Subgroups — #30670996

Order 16768.

452 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16862238	C₂	1	2	yes	maximal
2	#22553246	C₂	1	4	yes	maximal
3	#22553210	C₂	1	2	yes	maximal
4	#22553208	C₂	1	2	yes	maximal
5	#22553182	C₂	1	1	yes	maximal, characteristic
6	#22553181	C₂	1	2	yes	maximal
7	C₄×C₄⋊Dic₁₃₁	C₂	1	1	yes	maximal, characteristic
8	C₄×C₄⋊C₄×C₁₃₁	C₂	1	1	yes	maximal, characteristic
9	#16862238	—	2	8	no	—
10	C₂×C₄⋊Dic₁₃₁	C₂²	1	2	yes	—
11	C₂×C₄⋊Dic₁₃₁	C₂²	1	2	yes	—
12	C₂×C₄⋊Dic₁₃₁	C₂²	1	2	yes	—
13	C₂×C₄⋊Dic₁₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Dic₁₃₁	C₂²	1	1	yes	characteristic
15	C₂×Dic₁₃₁⋊C₄	C₂²	1	4	yes	—
16	C₂×Dic₁₃₁⋊C₄	C₂²	1	4	yes	—
17	C₂×Dic₁₃₁⋊C₄	C₂²	1	4	yes	—
18	C₂×C₄×Dic₁₃₁	C₂²	1	2	yes	—
19	C₁₃₁⋊#95	C₂²	1	2	yes	—
20	C₁₃₁⋊#95	C₂²	1	4	yes	—
21	C₂×C₄⋊C₄×C₁₃₁	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₄×C₁₃₁	C₂²	1	1	yes	characteristic
23	C₂×C₄²×C₁₃₁	C₂²	1	2	yes	—
24	C₂×C₄²×C₁₃₁	C₂²	1	1	yes	characteristic
25	#95×C₁₃₁	C₂²	1	2	yes	—
26	C₂²×Dic₁₃₁	C₂³	1	4	yes	—
27	C₂²×Dic₁₃₁	C₂³	1	2	yes	—
28	C₂²×Dic₁₃₁	C₂³	1	2	yes	—
29	C₄⋊Dic₁₃₁	—	2	4	no	—
30	C₄⋊Dic₁₃₁	—	2	4	no	—
31	C₄⋊Dic₁₃₁	—	4	2	no	—
32	C₄⋊Dic₁₃₁	D₄	1	4	yes	—
33	C₄⋊Dic₁₃₁	Q₈	1	4	yes	—
34	Dic₁₃₁⋊C₄	—	2	8	no	—
35	Dic₁₃₁⋊C₄	—	2	8	no	—
36	Dic₁₃₁⋊C₄	—	4	4	no	—
37	C₄×Dic₁₃₁	—	2	4	no	—
38	C₂²×C₄×C₁₃₁	C₂³	1	2	yes	—
39	C₂²×C₄×C₁₃₁	C₂³	1	2	yes	—
40	C₂²×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
41	C₂²×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
43	C₄⋊C₄×C₁₃₁	—	2	2	no	—
44	C₄⋊C₄×C₁₃₁	—	2	2	no	—
45	C₄²×C₁₃₁	—	2	4	no	—
46	C₄²×C₁₃₁	Q₈	1	4	yes	—
47	C₂×Dic₁₃₁	—	2	2	no	—
48	C₂×Dic₁₃₁	—	2	2	no	—
49	C₂×Dic₁₃₁	—	2	2	no	—
50	C₂×Dic₁₃₁	—	2	2	no	—
51	C₂×Dic₁₃₁	—	2	4	no	—
52	C₂×Dic₁₃₁	—	2	4	no	—
53	C₂×Dic₁₃₁	—	2	4	no	—
54	C₂×Dic₁₃₁	C₂×D₄	1	4	yes	—
55	C₂×Dic₁₃₁	C₂×Q₈	1	4	yes	—
56	C₂³×C₁₃₁	C₂⁴	1	1	yes	characteristic
57	C₂×C₄×C₁₃₁	—	2	1	no	—
58	C₂×C₄×C₁₃₁	—	2	1	no	—
59	C₂×C₄×C₁₃₁	—	2	2	no	—
60	C₂×C₄×C₁₃₁	—	2	2	no	—
61	C₂×C₄×C₁₃₁	—	2	2	no	—
62	C₂×C₄×C₁₃₁	—	2	2	no	—
