Subgroups — #16017578

Order 3328.

534 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₅₂	C₂	1	1	yes	maximal, characteristic
2	#13116997	C₂	1	4	yes	maximal, complemented
3	#13116985	C₂	1	2	yes	maximal, complemented
4	#13116975	C₂	1	2	yes	maximal, complemented
5	#13116625	C₂	1	2	yes	maximal
6	#13116592	C₂	1	1	yes	maximal, characteristic
7	#13116555	C₂	1	2	yes	maximal
8	C₁₃×#3160	C₂	1	1	yes	maximal, characteristic
9	C₁₃×#533	—	2	1	no	—
10	C₂²×Dic₅₂	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₂₆	—	2	1	no	—
12	C₁₃×#500	C₂²	1	1	yes	characteristic
13	C₁₃×#500	C₂²	1	1	yes	characteristic
14	C₁₃×#499	—	2	1	no	—
15	C₁₃×#499	C₂²	1	2	yes	—
16	C₁₃×#498	—	2	1	no	—
17	C₁₃×#427	—	2	1	no	—
18	C₁₃×#427	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
20	C₂×#88058	C₂²	1	1	yes	characteristic
21	C₂×#88056	—	2	1	no	—
22	C₂×#88052	C₂²	1	2	yes	—
23	C₄×Dic₅₂	—	2	1	no	—
24	C₄×Dic₅₂	—	2	1	no	—
25	C₄×Dic₅₂	C₂²	1	2	yes	—
26	C₄×Dic₅₂	C₂²	1	2	yes	—
27	#11718084	C₂²	1	4	yes	—
28	#11718084	C₂²	1	4	yes	—
29	#11718082	C₂²	1	4	yes	—
30	#11718081	C₂²	1	4	yes	—
31	C₄⋊#6090	C₂²	1	2	yes	—
32	C₄⋊#6090	C₂²	1	2	yes	—
33	C₂²×Dic₂₆	—	2	1	no	—
34	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
36	C₁₃×C₄⋊Q₈	—	2	1	no	—
37	C₁₃×C₄⋊Q₈	—	2	1	no	—
38	C₁₃×#125	—	2	2	no	—
39	C₁₃×C₂²⋊Q₈	—	4	1	no	—
40	C₁₃×C₂²⋊Q₈	—	4	1	no	—
41	C₄×Q₈×C₁₃	—	4	1	no	—
42	C₄×Q₈×C₁₃	—	4	1	no	—
43	C₁₃×#117	—	2	2	no	—
44	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
45	C₂×C₄⋊Dic₁₃	—	2	1	no	—
46	C₂×#6101	—	4	1	no	—
47	C₂×C₄×Dic₁₃	—	4	1	no	—
48	C₂×Dic₅₂	—	2	1	no	—
49	C₂×Dic₅₂	—	2	2	no	—
50	C₂×Dic₅₂	Q₈	1	4	yes	complemented
51	C₂×Dic₅₂	C₂³	1	2	yes	—
52	C₄×Dic₂₆	—	2	2	no	—
53	C₄×Dic₂₆	—	2	2	no	—
54	C₄×Dic₂₆	—	4	1	no	—
55	C₄×Dic₂₆	—	4	1	no	—
56	C₁₃×#107	—	2	1	no	—
57	C₁₃×#107	—	2	1	no	—
58	C₁₃×#107	C₂³	1	2	yes	—
59	C₁₃×#107	C₂³	1	2	yes	—
60	C₁₃×#106	—	2	2	no	—
61	C₁₃×#106	—	2	2	no	—
62	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
63	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
64	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
65	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
66	#88058	—	2	1	no	—
67	#88058	C₂³	1	2	yes	—
68	#88056	—	2	2	no	—
69	#88056	—	2	2	no	—
70	#88052	—	2	4	no	—
71	C₂×#6090	C₂³	1	4	yes	—
72	#33628	—	13	1	no	maximal, Sylow
73	C₂×Q₈×C₁₃	—	4	1	no	—
74	C₂×Q₈×C₁₃	—	4	1	no	—
75	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
76	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
77	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
78	C₂²×Dic₁₃	—	4	1	no	—
