Subgroups — #16015624

Order 3328.

1114 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13129547	C₂	1	1	yes	maximal, characteristic
2	#13127939	C₂	1	1	yes	maximal, characteristic, complemented
3	#13116463	C₂	1	6	yes	maximal, complemented
4	C₂×#11718100	C₂	1	6	yes	maximal, complemented
5	C₂×C₁₃×#504	C₂	1	1	yes	maximal, characteristic, complemented
6	#11719648	C₄	1	1	yes	characteristic
7	C₂×C₁₃×#131	C₂²	1	1	yes	characteristic
8	#11719524	C₄	1	1	yes	characteristic
9	#11719455	—	2	4	no	—
10	D₁₃×#131	—	2	4	no	—
11	#11719450	C₄	1	6	yes	—
12	C₂×#88155	C₂²	1	6	yes	—
13	C₁₃×#504	—	2	4	no	—
14	C₂×C₁₃×M₃₂	C₂²	1	3	yes	complemented
15	C₂²×C₁₃×C₁₆	C₂²	1	3	yes	complemented
16	#11718801	—	2	4	no	—
17	C₂×#88052	C₂²	1	6	yes	complemented
18	#11718100	C₂²	1	4	yes	complemented
19	#11718100	C₂²	1	12	yes	complemented
20	C₂×C₁₃×#41	C₂×C₄	1	1	yes	characteristic
21	#88257	C₈	1	4	yes	—
22	#88256	C₈	1	4	yes	—
23	#41×D₁₃	—	2	4	no	—
24	#88254	—	2	1	no	—
25	#88251	—	2	3	no	—
26	C₂×#6127	C₂×C₄	1	6	yes	—
27	C₁₃×#131	D₄	1	4	yes	—
28	C₁₃×#131	D₄	1	4	yes	—
29	C₂×C₁₃×M₁₆	C₂×C₄	1	3	yes	—
30	C₂²×C₈×C₁₃	C₂³	1	3	yes	—
31	#88202	—	2	4	no	—
32	C₂×#6099	C₂×C₄	1	6	yes	—
33	#88161	—	2	12	no	—
34	M₁₆×D₁₃	—	2	12	no	—
35	#88155	—	2	6	no	—
36	#88155	—	2	6	no	—
37	#88155	C₂×C₄	1	12	yes	—
38	#88155	C₂³	1	4	yes	—
39	C₂×#6094	—	2	6	no	—
40	C₂×C₈×D₁₃	—	2	6	no	—
41	C₁₃×M₃₂	—	2	6	no	—
42	C₂×C₁₃×C₁₆	—	2	6	no	—
43	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
44	C₂×C₁₃×C₁₆	C₂³	1	3	yes	—
45	#88052	—	2	6	no	—
46	#88052	C₂³	1	4	yes	—
47	C₂×#6090	—	2	6	no	—
48	#33340	—	13	1	no	maximal, Sylow
49	C₁₃×#41	C₂²⋊C₄	1	4	yes	—
50	C₁₃×#41	C₂×C₈	1	4	yes	—
51	C₂×Q₈×C₁₃	M₁₆	1	1	yes	characteristic
52	C₂×D₄×C₁₃	M₁₆	1	3	yes	—
53	C₂²×C₄×C₁₃	C₂²×C₄	1	3	yes	—
54	C₂×C₂²⋊D₁₃	—	2	6	no	—
55	#6131	—	2	4	no	—
56	Q₈×D₁₃	—	2	4	no	—
57	#6129	—	2	12	no	—
58	D₄×D₁₃	—	2	12	no	—
59	#6127	—	2	6	no	—
60	#6127	—	2	6	no	—
61	#6127	C₂×C₈	1	12	yes	—
62	#6127	C₂²×C₄	1	4	yes	—
63	C₂×D₅₂	C₂×C₈	1	6	yes	—
64	C₂×C₄×D₁₃	—	2	6	no	—
65	C₂×Dic₂₆	C₂×C₈	1	6	yes	—
66	C₁₃×M₁₆	C₂²⋊C₄	1	12	yes	—
67	C₂×C₈×C₁₃	C₂²×C₄	1	3	yes	—
68	C₂×C₈×C₁₃	C₂×D₄	1	6	yes	—
69	C₂×C₈×C₁₃	C₂×D₄	1	6	yes	—
70	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
71	#6099	—	2	6	no	—
72	#6099	C₂²×C₄	1	4	yes	—
73	C₂×#1055	—	2	6	no	—
74	#6094	—	2	12	no	—
75	#6094	—	2	4	no	—
76	C₈×D₁₃	—	2	12	no	—
77	C₈×D₁₃	—	2	4	no	—
78	C₁₃×C₁₆	—	2	1	no	—
79	C₁₃×C₁₆	—	2	3	no	—
80	#6090	—	2	4	no	—
81	#3573	—	13	1	no	—
82	#3407	—	13	1	no	—
83	C₂×#504	D₁₃	1	1	yes	characteristic, complemented
