Subgroups — #30675061

Order 16768.

330 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22553866	C₂	1	1	yes	maximal, characteristic
2	#22553443	C₂	1	1	yes	maximal, characteristic, complemented
3	#22553442	C₂	1	1	yes	maximal, characteristic, complemented
4	#22553441	C₂	1	2	yes	maximal, complemented
5	C₈×Dic₂₆₂	C₂	1	2	yes	maximal, complemented
6	#22553436	C₂	1	1	yes	maximal, characteristic, complemented
7	#22553372	C₂	1	2	yes	maximal
8	#22553371	C₂	1	2	yes	maximal
9	#22553364	C₂	1	1	yes	maximal, characteristic
10	#22553363	C₂	1	1	yes	maximal, characteristic
11	C₈×Q₈×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
12	#16862245	C₄	1	1	yes	characteristic
13	#16862244	C₄	1	1	yes	characteristic
14	#16862236	C₄	1	2	yes	—
15	#16862226	C₄	1	1	yes	characteristic
16	#16862224	C₄	1	2	yes	—
17	#16862222	C₄	1	1	yes	characteristic
18	C₄×Dic₂₆₂	C₂²	1	1	yes	characteristic
19	C₄×Dic₂₆₂	C₂²	1	2	yes	—
20	#16862203	C₂²	1	2	yes	—
21	C₄²×D₁₃₁	C₂²	1	1	yes	characteristic
22	C₄⋊(C₁₃₁⋊C₈)	C₂²	1	1	yes	characteristic
23	C₄⋊(C₁₃₁⋊C₈)	C₂²	1	2	yes	—
24	Dic₁₃₁⋊C₈	C₂²	1	1	yes	characteristic
25	Dic₁₃₁⋊C₈	C₂²	1	1	yes	characteristic
26	Dic₁₃₁⋊C₈	C₂²	1	2	yes	—
27	Dic₁₃₁⋊C₈	C₂²	1	2	yes	—
28	(C₂×D₁₃₁)⋊C₈	C₂²	1	1	yes	characteristic
29	(C₂×D₁₃₁)⋊C₈	C₂²	1	1	yes	characteristic
30	(C₂×D₁₃₁)⋊C₈	C₂²	1	2	yes	—
31	#16862157	C₂²	1	2	yes	—
32	#16862156	C₂²	1	2	yes	—
33	C₄×C₁₃₁⋊C₈	C₂²	1	1	yes	characteristic
34	C₈×Dic₁₃₁	C₂²	1	2	yes	—
35	C₈×Dic₁₃₁	C₂²	1	2	yes	—
36	C₄×Q₈×C₁₃₁	C₂²	1	1	yes	characteristic
37	C₄⋊C₈×C₁₃₁	C₂²	1	1	yes	characteristic
38	C₄⋊C₈×C₁₃₁	C₂²	1	2	yes	—
39	C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
40	C₄×C₈×C₁₃₁	C₂²	1	2	yes	—
41	C₂×Dic₂₆₂	C₂×C₄	1	2	yes	—
42	C₂×Dic₂₆₂	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄×D₁₃₁	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×D₁₃₁	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×D₁₃₁	C₂³	1	1	yes	characteristic
46	C₂×C₁₃₁⋊C₈	C₂³	1	1	yes	characteristic
47	C₂×C₁₃₁⋊C₈	C₂³	1	2	yes	—
48	C₂×C₁₃₁⋊C₈	C₂³	1	1	yes	characteristic
49	C₄⋊Dic₁₃₁	C₂×C₄	1	1	yes	characteristic
50	C₄⋊Dic₁₃₁	C₂×C₄	1	2	yes	—
51	Dic₁₃₁⋊C₄	C₂×C₄	1	2	yes	—
52	Dic₁₃₁⋊C₄	C₂×C₄	1	2	yes	—
53	Dic₁₃₁⋊C₄	C₂×C₄	1	2	yes	—
54	Dic₁₃₁⋊C₄	C₂×C₄	1	2	yes	—
55	Dic₁₃₁⋊C₄	C₂×C₄	1	2	yes	—
56	(C₂×D₁₃₁)⋊C₄	C₂×C₄	1	2	yes	—
57	(C₂×D₁₃₁)⋊C₄	C₂×C₄	1	2	yes	—
58	C₄×Dic₁₃₁	C₂×C₄	1	1	yes	characteristic
59	C₄×Dic₁₃₁	C₂×C₄	1	1	yes	characteristic
60	C₄×Dic₁₃₁	C₂³	1	1	yes	characteristic
61	C₄×Dic₁₃₁	C₂³	1	2	yes	—
62	C₂×Q₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
63	C₂×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
64	C₂×C₈×C₁₃₁	C₂³	1	2	yes	—
