Subgroups — #30675123

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	#22553846	C₂	1	1	yes	maximal, characteristic
2	#22553442	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈×Dic₂₆₂	C₂	1	1	yes	maximal, characteristic, complemented
4	#22553391	C₂	1	1	yes	maximal, characteristic, complemented
5	#22553380	C₂	1	2	yes	maximal, complemented
6	C₈×D₅₂₄	C₂	1	1	yes	maximal, characteristic, complemented
7	(C₂²⋊D₁₃₁)⋊C₈	C₂	1	2	yes	maximal, complemented
8	#22553372	C₂	1	2	yes	maximal, complemented
9	#22553337	C₂	1	2	yes	maximal, complemented
10	C₂×C₄⋊(C₁₃₁⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
11	#431×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14523496	C₂²	1	2	yes	—
13	C₂²×C₁₃₁⋊C₈	C₂²	1	2	yes	—
14	C₂×C₈×D₁₃₁	C₂²	1	2	yes	—
15	#16862240	C₄	1	2	yes	—
16	#16862238	C₄	1	1	yes	characteristic
17	#16862234	C₄	1	1	yes	characteristic
18	#16862229	C₄	1	2	yes	—
19	#16862225	C₄	1	1	yes	characteristic
20	#16862220	C₄	1	1	yes	characteristic
21	C₄×Dic₂₆₂	C₂²	1	1	yes	characteristic
22	C₄×D₅₂₄	C₂²	1	1	yes	characteristic
23	C₄×C₂²⋊D₁₃₁	C₂²	1	2	yes	—
24	#16862203	C₂²	1	2	yes	—
25	C₄⋊(C₁₃₁⋊C₈)	C₂²	1	2	yes	—
26	C₄⋊(C₁₃₁⋊C₈)	C₂²	1	1	yes	characteristic
27	C₄⋊(C₁₃₁⋊C₈)	C₂²	1	1	yes	characteristic
28	Dic₁₃₁⋊C₈	C₂²	1	2	yes	—
29	Dic₁₃₁⋊C₈	C₂²	1	2	yes	—
30	(C₂×D₁₃₁)⋊C₈	C₂²	1	2	yes	—
31	(C₂×D₁₃₁)⋊C₈	C₂²	1	2	yes	—
32	#16862156	C₂²	1	2	yes	—
33	C₈×Dic₁₃₁	C₂²	1	2	yes	—
34	C₂×C₄²×C₁₃₁	C₂²	1	1	yes	characteristic
35	C₄⋊C₈×C₁₃₁	C₂²	1	1	yes	characteristic
36	C₄⋊C₈×C₁₃₁	C₂²	1	1	yes	characteristic
37	C₂²⋊C₈×C₁₃₁	C₂²	1	1	yes	characteristic
38	C₂²⋊C₈×C₁₃₁	C₂²	1	1	yes	characteristic
39	C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
40	C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
41	C₂×Dic₂₆₂	C₂×C₄	1	1	yes	characteristic
42	C₂×D₅₂₄	C₂×C₄	1	1	yes	characteristic
43	C₂×C₂²⋊D₁₃₁	C₂×C₄	1	2	yes	—
44	C₂×C₄×D₁₃₁	C₂³	1	2	yes	—
45	#14523496	—	4	2	no	—
46	C₂×C₁₃₁⋊C₈	—	4	2	no	—
47	C₂×C₁₃₁⋊C₈	C₂³	1	2	yes	—
48	C₂×C₁₃₁⋊C₈	C₂³	1	2	yes	—
49	C₈×D₁₃₁	—	4	2	no	—
50	C₄⋊Dic₁₃₁	C₂×C₄	1	1	yes	characteristic
51	C₄⋊Dic₁₃₁	C₂×C₄	1	1	yes	characteristic
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	C₂²⋊Dic₁₃₁	C₂×C₄	1	2	yes	—
57	(C₂×D₁₃₁)⋊C₄	C₂×C₄	1	2	yes	—
58	(C₂×D₁₃₁)⋊C₄	C₂×C₄	1	2	yes	—
59	(C₂×D₁₃₁)⋊C₄	C₂×C₄	1	2	yes	—
60	(C₂×D₁₃₁)⋊C₄	C₂×C₄	1	2	yes	—
61	C₄×Dic₁₃₁	C₂³	1	2	yes	—
62	C₂²×C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
65	C₂×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
66	C₂×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
67	C₂×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
68	C₂×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
69	C₄²×C₁₃₁	C₂×C₄	1	1	yes	characteristic
70	C₄²×C₁₃₁	C₂×C₄	1	1	yes	characteristic
71	C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
