Subgroups — #13123840

Order 1664.

370 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11719184	C₂	1	1	yes	maximal, characteristic
2	C₄×C₁₃×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#11718789	C₂	1	2	yes	maximal, complemented
4	#11718788	C₂	1	2	yes	maximal, complemented
5	#11718586	C₂	1	1	yes	maximal, characteristic, complemented
6	#11718370	C₂	1	2	yes	maximal, complemented
7	#11718369	C₂	1	2	yes	maximal, complemented
8	#11718349	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈×D₅₂	C₂	1	1	yes	maximal, characteristic, complemented
10	#11718340	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×Dic₂₆	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
15	#88184	C₄	1	1	yes	characteristic
16	#88183	C₄	1	2	yes	—
17	C₄×C₂²⋊D₁₃	C₂²	1	2	yes	—
18	#88178	C₄	1	1	yes	characteristic
19	C₂×#6094	C₂²	1	2	yes	—
20	C₂×C₈×D₁₃	C₂²	1	2	yes	—
21	#88130	C₄	1	2	yes	—
22	#88129	C₄	1	1	yes	characteristic
23	C₄×D₅₂	C₂²	1	1	yes	characteristic
24	#88126	C₂²	1	2	yes	—
25	#88124	C₄	1	1	yes	characteristic
26	C₄×Dic₂₆	C₂²	1	1	yes	characteristic
27	C₁₃×#97	C₂²	1	1	yes	characteristic
28	C₁₃×#97	C₂²	1	1	yes	characteristic
29	C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
30	C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
31	C₂²⋊#1055	C₂²	1	2	yes	—
32	#88060	C₂²	1	2	yes	—
33	#88060	C₂²	1	2	yes	—
34	#88055	C₂²	1	2	yes	—
35	#88054	C₂²	1	2	yes	—
36	#88054	C₂²	1	2	yes	—
37	C₈×Dic₁₃	C₂²	1	2	yes	—
38	C₄⋊#1055	C₂²	1	1	yes	characteristic
39	C₄⋊#1055	C₂²	1	1	yes	characteristic
40	C₄×#1055	C₂²	1	2	yes	—
41	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
44	C₂×C₂²⋊D₁₃	C₂×C₄	1	2	yes	—
45	C₂×D₅₂	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×D₁₃	C₂³	1	2	yes	—
47	C₂×Dic₂₆	C₂×C₄	1	1	yes	characteristic
48	C₁₃×M₁₆	—	2	2	no	—
49	C₁₃×M₁₆	—	2	2	no	—
50	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
51	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
52	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
53	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
54	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
55	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
56	C₄²×C₁₃	C₂³	1	1	yes	characteristic
57	C₄²×C₁₃	C₂³	1	1	yes	characteristic
58	C₂²⋊Dic₁₃	C₂×C₄	1	2	yes	—
59	#6103	C₂×C₄	1	2	yes	—
60	#6103	C₂×C₄	1	2	yes	—
61	#6103	C₂×C₄	1	2	yes	—
62	#6103	C₂×C₄	1	2	yes	—
63	C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
64	C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
65	#6101	C₂×C₄	1	2	yes	—
66	#6101	C₂×C₄	1	2	yes	—
67	#6101	C₂×C₄	1	2	yes	—
68	#6101	C₂×C₄	1	2	yes	—
69	C₄×Dic₁₃	C₂³	1	2	yes	—
70	C₂×#1055	C₂³	1	2	yes	—
71	C₂×#1055	C₂³	1	2	yes	—
72	#6094	—	2	4	no	—
73	C₈×D₁₃	—	2	4	no	—
74	#2969	—	13	1	no	maximal, Sylow
75	C₂³×C₁₃	C₂²×C₄	1	1	yes	characteristic
76	C₂²×D₁₃	C₂²×C₄	1	2	yes	—
