Subgroups — #15750866

Order 3328.

462 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊#1055	C₂	1	1	yes	maximal, characteristic
2	#13114582	C₂	1	1	yes	maximal, characteristic
3	#13112662	C₂	1	2	yes	maximal
4	#13112659	C₂	1	2	yes	maximal
5	C₁₃×#2295	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Q₈×C₁₃	C₄	1	1	yes	characteristic
7	C₁₃×#391	C₄	1	1	yes	characteristic
8	C₁₃×#391	C₂²	1	1	yes	characteristic
9	C₁₃×#389	C₄	1	1	yes	characteristic
10	C₁₃×#389	C₂²	1	1	yes	characteristic
11	C₁₃×#387	—	2	1	no	—
12	C₁₃×#387	—	2	1	no	—
13	C₄×C₁₃×C₄⋊C₄	—	2	1	no	—
14	C₄×C₁₃×C₄⋊C₄	C₄	1	1	yes	characteristic
15	C₁₃×#376	—	2	1	no	—
16	C₄³×C₁₃	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊#1055	—	2	1	no	—
18	C₂×C₄×#1055	—	2	1	no	—
19	C₂×C₄×#1055	—	2	1	no	—
20	#11718116	—	2	1	no	—
21	#11718116	C₂²	1	2	yes	—
22	#11718116	C₂²	1	2	yes	—
23	C₂²×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₄×Q₈×C₁₃	—	2	2	no	—
25	C₄×Q₈×C₁₃	C₈	1	4	yes	complemented
26	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
27	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
28	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
29	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
30	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
31	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₁₃	—	2	1	no	—
33	C₂×C₄²×C₁₃	—	2	1	no	—
34	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
41	C₂²×#1055	—	2	2	no	—
42	C₂²×#1055	—	2	2	no	—
43	C₁₃×#95	—	2	1	no	—
44	C₁₃×#95	—	2	1	no	—
45	C₁₃×#95	—	2	1	no	—
46	C₁₃×#95	—	2	1	no	—
47	C₁₃×#95	—	2	1	no	—
48	C₁₃×#95	—	2	1	no	—
49	C₁₃×#95	—	2	1	no	—
50	C₄⋊#1055	—	2	4	no	—
51	C₄×#1055	—	2	4	no	—
52	C₄×#1055	—	4	2	no	—
53	#7562	—	13	1	no	maximal, Sylow
54	C₂×Q₈×C₁₃	—	2	1	no	—
55	C₂×Q₈×C₁₃	—	4	1	no	—
56	C₂×Q₈×C₁₃	—	4	1	no	—
57	C₂×Q₈×C₁₃	C₂×C₈	1	2	yes	—
58	C₂²×C₄×C₁₃	—	2	1	no	—
59	C₂²×C₄×C₁₃	—	2	1	no	—
60	C₂²×C₄×C₁₃	—	2	1	no	—
61	C₂²×C₄×C₁₃	—	2	1	no	—
62	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
69	C₁₃×C₄⋊C₄	—	2	4	no	—
70	C₁₃×C₄⋊C₄	—	2	1	no	—
71	C₁₃×C₄⋊C₄	—	4	1	no	—
72	C₁₃×C₄⋊C₄	—	4	1	no	—
73	C₁₃×C₄⋊C₄	—	4	2	no	—
74	C₁₃×C₄⋊C₄	—	4	2	no	—
75	C₁₃×C₄⋊C₄	C₂×C₈	1	2	yes	—
76	C₄²×C₁₃	—	2	4	no	—
77	C₄²×C₁₃	—	2	2	no	—
78	C₄²×C₁₃	—	2	2	no	—
79	C₄²×C₁₃	—	2	2	no	—
80	C₄²×C₁₃	—	4	1	no	—
81	C₄²×C₁₃	—	4	1	no	—
82	C₄²×C₁₃	—	4	2	no	—
83	C₄²×C₁₃	C₂×C₈	1	2	yes	—
84	C₄²×C₁₃	M₁₆	1	2	yes	—
85	C₄²×C₁₃	SD₁₆	1	2	yes	—
