Subgroups — #30675121

Order 16768.

364 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22553443	C₂	1	1	yes	maximal, characteristic, complemented
2	Q₈×C₁₃₁⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#22553390	C₂	1	1	yes	maximal, characteristic, complemented
4	#22553384	C₂	1	2	yes	maximal, complemented
5	D₄×C₁₃₁⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	D₄×C₁₃₁⋊C₈	C₂	1	2	yes	maximal, complemented
7	#22553373	C₂	1	2	yes	maximal, complemented
8	#22553365	C₂	1	1	yes	maximal, characteristic, complemented
9	#22553339	C₂	1	2	yes	maximal, complemented
10	C₂×C₄⋊(C₁₃₁⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
11	#529×C₁₃₁	C₂	1	1	yes	maximal, characteristic
12	C₂×#14523497	C₂²	1	2	yes	—
13	C₂²×C₁₃₁⋊C₈	C₂²	1	2	yes	—
14	C₂²×C₁₃₁⋊C₈	C₂²	1	2	yes	—
15	C₄⋊(C₁₃₁⋊C₈)	C₂²	1	1	yes	characteristic
16	C₄⋊(C₁₃₁⋊C₈)	C₂²	1	1	yes	characteristic
17	C₄⋊(C₁₃₁⋊C₈)	C₂²	1	2	yes	—
18	C₄⋊(C₁₃₁⋊C₈)	C₂²	1	1	yes	characteristic
19	C₄⋊(C₁₃₁⋊C₈)	C₂²	1	2	yes	—
20	C₄⋊(C₁₃₁⋊C₈)	C₂²	1	2	yes	—
21	C₄⋊(C₁₃₁⋊C₈)	C₂²	1	1	yes	characteristic
22	C₂²⋊(C₁₃₁⋊C₈)	C₂²	1	1	yes	characteristic
23	C₂²⋊(C₁₃₁⋊C₈)	C₂²	1	1	yes	characteristic
24	C₂²⋊(C₁₃₁⋊C₈)	C₂²	1	2	yes	—
25	C₂²⋊(C₁₃₁⋊C₈)	C₂²	1	2	yes	—
26	#16862157	C₂²	1	2	yes	—
27	C₄×C₁₃₁⋊C₈	C₂²	1	1	yes	characteristic
28	C₄×C₁₃₁⋊C₈	C₂²	1	1	yes	characteristic
29	C₄×C₁₃₁⋊C₈	C₂²	1	2	yes	—
30	#126×C₁₃₁	C₄	1	2	yes	—
31	#125×C₁₃₁	C₄	1	1	yes	characteristic
32	#124×C₁₃₁	C₄	1	1	yes	characteristic
33	#123×C₁₃₁	C₄	1	2	yes	—
34	C₂²⋊Q₈×C₁₃₁	C₄	1	1	yes	characteristic
35	#121×C₁₃₁	C₄	1	1	yes	characteristic
36	C₄×Q₈×C₁₃₁	C₂²	1	1	yes	characteristic
37	C₄×D₄×C₁₃₁	C₂²	1	2	yes	—
38	C₄×D₄×C₁₃₁	C₂²	1	1	yes	characteristic
39	#117×C₁₃₁	C₂²	1	2	yes	—
40	C₂×C₄²×C₁₃₁	C₂²	1	1	yes	characteristic
41	#14523497	—	4	2	no	—
42	C₂×C₁₃₁⋊C₈	—	4	2	no	—
43	C₂×C₁₃₁⋊C₈	—	4	2	no	—
44	C₂×C₁₃₁⋊C₈	C₂³	1	1	yes	characteristic
45	C₂×C₁₃₁⋊C₈	C₂³	1	2	yes	—
46	C₂×C₁₃₁⋊C₈	C₂³	1	2	yes	—
47	C₂×C₁₃₁⋊C₈	C₂³	1	1	yes	characteristic
48	C₂×C₁₃₁⋊C₈	C₂³	1	1	yes	characteristic
49	C₂×C₁₃₁⋊C₈	C₂³	1	1	yes	characteristic
50	C₂×Q₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
