Subgroups — #15756103

Order 3328.

408 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	C₄×#88059	C₂	1	1	yes	maximal, characteristic, complemented
2	#13113592	C₂	1	1	yes	maximal, characteristic, complemented
3	#13113275	C₂	1	2	yes	maximal
4	#13112727	C₂	1	2	yes	maximal, complemented
5	C₈×C₁₃×#41	C₂	1	1	yes	maximal, characteristic
6	C₄×C₁₃×#41	C₂²	1	1	yes	characteristic
7	C₈×Q₈×C₁₃	C₄	1	1	yes	characteristic
8	C₈×Q₈×C₁₃	C₄	1	1	yes	characteristic
9	C₈×D₄×C₁₃	—	2	1	no	—
10	C₈×D₄×C₁₃	—	2	1	no	—
11	C₈×D₄×C₁₃	C₄	1	1	yes	characteristic
12	C₈×D₄×C₁₃	C₄	1	1	yes	characteristic
13	C₁₃×#431	—	2	1	no	—
14	C₁₃×#431	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₁₃	—	2	1	no	—
16	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
17	C₂×#88059	—	2	1	no	—
18	#11718585	C₂²	1	2	yes	—
19	C₄×#6099	C₂²	1	2	yes	—
20	#11718144	—	2	1	no	—
21	Q₈⋊#1055	—	2	2	no	—
22	D₄⋊#1055	—	2	2	no	—
23	#11718049	—	2	1	no	—
24	#11718049	—	2	1	no	—
25	C₂×C₁₃×#41	Q₈	1	1	yes	characteristic
26	C₂²×C₈×C₁₃	—	2	1	no	—
27	C₂²×C₈×C₁₃	—	2	1	no	—
28	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
30	C₄×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
31	C₄×Q₈×C₁₃	D₄	1	1	yes	characteristic
32	C₄×D₄×C₁₃	—	2	1	no	—
33	C₄×D₄×C₁₃	—	2	1	no	—
34	C₄×D₄×C₁₃	C₂×C₄	1	1	yes	characteristic
35	C₄×D₄×C₁₃	D₄	1	1	yes	characteristic
36	C₁₃×#117	—	2	1	no	—
37	C₁₃×#117	Q₈	1	1	yes	characteristic
38	C₂×C₄²×C₁₃	—	2	1	no	—
39	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
40	C₂×#6099	—	2	2	no	—
41	C₂×#6099	—	2	2	no	—
42	C₁₃×C₄⋊C₈	—	2	1	no	—
43	C₁₃×C₄⋊C₈	—	2	1	no	—
44	C₁₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
45	C₁₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
46	C₁₃×C₂²⋊C₈	—	2	1	no	—
47	C₁₃×C₂²⋊C₈	—	2	1	no	—
48	C₁₃×C₂²⋊C₈	—	2	1	no	—
49	C₄×C₈×C₁₃	—	2	1	no	—
50	C₄×C₈×C₁₃	—	2	1	no	—
51	C₄×C₈×C₁₃	—	2	1	no	—
52	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
53	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
54	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
55	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
56	#88059	—	2	4	no	—
57	C₄⋊#1055	—	2	2	no	—
58	C₄⋊#1055	—	2	2	no	—
59	#88043	—	2	2	no	—
60	C₄×#1055	—	2	2	no	—
61	#8670	—	13	1	no	maximal, Sylow
62	C₁₃×#41	—	2	2	no	—
63	C₁₃×#41	—	2	2	no	—
64	C₂×Q₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
65	C₂×D₄×C₁₃	—	2	1	no	—
66	C₂×D₄×C₁₃	C₄⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₃	—	2	1	no	—
68	C₂²×C₄×C₁₃	—	2	1	no	—
69	C₂²×C₄×C₁₃	—	2	1	no	—
70	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
72	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
73	C₂×C₈×C₁₃	—	2	2	no	—