86	C₄²×C₁₃	Q₁₆	1	2	yes	—
87	C₂×#1055	—	2	4	no	—
88	C₂×#1055	—	2	4	no	—
89	C₂×#1055	—	4	2	no	—
90	C₂×#1055	—	4	2	no	—
91	C₂×#1055	—	4	2	no	—
92	C₂×#1055	—	4	2	no	—
93	#2295	D₁₃	1	1	yes	characteristic
94	C₄×C₄⋊C₈	—	13	1	no	—
95	#1744	—	13	1	no	—
96	#1272	—	13	2	no	—
97	#1271	—	13	2	no	—
98	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	Q₈×C₁₃	—	4	2	no	—
100	Q₈×C₁₃	—	4	2	no	—
101	C₂×C₄×C₁₃	—	2	1	no	—
102	C₂×C₄×C₁₃	—	2	1	no	—
103	C₂×C₄×C₁₃	—	2	1	no	—
104	C₂×C₄×C₁₃	—	2	1	no	—
105	C₂×C₄×C₁₃	—	2	1	no	—
106	C₂×C₄×C₁₃	—	2	1	no	—
107	C₂×C₄×C₁₃	—	2	1	no	—
108	C₂×C₄×C₁₃	—	2	1	no	—
109	C₂×C₄×C₁₃	—	2	1	no	—
110	C₂×C₄×C₁₃	—	2	1	no	—
111	C₂×C₄×C₁₃	—	2	1	no	—
112	C₂×C₄×C₁₃	—	2	1	no	—
113	C₂×C₄×C₁₃	—	2	2	no	—
114	C₂×C₄×C₁₃	—	2	2	no	—
115	C₂×C₄×C₁₃	—	4	1	no	—
116	C₂×C₄×C₁₃	—	4	1	no	—
117	C₂×C₄×C₁₃	—	4	1	no	—
118	C₂×C₄×C₁₃	—	4	1	no	—
119	C₂×C₄×C₁₃	—	4	1	no	—
120	C₂×C₄×C₁₃	—	4	1	no	—
121	C₂×C₄×C₁₃	—	4	1	no	—
122	C₂×C₄×C₁₃	—	4	1	no	—
123	C₂×C₄×C₁₃	—	4	1	no	—
124	C₂×C₄×C₁₃	—	4	1	no	—
125	C₂×C₄×C₁₃	C₂²⋊C₈	1	2	yes	—
126	C₂×C₄×C₁₃	C₂²⋊C₈	1	2	yes	—
127	C₂×C₄×C₁₃	#103	1	2	yes	—
128	C₂×C₄×C₁₃	#103	1	2	yes	—
129	C₂×C₄×C₁₃	#104	1	2	yes	—
130	C₂×C₄×C₁₃	#104	1	2	yes	—
131	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
132	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
133	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
134	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
135	C₂×C₄×C₁₃	#135	1	2	yes	—
136	#1055	—	4	4	no	—
137	#1055	—	4	4	no	—
138	C₂×C₄×Q₈	Dic₁₃	1	1	yes	characteristic
139	C₂×C₄⋊C₈	—	26	1	no	—
140	C₂×C₄×C₈	—	26	1	no	—
141	C₂×C₄×C₈	—	26	1	no	—
142	#391	Dic₁₃	1	1	yes	characteristic
143	#391	C₂×D₁₃	1	1	yes	characteristic
144	#389	Dic₁₃	1	1	yes	characteristic
145	#389	C₂×D₁₃	1	1	yes	characteristic
146	#387	—	2	1	no	—
147	#387	—	2	1	no	—
148	C₄×C₄⋊C₄	—	2	1	no	—
149	C₄×C₄⋊C₄	Dic₁₃	1	1	yes	characteristic
150	#376	—	2	1	no	—
151	C₄³	C₂×D₁₃	1	1	yes	characteristic
152	#336	—	13	2	no	—
153	#336	—	13	2	no	—
154	#336	—	26	1	no	—
155	C₂²×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
156	C₂²×C₁₃	#410	1	1	yes	characteristic
157	C₂²×C₁₃	#413	1	1	yes	characteristic
158	C₂²×C₁₃	C₂×#103	1	1	yes	characteristic
159	C₂²×C₁₃	#416	1	1	yes	characteristic
160	C₂²×C₁₃	C₂×#104	1	1	yes	characteristic
161	C₂²×C₁₃	C₂×#104	1	1	yes	characteristic
162	C₄×C₁₃	—	2	2	no	—
163	C₄×C₁₃	—	2	1	no	—
164	C₄×C₁₃	—	2	1	no	—
165	C₄×C₁₃	—	2	2	no	—