79	C₂²×Dic₁₃	—	4	1	no	—
80	C₂×Dic₂₆	—	2	4	no	—
81	C₂×Dic₂₆	—	2	2	no	—
82	C₂×Dic₂₆	—	4	1	no	—
83	C₂×Dic₂₆	—	4	2	no	—
84	C₂×C₈×C₁₃	—	2	2	no	—
85	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
89	C₂×C₈×C₁₃	#41	1	2	yes	—
90	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
91	C₁₃×C₄⋊C₄	—	2	2	no	—
92	C₁₃×C₄⋊C₄	—	2	2	no	—
93	C₁₃×C₄⋊C₄	—	4	2	no	—
94	C₁₃×C₄⋊C₄	—	4	2	no	—
95	C₁₃×C₄⋊C₄	—	4	1	no	—
96	C₁₃×C₄⋊C₄	—	4	1	no	—
97	C₁₃×C₂²⋊C₄	—	4	2	no	—
98	C₄²×C₁₃	—	4	2	no	—
99	C₄²×C₁₃	D₈	1	1	yes	characteristic
100	C₄²×C₁₃	D₈	1	1	yes	characteristic
101	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
102	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
103	C₄⋊Dic₁₃	—	2	2	no	—
104	C₄⋊Dic₁₃	—	4	1	no	—
105	#6101	—	4	2	no	—
106	#6101	—	4	2	no	—
107	C₄×Dic₁₃	—	4	2	no	—
108	C₄×Dic₁₃	—	4	2	no	—
109	Dic₅₂	—	2	4	no	—
110	Dic₅₂	—	2	2	no	—
111	Dic₅₂	C₂×Q₈	1	4	yes	—
112	#6090	—	2	4	no	—
113	#3160	D₁₃	1	1	yes	characteristic
114	C₂×C₄×Q₁₆	—	13	1	no	—
115	#2229	—	13	4	no	—
116	#2227	—	13	2	no	—
117	#2225	—	13	2	no	—
118	#2159	—	13	2	no	—
119	#2150	—	13	1	no	—
120	#2141	—	13	2	no	—
121	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
122	Q₈×C₁₃	—	8	2	no	—
123	Q₈×C₁₃	—	8	2	no	—
124	C₂×C₄×C₁₃	—	2	2	no	—
125	C₂×C₄×C₁₃	—	4	1	no	—
126	C₂×C₄×C₁₃	—	4	2	no	—
127	C₂×C₄×C₁₃	—	4	2	no	—
128	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
129	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
130	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
131	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
132	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
134	C₂×C₄×C₁₃	#137	1	2	yes	—
135	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
136	C₂×Dic₁₃	—	4	2	no	—
137	C₂×Dic₁₃	—	4	4	no	—
138	C₂×Dic₁₃	—	4	2	no	—
139	C₂×Dic₁₃	—	8	2	no	—
140	Dic₂₆	—	2	4	no	—
141	Dic₂₆	—	4	2	no	—
142	Dic₂₆	—	4	4	no	—
143	C₈×C₁₃	—	2	1	no	—
144	C₈×C₁₃	—	2	1	no	—
145	C₈×C₁₃	C₂²⋊Q₈	1	2	yes	—
146	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
147	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
148	C₂²×Q₁₆	—	13	1	no	—
149	#533	—	2	1	no	—
150	C₂×C₄×Q₈	—	26	1	no	—
151	C₂×M₃₂	—	13	2	no	—
152	#500	C₂×D₁₃	1	1	yes	characteristic
153	#500	C₂×D₁₃	1	1	yes	characteristic
154	#499	—	2	1	no	—
155	#499	C₂×D₁₃	1	2	yes	—
156	#498	—	2	1	no	—
157	C₄×Q₁₆	—	13	2	no	—
158	C₄×Q₁₆	—	13	2	no	—
159	C₄×Q₁₆	—	26	1	no	—
160	C₄×Q₁₆	—	26	1	no	—
161	#427	—	2	1	no	—
162	#427	C₂×D₁₃	1	2	yes	—
163	C₂×#107	—	13	1	no	—
164	C₂×#103	—	26	1	no	—
165	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
166	#367	—	13	4	no	—