74	C₂×C₈×C₁₃	—	2	1	no	—
75	C₂×C₈×C₁₃	—	2	1	no	—
76	C₂×C₈×C₁₃	—	4	1	no	—
77	C₂×C₈×C₁₃	—	4	2	no	—
78	C₂×C₈×C₁₃	—	4	2	no	—
79	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
83	C₁₃×C₄⋊C₄	—	2	1	no	—
84	C₁₃×C₄⋊C₄	—	2	1	no	—
85	C₁₃×C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
86	C₁₃×C₄⋊C₄	C₄⋊C₄	1	1	yes	characteristic
87	C₁₃×C₂²⋊C₄	—	2	1	no	—
88	C₁₃×C₂²⋊C₄	—	2	1	no	—
89	C₁₃×C₂²⋊C₄	—	2	1	no	—
90	C₄²×C₁₃	—	2	1	no	—
91	C₄²×C₁₃	—	2	1	no	—
92	C₄²×C₁₃	—	2	1	no	—
93	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
94	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
95	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
96	C₄²×C₁₃	#41	1	1	yes	characteristic
97	#6099	—	2	4	no	—
98	#6099	—	4	2	no	—
99	#6099	—	8	2	no	—
100	C₂×#1055	—	4	2	no	—
101	C₂×#1055	—	4	2	no	—
102	C₈×#41	D₁₃	1	1	yes	characteristic
103	C₄×#108	—	13	1	no	—
104	#1566	—	13	1	no	—
105	#1489	—	13	2	no	—
106	#1299	—	13	2	no	—
107	C₂³×C₁₃	—	2	1	no	—
108	C₂³×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
109	Q₈×C₁₃	—	2	1	no	—
110	Q₈×C₁₃	#108	1	2	yes	complemented
111	D₄×C₁₃	—	2	4	no	—
112	D₄×C₁₃	—	2	1	no	—
113	D₄×C₁₃	#108	1	2	yes	complemented
114	C₂×C₄×C₁₃	—	2	4	no	—
115	C₂×C₄×C₁₃	—	2	1	no	—
116	C₂×C₄×C₁₃	—	2	1	no	—
117	C₂×C₄×C₁₃	—	2	1	no	—
118	C₂×C₄×C₁₃	—	2	1	no	—
119	C₂×C₄×C₁₃	—	2	1	no	—
120	C₂×C₄×C₁₃	—	2	2	no	—
121	C₂×C₄×C₁₃	—	2	1	no	—
122	C₂×C₄×C₁₃	—	2	1	no	—
123	C₂×C₄×C₁₃	—	4	1	no	—
124	C₂×C₄×C₁₃	—	4	1	no	—
125	C₂×C₄×C₁₃	—	4	2	no	—
126	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
128	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
129	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
130	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
131	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
132	C₈×C₁₃	—	2	1	no	—
133	C₈×C₁₃	—	2	1	no	—
134	C₈×C₁₃	—	2	1	no	—
135	C₈×C₁₃	—	4	1	no	—
136	C₈×C₁₃	—	4	1	no	—
137	C₈×C₁₃	#104	1	2	yes	—
138	#1055	—	4	4	no	—
139	#1055	—	8	2	no	—
140	C₄×#41	C₂×D₁₃	1	1	yes	characteristic
141	C₈×Q₈	Dic₁₃	1	1	yes	characteristic
142	C₈×Q₈	Dic₁₃	1	1	yes	characteristic
143	C₈×D₄	—	2	1	no	—
144	C₈×D₄	—	2	1	no	—
145	C₈×D₄	Dic₁₃	1	1	yes	characteristic
146	C₈×D₄	Dic₁₃	1	1	yes	characteristic
147	#431	—	2	1	no	—
148	#431	C₂×D₁₃	1	1	yes	characteristic
149	C₂×#108	—	26	1	no	—
150	#423	—	13	2	no	—
151	C₄×M₁₆	—	13	2	no	—
152	C₂×C₄×C₈	—	2	1	no	—
153	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
154	#341	—	26	1	no	—
155	#334	—	26	1	no	—
156	#334	—	26	1	no	—
157	Q₈⋊C₈	—	26	2	no	—
158	D₄⋊C₈	—	26	2	no	—
159	C₂²×C₁₃	—	2	1	no	—
160	C₂²×C₁₃	—	2	1	no	—
161	C₂²×C₁₃	—	2	2	no	—
162	C₂²×C₁₃	—	2	1	no	—
