Subgroups — #30675091

Order 16768.

394 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14523513	C₂	1	1	yes	maximal, characteristic
2	#22553436	C₂	1	2	yes	maximal, complemented
3	#22553389	C₂	1	2	yes	maximal, complemented
4	#22553388	C₂	1	4	yes	maximal, complemented
5	#22553365	C₂	1	1	yes	maximal, characteristic, complemented
6	#22553364	C₂	1	2	yes	maximal, complemented
7	#22553334	C₂	1	2	yes	maximal, complemented
8	#423×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14523513	C₄	1	1	yes	characteristic
10	C₂×#14523496	C₂²	1	4	yes	—
11	C₄×Dic₂₆₂	C₄	1	1	yes	characteristic
12	C₄×Dic₂₆₂	C₂²	1	1	yes	characteristic
13	C₄×D₅₂₄	C₄	1	1	yes	characteristic
14	C₄×D₅₂₄	C₂²	1	1	yes	characteristic
15	C₄×C₂²⋊D₁₃₁	C₄	1	2	yes	—
16	C₄×C₂²⋊D₁₃₁	C₂²	1	2	yes	—
17	#16862205	C₄	1	1	yes	characteristic
18	#16862203	C₄	1	2	yes	—
19	C₄²×D₁₃₁	C₂²	1	2	yes	—
20	C₄⋊(C₁₃₁⋊C₈)	C₂²	1	1	yes	characteristic
21	C₄⋊(C₁₃₁⋊C₈)	C₂²	1	1	yes	characteristic
22	Dic₁₃₁⋊C₈	C₂²	1	4	yes	—
23	C₂²⋊(C₁₃₁⋊C₈)	C₂²	1	2	yes	—
24	(C₂×D₁₃₁)⋊C₈	C₂²	1	4	yes	—
25	C₄×C₁₃₁⋊C₈	C₂²	1	2	yes	—
26	C₈×Dic₁₃₁	C₂²	1	4	yes	—
27	C₂×M₁₆×C₁₃₁	C₂²	1	2	yes	—
28	C₂×C₄²×C₁₃₁	C₂²	1	1	yes	characteristic
29	C₄⋊C₈×C₁₃₁	C₂²	1	2	yes	—
30	C₄⋊C₈×C₁₃₁	C₂²	1	2	yes	—
31	#14523513	—	2	4	no	—
32	C₂×Dic₂₆₂	C₂×C₄	1	1	yes	characteristic
33	C₂×D₅₂₄	C₂×C₄	1	1	yes	characteristic
34	C₂×C₂²⋊D₁₃₁	C₂×C₄	1	2	yes	—
35	C₂×C₄×D₁₃₁	C₂×C₄	1	2	yes	—
36	C₂×C₄×D₁₃₁	C₂×C₄	1	2	yes	—
37	C₂×C₄×D₁₃₁	C₂³	1	2	yes	—
38	#14523496	—	4	4	no	—
39	C₂×C₁₃₁⋊C₈	C₂³	1	2	yes	—
40	C₂×C₁₃₁⋊C₈	C₂³	1	2	yes	—
41	C₄⋊Dic₁₃₁	C₂×C₄	1	1	yes	characteristic
42	C₄⋊Dic₁₃₁	C₂×C₄	1	1	yes	characteristic
43	Dic₁₃₁⋊C₄	C₂×C₄	1	2	yes	—
44	Dic₁₃₁⋊C₄	C₂×C₄	1	2	yes	—
45	C₂²⋊Dic₁₃₁	C₂×C₄	1	2	yes	—
46	(C₂×D₁₃₁)⋊C₄	C₂×C₄	1	2	yes	—
47	(C₂×D₁₃₁)⋊C₄	C₂×C₄	1	2	yes	—
48	C₄×Dic₁₃₁	C₂×C₄	1	2	yes	—
49	C₄×Dic₁₃₁	C₂×C₄	1	2	yes	—
50	C₄×Dic₁₃₁	C₂³	1	2	yes	—
51	C₂²×C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
54	M₁₆×C₁₃₁	—	4	2	no	—
55	C₂×C₈×C₁₃₁	C₂³	1	4	yes	—
56	C₄²×C₁₃₁	C₂×C₄	1	1	yes	characteristic
57	C₄²×C₁₃₁	C₂×C₄	1	1	yes	characteristic
58	C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
59	C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
60	C₂²×D₁₃₁	C₂²×C₄	1	2	yes	—
61	Dic₂₆₂	M₁₆	1	4	yes	complemented
62	D₅₂₄	—	2	2	no	—
63	C₂²⋊D₁₃₁	—	2	4	no	—
64	C₂×Dic₁₃₁	C₂²×C₄	1	2	yes	—
65	C₂×Dic₁₃₁	C₂²×C₄	1	2	yes	—
