Subgroups — #15761793

Order 3328.

590 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13115631	C₂	1	2	yes	maximal, complemented
2	C₂×#11718049	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×D₄⋊#1055	C₂	1	2	yes	maximal, complemented
4	C₂×#11718047	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄×C₁₃×D₈	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₃×D₈	C₄	1	1	yes	characteristic
7	C₂×C₄×D₄×C₁₃	C₂²	1	2	yes	—
8	C₄×C₁₃×D₈	—	2	2	no	—
9	C₄×C₁₃×D₈	—	2	2	no	—
10	C₂×C₁₃×#107	C₄	1	1	yes	characteristic
11	C₂×C₁₃×#102	C₄	1	2	yes	—
12	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
13	C₂×C₂²⋊#1055	—	2	4	no	—
14	C₂×C₄⋊#1055	C₂²	1	2	yes	—
15	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
16	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
17	D₄⋊#1055	—	2	4	no	—
18	#11718049	—	2	2	no	—
19	#11718047	—	2	2	no	—
20	C₂²×D₄×C₁₃	C₂×C₄	1	2	yes	—
21	C₂³×C₄×C₁₃	—	2	2	no	—
22	C₂×C₁₃×D₈	—	2	2	no	—
23	C₂×C₁₃×D₈	—	2	2	no	—
24	C₂×C₁₃×D₈	C₈	1	4	yes	complemented
25	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
27	C₄×D₄×C₁₃	—	2	2	no	—
28	C₄×D₄×C₁₃	—	2	2	no	—
29	C₄×D₄×C₁₃	—	2	2	no	—
30	C₄×D₄×C₁₃	—	2	2	no	—
31	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
32	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
33	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
34	C₂²×#1055	—	2	1	no	—
35	C₂²×#1055	—	2	2	no	—
36	C₂²×#1055	D₄	1	2	yes	—
37	C₁₃×#107	—	2	2	no	—
38	C₁₃×#102	—	2	4	no	—
39	C₄×C₈×C₁₃	D₄	1	2	yes	—
40	C₄×C₈×C₁₃	D₄	1	2	yes	—
41	C₂²⋊#1055	—	4	8	no	—
42	C₄⋊#1055	—	2	2	no	—
43	C₄⋊#1055	—	2	2	no	—
44	C₄×#1055	—	2	1	no	—
45	C₄×#1055	—	2	1	no	—
46	C₄×#1055	—	2	1	no	—
47	C₄×#1055	D₄	1	2	yes	—
48	#9659	—	13	1	no	maximal, Sylow
49	C₂⁴×C₁₃	—	2	2	no	—
50	C₂×D₄×C₁₃	—	2	2	no	—
51	C₂×D₄×C₁₃	—	2	2	no	—
52	C₂×D₄×C₁₃	—	2	2	no	—
53	C₂×D₄×C₁₃	—	2	2	no	—
54	C₂×D₄×C₁₃	—	2	2	no	—
55	C₂×D₄×C₁₃	C₂×C₈	1	4	yes	—
56	C₂²×C₄×C₁₃	—	2	2	no	—
57	C₂²×C₄×C₁₃	—	4	2	no	—
58	C₂²×C₄×C₁₃	—	4	2	no	—
59	C₂²×C₄×C₁₃	—	4	4	no	—
60	C₂²×C₄×C₁₃	—	4	4	no	—
61	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
64	C₁₃×D₈	—	2	8	no	—
65	C₂×C₈×C₁₃	—	2	1	no	—
66	C₂×C₈×C₁₃	—	2	1	no	—
67	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
68	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
69	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
70	C₂×C₈×C₁₃	M₁₆	1	2	yes	—
71	C₁₃×C₄⋊C₄	—	2	2	no	—
72	C₁₃×C₄⋊C₄	—	2	2	no	—
73	C₁₃×C₂²⋊C₄	—	4	2	no	—
74	C₁₃×C₂²⋊C₄	—	4	2	no	—
75	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
76	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
77	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
78	C₄²×C₁₃	#41	1	1	yes	characteristic
79	C₂×#1055	—	2	2	no	—
80	C₂×#1055	—	2	2	no	—
81	C₂×#1055	—	2	4	no	—
82	C₂×#1055	—	4	2	no	—
83	C₂×#1055	—	4	2	no	—
84	C₂×#1055	—	4	4	no	—
85	C₂×C₄×D₈	D₁₃	1	1	yes	characteristic
86	#1954	—	13	2	no	—
87	C₂×#334	—	13	1	no	—
88	C₂×D₄⋊C₈	—	13	2	no	—
89	C₂×#322	—	13	1	no	—
90	C₂³×C₁₃	—	2	4	no	—
91	C₂³×C₁₃	—	4	2	no	—
92	C₂³×C₁₃	—	4	2	no	—
93	C₂³×C₁₃	—	4	4	no	—
94	C₂³×C₁₃	—	4	4	no	—
95	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
96	D₄×C₁₃	—	2	4	no	—
97	D₄×C₁₃	—	2	4	no	—
98	D₄×C₁₃	—	4	2	no	—
99	D₄×C₁₃	—	4	2	no	—
100	C₂×C₄×C₁₃	—	2	2	no	—
101	C₂×C₄×C₁₃	—	4	2	no	—
102	C₂×C₄×C₁₃	—	4	4	no	—