84	#2117	—	13	6	no	—
85	C₂×#350	—	13	6	no	—
86	C₂³×C₁₃	C₂×M₁₆	1	3	yes	—
87	C₂²×D₁₃	—	2	6	no	—
88	Q₈×C₁₃	C₂²⋊C₈	1	4	yes	—
89	D₄×C₁₃	C₂²⋊C₈	1	12	yes	—
90	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	6	yes	—
91	C₂×C₄×C₁₃	C₂²×C₈	1	6	yes	—
92	C₂×C₄×C₁₃	C₂×M₁₆	1	3	yes	—
93	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
94	C₂²⋊D₁₃	—	2	4	no	—
95	C₂²⋊D₁₃	—	2	12	no	—
96	C₂×Dic₁₃	—	2	6	no	—
97	D₅₂	—	2	6	no	—
98	D₅₂	C₂²×C₈	1	4	yes	—
99	C₄×D₁₃	—	2	4	no	—
100	C₄×D₁₃	—	2	12	no	—
101	Dic₂₆	—	2	6	no	—
102	Dic₂₆	C₂²×C₈	1	4	yes	—
103	C₈×C₁₃	C₂×C₂²⋊C₄	1	6	yes	—
104	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
105	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
106	#1055	—	2	4	no	—
107	#585	—	13	1	no	—
108	#568	—	13	1	no	—
109	#568	—	13	6	no	—
110	#568	—	26	4	no	—
111	C₂×#131	—	13	6	no	—
112	C₂×#131	—	26	4	no	—
113	C₂×#131	C₂×D₁₃	1	1	yes	characteristic
114	#504	—	2	4	no	—
115	#504	—	26	4	no	—
116	C₂×M₃₂	—	13	6	no	—
117	C₂×M₃₂	C₂×D₁₃	1	3	yes	complemented
118	C₂²×C₁₆	C₂×D₁₃	1	3	yes	complemented
119	#350	—	13	4	no	—
120	#350	—	13	12	no	—
121	C₂²×C₁₃	—	4	3	no	—
122	C₂²×C₁₃	C₂×C₂²⋊C₈	1	6	yes	—
123	C₂²×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
124	C₂×D₁₃	—	2	4	no	—
125	C₂×D₁₃	—	8	6	no	—
126	C₄×C₁₃	C₂×C₂²⋊C₈	1	6	yes	—
127	C₄×C₁₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
128	C₄×C₁₃	C₂³×C₈	1	1	yes	characteristic
129	Dic₁₃	—	2	4	no	—
130	2₋¹⁺⁴	—	13	4	no	—
131	2₋¹⁺⁴	—	26	1	no	—
132	H(5,𝔽₂)	—	13	4	no	—
133	H(5,𝔽₂)	—	26	3	no	—
134	C₂×#41	—	13	6	no	—
135	C₂×#41	—	26	4	no	—
136	C₂×#41	C₄×D₁₃	1	1	yes	characteristic
137	#131	—	13	12	no	—
138	#131	—	13	4	no	—
139	#131	—	26	4	no	—
140	#131	—	26	12	no	—
141	#131	—	26	6	no	—
142	#131	—	26	6	no	—
143	#131	D₅₂	1	4	yes	—
144	#131	C₂²⋊D₁₃	1	4	yes	—
145	C₂×M₁₆	—	13	6	no	—
146	C₂×M₁₆	—	26	12	no	—
147	C₂×M₁₆	—	26	6	no	—
148	C₂×M₁₆	C₄×D₁₃	1	3	yes	—
149	C₂²×C₈	—	26	6	no	—
150	C₂²×C₈	C₂²×D₁₃	1	3	yes	—
151	M₃₂	—	2	6	no	—
152	M₃₂	—	13	4	no	—
153	M₃₂	—	26	6	no	—
154	C₂×C₁₆	—	2	6	no	—
155	C₂×C₁₆	—	26	6	no	—
156	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
157	C₂×C₁₆	C₂²×D₁₃	1	3	yes	—
158	C₂×C₁₃	—	2	6	no	—
159	C₂×C₁₃	—	2	1	no	—
160	C₂×C₁₃	C₂²×C₂²⋊C₈	1	1	yes	characteristic
161	D₁₃	—	4	4	no	—
162	#41	—	13	12	no	—
163	#41	—	13	4	no	—
164	#41	—	26	4	no	—
165	#41	—	26	12	no	—
166	#41	—	26	6	no	—
167	#41	—	26	6	no	—
168	#41	C₈×D₁₃	1	4	yes	—
169	#41	#6103	1	4	yes	—
170	C₂×Q₈	—	13	6	no	—
171	C₂×Q₈	—	26	4	no	—
172	C₂×Q₈	#6094	1	1	yes	characteristic