167	#366	—	13	4	no	—
168	C₄⋊C₁₆	—	13	2	no	—
169	C₄⋊C₁₆	—	13	2	no	—
170	#358	—	13	4	no	—
171	#358	—	13	4	no	—
172	C₂²×C₁₃	—	2	1	no	—
173	C₂²×C₁₃	—	2	1	no	—
174	C₂²×C₁₃	#474	1	2	yes	—
175	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
176	C₄×C₁₃	—	2	1	no	—
177	C₄×C₁₃	—	2	1	no	—
178	C₄×C₁₃	—	2	1	no	—
179	C₄×C₁₃	—	8	2	no	—
180	C₄×C₁₃	—	8	2	no	—
181	C₄×C₁₃	#474	1	2	yes	—
182	C₄×C₁₃	#510	1	2	yes	—
183	C₄×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
184	C₄×C₁₃	C₂×#137	1	1	yes	characteristic
185	Dic₁₃	—	4	4	no	—
186	Dic₁₃	—	8	2	no	—
187	C₂²×Q₈	—	26	1	no	—
188	C₂×Q₁₆	—	13	4	no	—
189	C₂×Q₁₆	—	13	2	no	—
190	C₂×Q₁₆	—	26	2	no	—
191	C₂×Q₁₆	—	26	1	no	—
192	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
193	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
194	C₄⋊Q₈	—	2	1	no	—
195	C₄⋊Q₈	—	2	1	no	—
196	#125	—	2	2	no	—
197	C₂²⋊Q₈	—	4	1	no	—
198	C₂²⋊Q₈	—	4	1	no	—
199	C₄×Q₈	—	4	1	no	—
200	C₄×Q₈	—	4	1	no	—
201	C₄×Q₈	—	26	2	no	—
202	C₄×Q₈	—	26	2	no	—
203	C₄×Q₈	—	52	1	no	—
204	C₄×Q₈	—	52	1	no	—
205	#117	—	2	2	no	—
206	C₂×C₄⋊C₄	—	26	1	no	—
207	C₂×C₄⋊C₄	—	52	1	no	—
208	C₂×C₄²	—	52	1	no	—
209	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
210	M₃₂	—	26	4	no	—
211	C₂×C₁₆	—	13	4	no	—
212	#107	—	2	1	no	—
213	#107	—	2	1	no	—
214	#107	—	13	2	no	—
215	#107	—	26	1	no	—
216	#107	C₂²×D₁₃	1	2	yes	—
217	#107	C₂²×D₁₃	1	2	yes	—
218	#106	—	2	2	no	—
219	#106	—	2	2	no	—
220	#103	—	26	2	no	—
221	#103	—	26	2	no	—
222	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
223	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
224	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
225	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
226	C₂×C₁₃	—	2	2	no	—
227	C₂×C₁₃	C₂×#474	1	1	yes	characteristic
228	C₂×C₁₃	C₂×#510	1	1	yes	characteristic
229	C₂×C₁₃	#3414	1	1	yes	characteristic
230	C₂×Q₈	—	4	1	no	—
231	C₂×Q₈	—	4	1	no	—
232	C₂×Q₈	—	26	4	no	—
233	C₂×Q₈	—	26	2	no	—
234	C₂×Q₈	—	52	1	no	—
235	C₂×Q₈	—	52	2	no	—
236	C₂²×C₄	—	52	1	no	—
237	C₂²×C₄	—	52	1	no	—
238	C₂²×C₄	#6104	1	1	yes	characteristic
239	C₂²×C₄	#6104	1	1	yes	characteristic
240	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
241	Q₁₆	—	13	4	no	—
242	Q₁₆	—	26	4	no	—
243	Q₁₆	—	26	2	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	Q₈×D₁₃	1	2	yes	—
246	C₂×C₈	#6131	1	2	yes	—
247	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
248	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
249	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
250	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
