Subgroups — #15761706

Order 3328.

582 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13114988	C₂	1	1	yes	maximal, characteristic, complemented
2	#13114582	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×Q₈⋊#1055	C₂	1	2	yes	maximal, complemented
4	C₂×D₄⋊#1055	C₂	1	2	yes	maximal
5	C₄×C₁₃×#124	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×#124	C₄	1	1	yes	characteristic
7	C₂×C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
9	C₁₃×#391	C₄	1	1	yes	characteristic
10	C₁₃×#390	C₄	1	1	yes	characteristic
11	C₁₃×#388	—	2	2	no	—
12	C₁₃×#382	C₄	1	1	yes	characteristic
13	C₄×C₁₃×C₂²⋊C₄	—	2	2	no	—
14	C₄³×C₁₃	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊#1055	—	2	2	no	—
16	C₂×C₄⋊#1055	C₂²	1	2	yes	—
17	C₂×C₄×#1055	C₂²	1	2	yes	—
18	Q₈⋊#1055	—	2	4	no	—
19	D₄⋊#1055	—	2	4	no	—
20	#11718116	—	2	2	no	—
21	C₂²×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×D₄×C₁₃	C₂×C₄	1	1	yes	characteristic
23	C₂³×C₄×C₁₃	—	2	1	no	—
24	C₁₃×#124	—	2	2	no	—
25	C₁₃×#124	C₈	1	4	yes	complemented
26	C₄×Q₈×C₁₃	—	2	2	no	—
27	C₄×Q₈×C₁₃	—	2	2	no	—
28	C₄×D₄×C₁₃	—	2	2	no	—
29	C₄×D₄×C₁₃	—	2	2	no	—
30	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
31	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
32	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
33	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
34	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
35	C₂×C₁₃×C₂²⋊C₄	—	2	1	no	—
36	C₂×C₄²×C₁₃	—	2	1	no	—
37	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₁₃	D₄	1	2	yes	—
40	C₂×C₄²×C₁₃	D₄	1	2	yes	—
41	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
42	C₂²×#1055	—	2	2	no	—
43	C₂²×#1055	—	2	2	no	—
44	C₁₃×#95	—	2	2	no	—
45	C₁₃×#95	—	2	2	no	—
46	C₄⋊#1055	—	2	2	no	—
47	C₄⋊#1055	—	2	2	no	—
48	C₄⋊#1055	—	4	4	no	—
49	C₄×#1055	—	2	2	no	—
50	C₄×#1055	D₄	1	4	yes	—
51	#9640	—	13	1	no	maximal, Sylow
52	C₂⁴×C₁₃	—	2	1	no	—
53	C₂×Q₈×C₁₃	—	2	1	no	—
54	C₂×Q₈×C₁₃	—	2	2	no	—
55	C₂×Q₈×C₁₃	—	2	2	no	—
56	C₂×Q₈×C₁₃	C₂×C₈	1	2	yes	—
57	C₂×D₄×C₁₃	—	2	2	no	—
58	C₂×D₄×C₁₃	—	2	1	no	—
59	C₂×D₄×C₁₃	—	2	2	no	—
60	C₂×D₄×C₁₃	C₂×C₈	1	2	yes	—
61	C₂²×C₄×C₁₃	—	2	1	no	—
62	C₂²×C₄×C₁₃	—	2	1	no	—
63	C₂²×C₄×C₁₃	—	2	1	no	—
64	C₂²×C₄×C₁₃	—	4	1	no	—
65	C₂²×C₄×C₁₃	—	4	1	no	—
66	C₂²×C₄×C₁₃	—	4	2	no	—
67	C₂²×C₄×C₁₃	—	4	2	no	—
68	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
69	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
70	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