72	C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
73	C₂²×D₁₃₁	C₂²×C₄	1	2	yes	—
74	Dic₂₆₂	—	2	2	no	—
75	D₅₂₄	—	2	2	no	—
76	C₂²⋊D₁₃₁	—	2	4	no	—
77	C₂×Dic₁₃₁	C₂²×C₄	1	2	yes	—
78	C₂×Dic₁₃₁	C₂²×C₄	1	2	yes	—
79	C₂×Dic₁₃₁	C₂²×C₄	1	2	yes	—
80	C₄×D₁₃₁	—	2	2	no	—
81	C₄×D₁₃₁	—	2	2	no	—
82	C₁₃₁⋊C₈	—	2	2	no	—
83	C₁₃₁⋊C₈	—	2	2	no	—
84	C₂³×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₄×C₁₃₁	—	2	2	no	—
86	C₂×C₄×C₁₃₁	—	2	2	no	—
87	C₂×C₄×C₁₃₁	—	2	1	no	—
88	C₂×C₄×C₁₃₁	—	2	1	no	—
89	C₂×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
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	C₂⁴	1	1	yes	characteristic
95	C₈×C₁₃₁	—	2	1	no	—
96	C₈×C₁₃₁	—	2	1	no	—
97	C₈×C₁₃₁	—	2	1	no	—
98	C₈×C₁₃₁	—	2	1	no	—
99	C₂²×C₁₃₁	—	2	1	no	—
100	C₂²×C₁₃₁	—	2	1	no	—
101	C₂²×C₁₃₁	#131	1	2	yes	—
102	C₂²×C₁₃₁	C₂³×C₄	1	1	yes	characteristic
103	C₂×D₁₃₁	—	2	2	no	—
104	C₂×D₁₃₁	—	2	2	no	—
105	C₂×D₁₃₁	—	2	2	no	—
106	C₄×C₁₃₁	—	2	1	no	—
107	C₄×C₁₃₁	—	2	1	no	—
108	C₄×C₁₃₁	—	2	1	no	—
109	C₄×C₁₃₁	—	2	1	no	—
110	C₄×C₁₃₁	#131	1	2	yes	—
111	C₄×C₁₃₁	H(5,𝔽₂)	1	1	yes	characteristic
112	C₄×C₁₃₁	2₋¹⁺⁴	1	1	yes	characteristic
113	Dic₁₃₁	—	2	2	no	—
114	Dic₁₃₁	—	2	2	no	—
115	Dic₁₃₁	—	2	2	no	—
116	C₂×C₁₃₁	—	2	2	no	—
117	C₂×C₁₃₁	#520	1	1	yes	characteristic
118	C₂×C₁₃₁	C₂×#131	1	1	yes	characteristic
119	C₂×C₁₃₁	C₂×#131	1	1	yes	characteristic
120	D₁₃₁	—	4	2	no	—
121	C₁₃₁	#2994	1	1	yes	characteristic, complemented, Sylow
122	#2994	—	131	1	no	maximal, Sylow
123	#529	—	131	1	no	—
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	2	no	—
129	C₈×D₄	—	131	1	no	—
130	#433	—	131	2	no	—
131	#431	D₁₃₁	1	1	yes	characteristic, complemented
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	—	131	2	no	—
138	#125	—	131	1	no	—
139	#124	—	131	1	no	—
140	#123	—	131	2	no	—
141	C₂²⋊Q₈	—	131	1	no	—
142	#121	—	131	1	no	—
143	C₄×Q₈	—	131	1	no	—
144	C₄×D₄	—	131	2	no	—
145	C₄×D₄	—	131	1	no	—
146	#117	—	131	2	no	—
147	C₂×C₄²	C₂×D₁₃₁	1	1	yes	characteristic
148	C₄⋊C₈	—	131	2	no	—
149	C₄⋊C₈	—	131	2	no	—
150	C₄⋊C₈	—	131	2	no	—
151	C₄⋊C₈	—	131	1	no	—
152	C₄⋊C₈	—	131	1	no	—
153	C₄⋊C₈	C₂×D₁₃₁	1	1	yes	characteristic
154	C₄⋊C₈	C₂×D₁₃₁	1	1	yes	characteristic
155	C₂²⋊C₈	—	131	2	no	—
156	C₂²⋊C₈	—	131	2	no	—
157	C₂²⋊C₈	C₂×D₁₃₁	1	1	yes	characteristic
158	C₂²⋊C₈	C₂×D₁₃₁	1	1	yes	characteristic
159	#97	—	131	2	no	—
160	C₄×C₈	—	131	2	no	—
161	C₄×C₈	C₂×D₁₃₁	1	1	yes	characteristic
162	C₄×C₈	C₂×D₁₃₁	1	1	yes	characteristic
163	C₂×Q₈	—	131	1	no	—
164	C₂×D₄	—	131	2	no	—
165	C₂×D₄	—	131	1	no	—
166	C₂²×C₄	—	131	2	no	—
167	C₂²×C₄	C₄×D₁₃₁	1	1	yes	characteristic
168	C₂²×C₄	C₄×D₁₃₁	1	1	yes	characteristic
169	C₂²×C₄	C₂²×D₁₃₁	1	1	yes	characteristic
170	M₁₆	—	524	2	no	—
171	C₂×C₈	—	524	2	no	—
172	C₂×C₈	—	524	2	no	—
173	C₂×C₈	—	131	2	no	—