66	C₂×Dic₁₃₁	C₂²×C₄	1	2	yes	—
67	C₄×D₁₃₁	—	2	4	no	—
68	C₄×D₁₃₁	—	2	2	no	—
69	C₄×D₁₃₁	—	2	2	no	—
70	C₄×D₁₃₁	—	2	4	no	—
71	C₁₃₁⋊C₈	—	2	2	no	—
72	C₁₃₁⋊C₈	—	2	2	no	—
73	C₂³×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₄×C₁₃₁	—	2	2	no	—
75	C₂×C₄×C₁₃₁	—	2	2	no	—
76	C₂×C₄×C₁₃₁	—	2	1	no	—
77	C₂×C₄×C₁₃₁	—	2	1	no	—
78	C₂×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
84	C₈×C₁₃₁	—	2	4	no	—
85	C₂²×C₁₃₁	—	2	1	no	—
86	C₂²×C₁₃₁	—	2	1	no	—
87	C₂²×C₁₃₁	—	2	1	no	—
88	C₂²×C₁₃₁	C₂³×C₄	1	1	yes	characteristic
89	C₂×D₁₃₁	—	2	2	no	—
90	C₂×D₁₃₁	—	2	2	no	—
91	C₂×D₁₃₁	—	2	2	no	—
92	C₄×C₁₃₁	—	2	1	no	—
93	C₄×C₁₃₁	—	2	1	no	—
94	C₄×C₁₃₁	—	2	1	no	—
95	C₄×C₁₃₁	C₂×M₁₆	1	2	yes	—
96	C₄×C₁₃₁	#131	1	2	yes	—
97	C₄×C₁₃₁	H(5,𝔽₂)	1	1	yes	characteristic
98	C₄×C₁₃₁	2₋¹⁺⁴	1	1	yes	characteristic
99	Dic₁₃₁	—	2	2	no	—
100	Dic₁₃₁	—	2	2	no	—
101	Dic₁₃₁	C₂×M₁₆	1	4	yes	—
102	C₂×C₁₃₁	—	4	1	no	—
103	C₂×C₁₃₁	#520	1	1	yes	characteristic
104	C₂×C₁₃₁	C₂²×M₁₆	1	1	yes	characteristic
105	C₂×C₁₃₁	C₂×#131	1	1	yes	characteristic
106	D₁₃₁	—	4	2	no	—
107	C₁₃₁	#2991	1	1	yes	characteristic, complemented, Sylow
108	#2991	—	131	1	no	maximal, Sylow
109	C₄×#41	—	131	1	no	—
110	C₈×Q₈	—	131	2	no	—
111	#436	—	131	4	no	—
112	#436	—	131	2	no	—
113	#431	—	131	1	no	—
114	#431	—	131	2	no	—
115	#423	—	131	2	no	—
116	#423	D₁₃₁	1	1	yes	characteristic, complemented
117	C₂×#41	—	131	1	no	—
118	C₂×M₁₆	—	131	4	no	—
119	C₂×M₁₆	C₂×D₁₃₁	1	2	yes	—
120	C₄×Q₈	—	131	1	no	—
121	C₄×Q₈	—	131	1	no	—
122	C₄×D₄	—	131	2	no	—
123	C₄×D₄	—	131	2	no	—
124	C₄×D₄	—	131	1	no	—
125	C₄×D₄	—	131	1	no	—
126	#117	—	131	1	no	—
127	#117	—	131	2	no	—
128	C₂×C₄²	—	131	2	no	—
129	C₂×C₄²	C₂×D₁₃₁	1	1	yes	characteristic
130	C₄⋊C₈	—	131	4	no	—
131	C₄⋊C₈	—	131	1	no	—
132	C₄⋊C₈	—	131	1	no	—
133	C₄⋊C₈	C₂×D₁₃₁	1	2	yes	—
134	C₄⋊C₈	C₂×D₁₃₁	1	2	yes	—
135	C₂²⋊C₈	—	131	2	no	—
136	C₂²⋊C₈	—	131	4	no	—
137	C₄×C₈	—	131	4	no	—
138	C₄×C₈	—	131	2	no	—
139	#41	—	262	4	no	—
140	C₂×Q₈	—	131	1	no	—
141	C₂×D₄	—	131	2	no	—
142	C₂×D₄	—	131	1	no	—
143	C₂²×C₄	—	131	2	no	—
144	C₂²×C₄	—	131	2	no	—
145	C₂²×C₄	—	131	2	no	—
146	C₂²×C₄	C₄×D₁₃₁	1	1	yes	characteristic
147	C₂²×C₄	C₄×D₁₃₁	1	1	yes	characteristic
148	C₂²×C₄	C₂²×D₁₃₁	1	1	yes	characteristic
149	M₁₆	—	4	2	no	—
150	M₁₆	—	524	4	no	—
151	C₂×C₈	—	131	2	no	—
152	C₂×C₈	—	131	2	no	—
153	C₂×C₈	C₂²×D₁₃₁	1	4	yes	—
154	C₄⋊C₄	—	131	2	no	—
155	C₄⋊C₄	—	131	2	no	—
