Subgroups — #15828639

Order 3328.

814 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13123802	C₂	1	2	yes	maximal
2	C₄×#88155	C₂	1	1	yes	maximal, characteristic
3	#13113138	C₂	1	2	yes	maximal, complemented
4	#13113130	C₂	1	2	yes	maximal, complemented
5	#13113125	C₂	1	2	yes	maximal, complemented
6	#13113119	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×#6094	C₂	1	4	yes	maximal, complemented
8	C₂×C₈²×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#88155	C₄	1	2	yes	—
10	C₄×#6127	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
13	#11718772	C₄	1	8	yes	—
14	#11718762	C₄	1	2	yes	—
15	#11718586	C₂²	1	2	yes	—
16	C₄×#6099	C₂²	1	1	yes	characteristic
17	#11718350	C₄	1	4	yes	—
18	#11718349	C₄	1	4	yes	—
19	C₄×#6094	C₂²	1	4	yes	—
20	C₄×#6094	C₂²	1	2	yes	—
21	#11718346	C₂²	1	4	yes	—
22	C₄×C₈×D₁₃	C₂²	1	2	yes	—
23	#11718340	C₄	1	4	yes	—
24	C₈²×C₁₃	C₂²	1	2	yes	—
25	C₈²×C₁₃	C₂²	1	1	yes	characteristic
26	C₈²×C₁₃	C₂²	1	1	yes	characteristic
27	#11718047	C₂²	1	4	yes	—
28	#11718047	C₂²	1	2	yes	—
29	#11718046	C₂²	1	4	yes	—
30	C₈×#1055	C₂²	1	2	yes	—
31	C₂×#6127	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
33	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
35	C₄×C₂²⋊D₁₃	C₂×C₄	1	4	yes	—
36	#88180	C₂×C₄	1	1	yes	characteristic
37	C₂×#6099	C₂×C₄	1	2	yes	—
38	#88155	—	2	8	no	—
39	C₂×#6094	C₂×C₄	1	4	yes	—
40	C₂×#6094	C₂×C₄	1	8	yes	—
41	C₂×C₈×D₁₃	C₂×C₄	1	4	yes	—
42	C₄×D₅₂	C₂×C₄	1	2	yes	—
43	#88126	C₂×C₄	1	2	yes	—
44	C₄²×D₁₃	C₂³	1	2	yes	—
45	C₄×Dic₂₆	C₂×C₄	1	2	yes	—
46	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
47	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
48	C₄×C₈×C₁₃	C₂³	1	2	yes	—
49	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
50	C₄×C₈×C₁₃	C₂³	1	2	yes	—
51	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
52	C₂²⋊#1055	C₂×C₄	1	4	yes	—
53	#88060	C₂×C₄	1	8	yes	—
54	#88055	C₂×C₄	1	4	yes	—
55	#88055	C₂×C₄	1	8	yes	—
56	#88054	C₂×C₄	1	8	yes	—
57	C₈×Dic₁₃	C₂×C₄	1	4	yes	—
58	C₄⋊#1055	C₂×C₄	1	4	yes	—
59	#88043	C₂×C₄	1	2	yes	—
60	C₄×#1055	C₂³	1	4	yes	—
61	C₄×#1055	C₂³	1	2	yes	—
62	#17335	—	13	1	no	maximal, Sylow
63	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₂²⋊D₁₃	C₄²	1	2	yes	—
66	#6127	—	2	4	no	—
67	C₂×D₅₂	C₄²	1	1	yes	characteristic
68	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
69	C₂×C₄×D₁₃	C₂²×C₄	1	4	yes	—
70	C₂×Dic₂₆	C₄²	1	1	yes	characteristic
71	C₂×C₈×C₁₃	—	2	8	no	—
72	C₂×C₈×C₁₃	M₁₆	1	8	yes	complemented
73	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
74	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
75	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
76	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
77	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
78	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
79	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
80	C₂²⋊Dic₁₃	C₄²	1	2	yes	—