163	C₂²×C₁₃	—	4	1	no	—
164	C₂²×C₁₃	—	4	1	no	—
165	C₂²×C₁₃	#380	1	1	yes	characteristic
166	C₄×C₁₃	—	2	1	no	—
167	C₄×C₁₃	—	2	1	no	—
168	C₄×C₁₃	—	2	2	no	—
169	C₄×C₁₃	—	2	1	no	—
170	C₄×C₁₃	—	4	1	no	—
171	C₄×C₁₃	—	4	1	no	—
172	C₄×C₁₃	C₂×#104	1	1	yes	characteristic
173	C₄×C₁₃	#421	1	1	yes	characteristic
174	C₄×C₁₃	C₂×#108	1	1	yes	characteristic
175	C₄×C₁₃	C₂×#108	1	1	yes	characteristic
176	C₄×C₁₃	#471	1	2	yes	—
177	C₂×#41	Dic₂₆	1	1	yes	characteristic
178	C₂×M₁₆	—	26	2	no	—
179	C₂×M₁₆	—	26	2	no	—
180	C₂²×C₈	—	2	1	no	—
181	C₂²×C₈	—	2	1	no	—
182	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
183	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
184	C₄×Q₈	D₅₂	1	1	yes	characteristic
185	C₄×Q₈	C₂×Dic₁₃	1	1	yes	characteristic
186	C₄×D₄	—	2	1	no	—
187	C₄×D₄	—	2	1	no	—
188	C₄×D₄	D₅₂	1	1	yes	characteristic
189	C₄×D₄	C₂×Dic₁₃	1	1	yes	characteristic
190	#117	—	2	1	no	—
191	#117	Dic₂₆	1	1	yes	characteristic
192	C₂×C₄²	—	2	1	no	—
193	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
194	#108	—	26	4	no	—
195	C₄⋊C₈	—	2	1	no	—
196	C₄⋊C₈	—	2	1	no	—
197	C₄⋊C₈	—	26	2	no	—
198	C₄⋊C₈	—	26	2	no	—
199	C₄⋊C₈	C₂×Dic₁₃	1	1	yes	characteristic
200	C₄⋊C₈	C₂×Dic₁₃	1	1	yes	characteristic
201	C₂²⋊C₈	—	2	1	no	—
202	C₂²⋊C₈	—	2	1	no	—
203	C₂²⋊C₈	—	2	1	no	—
204	#97	—	26	2	no	—
205	C₄×C₈	—	2	1	no	—
206	C₄×C₈	—	2	1	no	—
207	C₄×C₈	—	2	1	no	—
208	C₄×C₈	—	26	2	no	—
209	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
210	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
211	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
212	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
213	C₂×C₁₃	—	4	1	no	—
214	C₂×C₁₃	—	4	2	no	—
215	C₂×C₁₃	#1806	1	1	yes	characteristic
216	C₂×C₁₃	#1810	1	1	yes	characteristic
217	C₂×C₁₃	#1815	1	1	yes	characteristic
218	#41	—	2	2	no	—
219	#41	—	2	2	no	—
220	C₂×Q₈	C₄⋊Dic₁₃	1	1	yes	characteristic
221	C₂×D₄	—	2	1	no	—
222	C₂×D₄	C₄⋊Dic₁₃	1	1	yes	characteristic
223	C₂²×C₄	—	2	1	no	—
224	C₂²×C₄	—	2	1	no	—
225	C₂²×C₄	—	2	1	no	—
226	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
227	C₂²×C₄	#6127	1	1	yes	characteristic
228	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
229	M₁₆	—	26	4	no	—
230	M₁₆	—	52	2	no	—
231	M₁₆	—	104	2	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	1	no	—
234	C₂×C₈	—	2	1	no	—
235	C₂×C₈	—	4	1	no	—
236	C₂×C₈	—	4	2	no	—
237	C₂×C₈	—	4	2	no	—
238	C₂×C₈	—	52	2	no	—
239	C₂×C₈	—	52	2	no	—
240	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
241	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
242	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
243	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