174	C₂×C₈	—	131	2	no	—
175	C₂×C₈	C₂²×D₁₃₁	1	1	yes	characteristic
176	C₂×C₈	C₂²×D₁₃₁	1	1	yes	characteristic
177	C₂×C₈	C₂²×D₁₃₁	1	1	yes	characteristic
178	C₂×C₈	C₂²×D₁₃₁	1	1	yes	characteristic
179	C₄⋊C₄	—	131	2	no	—
180	C₄⋊C₄	—	131	2	no	—
181	C₄⋊C₄	—	131	2	no	—
182	C₄⋊C₄	—	131	2	no	—
183	C₄⋊C₄	—	131	1	no	—
184	C₄⋊C₄	—	131	1	no	—
185	C₂²⋊C₄	—	131	2	no	—
186	C₂²⋊C₄	—	131	2	no	—
187	C₂²⋊C₄	—	131	2	no	—
188	C₂²⋊C₄	—	131	2	no	—
189	C₂²⋊C₄	—	131	2	no	—
190	C₄²	—	131	2	no	—
191	C₄²	C₄×D₁₃₁	1	1	yes	characteristic
192	C₄²	C₄×D₁₃₁	1	1	yes	characteristic
193	C₄²	C₂²×D₁₃₁	1	1	yes	characteristic
194	C₄²	C₂²×D₁₃₁	1	1	yes	characteristic
195	C₂³	—	131	2	no	—
196	C₂³	C₂×C₄×D₁₃₁	1	1	yes	characteristic
197	Q₈	—	262	2	no	—
198	D₄	—	262	4	no	—
199	D₄	—	262	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	262	2	no	—
205	C₂×C₄	—	262	2	no	—
206	C₂×C₄	—	131	2	no	—
207	C₂×C₄	—	131	2	no	—
208	C₂×C₄	—	131	2	no	—
209	C₂×C₄	C₂×C₄×D₁₃₁	1	1	yes	characteristic
210	C₂×C₄	C₂×C₄×D₁₃₁	1	1	yes	characteristic
211	C₂×C₄	C₂×C₄×D₁₃₁	1	1	yes	characteristic
212	C₂×C₄	C₂×C₄×D₁₃₁	1	1	yes	characteristic
213	C₂×C₄	C₂×C₄×D₁₃₁	1	1	yes	characteristic
214	C₂×C₄	C₂³×D₁₃₁	1	1	yes	characteristic
215	C₈	—	2	1	no	—
216	C₈	—	2	1	no	—
217	C₈	—	2	1	no	—
218	C₈	—	2	1	no	—
219	C₈	—	262	2	no	—
220	C₈	—	262	2	no	—
221	C₂²	—	2	1	no	—
222	C₂²	—	2	1	no	—
223	C₂²	—	262	2	no	—
224	C₂²	—	262	2	no	—
225	C₂²	—	262	2	no	—
226	C₂²	#16862252	1	2	yes	—
227	C₂²	C₂²×C₄×D₁₃₁	1	1	yes	characteristic
228	C₄	—	2	1	no	—
229	C₄	—	2	1	no	—
230	C₄	—	2	1	no	—
231	C₄	—	2	1	no	—
232	C₄	—	262	2	no	—
233	C₄	—	262	2	no	—
234	C₄	—	262	2	no	—
235	C₄	#16862254	1	2	yes	—
236	C₄	#16862292	1	1	yes	characteristic
237	C₄	#16862293	1	1	yes	characteristic
238	C₂	—	2	2	no	—
239	C₂	—	524	2	no	—
240	C₂	#22553792	1	1	yes	characteristic
241	C₂	C₂×#16862252	1	1	yes	characteristic
242	C₂	C₂×#16862254	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₃₁
commutator, socle	C₂²×C₁₃₁	C₂³×C₄
Fitting	#431×C₁₃₁	C₂
prime radical	#16862220	C₄
cosocle	C₂×C₄×C₁₃₁	C₂⁴
hypercenter	#431	D₁₃₁
hypocenter	C₁₃₁	#2994
LCS	G ▷ C₂²×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂×C₄ ◁ #431

Complements

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

N	H ≅ G/N	type
2. #22553442	C₂	semidirect
3. C₈×Dic₂₆₂	C₂	semidirect
4. #22553391	C₂	semidirect
5. #22553380	C₂	semidirect
6. C₈×D₅₂₄	C₂	semidirect
7. (C₂²⋊D₁₃₁)⋊C₈	C₂	semidirect
8. #22553372	C₂	semidirect
9. #22553337	C₂	semidirect
10. C₂×C₄⋊(C₁₃₁⋊C₈)	C₂	semidirect
11. #431×C₁₃₁	C₂	semidirect
121. C₁₃₁	#2994	semidirect
131. #431	D₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
#22553846	2
#22553442	2
C₈×Dic₂₆₂	2
#22553391	2
#22553380	2
C₈×D₅₂₄	2
(C₂²⋊D₁₃₁)⋊C₈	2
#22553372	2
#22553337	2
C₂×C₄⋊(C₁₃₁⋊C₈)	2
#431×C₁₃₁	2
#2994	131