Subgroups — #15778264

Order 3328.

398 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#6090	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₆×Dic₁₃	C₂	1	2	yes	maximal
3	C₈×C₄⋊Dic₁₃	C₂	1	1	yes	maximal, characteristic
4	#13112913	C₂	1	2	yes	maximal
5	C₂×C₁₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
7	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×C₈×Dic₁₃	C₂²	1	2	yes	—
9	C₂²×#6090	C₂²	1	1	yes	characteristic
10	C₂²×#6090	C₂²	1	1	yes	characteristic
11	C₄×C₄⋊Dic₁₃	C₄	1	1	yes	characteristic
12	C₂×C₄⋊#1055	C₄	1	1	yes	characteristic
13	C₁₃×C₄⋊C₁₆	C₄	1	4	yes	complemented
14	#11718142	C₄	1	2	yes	—
15	C₁₆×Dic₁₃	—	2	4	no	—
16	C₄×#6090	C₄	1	4	yes	complemented
17	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
19	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄⋊Dic₁₃	C₈	1	2	yes	—
21	C₂×C₄×Dic₁₃	C₂×C₄	1	2	yes	—
22	C₂²×#1055	C₂×C₄	1	2	yes	—
23	C₂×C₁₃×C₁₆	—	2	2	no	—
24	C₂×C₁₃×C₁₆	—	2	2	no	—
25	C₂×C₁₃×C₁₆	C₂×C₄	1	4	yes	—
26	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
27	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
28	#88071	C₈	1	2	yes	—
29	C₈×Dic₁₃	—	2	2	no	—
30	C₈×Dic₁₃	—	2	2	no	—
31	C₂×#6090	—	2	1	no	—
32	C₂×#6090	—	2	1	no	—
33	C₂×#6090	C₂×C₄	1	2	yes	—
34	C₂×#6090	C₂×C₄	1	2	yes	—
35	C₂×#6090	D₄	1	2	yes	—
36	C₂×#6090	Q₈	1	2	yes	—
37	C₄⋊#1055	C₈	1	4	yes	—
38	C₁₆×C₄⋊C₄	—	13	1	no	maximal, Sylow
39	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
40	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
41	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
42	C₂²×Dic₁₃	C₂×C₈	1	4	yes	—
43	C₂×C₈×C₁₃	—	2	1	no	—
44	C₂×C₈×C₁₃	—	2	1	no	—
45	C₂×C₈×C₁₃	C₄²	1	2	yes	—
46	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
49	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
51	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
52	C₄²×C₁₃	C₄²	1	2	yes	—
53	C₄²×C₁₃	C₂×C₈	1	2	yes	—
54	C₄⋊Dic₁₃	C₁₆	1	8	yes	complemented
55	C₄×Dic₁₃	—	2	2	no	—
56	C₄×Dic₁₃	—	2	2	no	—
57	C₂×#1055	—	2	2	no	—
58	C₂×#1055	—	2	2	no	—
59	C₂×#1055	C₂×C₈	1	4	yes	—
60	C₁₃×C₁₆	—	2	4	no	—
61	#6090	—	2	2	no	—
62	#6090	C₄⋊C₄	1	4	yes	complemented
63	C₂×C₄⋊C₁₆	D₁₃	1	1	yes	characteristic
64	C₂×C₄×C₁₆	—	13	1	no	—
65	C₂×C₄×C₁₆	—	13	2	no	—
66	C₈×C₄⋊C₄	—	13	1	no	—
67	#1374	—	13	2	no	—
68	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
69	C₂×C₄×C₁₃	—	2	2	no	—
70	C₂×C₄×C₁₃	C₄×C₈	1	2	yes	—
71	C₂×C₄×C₁₃	C₄×C₈	1	2	yes	—
72	C₂×C₄×C₁₃	C₂×C₁₆	1	4	yes	—
73	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