63	C₂×C₄×C₁₃₁	—	2	2	no	—
64	C₂×C₄×C₁₃₁	—	2	2	no	—
65	C₂×C₄×C₁₃₁	C₂×D₄	1	2	yes	—
66	C₂×C₄×C₁₃₁	C₂×Q₈	1	2	yes	—
67	C₂×C₄×C₁₃₁	C₂×Q₈	1	2	yes	—
68	C₂×C₄×C₁₃₁	C₂×Q₈	1	2	yes	—
69	C₂×C₄×C₁₃₁	C₂×Q₈	1	2	yes	—
70	C₂×C₄×C₁₃₁	#41	1	2	yes	—
71	C₂×C₄×C₁₃₁	#41	1	2	yes	—
72	C₂²×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
73	C₂²×C₁₃₁	C₂²×Q₈	1	1	yes	characteristic
74	C₂²×C₁₃₁	C₂²×Q₈	1	1	yes	characteristic
75	C₂²×C₁₃₁	C₂×#41	1	1	yes	characteristic
76	C₂²×C₁₃₁	C₂×#41	1	1	yes	characteristic
77	C₂²×C₁₃₁	H(5,𝔽₂)	1	1	yes	characteristic
78	C₂²×C₁₃₁	2₋¹⁺⁴	1	1	yes	characteristic
79	C₄×C₁₃₁	—	2	2	no	—
80	C₄×C₁₃₁	—	4	2	no	—
81	C₄×C₁₃₁	—	4	2	no	—
82	C₄×C₁₃₁	#125	1	4	yes	complemented
83	C₄×C₁₃₁	C₄⋊Q₈	1	4	yes	—
84	Dic₁₃₁	—	2	4	no	—
85	Dic₁₃₁	—	2	4	no	—
86	Dic₁₃₁	—	4	4	no	—
87	C₂×C₁₃₁	C₂×#125	1	1	yes	characteristic
88	C₂×C₁₃₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
89	C₂×C₁₃₁	#544	1	1	yes	characteristic
90	C₂×C₁₃₁	#547	1	1	yes	characteristic
91	C₂×C₁₃₁	#550	1	1	yes	characteristic
92	C₂×C₁₃₁	#554	1	1	yes	characteristic
93	C₂×C₁₃₁	#557	1	1	yes	characteristic
94	C₁₃₁	#2552	1	1	yes	characteristic, complemented, Sylow
95	#2552	—	131	1	no	maximal, Sylow
96	C₂×#125	—	131	2	no	—
97	#398	—	131	4	no	—
98	#389	—	131	2	no	—
99	#389	—	131	2	no	—
100	#384	—	131	2	no	—
101	#384	—	131	1	no	—
102	C₄×C₄⋊C₄	—	131	1	no	—
103	C₄×C₄⋊C₄	D₁₃₁	1	1	yes	characteristic
104	#125	—	262	8	no	—
105	C₂×C₄⋊C₄	—	131	2	no	—
106	C₂×C₄⋊C₄	—	131	4	no	—
107	C₂×C₄⋊C₄	—	131	4	no	—
108	C₂×C₄⋊C₄	—	131	2	no	—
109	C₂×C₄⋊C₄	—	131	4	no	—
110	C₂×C₄⋊C₄	—	131	2	no	—
111	C₂×C₄⋊C₄	—	131	1	no	—
112	C₂×C₄⋊C₄	—	131	1	no	—
113	C₂×C₄⋊C₄	C₂×D₁₃₁	1	1	yes	characteristic
114	C₂×C₄⋊C₄	C₂×D₁₃₁	1	1	yes	characteristic
115	C₂×C₄²	—	131	2	no	—
116	C₂×C₄²	C₂×D₁₃₁	1	2	yes	—
117	C₂×C₄²	C₂×D₁₃₁	1	1	yes	characteristic
118	#95	—	131	2	no	—
119	#95	—	131	4	no	—
120	#95	C₂×D₁₃₁	1	2	yes	—
121	C₂²×C₄	—	131	4	no	—
122	C₂²×C₄	—	131	2	no	—
123	C₂²×C₄	—	131	2	no	—
124	C₂²×C₄	C₂²×D₁₃₁	1	2	yes	—
125	C₂²×C₄	C₂²×D₁₃₁	1	2	yes	—