65	C₂×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
66	C₄⋊C₄×C₁₃₁	C₂×C₄	1	2	yes	—
67	C₄⋊C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
68	C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
69	C₄²×C₁₃₁	C₂³	1	2	yes	—
70	C₂²×D₁₃₁	C₂²×C₄	1	1	yes	characteristic
71	Dic₂₆₂	—	2	2	no	—
72	Dic₂₆₂	—	2	4	no	—
73	C₂×Dic₁₃₁	C₂²×C₄	1	1	yes	characteristic
74	C₂×Dic₁₃₁	C₂²×C₄	1	1	yes	characteristic
75	C₂×Dic₁₃₁	C₂²×C₄	1	2	yes	—
76	C₂×Dic₁₃₁	C₂²×C₄	1	2	yes	—
77	C₂×Dic₁₃₁	C₂²×C₄	1	1	yes	characteristic
78	C₄×D₁₃₁	—	2	2	no	—
79	C₄×D₁₃₁	—	2	1	no	—
80	C₄×D₁₃₁	—	2	1	no	—
81	C₄×D₁₃₁	—	2	2	no	—
82	C₁₃₁⋊C₈	—	2	1	no	—
83	C₁₃₁⋊C₈	—	2	1	no	—
84	C₁₃₁⋊C₈	—	2	2	no	—
85	Q₈×C₁₃₁	—	2	2	no	—
86	C₂×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₄×C₁₃₁	C₂²×C₄	1	2	yes	—
88	C₂×C₄×C₁₃₁	C₂²×C₄	1	2	yes	—
89	C₂×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
91	C₈×C₁₃₁	—	2	2	no	—
92	C₈×C₁₃₁	—	2	1	no	—
93	C₈×C₁₃₁	—	2	1	no	—
94	C₂²×C₁₃₁	C₂³×C₄	1	1	yes	characteristic
95	C₂×D₁₃₁	—	2	1	no	—
96	C₂×D₁₃₁	—	2	1	no	—
97	C₂×D₁₃₁	—	2	1	no	—
98	C₄×C₁₃₁	—	2	2	no	—
99	C₄×C₁₃₁	—	2	1	no	—
100	C₄×C₁₃₁	—	2	2	no	—
101	C₄×C₁₃₁	#131	1	2	yes	—
102	C₄×C₁₃₁	2₋¹⁺⁴	1	1	yes	characteristic
103	C₄×C₁₃₁	2₋¹⁺⁴	1	1	yes	characteristic
104	Dic₁₃₁	—	2	1	no	—
105	Dic₁₃₁	—	2	1	no	—
106	Dic₁₃₁	—	2	2	no	—
107	Dic₁₃₁	—	2	2	no	—
108	Dic₁₃₁	#131	1	2	yes	—
109	C₂×C₁₃₁	#519	1	1	yes	characteristic
110	C₂×C₁₃₁	C₂×#131	1	1	yes	characteristic
111	C₂×C₁₃₁	C₂×#131	1	1	yes	characteristic
112	D₁₃₁	—	4	1	no	—
113	C₁₃₁	#2986	1	1	yes	characteristic, complemented, Sylow
114	#2986	—	131	1	no	maximal, Sylow
115	#533	—	131	1	no	—
116	#446	—	131	1	no	—
117	#446	—	131	2	no	—
118	#446	—	131	1	no	—
119	C₈×Q₈	—	131	1	no	—
120	C₈×Q₈	—	131	2	no	—
121	C₈×Q₈	D₁₃₁	1	1	yes	characteristic, complemented
122	#433	—	131	2	no	—
123	#433	—	131	2	no	—
124	#431	—	131	1	no	—
125	#431	—	131	1	no	—
126	C₄⋊Q₈	—	131	1	no	—
127	C₄⋊Q₈	—	131	1	no	—
128	#125	—	131	2	no	—
129	C₂²⋊Q₈	—	131	1	no	—
130	C₂²⋊Q₈	—	131	2	no	—
131	C₂²⋊Q₈	—	131	1	no	—
132	C₄×Q₈	—	131	2	no	—
133	C₄×Q₈	—	131	1	no	—
134	C₄×Q₈	C₂×D₁₃₁	1	1	yes	characteristic
135	#117	—	131	2	no	—
136	C₂×C₄²	—	131	1	no	—
137	C₄⋊C₈	—	131	1	no	—
138	C₄⋊C₈	—	131	1	no	—
139	C₄⋊C₈	—	131	2	no	—
140	C₄⋊C₈	—	131	2	no	—
141	C₄⋊C₈	—	131	1	no	—
142	C₄⋊C₈	—	131	2	no	—
143	C₄⋊C₈	C₂×D₁₃₁	1	1	yes	characteristic
144	C₄⋊C₈	C₂×D₁₃₁	1	2	yes	—
145	C₂²⋊C₈	—	131	1	no	—
146	C₂²⋊C₈	—	131	2	no	—
147	C₂²⋊C₈	—	131	1	no	—
148	#97	—	131	2	no	—
149	#97	—	131	2	no	—
150	C₄×C₈	—	131	2	no	—
151	C₄×C₈	—	131	2	no	—
152	C₄×C₈	—	131	1	no	—
153	C₄×C₈	C₂×D₁₃₁	1	1	yes	characteristic
154	C₄×C₈	C₂×D₁₃₁	1	2	yes	—
155	C₂×Q₈	—	131	1	no	—