103	C₂×C₄×C₁₃	—	8	4	no	—
104	C₂×C₄×C₁₃	—	8	2	no	—
105	C₂×C₄×C₁₃	—	8	2	no	—
106	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
108	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
109	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
110	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
111	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
112	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
113	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
114	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
115	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
116	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	4	yes	—
117	C₈×C₁₃	—	2	2	no	—
118	C₈×C₁₃	C₂²⋊C₈	1	4	yes	complemented
119	#1055	—	4	2	no	—
120	#1055	—	4	2	no	—
121	#1055	—	8	2	no	—
122	C₂²×D₈	Dic₁₃	1	1	yes	characteristic
123	C₂×C₄×D₄	C₂×D₁₃	1	2	yes	—
124	C₄×D₈	—	2	2	no	—
125	C₄×D₈	—	2	2	no	—
126	C₂×#107	Dic₁₃	1	1	yes	characteristic
127	C₂×C₄⋊C₈	—	13	2	no	—
128	C₂×#102	Dic₁₃	1	2	yes	—
129	C₂×C₂²⋊C₈	—	26	4	no	—
130	C₂×C₄×C₈	—	13	1	no	—
131	C₂×C₄×C₈	—	13	1	no	—
132	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
133	#334	—	26	2	no	—
134	D₄⋊C₈	—	26	4	no	—
135	#322	—	26	2	no	—
136	C₂²×C₁₃	—	4	8	no	—
137	C₂²×C₁₃	—	4	4	no	—
138	C₂²×C₁₃	—	8	4	no	—
139	C₂²×C₁₃	—	8	2	no	—
140	C₂²×C₁₃	—	8	2	no	—
141	C₂²×C₁₃	#390	1	1	yes	characteristic
142	C₂²×C₁₃	C₈×D₄	1	1	yes	characteristic
143	C₂²×C₁₃	#436	1	1	yes	characteristic
144	C₂²×C₁₃	#442	1	2	yes	—
145	C₂²×C₁₃	#444	1	2	yes	—
146	C₄×C₁₃	—	2	1	no	—
147	C₄×C₁₃	—	2	1	no	—
148	C₄×C₁₃	—	8	2	no	—
149	C₄×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
150	C₄×C₁₃	#408	1	1	yes	characteristic
151	C₄×C₁₃	C₈×D₄	1	1	yes	characteristic
152	C₄×C₁₃	#436	1	1	yes	characteristic
153	C₄×C₁₃	#469	1	2	yes	—
154	C₄×C₁₃	#496	1	2	yes	—
155	C₂²×D₄	C₂×Dic₁₃	1	2	yes	—
156	C₂³×C₄	—	2	2	no	—
157	C₂×D₈	—	2	2	no	—
158	C₂×D₈	—	2	2	no	—
159	C₂×D₈	#1055	1	4	yes	complemented
160	C₂²×C₈	—	13	2	no	—
161	C₂²×C₈	—	26	1	no	—
162	C₂²×C₈	—	26	2	no	—
163	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
164	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
165	C₄×D₄	—	2	2	no	—
166	C₄×D₄	—	2	2	no	—
167	C₄×D₄	—	2	2	no	—
168	C₄×D₄	—	2	2	no	—
169	C₂×C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
170	C₂×C₂²⋊C₄	—	2	2	no	—
171	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
172	#107	—	2	2	no	—
173	C₄⋊C₈	—	26	2	no	—
174	C₄⋊C₈	—	26	2	no	—
175	#102	—	2	4	no	—
176	C₂²⋊C₈	—	52	8	no	—
177	C₄×C₈	—	13	2	no	—
178	C₄×C₈	—	26	1	no	—
179	C₄×C₈	—	26	1	no	—
180	C₄×C₈	—	26	1	no	—
181	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
182	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
183	C₂×C₁₃	—	8	4	no	—
184	C₂×C₁₃	#1580	1	2	yes	—
185	C₂×C₁₃	#1594	1	2	yes	—
186	C₂×C₁₃	#1954	1	1	yes	characteristic
187	C₂×C₁₃	#1972	1	1	yes	characteristic
188	C₂×C₁₃	#1975	1	1	yes	characteristic
189	C₂⁴	—	2	2	no	—
190	C₂×D₄	—	2	2	no	—
191	C₂×D₄	—	2	2	no	—
192	C₂×D₄	—	2	2	no	—
193	C₂×D₄	—	2	2	no	—
194	C₂×D₄	—	2	2	no	—
195	C₂×D₄	C₂×#1055	1	4	yes	—
196	C₂²×C₄	—	2	2	no	—
197	C₂²×C₄	—	4	4	no	—
198	C₂²×C₄	—	4	4	no	—
199	C₂²×C₄	—	4	2	no	—
200	C₂²×C₄	—	4	2	no	—
201	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
202	C₂²×C₄	#6129	1	1	yes	characteristic