73	C₁₃×C₄⋊C₄	—	2	2	no	—
74	C₁₃×C₄⋊C₄	—	2	2	no	—
75	C₁₃×C₄⋊C₄	—	4	2	no	—
76	C₁₃×C₂²⋊C₄	—	2	8	no	—
77	C₁₃×C₂²⋊C₄	—	4	4	no	—
78	C₁₃×C₂²⋊C₄	—	4	2	no	—
79	C₄²×C₁₃	—	2	2	no	—
80	C₄²×C₁₃	—	2	2	no	—
81	C₄²×C₁₃	—	2	2	no	—
82	C₄²×C₁₃	—	4	2	no	—
83	C₄²×C₁₃	—	4	1	no	—
84	C₄²×C₁₃	—	4	1	no	—
85	C₄²×C₁₃	C₂×C₈	1	2	yes	—
86	C₄²×C₁₃	M₁₆	1	2	yes	—
87	C₄²×C₁₃	C₂×D₄	1	2	yes	—
88	C₄²×C₁₃	#41	1	2	yes	—
89	C₂×#1055	—	2	4	no	—
90	C₂×#1055	—	4	4	no	—
91	C₂×#1055	—	4	2	no	—
92	C₂×#1055	—	4	4	no	—
93	C₄×#124	D₁₃	1	1	yes	characteristic
94	#1842	—	13	1	no	—
95	#1744	—	13	1	no	—
96	C₂×Q₈⋊C₈	—	13	2	no	—
97	C₂×D₄⋊C₈	—	13	2	no	—
98	C₂³×C₁₃	—	2	2	no	—
99	C₂³×C₁₃	—	4	1	no	—
100	C₂³×C₁₃	—	4	1	no	—
101	C₂³×C₁₃	—	4	2	no	—
102	C₂³×C₁₃	—	4	2	no	—
103	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	Q₈×C₁₃	—	2	4	no	—
105	Q₈×C₁₃	—	4	2	no	—
106	D₄×C₁₃	—	2	4	no	—
107	D₄×C₁₃	—	4	2	no	—
108	C₂×C₄×C₁₃	—	2	2	no	—
109	C₂×C₄×C₁₃	—	2	2	no	—
110	C₂×C₄×C₁₃	—	2	2	no	—
111	C₂×C₄×C₁₃	—	2	2	no	—
112	C₂×C₄×C₁₃	—	2	2	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	2	no	—
120	C₂×C₄×C₁₃	—	4	2	no	—
121	C₂×C₄×C₁₃	—	8	2	no	—
122	C₂×C₄×C₁₃	—	8	1	no	—
123	C₂×C₄×C₁₃	—	8	1	no	—
124	C₂×C₄×C₁₃	C₂²⋊C₈	1	4	yes	—
125	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
126	C₂×C₄×C₁₃	#121	1	2	yes	—
127	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
128	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
129	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
130	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
131	C₂×C₄×C₁₃	#131	1	2	yes	—
132	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
133	C₂×C₄×C₁₃	#137	1	2	yes	—
134	#1055	—	4	4	no	—
135	#1055	—	8	2	no	—
136	C₂×#124	Dic₁₃	1	1	yes	characteristic
137	C₂×C₄×Q₈	C₂×D₁₃	1	1	yes	characteristic
138	C₂×C₄×D₄	C₂×D₁₃	1	1	yes	characteristic
139	C₂×C₄⋊C₈	—	13	2	no	—
140	C₂×C₄⋊C₈	—	26	2	no	—
141	C₂×C₄×C₈	—	13	2	no	—
142	#391	Dic₁₃	1	1	yes	characteristic
143	#390	Dic₁₃	1	1	yes	characteristic
144	#388	—	2	2	no	—
145	#382	Dic₁₃	1	1	yes	characteristic
146	C₄×C₂²⋊C₄	—	2	2	no	—
147	C₄³	C₂×D₁₃	1	1	yes	characteristic
148	#336	—	26	2	no	—
149	Q₈⋊C₈	—	26	4	no	—
150	D₄⋊C₈	—	26	4	no	—
151	C₂²×C₁₃	—	4	2	no	—
152	C₂²×C₁₃	—	4	4	no	—
153	C₂²×C₁₃	—	8	1	no	—
154	C₂²×C₁₃	—	8	1	no	—
155	C₂²×C₁₃	—	8	2	no	—
156	C₂²×C₁₃	#390	1	1	yes	characteristic
157	C₂²×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
158	C₂²×C₁₃	#408	1	1	yes	characteristic
