	no	—
194	#106	—	13	1	no	—
195	#106	—	13	1	no	—
196	#106	—	13	1	no	—
197	#106	—	13	1	no	—
198	C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
199	C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
200	C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
201	#103	—	13	1	no	—
202	#103	—	13	1	no	—
203	#103	—	13	1	no	—
204	#102	—	13	1	no	—
205	#102	—	13	1	no	—
206	#102	—	13	1	no	—
207	#97	C₂×D₁₃	1	1	yes	characteristic
208	#97	C₂×D₁₃	1	1	yes	characteristic
209	C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
210	C₂×C₁₃	D₄×Q₈	1	1	yes	characteristic
211	C₂×C₁₃	#575	1	1	yes	characteristic
212	C₂×C₁₃	#577	1	1	yes	characteristic
213	D₁₃	—	4	2	no	—
214	C₂×Q₈	—	13	1	no	—
215	C₂×Q₈	—	26	1	no	—
216	C₂×Q₈	—	26	1	no	—
217	C₂×Q₈	D₅₂	1	1	yes	characteristic
218	C₂×D₄	—	13	1	no	—
219	C₂²×C₄	—	26	1	no	—
220	C₂²×C₄	—	26	1	no	—
221	C₂²×C₄	—	26	1	no	—
222	SD₁₆	—	13	4	no	—
223	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
224	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
225	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
226	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
227	C₄⋊C₄	—	13	1	no	—
228	C₄⋊C₄	—	13	1	no	—
229	C₄⋊C₄	—	13	1	no	—
230	C₄⋊C₄	—	13	1	no	—
231	C₄⋊C₄	—	13	1	no	—
232	C₄⋊C₄	—	13	1	no	—
233	C₄⋊C₄	—	26	1	no	—
234	C₄⋊C₄	—	26	1	no	—
235	C₄⋊C₄	—	26	1	no	—
236	C₄⋊C₄	—	26	1	no	—
237	C₄⋊C₄	—	26	1	no	—
238	C₄⋊C₄	—	26	1	no	—
239	C₄⋊C₄	—	26	1	no	—
240	C₄⋊C₄	—	26	1	no	—
241	C₄⋊C₄	—	26	1	no	—
242	C₄⋊C₄	—	26	1	no	—
243	C₄⋊C₄	—	26	1	no	—
244	C₄⋊C₄	—	26	1	no	—
245	C₄⋊C₄	—	26	1	no	—
246	C₄⋊C₄	D₅₂	1	1	yes	characteristic
247	C₄⋊C₄	D₅₂	1	1	yes	characteristic
248	C₄⋊C₄	D₅₂	1	1	yes	characteristic
249	C₂²⋊C₄	—	26	1	no	—
250	C₂²⋊C₄	—	26	1	no	—
251	C₂²⋊C₄	—	26	1	no	—
252	C₄²	—	26	1	no	—
253	C₄²	—	26	1	no	—
254	C₄²	—	26	1	no	—
255	C₄²	C₂²×D₁₃	1	1	yes	characteristic
256	C₄²	C₂²×D₁₃	1	1	yes	characteristic
257	C₄²	C₂²×D₁₃	1	1	yes	characteristic
258	C₁₃	#3388	1	1	yes	characteristic, complemented, Sylow
259	C₂³	—	26	1	no	—
260	Q₈	—	4	1	no	—
261	Q₈	—	13	2	no	—
262	Q₈	—	26	1	no	—
263	Q₈	—	52	2	no	—
264	Q₈	—	104	1	no	—
265	D₄	—	13	2	no	—
266	D₄	—	26	1	no	—
267	C₂×C₄	—	26	1	no	—
268	C₂×C₄	—	26	1	no	—
269	C₂×C₄	—	26	1	no	—
270	C₂×C₄	—	26	1	no	—
271	C₂×C₄	—	26	1	no	—
272	C₂×C₄	—	26	1	no	—
273	C₂×C₄	—	26	1	no	—
274	C₂×C₄	—	52	2	no	—
275	C₂×C₄	—	104	1	no	—
276	C₂×C₄	—	104	1	no	—
277	C₂×C₄	C₂×D₅₂	1	1	yes	characteristic
278	C₂×C₄	C₂×D₅₂	1	1	yes	characteristic
279	C₂×C₄	C₂×D₅₂	1	1	yes	characteristic
280	C₂×C₄	C₂×D₅₂	1	1	yes	characteristic
281	C₂×C₄	C₂×D₅₂	1	1	yes	characteristic
282	C₂×C₄	C₂×D₅₂	1	1	yes	characteristic
283	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
284	C₈	—	2	1	no	—
285	C₈	—	2	1	no	—
286	C₈	—	2	1	no	—
287	C₈	Q₈×D₁₃	1	2	yes	—
288	C₂²	—	26	2	no	—
289	C₂²	—	52	1	no	—
290	C₂²	—	52	1	no	—
291	C₂²	C₂²×D₅₂	1	1	yes	characteristic
292	C₄	—	2	1	no	—
293	C₄	—	2	1	no	—
294	C₄	—	2	1	no	—
295	C₄	—	2	1	no	—
296	C₄	—	2	1	no	—
297	C₄	—	2	1	no	—
298	C₄	—	26	2	no	—
299	C₄	—	52	1	no	—
300	C₄	—	52	1	no	—
301	C₄	—	52	1	no	—
302	C₄	—	52	1	no	—
303	C₄	—	52	1	no	—
304	C₄	—	52	1	no	—
305	C₄	C₂×Q₈×D₁₃	1	1	yes	characteristic
306	C₄	#88257	1	1	yes	characteristic
307	C₂	—	52	2	no	—
308	C₂	Q₈×D₅₂	1	1	yes	characteristic
309	C₂	#11719453	1	1	yes	characteristic
310	C₂	#11719456	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₅₂
commutator, cosocle	C₂×C₄×C₁₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃
Fitting	C₁₃×#446	C₂
socle	C₂²×C₁₃	C₂²×D₄
prime radical	C₂×D₅₂	C₂³
hypercenter	#446	D₁₃
hypocenter	C₁₃	#3388
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #446

Complements

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

N	H ≅ G/N	type
2. #11719272	C₂	semidirect
3. C₁₃×#446	C₂	semidirect
4. #11718512	C₂	semidirect
8. #11718490	C₂	semidirect
9. #11718490	C₂	semidirect
12. #11718363	C₂	semidirect
13. #11718363	C₂	semidirect
15. #11718344	C₂	semidirect
102. #6095	Q₈	semidirect
144. #446	D₁₃	semidirect
258. C₁₃	#3388	semidirect

Maximal subgroups

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

subgroup	index
#11719284	2
#11719272	2
C₁₃×#446	2
#11718512	2
#11718508	2
#11718503	2
#11718490	2
#11718371	2
#11718363	2
C₄×#6095	2
#11718344	2
#3388	13