51	C₂×D₄×C₁₃₁	C₂×C₄	1	2	yes	—
52	C₂×D₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
56	C₂²×C₄×C₁₃₁	C₂³	1	2	yes	—
57	C₄⋊C₄×C₁₃₁	C₂×C₄	1	2	yes	—
58	C₄⋊C₄×C₁₃₁	C₂×C₄	1	2	yes	—
59	C₄⋊C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
60	C₄⋊C₄×C₁₃₁	C₂×C₄	1	2	yes	—
61	C₄⋊C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
62	C₄⋊C₄×C₁₃₁	C₂×C₄	1	2	yes	—
63	C₂²⋊C₄×C₁₃₁	C₂×C₄	1	2	yes	—
64	C₂²⋊C₄×C₁₃₁	C₂×C₄	1	2	yes	—
65	C₂²⋊C₄×C₁₃₁	C₂×C₄	1	2	yes	—
66	C₂²⋊C₄×C₁₃₁	C₂×C₄	1	2	yes	—
67	C₂²⋊C₄×C₁₃₁	C₂×C₄	1	2	yes	—
68	C₄²×C₁₃₁	C₂×C₄	1	1	yes	characteristic
69	C₄²×C₁₃₁	C₂×C₄	1	1	yes	characteristic
70	C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
71	C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
72	C₄²×C₁₃₁	C₂³	1	2	yes	—
73	C₁₃₁⋊C₈	—	2	2	no	—
74	C₁₃₁⋊C₈	—	2	1	no	—
75	C₁₃₁⋊C₈	—	2	2	no	—
76	C₁₃₁⋊C₈	—	2	1	no	—
77	C₁₃₁⋊C₈	—	2	1	no	—
78	C₁₃₁⋊C₈	—	2	1	no	—
79	C₂³×C₁₃₁	C₂²×C₄	1	2	yes	—
80	C₂³×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
81	Q₈×C₁₃₁	—	2	2	no	—
82	D₄×C₁₃₁	—	2	4	no	—
83	D₄×C₁₃₁	—	2	2	no	—
84	C₂×C₄×C₁₃₁	—	2	2	no	—
85	C₂×C₄×C₁₃₁	—	2	2	no	—
86	C₂×C₄×C₁₃₁	—	2	1	no	—
87	C₂×C₄×C₁₃₁	—	2	2	no	—
88	C₂×C₄×C₁₃₁	—	2	1	no	—
89	C₂×C₄×C₁₃₁	—	2	2	no	—
90	C₂×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₄×C₁₃₁	C₂²×C₄	1	2	yes	—
95	C₂×C₄×C₁₃₁	C₂²×C₄	1	2	yes	—
96	C₂×C₄×C₁₃₁	C₂²×C₄	1	2	yes	—
97	C₂×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
99	C₂²×C₁₃₁	—	2	1	no	—
100	C₂²×C₁₃₁	—	2	1	no	—
101	C₂²×C₁₃₁	—	2	2	no	—
102	C₂²×C₁₃₁	—	2	2	no	—
103	C₂²×C₁₃₁	—	2	2	no	—
104	C₂²×C₁₃₁	#131	1	2	yes	—
105	C₂²×C₁₃₁	C₂³×C₄	1	1	yes	characteristic
106	C₄×C₁₃₁	—	2	1	no	—
107	C₄×C₁₃₁	—	2	2	no	—
108	C₄×C₁₃₁	—	2	1	no	—
109	C₄×C₁₃₁	—	2	1	no	—
110	C₄×C₁₃₁	—	2	1	no	—
111	C₄×C₁₃₁	—	2	2	no	—
112	C₄×C₁₃₁	—	2	2	no	—
113	C₄×C₁₃₁	#131	1	2	yes	—
114	C₄×C₁₃₁	H(5,𝔽₂)	1	1	yes	characteristic
115	C₄×C₁₃₁	2₋¹⁺⁴	1	1	yes	characteristic
116	C₂×C₁₃₁	—	2	2	no	—
117	C₂×C₁₃₁	—	4	2	no	—
118	C₂×C₁₃₁	#520	1	1	yes	characteristic
119	C₂×C₁₃₁	C₂×#131	1	1	yes	characteristic
120	C₂×C₁₃₁	C₂×#131	1	1	yes	characteristic
121	C₁₃₁	#2994	1	1	yes	characteristic, complemented, Sylow
122	#2994	—	131	1	no	maximal, Sylow