203	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
204	D₈	—	2	8	no	—
205	C₂×C₈	—	2	1	no	—
206	C₂×C₈	—	2	1	no	—
207	C₂×C₈	—	26	4	no	—
208	C₂×C₈	—	26	2	no	—
209	C₂×C₈	—	26	2	no	—
210	C₂×C₈	—	52	2	no	—
211	C₂×C₈	—	52	4	no	—
212	C₂×C₈	—	52	2	no	—
213	C₂×C₈	C₂×#1055	1	2	yes	—
214	C₂×C₈	#6099	1	2	yes	—
215	C₂×C₈	C₂²⋊Dic₁₃	1	2	yes	—
216	C₂×C₈	C₂²⋊Dic₁₃	1	2	yes	—
217	C₄⋊C₄	—	2	2	no	—
218	C₄⋊C₄	—	2	2	no	—
219	C₂²⋊C₄	—	4	2	no	—
220	C₂²⋊C₄	—	4	2	no	—
221	C₄²	D₄×D₁₃	1	1	yes	characteristic
222	C₄²	#6129	1	1	yes	characteristic
223	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
224	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
225	C₁₃	#9659	1	1	yes	characteristic, complemented, Sylow
226	C₂³	—	2	4	no	—
227	C₂³	—	4	2	no	—
228	C₂³	—	4	4	no	—
229	C₂³	—	4	4	no	—
230	C₂³	—	4	2	no	—
231	C₂³	D₄×Dic₁₃	1	1	yes	characteristic
232	D₄	—	2	4	no	—
233	D₄	—	2	4	no	—
234	D₄	—	4	2	no	—
235	D₄	—	4	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	4	4	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	8	2	no	—
240	C₂×C₄	—	8	2	no	—
241	C₂×C₄	—	8	4	no	—
242	C₂×C₄	#88165	1	4	yes	—
243	C₂×C₄	C₂²×#1055	1	1	yes	characteristic
244	C₂×C₄	C₂×#6099	1	1	yes	characteristic
245	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
246	C₂×C₄	#88190	1	1	yes	characteristic
247	C₂×C₄	#88192	1	1	yes	characteristic
248	C₂×C₄	#88192	1	1	yes	characteristic
249	C₂×C₄	#88194	1	1	yes	characteristic
250	C₂×C₄	#88202	1	1	yes	characteristic
251	C₂×C₄	#88202	1	1	yes	characteristic
252	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
253	C₈	—	2	2	no	—
254	C₈	—	52	2	no	—
255	C₈	—	52	2	no	—
256	C₈	—	104	2	no	—
257	C₈	C₂²⋊#1055	1	4	yes	complemented
258	C₂²	—	4	8	no	—
259	C₂²	—	4	4	no	—
260	C₂²	—	8	2	no	—
261	C₂²	—	8	2	no	—
262	C₂²	—	8	4	no	—
263	C₂²	D₄×#1055	1	1	yes	characteristic
264	C₂²	#11718673	1	1	yes	characteristic
265	C₂²	#11718813	1	2	yes	—
266	C₂²	#11718858	1	2	yes	—
267	C₂²	#11718882	1	1	yes	characteristic
268	C₄	—	2	1	no	—
269	C₄	—	2	1	no	—
270	C₄	—	8	2	no	—
271	C₄	D₄×#1055	1	1	yes	characteristic
272	C₄	#11718673	1	1	yes	characteristic
273	C₄	#11718815	1	2	yes	—
274	C₄	#11718820	1	2	yes	—
275	C₄	C₂×C₂²⋊#1055	1	1	yes	characteristic
276	C₄	#11718895	1	1	yes	characteristic
277	C₂	—	8	4	no	—
278	C₂	#13113660	1	2	yes	—
279	C₂	#13113750	1	2	yes	—
280	C₂	#13115631	1	1	yes	characteristic
281	C₂	#13115713	1	1	yes	characteristic
282	C₂	#13115730	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₁₃
commutator	C₂×C₄×C₁₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₂×C₄×C₁₃×D₈	C₂
socle	C₂³×C₁₃	C₄×D₄
prime radical	C₂²×C₁₃×D₈	C₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	C₂×C₄×D₈	D₁₃
hypocenter	C₁₃	#9659
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×D₈

Complements

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

N	H ≅ G/N	type
1. #13115631	C₂	semidirect
2. C₂×#11718049	C₂	semidirect
3. C₂×D₄⋊#1055	C₂	semidirect
4. C₂×#11718047	C₂	semidirect
24. C₂×C₁₃×D₈	C₈	semidirect
118. C₈×C₁₃	C₂²⋊C₈	semidirect
159. C₂×D₈	#1055	semidirect
225. C₁₃	#9659	semidirect
257. C₈	C₂²⋊#1055	semidirect

Maximal subgroups

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

subgroup	index
#13115631	2
C₂×#11718049	2
C₂×D₄⋊#1055	2
C₂×#11718047	2
C₂×C₄×C₁₃×D₈	2
#9659	13