74	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
75	C₂×C₄×C₁₃	C₄×Q₈	1	1	yes	characteristic
76	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
77	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
78	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
79	C₂×Dic₁₃	—	2	4	no	—
80	C₂×Dic₁₃	—	2	4	no	—
81	C₂×Dic₁₃	C₂×C₁₆	1	8	yes	—
82	C₈×C₁₃	—	2	2	no	—
83	C₈×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₈×C₁₃	#117	1	1	yes	characteristic
85	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
86	C₈×C₁₃	C₄×Q₈	1	1	yes	characteristic
87	#1055	—	2	4	no	—
88	C₂²×C₁₆	—	13	1	no	—
89	C₂²×C₁₆	—	13	1	no	—
90	C₂²×C₁₆	C₂×D₁₃	1	2	yes	—
91	C₂×C₄⋊C₈	—	13	1	no	—
92	C₂×C₄×C₈	—	13	2	no	—
93	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
94	C₄×C₄⋊C₄	—	13	1	no	—
95	C₄⋊C₁₆	Dic₁₃	1	4	yes	complemented
96	C₄×C₁₆	—	13	4	no	—
97	C₄×C₁₆	—	26	4	no	—
98	#336	—	13	2	no	—
99	C₂²×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
100	C₂²×C₁₃	C₈×D₄	1	1	yes	characteristic
101	C₂²×C₁₃	C₈×Q₈	1	1	yes	characteristic
102	C₂²×C₁₃	C₂²×C₁₆	1	2	yes	—
103	C₂²×C₁₃	#504	1	2	yes	—
104	C₄×C₁₃	—	2	2	no	—
105	C₄×C₁₃	C₄×C₁₆	1	4	yes	complemented
106	C₄×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
107	C₄×C₁₃	#405	1	1	yes	characteristic
108	C₄×C₁₃	C₈×D₄	1	1	yes	characteristic
109	C₄×C₁₃	C₈×Q₈	1	1	yes	characteristic
110	Dic₁₃	—	2	8	no	—
111	C₂²×C₈	—	13	2	no	—
112	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
113	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
114	C₂×C₄⋊C₄	—	13	2	no	—
115	C₂×C₄²	—	13	2	no	—
116	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
117	C₂×C₁₆	—	2	2	no	—
118	C₂×C₁₆	—	2	2	no	—
119	C₂×C₁₆	—	13	2	no	—
120	C₂×C₁₆	—	13	2	no	—
121	C₂×C₁₆	—	13	2	no	—
122	C₂×C₁₆	—	13	2	no	—
123	C₂×C₁₆	—	26	1	no	—
124	C₂×C₁₆	—	26	1	no	—
125	C₂×C₁₆	C₂×Dic₁₃	1	4	yes	—
126	C₄⋊C₈	—	13	4	no	—
127	C₄×C₈	—	26	2	no	—
128	C₄×C₈	—	26	2	no	—
129	C₄×C₈	C₄×D₁₃	1	2	yes	—
130	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
131	#95	—	13	2	no	—
132	C₂×C₁₃	C₈×C₄⋊C₄	1	1	yes	characteristic
133	C₂×C₁₃	C₂×C₄×C₁₆	1	1	yes	characteristic
134	C₂×C₁₃	#2108	1	1	yes	characteristic
135	C₂×C₁₃	D₄×C₁₆	1	2	yes	—
136	C₂×C₁₃	Q₈×C₁₆	1	2	yes	—
137	C₂²×C₄	—	13	4	no	—
138	C₂²×C₄	C₈×D₁₃	1	1	yes	characteristic
139	C₂²×C₄	C₈×D₁₃	1	1	yes	characteristic
140	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
141	C₂×C₈	—	2	1	no	—
142	C₂×C₈	—	2	1	no	—
143	C₂×C₈	—	13	4	no	—
144	C₂×C₈	—	26	2	no	—
145	C₂×C₈	—	26	2	no	—
146	C₂×C₈	C₄×Dic₁₃	1	2	yes	—