77	C₂×C₄×C₁₃	—	2	2	no	—
78	C₂×C₄×C₁₃	—	2	1	no	—
79	C₂×C₄×C₁₃	—	2	1	no	—
80	C₂×C₄×C₁₃	—	4	1	no	—
81	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
87	C₂²⋊D₁₃	—	4	2	no	—
88	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
89	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
90	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
91	D₅₂	—	2	2	no	—
92	C₄×D₁₃	—	2	2	no	—
93	C₄×D₁₃	—	2	2	no	—
94	Dic₂₆	—	2	2	no	—
95	C₈×C₁₃	—	2	1	no	—
96	C₈×C₁₃	—	2	1	no	—
97	C₈×C₁₃	#41	1	2	yes	—
98	C₈×C₁₃	#41	1	2	yes	—
99	#1055	—	2	2	no	—
100	#1055	—	2	2	no	—
101	#529	—	13	1	no	—
102	#446	—	13	1	no	—
103	C₈×Q₈	—	13	1	no	—
104	#436	—	13	2	no	—
105	#436	—	13	1	no	—
106	#435	—	13	2	no	—
107	C₈×D₄	—	13	1	no	—
108	#433	—	13	2	no	—
109	#431	—	13	1	no	—
110	#405	—	13	2	no	—
111	C₄×M₁₆	D₁₃	1	1	yes	characteristic, complemented
112	C₂²×C₁₃	—	2	1	no	—
113	C₂²×C₁₃	—	2	1	no	—
114	C₂²×C₁₃	—	2	1	no	—
115	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
116	C₂×D₁₃	—	2	2	no	—
117	C₂×D₁₃	—	2	2	no	—
118	C₂×D₁₃	—	2	2	no	—
119	C₄×C₁₃	—	2	1	no	—
120	C₄×C₁₃	—	2	1	no	—
121	C₄×C₁₃	—	2	1	no	—
122	C₄×C₁₃	—	2	1	no	—
123	C₄×C₁₃	#131	1	2	yes	—
124	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
125	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
126	Dic₁₃	—	2	2	no	—
127	Dic₁₃	—	2	2	no	—
128	Dic₁₃	—	2	2	no	—
129	C₂×M₁₆	—	13	2	no	—
130	C₂×M₁₆	C₂×D₁₃	1	1	yes	characteristic
131	C₂×M₁₆	C₂×D₁₃	1	1	yes	characteristic
132	C₂²×C₈	—	13	2	no	—
133	#126	—	13	2	no	—
134	#125	—	13	1	no	—
135	#124	—	13	1	no	—
136	#123	—	13	2	no	—
137	C₂²⋊Q₈	—	13	1	no	—
138	#121	—	13	1	no	—
139	C₄×Q₈	—	13	1	no	—
140	C₄×D₄	—	13	2	no	—
141	C₄×D₄	—	13	1	no	—
142	#117	—	13	2	no	—
143	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
144	C₄⋊C₈	—	13	2	no	—
145	C₄⋊C₈	—	13	2	no	—
146	C₄⋊C₈	—	13	1	no	—
147	C₄⋊C₈	—	13	1	no	—
148	C₂²⋊C₈	—	13	2	no	—
149	C₂²⋊C₈	—	13	2	no	—
150	C₂²⋊C₈	—	13	2	no	—
151	#97	—	13	2	no	—
152	#97	C₂×D₁₃	1	1	yes	characteristic
153	#97	C₂×D₁₃	1	1	yes	characteristic
154	C₄×C₈	—	13	2	no	—
155	C₄×C₈	—	13	2	no	—
156	C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
157	C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
158	C₂×C₁₃	—	4	1	no	—
159	C₂×C₁₃	C₄×#41	1	1	yes	characteristic
160	C₂×C₁₃	C₂×#131	1	1	yes	characteristic
161	C₂×C₁₃	#568	1	1	yes	characteristic
162	D₁₃	—	4	2	no	—
163	C₂×Q₈	—	13	1	no	—
164	C₂×D₄	—	13	2	no	—
165	C₂×D₄	—	13	1	no	—
166	C₂²×C₄	—	13	2	no	—
167	C₂²×C₄	C₄×D₁₃	1	1	yes	characteristic
168	C₂²×C₄	C₄×D₁₃	1	1	yes	characteristic
169	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
170	M₁₆	—	2	2	no	—
171	M₁₆	—	2	2	no	—
172	M₁₆	—	26	4	no	—
173	C₂×C₈	—	13	2	no	—
174	C₂×C₈	—	13	2	no	—
175	C₂×C₈	—	26	4	no	—
176	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
177	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