173	C₂×D₄	—	13	6	no	—
174	C₂×D₄	—	26	6	no	—
175	C₂×D₄	—	26	12	no	—
176	C₂×D₄	#6094	1	3	yes	—
177	C₂²×C₄	—	26	6	no	—
178	C₂²×C₄	C₂×C₄×D₁₃	1	3	yes	—
179	M₁₆	—	13	4	no	—
180	M₁₆	—	26	6	no	—
181	M₁₆	—	26	12	no	—
182	M₁₆	—	26	4	no	—
183	M₁₆	#6103	1	12	yes	—
184	C₂×C₈	—	26	6	no	—
185	C₂×C₈	—	26	12	no	—
186	C₂×C₈	—	26	4	no	—
187	C₂×C₈	C₂×C₄×D₁₃	1	3	yes	—
188	C₂×C₈	C₂×D₅₂	1	6	yes	—
189	C₂×C₈	C₂×C₂²⋊D₁₃	1	6	yes	—
190	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
191	C₁₆	—	2	1	no	—
192	C₁₆	—	2	3	no	—
193	C₁₆	—	26	4	no	—
194	C₁₃	#33340	1	1	yes	characteristic, complemented, Sylow
195	C₂³	—	26	6	no	—
196	C₂³	C₂×#6094	1	3	yes	—
197	Q₈	—	13	4	no	—
198	Q₈	—	26	6	no	—
199	Q₈	#88060	1	4	yes	—
200	D₄	—	13	4	no	—
201	D₄	—	26	4	no	—
202	D₄	—	26	12	no	—
203	D₄	—	26	6	no	—
204	D₄	#88060	1	12	yes	—
205	C₂×C₄	—	26	6	no	—
206	C₂×C₄	—	26	4	no	—
207	C₂×C₄	—	26	12	no	—
208	C₂×C₄	C₂×C₈×D₁₃	1	6	yes	—
209	C₂×C₄	C₂×#6094	1	3	yes	—
210	C₂×C₄	C₂×#6103	1	6	yes	—
211	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
212	C₈	—	26	4	no	—
213	C₈	C₂×#6103	1	6	yes	—
214	C₈	C₂²×D₅₂	1	1	yes	characteristic
215	C₈	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
216	C₂²	—	4	3	no	—
217	C₂²	—	26	4	no	—
218	C₂²	—	104	6	no	—
219	C₂²	C₂×#88060	1	6	yes	—
220	C₂²	C₂²×#6094	1	1	yes	characteristic
221	C₄	—	26	4	no	—
222	C₄	C₂×#88060	1	6	yes	—
223	C₄	C₂²×C₈×D₁₃	1	1	yes	characteristic
224	C₄	C₂²×#6103	1	1	yes	characteristic
225	C₂	—	2	1	no	—
226	C₂	—	2	6	no	—
227	C₂	—	52	4	no	—
228	C₂	C₂²×#88060	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂²×D₅₂
commutator	C₄×C₁₃	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₂×C₁₃×#504	C₂
socle	C₂×C₁₃	C₂²×C₂²⋊C₈
prime radical	#11719648	C₄
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	C₂×#504	D₁₃
hypocenter	C₁₃	#33340
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₈ ◁ C₂×#131 ◁ C₂×#504

Complements

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

N	H ≅ G/N	type
2. #13127939	C₂	semidirect
3. #13116463	C₂	semidirect
4. C₂×#11718100	C₂	semidirect
5. C₂×C₁₃×#504	C₂	semidirect
14. C₂×C₁₃×M₃₂	C₂²	semidirect
15. C₂²×C₁₃×C₁₆	C₂²	semidirect
17. C₂×#88052	C₂²	semidirect
18. #11718100	C₂²	semidirect
19. #11718100	C₂²	semidirect
83. C₂×#504	D₁₃	semidirect
117. C₂×M₃₂	C₂×D₁₃	semidirect
118. C₂²×C₁₆	C₂×D₁₃	semidirect
194. C₁₃	#33340	semidirect

Maximal subgroups

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

subgroup	index
#13129547	2
#13127939	2
#13116463	2
C₂×#11718100	2
C₂×C₁₃×#504	2
#33340	13