159	C₂²×C₁₃	#417	1	2	yes	—
160	C₂²×C₁₃	#421	1	2	yes	—
161	C₄×C₁₃	—	2	2	no	—
162	C₄×C₁₃	—	2	4	no	—
163	C₄×C₁₃	—	4	2	no	—
164	C₄×C₁₃	—	4	2	no	—
165	C₄×C₁₃	—	8	1	no	—
166	C₄×C₁₃	—	8	1	no	—
167	C₄×C₁₃	C₈×D₄	1	2	yes	—
168	C₄×C₁₃	#436	1	2	yes	—
169	C₄×C₁₃	#489	1	2	yes	—
170	C₄×C₁₃	#496	1	1	yes	characteristic
171	C₄×C₁₃	#497	1	1	yes	characteristic
172	C₂²×Q₈	C₂×Dic₁₃	1	1	yes	characteristic
173	C₂²×D₄	C₂×Dic₁₃	1	1	yes	characteristic
174	C₂³×C₄	—	2	1	no	—
175	C₂²×C₈	—	26	2	no	—
176	C₂²×C₈	—	26	2	no	—
177	#124	—	2	2	no	—
178	#124	#1055	1	4	yes	complemented
179	C₄×Q₈	—	2	2	no	—
180	C₄×Q₈	—	2	2	no	—
181	C₄×D₄	—	2	2	no	—
182	C₄×D₄	—	2	2	no	—
183	C₂×C₄⋊C₄	—	2	1	no	—
184	C₂×C₄⋊C₄	C₂×Dic₁₃	1	1	yes	characteristic
185	C₂×C₄⋊C₄	C₂×Dic₁₃	1	1	yes	characteristic
186	C₂×C₂²⋊C₄	—	2	2	no	—
187	C₂×C₂²⋊C₄	—	2	2	no	—
188	C₂×C₂²⋊C₄	—	2	1	no	—
189	C₂×C₄²	—	2	1	no	—
190	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
191	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
192	C₂×C₄²	C₂²⋊D₁₃	1	2	yes	—
193	C₂×C₄²	C₂²⋊D₁₃	1	2	yes	—
194	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
195	C₄⋊C₈	—	26	2	no	—
196	C₄⋊C₈	—	26	2	no	—
197	C₄⋊C₈	—	52	4	no	—
198	C₄×C₈	—	13	4	no	—
199	C₄×C₈	—	26	2	no	—
200	#95	—	2	2	no	—
201	#95	—	2	2	no	—
202	C₂×C₁₃	—	8	2	no	—
203	C₂×C₁₃	#1477	1	2	yes	—
204	C₂×C₁₃	#1495	1	2	yes	—
205	C₂×C₁₃	#1954	1	1	yes	characteristic
206	C₂×C₁₃	#1959	1	1	yes	characteristic
207	C₂×C₁₃	#1963	1	1	yes	characteristic
208	C₂⁴	—	2	1	no	—
209	C₂×Q₈	—	2	1	no	—
210	C₂×Q₈	—	2	2	no	—
211	C₂×Q₈	—	2	2	no	—
212	C₂×Q₈	C₂×#1055	1	2	yes	—
213	C₂×D₄	—	2	1	no	—
214	C₂×D₄	—	2	2	no	—
215	C₂×D₄	—	2	2	no	—
216	C₂×D₄	C₂×#1055	1	2	yes	—
217	C₂²×C₄	—	2	1	no	—
218	C₂²×C₄	—	2	1	no	—
219	C₂²×C₄	—	2	1	no	—
220	C₂²×C₄	—	4	1	no	—
221	C₂²×C₄	—	4	1	no	—
222	C₂²×C₄	—	4	2	no	—
223	C₂²×C₄	—	4	2	no	—
224	C₂²×C₄	C₂²⋊Dic₁₃	1	2	yes	—
225	C₂²×C₄	C₂²⋊Dic₁₃	1	2	yes	—
226	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
227	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
228	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
229	C₂×C₈	—	26	4	no	—
230	C₂×C₈	—	52	4	no	—
231	C₂×C₈	—	52	2	no	—
232	C₂×C₈	—	52	4	no	—
233	C₄⋊C₄	—	2	2	no	—
234	C₄⋊C₄	—	2	2	no	—
235	C₄⋊C₄	—	4	2	no	—
236	C₂²⋊C₄	—	2	8	no	—
237	C₂²⋊C₄	—	4	4	no	—
238	C₂²⋊C₄	—	4	2	no	—
239	C₄²	—	2	2	no	—
240	C₄²	—	2	2	no	—
241	C₄²	—	2	2	no	—
242	C₄²	—	4	2	no	—
243	C₄²	—	4	1	no	—
244	C₄²	—	4	1	no	—
245	C₄²	C₂×#1055	1	2	yes	—
246	C₄²	#6099	1	2	yes	—