147	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
148	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
149	C₂×C₈	#6129	1	1	yes	characteristic
150	C₂×C₈	Q₈×D₁₃	1	1	yes	characteristic
151	C₂×C₈	#6131	1	1	yes	characteristic
152	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
153	C₄⋊C₄	—	13	8	no	—
154	C₄²	—	26	2	no	—
155	C₄²	—	26	2	no	—
156	C₄²	C₈×D₁₃	1	2	yes	—
157	C₄²	C₄×Dic₁₃	1	2	yes	—
158	C₁₆	—	2	4	no	—
159	C₁₆	—	13	4	no	—
160	C₁₆	—	26	2	no	—
161	C₁₃	C₁₆×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
162	C₂³	C₂×C₈×D₁₃	1	1	yes	characteristic
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	13	8	no	—
165	C₂×C₄	—	26	4	no	—
166	C₂×C₄	—	26	4	no	—
167	C₂×C₄	C₁₆×D₁₃	1	4	yes	—
168	C₂×C₄	C₈×Dic₁₃	1	2	yes	—
169	C₂×C₄	C₈×Dic₁₃	1	2	yes	—
170	C₂×C₄	#88141	1	1	yes	characteristic
171	C₂×C₄	#88147	1	1	yes	characteristic
172	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
173	C₂×C₄	#88161	1	1	yes	characteristic
174	C₂×C₄	#88161	1	1	yes	characteristic
175	C₂×C₄	C₂×C₄×Dic₁₃	1	1	yes	characteristic
176	C₈	—	2	2	no	—
177	C₈	—	26	4	no	—
178	C₈	C₄⋊C₄×D₁₃	1	1	yes	characteristic
179	C₈	#88146	1	1	yes	characteristic
180	C₈	D₄×Dic₁₃	1	1	yes	characteristic
181	C₈	Q₈×Dic₁₃	1	1	yes	characteristic
182	C₂²	#11718491	1	1	yes	characteristic
183	C₂²	#11718495	1	1	yes	characteristic
184	C₂²	C₂×C₁₆×D₁₃	1	2	yes	—
185	C₂²	#11718568	1	2	yes	—
186	C₂²	C₂×C₈×Dic₁₃	1	1	yes	characteristic
187	C₄	—	2	2	no	—
188	C₄	—	26	8	no	—
189	C₄	C₁₆×Dic₁₃	1	4	yes	complemented
190	C₄	#11718491	1	1	yes	characteristic
191	C₄	#11718495	1	1	yes	characteristic
192	C₄	C₄⋊C₄×Dic₁₃	1	1	yes	characteristic
193	C₄	#11718783	1	1	yes	characteristic
194	C₂	#13114701	1	1	yes	characteristic
195	C₂	C₂×C₁₆×Dic₁₃	1	1	yes	characteristic
196	C₂	#13116435	1	1	yes	characteristic
197	C₂	#13116655	1	2	yes	—
198	C₂	#13116738	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₁₃
commutator	C₂×C₁₃	C₂×C₄×C₁₆
Fitting	C₂×C₁₃×C₄⋊C₁₆	C₂
socle, prime radical	C₂³×C₁₃	C₂²×C₈
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₁₃
hypocenter	C₁₃	C₁₆×C₄⋊C₄
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
13. C₁₃×C₄⋊C₁₆	C₄	semidirect
16. C₄×#6090	C₄	semidirect
54. C₄⋊Dic₁₃	C₁₆	semidirect
62. #6090	C₄⋊C₄	semidirect
95. C₄⋊C₁₆	Dic₁₃	semidirect
105. C₄×C₁₃	C₄×C₁₆	semidirect
161. C₁₃	C₁₆×C₄⋊C₄	semidirect
189. C₄	C₁₆×Dic₁₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#6090	2
C₂×C₁₆×Dic₁₃	2
C₈×C₄⋊Dic₁₃	2
#13112913	2
C₂×C₁₃×C₄⋊C₁₆	2
C₁₆×C₄⋊C₄	13