156	C₄⋊C₄	—	131	1	no	—
157	C₄⋊C₄	—	131	1	no	—
158	C₂²⋊C₄	—	131	2	no	—
159	C₂²⋊C₄	—	131	2	no	—
160	C₂²⋊C₄	—	131	2	no	—
161	C₄²	—	131	2	no	—
162	C₄²	—	131	2	no	—
163	C₄²	—	131	2	no	—
164	C₄²	C₄×D₁₃₁	1	1	yes	characteristic
165	C₄²	C₄×D₁₃₁	1	1	yes	characteristic
166	C₄²	C₂²×D₁₃₁	1	1	yes	characteristic
167	C₄²	C₂²×D₁₃₁	1	1	yes	characteristic
168	C₂³	—	131	2	no	—
169	C₂³	C₂×C₄×D₁₃₁	1	1	yes	characteristic
170	Q₈	—	131	4	no	—
171	D₄	—	262	4	no	—
172	D₄	—	262	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	262	4	no	—
178	C₂×C₄	—	262	4	no	—
179	C₂×C₄	—	262	2	no	—
180	C₂×C₄	—	262	2	no	—
181	C₂×C₄	—	131	2	no	—
182	C₂×C₄	—	131	2	no	—
183	C₂×C₄	—	131	2	no	—
184	C₂×C₄	C₂×C₄×D₁₃₁	1	1	yes	characteristic
185	C₂×C₄	C₂×C₄×D₁₃₁	1	1	yes	characteristic
186	C₂×C₄	C₂×C₄×D₁₃₁	1	1	yes	characteristic
187	C₂×C₄	C₂×C₄×D₁₃₁	1	1	yes	characteristic
188	C₂×C₄	C₂×C₄×D₁₃₁	1	1	yes	characteristic
189	C₂×C₄	C₂³×D₁₃₁	1	1	yes	characteristic
190	C₈	—	2	4	no	—
191	C₈	—	262	2	no	—
192	C₈	—	262	2	no	—
193	C₂²	—	2	1	no	—
194	C₂²	—	2	1	no	—
195	C₂²	—	2	1	no	—
196	C₂²	—	262	2	no	—
197	C₂²	—	262	2	no	—
198	C₂²	—	262	2	no	—
199	C₂²	C₂²×C₄×D₁₃₁	1	1	yes	characteristic
200	C₄	—	2	1	no	—
201	C₄	—	2	1	no	—
202	C₄	—	2	1	no	—
203	C₄	—	262	2	no	—
204	C₄	—	262	2	no	—
205	C₄	—	131	4	no	—
206	C₄	M₁₆×D₁₃₁	1	2	yes	—
207	C₄	#16862254	1	2	yes	—
208	C₄	#16862291	1	1	yes	characteristic
209	C₄	#16862294	1	1	yes	characteristic
210	C₂	—	4	1	no	—
211	C₂	—	524	2	no	—
212	C₂	#22553793	1	1	yes	characteristic
213	C₂	C₂×M₁₆×D₁₃₁	1	1	yes	characteristic
214	C₂	C₂×#16862254	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₃₁
commutator, socle	C₂²×C₁₃₁	C₂³×C₄
Fitting	#423×C₁₃₁	C₂
prime radical	C₂×#14523513	C₄
cosocle	C₂×C₄×C₁₃₁	C₂⁴
hypercenter	#423	D₁₃₁
hypocenter	C₁₃₁	#2991
LCS	G ▷ C₂²×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂×C₄ ◁ #423

Complements

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

N	H ≅ G/N	type
2. #22553436	C₂	semidirect
3. #22553389	C₂	semidirect
4. #22553388	C₂	semidirect
5. #22553365	C₂	semidirect
6. #22553364	C₂	semidirect
7. #22553334	C₂	semidirect
8. #423×C₁₃₁	C₂	semidirect
61. Dic₂₆₂	M₁₆	semidirect
107. C₁₃₁	#2991	semidirect
116. #423	D₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14523513	2
#22553436	2
#22553389	2
#22553388	2
#22553365	2
#22553364	2
#22553334	2
#423×C₁₃₁	2
#2991	131