166	C₄×C₁₃	—	2	2	no	—
167	C₄×C₁₃	—	2	1	no	—
168	C₄×C₁₃	—	2	1	no	—
169	C₄×C₁₃	—	2	2	no	—
170	C₄×C₁₃	—	4	2	no	—
171	C₄×C₁₃	—	4	2	no	—
172	C₄×C₁₃	—	8	1	no	—
173	C₄×C₁₃	—	8	1	no	—
174	C₄×C₁₃	Q₈⋊C₈	1	4	yes	—
175	C₂²×Q₈	C₂×Dic₁₃	1	1	yes	characteristic
176	C₂²×C₈	—	26	2	no	—
177	C₂²×C₈	—	26	2	no	—
178	C₄×Q₈	—	2	2	no	—
179	C₄×Q₈	#1055	1	4	yes	complemented
180	C₂×C₄⋊C₄	—	2	1	no	—
181	C₂×C₄⋊C₄	—	2	1	no	—
182	C₂×C₄⋊C₄	—	2	1	no	—
183	C₂×C₄⋊C₄	C₂×Dic₁₃	1	1	yes	characteristic
184	C₂×C₄⋊C₄	C₂×Dic₁₃	1	1	yes	characteristic
185	C₂×C₄⋊C₄	C₂×Dic₁₃	1	1	yes	characteristic
186	C₂×C₄²	—	2	1	no	—
187	C₂×C₄²	—	2	1	no	—
188	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
189	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
190	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
191	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
192	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
193	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
194	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
195	C₄⋊C₈	—	26	4	no	—
196	C₄×C₈	—	26	4	no	—
197	C₄×C₈	—	52	2	no	—
198	#95	—	2	1	no	—
199	#95	—	2	1	no	—
200	#95	—	2	1	no	—
201	#95	—	2	1	no	—
202	#95	—	2	1	no	—
203	#95	—	2	1	no	—
204	#95	—	2	1	no	—
205	C₂×C₁₃	#1458	1	1	yes	characteristic
206	C₂×C₁₃	C₂×Q₈⋊C₈	1	1	yes	characteristic
207	C₂×C₁₃	#1476	1	1	yes	characteristic
208	C₂×C₁₃	#1500	1	1	yes	characteristic
209	C₂×C₁₃	#1503	1	1	yes	characteristic
210	C₂×C₁₃	#1524	1	1	yes	characteristic
211	C₂×C₁₃	#1542	1	1	yes	characteristic
212	C₂×Q₈	—	2	1	no	—
213	C₂×Q₈	—	4	1	no	—
214	C₂×Q₈	—	4	1	no	—
215	C₂×Q₈	C₂×#1055	1	2	yes	—
216	C₂²×C₄	—	2	1	no	—
217	C₂²×C₄	—	2	1	no	—
218	C₂²×C₄	—	2	1	no	—
219	C₂²×C₄	—	2	1	no	—
220	C₂²×C₄	C₂²⋊Dic₁₃	1	1	yes	characteristic
221	C₂²×C₄	C₂²⋊Dic₁₃	1	1	yes	characteristic
222	C₂²×C₄	C₂²⋊Dic₁₃	1	1	yes	characteristic
223	C₂²×C₄	C₂²⋊Dic₁₃	1	1	yes	characteristic
224	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
225	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
226	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
227	C₂×C₈	—	26	4	no	—
228	C₂×C₈	—	26	4	no	—
229	C₂×C₈	—	52	2	no	—
230	C₂×C₈	—	52	2	no	—
231	C₂×C₈	—	52	2	no	—
232	C₂×C₈	—	52	2	no	—
233	C₄⋊C₄	—	2	4	no	—
234	C₄⋊C₄	—	2	1	no	—
235	C₄⋊C₄	—	4	1	no	—
236	C₄⋊C₄	—	4	1	no	—
237	C₄⋊C₄	—	4	2	no	—
238	C₄⋊C₄	—	4	2	no	—
239	C₄⋊C₄	C₂×#1055	1	2	yes	—
240	C₄²	—	2	4	no	—
241	C₄²	—	2	2	no	—
242	C₄²	—	2	2	no	—