247	C₄²	D₄×D₁₃	1	2	yes	—
248	C₄²	#6129	1	2	yes	—
249	C₁₃	#9640	1	1	yes	characteristic, complemented, Sylow
250	C₂³	—	2	2	no	—
251	C₂³	—	4	1	no	—
252	C₂³	—	4	2	no	—
253	C₂³	—	4	1	no	—
254	C₂³	—	4	2	no	—
255	C₂³	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
256	Q₈	—	2	4	no	—
257	Q₈	—	4	2	no	—
258	D₄	—	2	4	no	—
259	D₄	—	4	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	4	1	no	—
268	C₂×C₄	—	4	1	no	—
269	C₂×C₄	—	4	1	no	—
270	C₂×C₄	—	4	1	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	8	2	no	—
274	C₂×C₄	—	8	1	no	—
275	C₂×C₄	—	8	1	no	—
276	C₂×C₄	C₂²⋊#1055	1	4	yes	—
277	C₂×C₄	C₂²×#1055	1	1	yes	characteristic
278	C₂×C₄	C₂×#6099	1	1	yes	characteristic
279	C₂×C₄	D₄×Dic₁₃	1	2	yes	—
280	C₂×C₄	#88190	1	1	yes	characteristic
281	C₂×C₄	#88192	1	2	yes	—
282	C₂×C₄	#88194	1	1	yes	characteristic
283	C₂×C₄	#88202	1	2	yes	—
284	C₂×C₄	#88203	1	2	yes	—
285	C₂×C₄	#88205	1	2	yes	—
286	C₈	—	52	4	no	—
287	C₈	—	104	2	no	—
288	C₂²	—	4	4	no	—
289	C₂²	—	4	2	no	—
290	C₂²	—	8	1	no	—
291	C₂²	—	8	1	no	—
292	C₂²	—	8	2	no	—
293	C₂²	C₂×C₂²⋊#1055	1	1	yes	characteristic
294	C₂²	#11718882	1	1	yes	characteristic
295	C₂²	#11718893	1	2	yes	—
296	C₂²	#11718895	1	1	yes	characteristic
297	C₂²	#11718897	1	2	yes	—
298	C₄	—	2	2	no	—
299	C₄	—	2	4	no	—
300	C₄	—	4	2	no	—
301	C₄	—	4	2	no	—
302	C₄	—	8	1	no	—
303	C₄	—	8	1	no	—
304	C₄	D₄×#1055	1	2	yes	—
305	C₄	#11718673	1	2	yes	—
306	C₄	#11718716	1	2	yes	—
307	C₄	#11718719	1	1	yes	characteristic
308	C₄	#11718721	1	1	yes	characteristic
309	C₂	—	8	2	no	—
310	C₂	#13113223	1	2	yes	—
311	C₂	#13113301	1	2	yes	—
312	C₂	#13115631	1	1	yes	characteristic
313	C₂	#13115653	1	1	yes	characteristic
314	C₂	#13115669	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₁₃
commutator	C₂×C₄×C₁₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₄×C₁₃×#124	C₂
socle	C₂³×C₁₃	C₂×C₂²⋊C₄
prime radical	C₂²×D₄×C₁₃	C₂×C₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	C₄×#124	D₁₃
hypocenter	C₁₃	#9640
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×#124

Complements

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

N	H ≅ G/N	type
1. #13114988	C₂	semidirect
2. #13114582	C₂	semidirect
3. C₂×Q₈⋊#1055	C₂	semidirect
25. C₁₃×#124	C₈	semidirect
178. #124	#1055	semidirect
249. C₁₃	#9640	semidirect

Maximal subgroups

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

subgroup	index
#13114988	2
#13114582	2
C₂×Q₈⋊#1055	2
C₂×D₄⋊#1055	2
C₄×C₁₃×#124	2
#9640	13