123	#529	D₁₃₁	1	1	yes	characteristic
124	#446	—	131	1	no	—
125	C₈×Q₈	—	131	1	no	—
126	#436	—	131	1	no	—
127	#435	—	131	2	no	—
128	C₈×D₄	—	131	1	no	—
129	C₈×D₄	—	131	2	no	—
130	#433	—	131	2	no	—
131	#431	—	131	1	no	—
132	#424	—	131	2	no	—
133	C₂×C₄⋊C₈	—	131	1	no	—
134	C₂×M₁₆	—	131	2	no	—
135	C₂²×C₈	—	131	2	no	—
136	C₂²×C₈	—	131	2	no	—
137	#126	Dic₁₃₁	1	2	yes	—
138	#125	Dic₁₃₁	1	1	yes	characteristic
139	#124	Dic₁₃₁	1	1	yes	characteristic
140	#123	Dic₁₃₁	1	2	yes	—
141	C₂²⋊Q₈	Dic₁₃₁	1	1	yes	characteristic
142	#121	Dic₁₃₁	1	1	yes	characteristic
143	C₄×Q₈	C₂×D₁₃₁	1	1	yes	characteristic
144	C₄×D₄	C₂×D₁₃₁	1	2	yes	—
145	C₄×D₄	C₂×D₁₃₁	1	1	yes	characteristic
146	#117	C₂×D₁₃₁	1	2	yes	—
147	C₂×C₄²	C₂×D₁₃₁	1	1	yes	characteristic
148	C₄⋊C₈	—	131	2	no	—
149	C₄⋊C₈	—	131	1	no	—
150	C₄⋊C₈	—	131	1	no	—
151	C₄⋊C₈	—	131	1	no	—
152	C₄⋊C₈	—	131	2	no	—
153	C₄⋊C₈	—	131	2	no	—
154	C₄⋊C₈	—	131	1	no	—
155	C₂²⋊C₈	—	131	1	no	—
156	C₂²⋊C₈	—	131	1	no	—
157	C₂²⋊C₈	—	131	2	no	—
158	C₂²⋊C₈	—	131	2	no	—
159	#97	—	131	2	no	—
160	C₄×C₈	—	131	1	no	—
161	C₄×C₈	—	131	1	no	—
162	C₄×C₈	—	131	2	no	—
163	C₂×Q₈	C₂×Dic₁₃₁	1	1	yes	characteristic
164	C₂×D₄	C₂×Dic₁₃₁	1	2	yes	—
165	C₂×D₄	C₂×Dic₁₃₁	1	1	yes	characteristic
166	C₂²×C₄	C₂×Dic₁₃₁	1	1	yes	characteristic
167	C₂²×C₄	C₂×Dic₁₃₁	1	1	yes	characteristic
168	C₂²×C₄	C₂²×D₁₃₁	1	1	yes	characteristic
169	C₂²×C₄	C₂²×D₁₃₁	1	2	yes	—
170	M₁₆	—	524	2	no	—
171	C₂×C₈	—	524	2	no	—
172	C₂×C₈	—	524	2	no	—
173	C₂×C₈	—	131	1	no	—
174	C₂×C₈	—	131	1	no	—
175	C₂×C₈	—	131	1	no	—
176	C₂×C₈	—	131	2	no	—
177	C₂×C₈	—	131	1	no	—
178	C₂×C₈	—	131	2	no	—
179	C₄⋊C₄	C₂×Dic₁₃₁	1	2	yes	—
180	C₄⋊C₄	C₂×Dic₁₃₁	1	2	yes	—
181	C₄⋊C₄	C₂×Dic₁₃₁	1	1	yes	characteristic
182	C₄⋊C₄	C₂×Dic₁₃₁	1	2	yes	—
183	C₄⋊C₄	C₂×Dic₁₃₁	1	1	yes	characteristic
184	C₄⋊C₄	C₂×Dic₁₃₁	1	2	yes	—
185	C₂²⋊C₄	C₂×Dic₁₃₁	1	2	yes	—
186	C₂²⋊C₄	C₂×Dic₁₃₁	1	2	yes	—
187	C₂²⋊C₄	C₂×Dic₁₃₁	1	2	yes	—
188	C₂²⋊C₄	C₂×Dic₁₃₁	1	2	yes	—
189	C₂²⋊C₄	C₂×Dic₁₃₁	1	2	yes	—
190	C₄²	C₂×Dic₁₃₁	1	1	yes	characteristic
191	C₄²	C₂×Dic₁₃₁	1	1	yes	characteristic
192	C₄²	C₂²×D₁₃₁	1	1	yes	characteristic
193	C₄²	C₂²×D₁₃₁	1	2	yes	—
