Subgroups — #30670987

Order 16768.

556 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16862246	C₂	1	2	yes	maximal
2	C₂×#16862245	C₂	1	2	yes	maximal
3	#22553233	C₂	1	4	yes	maximal
4	#22553211	C₂	1	2	yes	maximal
5	#22553210	C₂	1	2	yes	maximal
6	#22553182	C₂	1	1	yes	maximal, characteristic
7	C₄×C₄⋊Dic₁₃₁	C₂	1	1	yes	maximal, characteristic
8	C₄×C₄⋊C₄×C₁₃₁	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₂₆₂	C₂²	1	4	yes	—
10	#16862246	—	2	8	no	—
11	#16862245	—	2	8	no	—
12	C₂×C₄⋊Dic₁₃₁	C₂²	1	2	yes	—
13	C₂×C₄⋊Dic₁₃₁	C₂²	1	4	yes	—
14	C₂×C₄⋊Dic₁₃₁	C₂²	1	2	yes	—
15	C₂×C₄⋊Dic₁₃₁	C₂²	1	2	yes	—
16	C₂×C₄⋊Dic₁₃₁	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊Dic₁₃₁	C₂²	1	1	yes	characteristic
18	C₂×Dic₁₃₁⋊C₄	C₂²	1	4	yes	—
19	C₂×C₄×Dic₁₃₁	C₂²	1	2	yes	—
20	C₁₃₁⋊#95	C₂²	1	2	yes	—
21	C₁₃₁⋊#95	C₂²	1	4	yes	—
22	C₂×C₄⋊C₄×C₁₃₁	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊C₄×C₁₃₁	C₂²	1	1	yes	characteristic
24	C₂×C₄²×C₁₃₁	C₂²	1	2	yes	—
25	C₂×C₄²×C₁₃₁	C₂²	1	1	yes	characteristic
26	#95×C₁₃₁	C₂²	1	2	yes	—
27	C₂×Dic₂₆₂	—	2	8	no	—
28	C₂×Dic₂₆₂	—	2	8	no	—
29	C₂×Dic₂₆₂	—	4	4	no	—
30	C₂²×Dic₁₃₁	C₂³	1	4	yes	—
31	C₂²×Dic₁₃₁	C₂³	1	2	yes	—
32	C₂²×Dic₁₃₁	C₂³	1	2	yes	—
33	C₄⋊Dic₁₃₁	—	2	8	no	—
34	C₄⋊Dic₁₃₁	—	2	4	no	—
35	C₄⋊Dic₁₃₁	—	2	4	no	—
36	C₄⋊Dic₁₃₁	—	4	2	no	—
37	C₄⋊Dic₁₃₁	D₄	1	4	yes	—
38	C₄⋊Dic₁₃₁	Q₈	1	4	yes	complemented
39	Dic₁₃₁⋊C₄	—	2	8	no	—
40	C₄×Dic₁₃₁	—	2	4	no	—
41	C₂²×C₄×C₁₃₁	C₂³	1	2	yes	—
42	C₂²×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₁₃₁	C₂³	1	2	yes	—
44	C₂²×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
45	C₂²×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
46	C₄⋊C₄×C₁₃₁	D₄	1	4	yes	—
47	C₄⋊C₄×C₁₃₁	Q₈	1	4	yes	complemented
48	C₄²×C₁₃₁	—	2	4	no	—
49	C₄²×C₁₃₁	Q₈	1	4	yes	complemented
50	Dic₂₆₂	—	4	16	no	—
51	C₂×Dic₁₃₁	—	2	2	no	—
52	C₂×Dic₁₃₁	—	2	2	no	—
53	C₂×Dic₁₃₁	—	2	2	no	—
54	C₂×Dic₁₃₁	—	2	2	no	—
55	C₂×Dic₁₃₁	—	2	4	no	—
56	C₂×Dic₁₃₁	—	2	4	no	—
57	C₂×Dic₁₃₁	—	2	4	no	—
58	C₂×Dic₁₃₁	C₂×D₄	1	4	yes	—
59	C₂×Dic₁₃₁	C₂×Q₈	1	4	yes	—
60	C₂³×C₁₃₁	C₂⁴	1	1	yes	characteristic
61	C₂×C₄×C₁₃₁	—	2	2	no	—
62	C₂×C₄×C₁₃₁	—	2	2	no	—
63	C₂×C₄×C₁₃₁	—	2	1	no	—
64	C₂×C₄×C₁₃₁	—	2	2	no	—
65	C₂×C₄×C₁₃₁	—	2	1	no	—
66	C₂×C₄×C₁₃₁	—	2	2	no	—
67	C₂×C₄×C₁₃₁	C₂×D₄	1	2	yes	—
68	C₂×C₄×C₁₃₁	C₂×D₄	1	4	yes	—
69	C₂×C₄×C₁₃₁	C₂×D₄	1	2	yes	—