178	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
179	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
180	C₄⋊C₄	—	13	2	no	—
181	C₄⋊C₄	—	13	2	no	—
182	C₄⋊C₄	—	13	1	no	—
183	C₄⋊C₄	—	13	2	no	—
184	C₄⋊C₄	—	13	1	no	—
185	C₄⋊C₄	—	13	2	no	—
186	C₂²⋊C₄	—	13	2	no	—
187	C₂²⋊C₄	—	13	2	no	—
188	C₂²⋊C₄	—	13	2	no	—
189	C₂²⋊C₄	—	13	2	no	—
190	C₂²⋊C₄	—	13	2	no	—
191	C₄²	—	13	2	no	—
192	C₄²	C₄×D₁₃	1	1	yes	characteristic
193	C₄²	C₄×D₁₃	1	1	yes	characteristic
194	C₄²	C₂²×D₁₃	1	1	yes	characteristic
195	C₄²	C₂²×D₁₃	1	1	yes	characteristic
196	C₁₃	#2969	1	1	yes	characteristic, complemented, Sylow
197	C₂³	—	13	2	no	—
198	C₂³	C₂×C₄×D₁₃	1	1	yes	characteristic
199	Q₈	—	26	2	no	—
200	D₄	—	26	2	no	—
201	D₄	—	52	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	4	1	no	—
206	C₂×C₄	—	13	2	no	—
207	C₂×C₄	—	13	2	no	—
208	C₂×C₄	—	13	2	no	—
209	C₂×C₄	—	26	2	no	—
210	C₂×C₄	—	26	2	no	—
211	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
212	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
213	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
214	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
215	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
216	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
217	C₈	—	2	1	no	—
218	C₈	—	2	1	no	—
219	C₈	—	26	2	no	—
220	C₈	—	26	2	no	—
221	C₈	#6127	1	2	yes	—
222	C₈	#6127	1	2	yes	—
223	C₂²	—	2	1	no	—
224	C₂²	—	2	1	no	—
225	C₂²	—	2	1	no	—
226	C₂²	—	26	2	no	—
227	C₂²	—	26	2	no	—
228	C₂²	—	26	2	no	—
229	C₂²	C₂²×C₄×D₁₃	1	1	yes	characteristic
230	C₄	—	2	1	no	—
231	C₄	—	2	1	no	—
232	C₄	—	2	1	no	—
233	C₄	—	2	1	no	—
234	C₄	—	26	2	no	—
235	C₄	—	26	2	no	—
236	C₄	—	26	2	no	—
237	C₄	#88161	1	2	yes	—
238	C₄	C₂×#6127	1	1	yes	characteristic
239	C₄	C₂×#6127	1	1	yes	characteristic
240	C₂	—	4	1	no	—
241	C₂	—	52	2	no	—
242	C₂	C₄×#6127	1	1	yes	characteristic
243	C₂	C₂×#88161	1	1	yes	characteristic
244	C₂	#11719450	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₃
commutator, socle	C₂²×C₁₃	C₂³×C₄
Fitting	C₄×C₁₃×M₁₆	C₂
prime radical	#88184	C₄
cosocle	C₂×C₄×C₁₃	C₂⁴
hypercenter	C₄×M₁₆	D₁₃
hypocenter	C₁₃	#2969
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆

Complements

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

N	H ≅ G/N	type
2. C₄×C₁₃×M₁₆	C₂	semidirect
3. #11718789	C₂	semidirect
4. #11718788	C₂	semidirect
5. #11718586	C₂	semidirect
6. #11718370	C₂	semidirect
7. #11718369	C₂	semidirect
8. #11718349	C₂	semidirect
9. C₈×D₅₂	C₂	semidirect
10. #11718340	C₂	semidirect
11. C₈×Dic₂₆	C₂	semidirect
111. C₄×M₁₆	D₁₃	semidirect
196. C₁₃	#2969	semidirect

Maximal subgroups

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

subgroup	index
#11719184	2
C₄×C₁₃×M₁₆	2
#11718789	2
#11718788	2
#11718586	2
#11718370	2
#11718369	2
#11718349	2
C₈×D₅₂	2
#11718340	2
C₈×Dic₂₆	2
#2969	13