81	#6103	C₄²	1	4	yes	—
82	C₄⋊Dic₁₃	C₄²	1	2	yes	—
83	#6101	C₄²	1	4	yes	—
84	C₄×Dic₁₃	C₂²×C₄	1	4	yes	—
85	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
86	#6099	—	2	4	no	—
87	C₂×#1055	C₂²×C₄	1	8	yes	—
88	C₂×#1055	C₂²×C₄	1	4	yes	—
89	#6094	—	2	8	no	—
90	#6094	—	2	16	no	—
91	C₈×D₁₃	—	2	8	no	—
92	#2965	—	13	2	no	—
93	C₄×#131	—	13	1	no	—
94	#1455	—	13	2	no	—
95	#1453	—	13	2	no	—
96	#1451	—	13	2	no	—
97	#1450	—	13	1	no	—
98	C₈×M₁₆	—	13	4	no	—
99	C₂×C₈²	D₁₃	1	1	yes	characteristic, complemented
100	C₂³×C₁₃	C₂×C₄²	1	1	yes	characteristic
101	C₂²×D₁₃	C₂×C₄²	1	2	yes	—
102	C₂×C₄×C₁₃	—	2	4	no	—
103	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
104	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
105	C₂×C₄×C₁₃	C₂×M₁₆	1	4	yes	—
106	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
107	C₂×C₄×C₁₃	C₂³×C₄	1	2	yes	—
108	C₂²⋊D₁₃	—	2	4	no	—
109	C₂×Dic₁₃	C₂×C₄²	1	2	yes	—
110	C₂×Dic₁₃	C₂×C₄²	1	4	yes	—
111	D₅₂	—	2	2	no	—
112	C₄×D₁₃	—	2	4	no	—
113	C₄×D₁₃	—	2	8	no	—
114	Dic₂₆	—	2	2	no	—
115	C₈×C₁₃	—	2	4	no	—
116	C₈×C₁₃	C₂×M₁₆	1	4	yes	—
117	C₈×C₁₃	C₂×M₁₆	1	4	yes	—
118	C₈×C₁₃	#131	1	8	yes	complemented
119	#1055	—	2	4	no	—
120	#1055	—	2	8	no	—
121	C₂×#131	—	13	2	no	—
122	C₄×#41	—	13	1	no	—
123	#446	—	13	4	no	—
124	#436	—	13	4	no	—
125	#435	—	13	8	no	—
126	#431	—	13	2	no	—
127	#431	—	13	4	no	—
128	#405	—	13	2	no	—
129	#405	—	13	4	no	—
130	C₄×M₁₆	—	13	1	no	—
131	C₄×M₁₆	—	13	2	no	—
132	C₄×M₁₆	—	13	4	no	—
133	C₂×C₄×C₈	—	13	2	no	—
134	C₂×C₄×C₈	C₂×D₁₃	1	2	yes	—
135	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
136	#322	—	13	4	no	—
137	#322	—	13	4	no	—
138	#322	—	13	2	no	—
139	C₈²	—	13	2	no	—
140	C₈²	C₂×D₁₃	1	1	yes	characteristic
141	C₈²	C₂×D₁₃	1	1	yes	characteristic
142	C₈²	C₂×D₁₃	1	2	yes	—
143	C₂²×C₁₃	—	2	2	no	—
144	C₂²×C₁₃	C₄×M₁₆	1	2	yes	—
145	C₂²×C₁₃	C₂²×C₄²	1	1	yes	characteristic
146	C₂×D₁₃	—	2	2	no	—
147	C₂×D₁₃	—	2	4	no	—
148	C₄×C₁₃	—	2	2	no	—
149	C₄×C₁₃	C₄×M₁₆	1	2	yes	—
150	C₄×C₁₃	C₂²×M₁₆	1	2	yes	—
151	C₄×C₁₃	C₂×#131	1	4	yes	—
152	Dic₁₃	—	2	2	no	—
153	Dic₁₃	—	2	4	no	—
154	C₂×#41	—	13	1	no	—
155	#131	—	26	8	no	—
156	C₂×M₁₆	—	13	2	no	—
157	C₂×M₁₆	—	13	4	no	—
158	C₂×M₁₆	—	13	8	no	—
159	C₂²×C₈	—	13	4	no	—
160	C₂²×C₈	C₄×D₁₃	1	4	yes	—
161	C₂²×C₈	C₄×D₁₃	1	2	yes	—
162	C₄×Q₈	—	13	2	no	—
163	C₄×D₄	—	13	4	no	—
164	C₄×D₄	—	13	2	no	—
165	#117	—	13	1	no	—
166	#117	—	13	2	no	—
167	C₂×C₄²	—	13	2	no	—
168	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
169	C₄⋊C₈	—	13	8	no	—
170	C₄⋊C₈	—	13	4	no	—
171	C₂²⋊C₈	—	13	4	no	—
172	C₂²⋊C₈	—	13	8	no	—
173	#97	—	13	4	no	—
174	#97	—	13	8	no	—
175	#97	—	13	2	no	—
176	C₄×C₈	—	13	4	no	—
177	C₄×C₈	—	13	4	no	—
178	C₄×C₈	—	13	2	no	—
179	C₄×C₈	C₄×D₁₃	1	2	yes	—
180	C₄×C₈	C₄×D₁₃	1	4	yes	—
181	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
182	C₄×C₈	C₂²×D₁₃	1	2	yes	—
183	C₄×C₈	C₂²×D₁₃	1	2	yes	—
184	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