70	C₂×C₄×C₁₃₁	C₂×Q₈	1	2	yes	—
71	C₂×C₄×C₁₃₁	C₂×Q₈	1	2	yes	—
72	C₂×C₄×C₁₃₁	C₂×Q₈	1	4	yes	—
73	C₂×C₄×C₁₃₁	C₂×Q₈	1	2	yes	—
74	C₂×C₄×C₁₃₁	C₂×Q₈	1	2	yes	—
75	C₂×C₄×C₁₃₁	C₂×Q₈	1	2	yes	—
76	C₂²×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
77	C₂²×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
78	C₂²×C₁₃₁	C₂²×Q₈	1	1	yes	characteristic
79	C₂²×C₁₃₁	C₂²×Q₈	1	1	yes	characteristic
80	C₂²×C₁₃₁	C₂²×Q₈	1	1	yes	characteristic
81	C₂²×C₁₃₁	H(5,𝔽₂)	1	1	yes	characteristic
82	C₂²×C₁₃₁	2₋¹⁺⁴	1	1	yes	characteristic
83	C₄×C₁₃₁	—	2	2	no	—
84	C₄×C₁₃₁	—	2	4	no	—
85	C₄×C₁₃₁	—	2	4	no	—
86	C₄×C₁₃₁	C₄⋊Q₈	1	4	yes	complemented
87	C₄×C₁₃₁	C₄⋊Q₈	1	4	yes	complemented
88	Dic₁₃₁	—	2	4	no	—
89	Dic₁₃₁	—	2	4	no	—
90	Dic₁₃₁	—	4	4	no	—
91	C₂×C₁₃₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
92	C₂×C₁₃₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
93	C₂×C₁₃₁	#544	1	1	yes	characteristic
94	C₂×C₁₃₁	D₄²	1	1	yes	characteristic
95	C₂×C₁₃₁	D₄×Q₈	1	1	yes	characteristic
96	C₂×C₁₃₁	D₄×Q₈	1	1	yes	characteristic
97	C₂×C₁₃₁	Q₈²	1	1	yes	characteristic
98	C₁₃₁	#2551	1	1	yes	characteristic, complemented, Sylow
99	#2551	—	131	1	no	maximal, Sylow
100	C₂×C₄⋊Q₈	—	131	2	no	—
101	C₂×C₄⋊Q₈	—	131	2	no	—
102	#395	—	131	4	no	—
103	#389	—	131	2	no	—
104	#389	—	131	2	no	—
105	#384	—	131	1	no	—
106	C₄×C₄⋊C₄	—	131	1	no	—
107	C₄×C₄⋊C₄	D₁₃₁	1	1	yes	characteristic
108	C₂²×Q₈	—	131	4	no	—
109	C₄⋊Q₈	—	262	8	no	—
110	C₄⋊Q₈	—	262	8	no	—
111	C₂×C₄⋊C₄	—	131	2	no	—
112	C₂×C₄⋊C₄	—	131	2	no	—
113	C₂×C₄⋊C₄	—	131	4	no	—
114	C₂×C₄⋊C₄	—	131	4	no	—
115	C₂×C₄⋊C₄	—	131	2	no	—
116	C₂×C₄⋊C₄	—	131	1	no	—
117	C₂×C₄⋊C₄	—	131	1	no	—
118	C₂×C₄⋊C₄	C₂×D₁₃₁	1	1	yes	characteristic
119	C₂×C₄⋊C₄	C₂×D₁₃₁	1	1	yes	characteristic
120	C₂×C₄²	—	131	2	no	—
121	C₂×C₄²	C₂×D₁₃₁	1	2	yes	—
122	C₂×C₄²	C₂×D₁₃₁	1	1	yes	characteristic
123	#95	—	131	2	no	—
124	#95	—	131	4	no	—
125	#95	C₂×D₁₃₁	1	2	yes	—
126	C₂×Q₈	—	262	8	no	—
127	C₂×Q₈	—	262	8	no	—
128	C₂×Q₈	—	524	4	no	—
129	C₂²×C₄	—	131	4	no	—
130	C₂²×C₄	—	131	2	no	—
131	C₂²×C₄	—	131	2	no	—
132	C₂²×C₄	C₂²×D₁₃₁	1	2	yes	—
133	C₂²×C₄	C₂²×D₁₃₁	1	1	yes	characteristic
134	C₂²×C₄	C₂²×D₁₃₁	1	2	yes	—
135	C₂²×C₄	C₂²×D₁₃₁	1	1	yes	characteristic
136	C₂²×C₄	C₂²×D₁₃₁	1	1	yes	characteristic
137	C₄⋊C₄	—	262	8	no	—
138	C₄⋊C₄	—	262	4	no	—
139	C₄⋊C₄	—	262	8	no	—
140	C₄⋊C₄	—	262	4	no	—