251	C₄⋊C₄	—	2	2	no	—
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	4	2	no	—
254	C₄⋊C₄	—	4	2	no	—
255	C₄⋊C₄	—	4	1	no	—
256	C₄⋊C₄	—	4	1	no	—
257	C₄⋊C₄	—	26	2	no	—
258	C₄⋊C₄	—	52	1	no	—
259	C₄⋊C₄	—	52	2	no	—
260	C₄⋊C₄	—	52	2	no	—
261	C₂²⋊C₄	—	4	2	no	—
262	C₄²	—	4	2	no	—
263	C₄²	—	52	2	no	—
264	C₄²	—	52	2	no	—
265	C₄²	#6104	1	1	yes	characteristic
266	C₄²	#6104	1	1	yes	characteristic
267	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
268	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
269	C₁₆	—	26	4	no	—
270	C₁₃	#33628	1	1	yes	characteristic, complemented, Sylow
271	C₂³	C₂×#6104	1	1	yes	characteristic
272	Q₈	—	8	2	no	—
273	Q₈	—	8	2	no	—
274	Q₈	—	26	4	no	—
275	Q₈	—	52	2	no	—
276	Q₈	—	52	4	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	4	1	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	52	2	no	—
282	C₂×C₄	—	52	4	no	—
283	C₂×C₄	—	52	2	no	—
284	C₂×C₄	—	104	2	no	—
285	C₂×C₄	C₂×#6104	1	1	yes	characteristic
286	C₂×C₄	C₂×#6104	1	1	yes	characteristic
287	C₂×C₄	C₂×#6104	1	1	yes	characteristic
288	C₂×C₄	C₂×#6104	1	1	yes	characteristic
289	C₂×C₄	C₂×#6104	1	1	yes	characteristic
290	C₂×C₄	#88196	1	2	yes	—
291	C₂×C₄	#88200	1	2	yes	—
292	C₂×C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
293	C₈	—	2	1	no	—
294	C₈	—	2	1	no	—
295	C₈	#88200	1	2	yes	—
296	C₈	C₂×Q₈×D₁₃	1	1	yes	characteristic
297	C₈	C₂×#6131	1	1	yes	characteristic
298	C₂²	—	2	1	no	—
299	C₂²	—	2	1	no	—
300	C₂²	#11718748	1	2	yes	—
301	C₂²	C₂²×#6104	1	1	yes	characteristic
302	C₄	—	2	1	no	—
303	C₄	—	2	1	no	—
304	C₄	—	2	1	no	—
305	C₄	—	8	2	no	—
306	C₄	—	8	2	no	—
307	C₄	—	52	4	no	—
308	C₄	—	104	2	no	—
309	C₄	#11718748	1	2	yes	—
310	C₄	#11718856	1	2	yes	—
311	C₄	C₂×#88196	1	1	yes	characteristic
312	C₄	C₂×#88200	1	1	yes	characteristic
313	C₂	—	2	2	no	—
314	C₂	C₂×#11718748	1	1	yes	characteristic
315	C₂	C₂×#11718856	1	1	yes	characteristic
316	C₂	#13127970	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#6104
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₁₃×#3160	C₂
socle	C₂²×C₁₃	C₂²×D₈
prime radical	C₂³×C₁₃	C₂×D₈
hypercenter	#3160	D₁₃
hypocenter	C₁₃	#33628
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3160

Complements

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

N	H ≅ G/N	type
2. #13116997	C₂	semidirect
3. #13116985	C₂	semidirect
4. #13116975	C₂	semidirect
50. C₂×Dic₅₂	Q₈	semidirect
270. C₁₃	#33628	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₅₂	2
#13116997	2
#13116985	2
#13116975	2
#13116625	2
#13116592	2
#13116555	2
C₁₃×#3160	2
#33628	13