185	C₂×C₁₃	—	2	2	no	—
186	C₂×C₁₃	C₂×C₄×M₁₆	1	1	yes	characteristic
187	C₂×C₁₃	C₄×#131	1	2	yes	—
188	D₁₃	—	4	2	no	—
189	#41	—	26	4	no	—
190	C₂×Q₈	—	13	1	no	—
191	C₂×D₄	—	13	2	no	—
192	C₂×D₄	—	13	1	no	—
193	C₂²×C₄	—	13	4	no	—
194	C₂²×C₄	—	13	2	no	—
195	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
196	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
197	M₁₆	—	26	4	no	—
198	M₁₆	—	26	8	no	—
199	M₁₆	—	26	16	no	—
200	C₂×C₈	—	2	8	no	—
201	C₂×C₈	—	13	4	no	—
202	C₂×C₈	—	13	8	no	—
203	C₂×C₈	—	26	8	no	—
204	C₂×C₈	#6094	1	8	yes	complemented
205	C₂×C₈	C₂×C₄×D₁₃	1	4	yes	—
206	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
207	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
208	C₂×C₈	C₂×C₄×D₁₃	1	4	yes	—
209	C₄⋊C₄	—	13	4	no	—
210	C₄⋊C₄	—	13	2	no	—
211	C₂²⋊C₄	—	13	2	no	—
212	C₂²⋊C₄	—	13	4	no	—
213	C₄²	—	13	2	no	—
214	C₄²	—	13	4	no	—
215	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
216	C₄²	C₂×C₄×D₁₃	1	2	yes	—
217	C₄²	C₂³×D₁₃	1	1	yes	characteristic
218	C₁₃	#17335	1	1	yes	characteristic, complemented, Sylow
219	C₂³	—	13	2	no	—
220	C₂³	C₄²×D₁₃	1	1	yes	characteristic
221	Q₈	—	26	2	no	—
222	D₄	—	26	4	no	—
223	D₄	—	26	2	no	—
224	C₂×C₄	—	2	4	no	—
225	C₂×C₄	—	13	4	no	—
226	C₂×C₄	—	13	2	no	—
227	C₂×C₄	—	26	4	no	—
228	C₂×C₄	—	26	8	no	—
229	C₂×C₄	C₄²×D₁₃	1	2	yes	—
230	C₂×C₄	C₄²×D₁₃	1	1	yes	characteristic
231	C₂×C₄	C₂×#6094	1	4	yes	—
232	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
233	C₂×C₄	C₂²×C₄×D₁₃	1	2	yes	—
234	C₈	—	2	4	no	—
235	C₈	—	26	4	no	—
236	C₈	—	26	8	no	—
237	C₈	C₂×#6094	1	4	yes	—
238	C₈	C₂×#6094	1	4	yes	—
239	C₈	#88155	1	8	yes	complemented
240	C₂²	—	2	2	no	—
241	C₂²	—	26	2	no	—
242	C₂²	—	26	4	no	—
243	C₂²	C₄×#6094	1	2	yes	—
244	C₂²	C₂×C₄²×D₁₃	1	1	yes	characteristic
245	C₄	—	2	2	no	—
246	C₄	—	26	2	no	—
247	C₄	—	26	4	no	—
248	C₄	C₄×#6094	1	2	yes	—
249	C₄	C₂²×#6094	1	2	yes	—
250	C₄	C₂×#88155	1	4	yes	—
251	C₂	—	2	2	no	—
252	C₂	—	52	2	no	—
253	C₂	C₂×C₄×#6094	1	1	yes	characteristic
254	C₂	C₄×#88155	1	2	yes	—

	subgroup	G/H
center	C₄×C₈	C₂²×D₁₃
commutator, socle	C₂²×C₁₃	C₂²×C₄²
Frattini	C₄²	C₂³×D₁₃
Fitting	C₂×C₈²×C₁₃	C₂
prime radical	C₂×#6127	C₂×C₄
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	C₂×C₈²	D₁₃
hypocenter	C₁₃	#17335
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄×C₈ ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
3. #13113138	C₂	semidirect
4. #13113130	C₂	semidirect
5. #13113125	C₂	semidirect
6. #13113119	C₂	semidirect
7. C₈×#6094	C₂	semidirect
8. C₂×C₈²×C₁₃	C₂	semidirect
72. C₂×C₈×C₁₃	M₁₆	semidirect
99. C₂×C₈²	D₁₃	semidirect
118. C₈×C₁₃	#131	semidirect
204. C₂×C₈	#6094	semidirect
218. C₁₃	#17335	semidirect
239. C₈	#88155	semidirect

Maximal subgroups

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

subgroup	index
#13123802	2
C₄×#88155	2
#13113138	2
#13113130	2
#13113125	2
#13113119	2
C₈×#6094	2
C₂×C₈²×C₁₃	2
#17335	13