244	C₄⋊C₄	—	2	1	no	—
245	C₄⋊C₄	—	2	1	no	—
246	C₄⋊C₄	C₄⋊Dic₁₃	1	1	yes	characteristic
247	C₄⋊C₄	C₄⋊Dic₁₃	1	1	yes	characteristic
248	C₂²⋊C₄	—	2	1	no	—
249	C₂²⋊C₄	—	2	1	no	—
250	C₂²⋊C₄	—	2	1	no	—
251	C₄²	—	2	1	no	—
252	C₄²	—	2	1	no	—
253	C₄²	—	2	1	no	—
254	C₄²	C₂×D₅₂	1	1	yes	characteristic
255	C₄²	#6127	1	1	yes	characteristic
256	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
257	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
258	C₁₃	#8670	1	1	yes	characteristic, complemented, Sylow
259	C₂³	—	2	1	no	—
260	C₂³	#88178	1	1	yes	characteristic
261	Q₈	—	2	1	no	—
262	Q₈	#88059	1	2	yes	complemented
263	D₄	—	2	4	no	—
264	D₄	—	2	1	no	—
265	D₄	#88059	1	2	yes	complemented
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	1	no	—
277	C₂×C₄	—	4	1	no	—
278	C₂×C₄	#88178	1	1	yes	characteristic
279	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
280	C₂×C₄	#88180	1	1	yes	characteristic
281	C₂×C₄	#88184	1	1	yes	characteristic
282	C₂×C₄	#88184	1	1	yes	characteristic
283	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
284	C₈	—	2	1	no	—
285	C₈	—	2	1	no	—
286	C₈	—	2	1	no	—
287	C₈	—	4	1	no	—
288	C₈	—	4	1	no	—
289	C₈	—	52	4	no	—
290	C₈	—	104	2	no	—
291	C₈	#88077	1	2	yes	—
292	C₂²	—	2	1	no	—
293	C₂²	—	2	1	no	—
294	C₂²	—	2	1	no	—
295	C₂²	—	2	2	no	—
296	C₂²	—	4	1	no	—
297	C₂²	—	4	1	no	—
298	C₂²	#11718873	1	1	yes	characteristic
299	C₄	—	2	1	no	—
300	C₄	—	2	1	no	—
301	C₄	—	2	1	no	—
302	C₄	—	2	2	no	—
303	C₄	—	4	1	no	—
304	C₄	—	4	1	no	—
305	C₄	C₂×#88059	1	1	yes	characteristic
306	C₄	C₂×#88059	1	1	yes	characteristic
307	C₄	#11718802	1	2	yes	—
308	C₄	C₂×#88077	1	1	yes	characteristic
309	C₄	#11718897	1	1	yes	characteristic
310	C₂	—	4	1	no	—
311	C₂	—	4	2	no	—
312	C₂	#13114856	1	1	yes	characteristic
313	C₂	#13114874	1	1	yes	characteristic
314	C₂	#13114895	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#88184
commutator	C₄²×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₈×C₁₃×#41	C₂
socle	C₂²×C₁₃	#380
prime radical	C₂×C₁₃×#41	Q₈
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	C₈×#41	D₁₃
hypocenter	C₁₃	#8670
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₈×#41

Complements

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

N	H ≅ G/N	type
1. C₄×#88059	C₂	semidirect
2. #13113592	C₂	semidirect
4. #13112727	C₂	semidirect
110. Q₈×C₁₃	#108	semidirect
113. D₄×C₁₃	#108	semidirect
258. C₁₃	#8670	semidirect
262. Q₈	#88059	semidirect
265. D₄	#88059	semidirect

Maximal subgroups

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

subgroup	index
C₄×#88059	2
#13113592	2
#13113275	2
#13112727	2
C₈×C₁₃×#41	2
#8670	13