141	C₄⋊C₄	—	524	2	no	—
142	C₄⋊C₄	—	131	4	no	—
143	C₄⋊C₄	—	131	4	no	—
144	C₄⋊C₄	D₅₂₄	1	4	yes	—
145	C₄⋊C₄	Dic₂₆₂	1	4	yes	complemented
146	C₄²	—	2	4	no	—
147	C₄²	—	262	4	no	—
148	C₄²	Dic₂₆₂	1	4	yes	complemented
149	C₂³	C₂³×D₁₃₁	1	1	yes	characteristic
150	Q₈	—	524	16	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	1	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	262	2	no	—
158	C₂×C₄	—	262	2	no	—
159	C₂×C₄	—	262	2	no	—
160	C₂×C₄	—	262	2	no	—
161	C₂×C₄	—	262	4	no	—
162	C₂×C₄	—	262	4	no	—
163	C₂×C₄	—	262	4	no	—
164	C₂×C₄	—	131	4	no	—
165	C₂×C₄	—	131	4	no	—
166	C₂×C₄	D₄×D₁₃₁	1	2	yes	—
167	C₂×C₄	C₂×D₅₂₄	1	4	yes	—
168	C₂×C₄	C₂×D₅₂₄	1	2	yes	—
169	C₂×C₄	Q₈×D₁₃₁	1	2	yes	—
170	C₂×C₄	C₂×Dic₂₆₂	1	2	yes	—
171	C₂×C₄	C₂×Dic₂₆₂	1	4	yes	—
172	C₂×C₄	C₂×Dic₂₆₂	1	2	yes	—
173	C₂×C₄	C₂×Dic₂₆₂	1	2	yes	—
174	C₂×C₄	C₂×Dic₂₆₂	1	2	yes	—
175	C₂²	C₂×D₄×D₁₃₁	1	1	yes	characteristic
176	C₂²	C₂²×D₅₂₄	1	1	yes	characteristic
177	C₂²	C₂×Q₈×D₁₃₁	1	1	yes	characteristic
178	C₂²	C₂²×Dic₂₆₂	1	1	yes	characteristic
179	C₂²	C₂²×Dic₂₆₂	1	1	yes	characteristic
180	C₂²	#16862292	1	1	yes	characteristic
181	C₂²	#16862293	1	1	yes	characteristic
182	C₄	—	2	2	no	—
183	C₄	—	2	4	no	—
184	C₄	—	2	4	no	—
185	C₄	—	262	4	no	—
186	C₄	—	262	4	no	—
187	C₄	—	524	4	no	—
188	C₄	#16862245	1	4	yes	complemented
189	C₄	#16862246	1	4	yes	complemented
190	C₂	C₂×#16862245	1	1	yes	characteristic
191	C₂	C₂×#16862246	1	1	yes	characteristic
192	C₂	#22553940	1	1	yes	characteristic
193	C₂	D₄×D₅₂₄	1	1	yes	characteristic
194	C₂	D₄×Dic₂₆₂	1	1	yes	characteristic
195	C₂	Q₈×D₅₂₄	1	1	yes	characteristic
196	C₂	Q₈×Dic₂₆₂	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₃₁
commutator, socle, prime radical, cosocle	C₂³×C₁₃₁	C₂⁴
Fitting	C₄×C₄⋊C₄×C₁₃₁	C₂
hypercenter	C₄×C₄⋊C₄	D₁₃₁
hypocenter	C₁₃₁	#2551
LCS	G ▷ C₂³×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
38. C₄⋊Dic₁₃₁	Q₈	semidirect
47. C₄⋊C₄×C₁₃₁	Q₈	semidirect
49. C₄²×C₁₃₁	Q₈	semidirect
86. C₄×C₁₃₁	C₄⋊Q₈	semidirect
87. C₄×C₁₃₁	C₄⋊Q₈	semidirect
98. C₁₃₁	#2551	semidirect
145. C₄⋊C₄	Dic₂₆₂	semidirect
148. C₄²	Dic₂₆₂	semidirect
188. C₄	#16862245	semidirect
189. C₄	#16862246	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16862246	2
C₂×#16862245	2
#22553233	2
#22553211	2
#22553210	2
#22553182	2
C₄×C₄⋊Dic₁₃₁	2
C₄×C₄⋊C₄×C₁₃₁	2
#2551	131