126	C₂²×C₄	C₂²×D₁₃₁	1	1	yes	characteristic
127	C₂²×C₄	C₂²×D₁₃₁	1	1	yes	characteristic
128	C₂²×C₄	C₂²×D₁₃₁	1	1	yes	characteristic
129	C₄⋊C₄	—	2	2	no	—
130	C₄⋊C₄	—	2	2	no	—
131	C₄⋊C₄	—	262	8	no	—
132	C₄⋊C₄	—	262	8	no	—
133	C₄⋊C₄	—	262	4	no	—
134	C₄⋊C₄	—	262	4	no	—
135	C₄⋊C₄	—	524	4	no	—
136	C₄⋊C₄	—	524	2	no	—
137	C₄⋊C₄	—	131	4	no	—
138	C₄⋊C₄	—	131	4	no	—
139	C₄²	—	2	4	no	—
140	C₄²	—	262	4	no	—
141	C₄²	Dic₂₆₂	1	4	yes	—
142	C₂³	C₂³×D₁₃₁	1	1	yes	characteristic
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	262	2	no	—
152	C₂×C₄	—	262	2	no	—
153	C₂×C₄	—	262	2	no	—
154	C₂×C₄	—	262	2	no	—
155	C₂×C₄	—	262	4	no	—
156	C₂×C₄	—	262	4	no	—
157	C₂×C₄	—	262	4	no	—
158	C₂×C₄	—	131	4	no	—
159	C₂×C₄	—	131	4	no	—
160	C₂×C₄	D₄×D₁₃₁	1	2	yes	—
161	C₂×C₄	Q₈×D₁₃₁	1	2	yes	—
162	C₂×C₄	C₂×Dic₂₆₂	1	2	yes	—
163	C₂×C₄	C₂×Dic₂₆₂	1	2	yes	—
164	C₂×C₄	C₂×Dic₂₆₂	1	2	yes	—
165	C₂×C₄	#14523513	1	2	yes	—
166	C₂×C₄	#14523513	1	2	yes	—
167	C₂²	C₂×D₄×D₁₃₁	1	1	yes	characteristic
168	C₂²	C₂×Q₈×D₁₃₁	1	1	yes	characteristic
169	C₂²	C₂²×Dic₂₆₂	1	1	yes	characteristic
170	C₂²	C₂×#14523513	1	1	yes	characteristic
171	C₂²	C₂×#14523513	1	1	yes	characteristic
172	C₂²	#16862292	1	1	yes	characteristic
173	C₂²	#16862293	1	1	yes	characteristic
174	C₄	—	2	2	no	—
175	C₄	—	4	2	no	—
176	C₄	—	4	2	no	—
177	C₄	—	262	4	no	—
178	C₄	—	262	4	no	—
179	C₄	—	524	4	no	—
180	C₄	#16862238	1	4	yes	complemented
181	C₄	#16862245	1	4	yes	—
182	C₂	C₂×#16862238	1	1	yes	characteristic
183	C₂	C₂×#16862245	1	1	yes	characteristic
184	C₂	#22553940	1	1	yes	characteristic
185	C₂	#22553967	1	1	yes	characteristic
186	C₂	#22553983	1	1	yes	characteristic
187	C₂	#22554022	1	1	yes	characteristic
188	C₂	#22554040	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	#2552
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
82. C₄×C₁₃₁	#125	semidirect
94. C₁₃₁	#2552	semidirect
180. C₄	#16862238	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16862238	2
#22553246	2
#22553210	2
#22553208	2
#22553182	2
#22553181	2
C₄×C₄⋊Dic₁₃₁	2
C₄×C₄⋊C₄×C₁₃₁	2
#2552	131