243	C₄²	—	2	2	no	—
244	C₄²	—	4	1	no	—
245	C₄²	—	4	1	no	—
246	C₄²	—	4	2	no	—
247	C₄²	C₂×#1055	1	2	yes	—
248	C₄²	#6099	1	2	yes	—
249	C₄²	#6106	1	2	yes	—
250	C₄²	#6107	1	2	yes	—
251	C₁₃	#7562	1	1	yes	characteristic, complemented, Sylow
252	C₂³	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
253	Q₈	—	4	2	no	—
254	Q₈	—	4	2	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	4	1	no	—
270	C₂×C₄	—	4	1	no	—
271	C₂×C₄	—	4	1	no	—
272	C₂×C₄	—	4	1	no	—
273	C₂×C₄	—	4	1	no	—
274	C₂×C₄	—	4	1	no	—
275	C₂×C₄	—	4	1	no	—
276	C₂×C₄	—	4	1	no	—
277	C₂×C₄	—	4	1	no	—
278	C₂×C₄	—	4	1	no	—
279	C₂×C₄	C₂²⋊#1055	1	2	yes	—
280	C₂×C₄	C₂²⋊#1055	1	2	yes	—
281	C₂×C₄	#88075	1	2	yes	—
282	C₂×C₄	#88075	1	2	yes	—
283	C₂×C₄	#88077	1	2	yes	—
284	C₂×C₄	#88077	1	2	yes	—
285	C₂×C₄	C₂²×#1055	1	1	yes	characteristic
286	C₂×C₄	C₂×#6099	1	1	yes	characteristic
287	C₂×C₄	C₂×#6106	1	1	yes	characteristic
288	C₂×C₄	C₂×#6107	1	1	yes	characteristic
289	C₂×C₄	#88204	1	2	yes	—
290	C₈	—	52	4	no	—
291	C₈	—	52	4	no	—
292	C₂²	C₂×C₂²⋊#1055	1	1	yes	characteristic
293	C₂²	C₂×#88075	1	1	yes	characteristic
294	C₂²	#11718894	1	1	yes	characteristic
295	C₂²	C₂×#88077	1	1	yes	characteristic
296	C₂²	C₂×#88077	1	1	yes	characteristic
297	C₂²	#11718898	1	1	yes	characteristic
298	C₂²	#11718901	1	1	yes	characteristic
299	C₄	—	2	1	no	—
300	C₄	—	2	1	no	—
301	C₄	—	2	2	no	—
302	C₄	—	2	2	no	—
303	C₄	—	2	2	no	—
304	C₄	—	2	2	no	—
305	C₄	—	2	1	no	—
306	C₄	—	2	1	no	—
307	C₄	—	4	2	no	—
308	C₄	—	4	2	no	—
309	C₄	—	8	1	no	—
310	C₄	—	8	1	no	—
311	C₄	Q₈⋊#1055	1	4	yes	—
312	C₂	#13113150	1	1	yes	characteristic
313	C₂	C₂×Q₈⋊#1055	1	1	yes	characteristic
314	C₂	#13113218	1	1	yes	characteristic
315	C₂	#13113322	1	1	yes	characteristic
316	C₂	#13113335	1	1	yes	characteristic
317	C₂	#13113424	1	1	yes	characteristic
318	C₂	#13113493	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²⋊Dic₁₃
commutator	C₂×C₄×C₁₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#2295	C₂
socle, prime radical	C₂³×C₁₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	#2295	D₁₃
hypocenter	C₁₃	#7562
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #2295

Complements

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

N	H ≅ G/N	type
25. C₄×Q₈×C₁₃	C₈	semidirect
179. C₄×Q₈	#1055	semidirect
251. C₁₃	#7562	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₄⋊#1055	2
#13114582	2
#13112662	2
#13112659	2
C₁₃×#2295	2
#7562	13