194	C₄²	C₂²×D₁₃₁	1	1	yes	characteristic
195	C₂³	C₂²×Dic₁₃₁	1	2	yes	—
196	C₂³	C₂²×Dic₁₃₁	1	1	yes	characteristic
197	Q₈	—	2	2	no	—
198	D₄	—	2	4	no	—
199	D₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	C₂²×Dic₁₃₁	1	1	yes	characteristic
207	C₂×C₄	C₂²×Dic₁₃₁	1	1	yes	characteristic
208	C₂×C₄	C₂²×Dic₁₃₁	1	1	yes	characteristic
209	C₂×C₄	C₂²×Dic₁₃₁	1	1	yes	characteristic
210	C₂×C₄	C₂²×Dic₁₃₁	1	2	yes	—
211	C₂×C₄	C₂²×Dic₁₃₁	1	2	yes	—
212	C₂×C₄	C₂²×Dic₁₃₁	1	2	yes	—
213	C₂×C₄	C₂²×Dic₁₃₁	1	1	yes	characteristic
214	C₂×C₄	C₂³×D₁₃₁	1	1	yes	characteristic
215	C₈	—	262	1	no	—
216	C₈	—	262	2	no	—
217	C₈	—	262	2	no	—
218	C₈	—	262	1	no	—
219	C₈	—	262	1	no	—
220	C₈	—	262	1	no	—
221	C₂²	—	2	1	no	—
222	C₂²	—	2	1	no	—
223	C₂²	—	2	2	no	—
224	C₂²	—	2	2	no	—
225	C₂²	—	2	2	no	—
226	C₂²	#16862253	1	2	yes	—
227	C₂²	C₂³×Dic₁₃₁	1	1	yes	characteristic
228	C₄	—	2	1	no	—
229	C₄	—	2	2	no	—
230	C₄	—	2	1	no	—
231	C₄	—	2	1	no	—
232	C₄	—	2	1	no	—
233	C₄	—	2	2	no	—
234	C₄	—	2	2	no	—
235	C₄	#16862253	1	2	yes	—
236	C₄	#16862292	1	1	yes	characteristic
237	C₄	#16862293	1	1	yes	characteristic
238	C₂	—	2	2	no	—
239	C₂	—	4	2	no	—
240	C₂	#22553795	1	1	yes	characteristic
241	C₂	C₂×#16862253	1	1	yes	characteristic
242	C₂	C₂×#16862253	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₃₁
commutator, socle	C₂²×C₁₃₁	C₂³×C₄
Fitting	#529×C₁₃₁	C₂
prime radical	#121×C₁₃₁	C₄
cosocle	C₂×C₄×C₁₃₁	C₂⁴
hypercenter	#529	D₁₃₁
hypocenter	C₁₃₁	#2994
LCS	G ▷ C₂²×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂×C₄ ◁ #529

Complements

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

N	H ≅ G/N	type
1. #22553443	C₂	semidirect
2. Q₈×C₁₃₁⋊C₈	C₂	semidirect
3. #22553390	C₂	semidirect
4. #22553384	C₂	semidirect
5. D₄×C₁₃₁⋊C₈	C₂	semidirect
6. D₄×C₁₃₁⋊C₈	C₂	semidirect
7. #22553373	C₂	semidirect
8. #22553365	C₂	semidirect
9. #22553339	C₂	semidirect
10. C₂×C₄⋊(C₁₃₁⋊C₈)	C₂	semidirect
121. C₁₃₁	#2994	semidirect

Maximal subgroups

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

subgroup	index
#22553443	2
Q₈×C₁₃₁⋊C₈	2
#22553390	2
#22553384	2
D₄×C₁₃₁⋊C₈	2
#22553373	2
#22553365	2
#22553339	2
C₂×C₄⋊(C₁₃₁⋊C₈)	2
#529×C₁₃₁	2
#2994	131