156	C₂×Q₈	—	131	2	no	—
157	C₂×Q₈	C₄×D₁₃₁	1	1	yes	characteristic
158	C₂²×C₄	—	131	1	no	—
159	C₂²×C₄	—	131	1	no	—
160	C₂²×C₄	—	131	1	no	—
161	C₂×C₈	—	131	1	no	—
162	C₂×C₈	—	131	1	no	—
163	C₂×C₈	—	131	2	no	—
164	C₂×C₈	C₂²×D₁₃₁	1	2	yes	—
165	C₂×C₈	C₂²×D₁₃₁	1	1	yes	characteristic
166	C₂×C₈	C₂²×D₁₃₁	1	1	yes	characteristic
167	C₄⋊C₄	—	131	2	no	—
168	C₄⋊C₄	—	131	2	no	—
169	C₄⋊C₄	—	131	2	no	—
170	C₄⋊C₄	—	131	2	no	—
171	C₄⋊C₄	—	131	2	no	—
172	C₄⋊C₄	—	131	2	no	—
173	C₄⋊C₄	—	131	1	no	—
174	C₄⋊C₄	C₄×D₁₃₁	1	2	yes	—
175	C₄⋊C₄	C₄×D₁₃₁	1	1	yes	characteristic
176	C₂²⋊C₄	—	131	2	no	—
177	C₂²⋊C₄	—	131	2	no	—
178	C₄²	—	131	1	no	—
179	C₄²	—	131	1	no	—
180	C₄²	—	131	1	no	—
181	C₄²	—	131	2	no	—
182	C₄²	C₂²×D₁₃₁	1	1	yes	characteristic
183	C₄²	C₂²×D₁₃₁	1	2	yes	—
184	C₂³	—	131	1	no	—
185	Q₈	—	2	2	no	—
186	Q₈	—	262	4	no	—
187	Q₈	—	262	2	no	—
188	C₂×C₄	—	262	2	no	—
189	C₂×C₄	—	262	2	no	—
190	C₂×C₄	—	262	1	no	—
191	C₂×C₄	—	262	1	no	—
192	C₂×C₄	—	131	1	no	—
193	C₂×C₄	—	131	1	no	—
194	C₂×C₄	—	131	2	no	—
195	C₂×C₄	—	131	2	no	—
196	C₂×C₄	—	131	1	no	—
197	C₂×C₄	C₂×C₄×D₁₃₁	1	1	yes	characteristic
198	C₂×C₄	C₂×C₄×D₁₃₁	1	2	yes	—
199	C₂×C₄	C₂×C₄×D₁₃₁	1	2	yes	—
200	C₂×C₄	C₂×C₄×D₁₃₁	1	1	yes	characteristic
201	C₂×C₄	C₂³×D₁₃₁	1	1	yes	characteristic
202	C₈	—	2	2	no	—
203	C₈	—	2	1	no	—
204	C₈	—	2	1	no	—
205	C₈	—	262	1	no	—
206	C₈	—	262	1	no	—
207	C₈	—	262	2	no	—
208	C₂²	—	262	1	no	—
209	C₂²	—	262	1	no	—
210	C₂²	—	262	1	no	—
211	C₂²	C₂²×C₄×D₁₃₁	1	1	yes	characteristic
212	C₄	—	2	1	no	—
213	C₄	—	2	2	no	—
214	C₄	—	2	2	no	—
215	C₄	—	262	1	no	—
216	C₄	—	262	2	no	—
217	C₄	—	262	1	no	—
218	C₄	—	262	2	no	—
219	C₄	—	131	2	no	—
220	C₄	#16862252	1	2	yes	—
221	C₄	#16862293	1	1	yes	characteristic
222	C₄	#16862294	1	1	yes	characteristic
223	C₂	—	524	1	no	—
224	C₂	#22553788	1	1	yes	characteristic
225	C₂	#131×D₁₃₁	1	1	yes	characteristic
226	C₂	C₂×#16862252	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₃₁
commutator, socle	C₂²×C₁₃₁	C₂³×C₄
Fitting	C₈×Q₈×C₁₃₁	C₂
prime radical	C₂²×D₁₃₁	C₂²×C₄
cosocle	C₂×C₄×C₁₃₁	C₂⁴
hypercenter	C₈×Q₈	D₁₃₁
hypocenter	C₁₃₁	#2986
LCS	G ▷ C₂²×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂×C₄ ◁ C₈×Q₈

Complements

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

N	H ≅ G/N	type
2. #22553443	C₂	semidirect
3. #22553442	C₂	semidirect
4. #22553441	C₂	semidirect
5. C₈×Dic₂₆₂	C₂	semidirect
6. #22553436	C₂	semidirect
11. C₈×Q₈×C₁₃₁	C₂	semidirect
113. C₁₃₁	#2986	semidirect
121. C₈×Q₈	D₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
#22553866	2
#22553443	2
#22553442	2
#22553441	2
C₈×Dic₂₆₂	2
#22553436	2
#22553372	2
#22553371	2
#22553364	2
#22553363	2
C₈×Q₈×C₁₃₁	2
#2986	131