Subgroups — #30675085

Order 16768.

398 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14523514	C₂	1	1	yes	maximal, characteristic
2	#22553443	C₂	1	1	yes	maximal, characteristic, complemented
3	#22553385	C₂	1	2	yes	maximal, complemented
4	#22553382	C₂	1	2	yes	maximal, complemented
5	C₈×D₅₂₄	C₂	1	1	yes	maximal, characteristic, complemented
6	#22553377	C₂	1	1	yes	maximal, characteristic, complemented
7	#22553367	C₂	1	2	yes	maximal, complemented
8	#22553363	C₂	1	1	yes	maximal, characteristic, complemented
9	#22553334	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄⋊C₈×D₁₃₁	C₂	1	2	yes	maximal, complemented
11	#446×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14523514	C₄	1	1	yes	characteristic
13	C₂×#14523496	C₂²	1	2	yes	—
14	C₂×C₈×D₁₃₁	C₂²	1	2	yes	—
15	C₂×C₈×D₁₃₁	C₂²	1	2	yes	—
16	Q₈×Dic₁₃₁	C₄	1	1	yes	characteristic
17	C₄×D₅₂₄	C₂²	1	1	yes	characteristic
18	C₄×D₅₂₄	C₂²	1	2	yes	—
19	#16862208	C₄	1	2	yes	—
20	#16862208	C₄	1	1	yes	characteristic
21	#16862204	C₄	1	2	yes	—
22	#16862204	C₄	1	1	yes	characteristic
23	C₄²×D₁₃₁	C₂²	1	1	yes	characteristic
24	C₄²×D₁₃₁	C₂²	1	2	yes	—
25	C₄⋊(C₁₃₁⋊C₈)	C₂²	1	1	yes	characteristic
26	C₄⋊(C₁₃₁⋊C₈)	C₂²	1	2	yes	—
27	Dic₁₃₁⋊C₈	C₂²	1	1	yes	characteristic
28	Dic₁₃₁⋊C₈	C₂²	1	1	yes	characteristic
29	Dic₁₃₁⋊C₈	C₂²	1	2	yes	—
30	(C₂×D₁₃₁)⋊C₈	C₂²	1	1	yes	characteristic
31	(C₂×D₁₃₁)⋊C₈	C₂²	1	1	yes	characteristic
32	(C₂×D₁₃₁)⋊C₈	C₂²	1	2	yes	—
33	(C₂×D₁₃₁)⋊C₈	C₂²	1	2	yes	—
34	#16862157	C₂²	1	2	yes	—
35	C₄×C₁₃₁⋊C₈	C₂²	1	1	yes	characteristic
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	#97×C₁₃₁	C₂²	1	2	yes	—
40	C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
41	#14523514	—	2	4	no	—
42	C₂×D₅₂₄	C₂×C₄	1	2	yes	—
43	C₂×D₅₂₄	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×D₁₃₁	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×D₁₃₁	C₂×C₄	1	2	yes	—
46	C₂×C₄×D₁₃₁	C₂×C₄	1	2	yes	—
47	C₂×C₄×D₁₃₁	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄×D₁₃₁	C₂³	1	2	yes	—
49	C₂×C₄×D₁₃₁	C₂³	1	1	yes	characteristic
50	#14523496	—	4	2	no	—
51	C₂×C₁₃₁⋊C₈	C₂³	1	1	yes	characteristic
52	C₂×C₁₃₁⋊C₈	C₂³	1	2	yes	—
53	C₂×C₁₃₁⋊C₈	C₂³	1	1	yes	characteristic
54	C₈×D₁₃₁	—	4	2	no	—
55	C₈×D₁₃₁	—	4	2	no	—
56	C₄⋊Dic₁₃₁	C₂×C₄	1	2	yes	—
57	C₄⋊Dic₁₃₁	C₂×C₄	1	1	yes	characteristic
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	C₂×C₄	1	2	yes	—
62	C₄×Dic₁₃₁	C₂×C₄	1	1	yes	characteristic
63	C₄×Dic₁₃₁	C₂×C₄	1	1	yes	characteristic
64	C₄×Dic₁₃₁	C₂×C₄	1	2	yes	—
65	C₄×Dic₁₃₁	C₂³	1	1	yes	characteristic
66	C₂×Q₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
67	C₂×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
68	C₂×C₈×C₁₃₁	C₂³	1	2	yes	—
69	C₂×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
70	C₄⋊C₄×C₁₃₁	C₂×C₄	1	2	yes	—
71	C₄⋊C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
72	C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
73	C₄²×C₁₃₁	C₂³	1	2	yes	—
74	C₂²×D₁₃₁	C₂²×C₄	1	1	yes	characteristic
75	C₂²×D₁₃₁	C₂²×C₄	1	2	yes	—
76	D₅₂₄	—	2	4	no	—
77	D₅₂₄	M₁₆	1	4	yes	complemented
78	C₂×Dic₁₃₁	C₂²×C₄	1	1	yes	characteristic
79	C₂×Dic₁₃₁	C₂²×C₄	1	2	yes	—
80	C₂×Dic₁₃₁	C₂²×C₄	1	1	yes	characteristic
81	C₂×Dic₁₃₁	C₂²×C₄	1	1	yes	characteristic
82	C₄×D₁₃₁	—	2	4	no	—
83	C₄×D₁₃₁	—	2	2	no	—
84	C₄×D₁₃₁	—	2	2	no	—
85	C₄×D₁₃₁	—	2	1	no	—
86	C₄×D₁₃₁	—	2	2	no	—
87	C₄×D₁₃₁	—	2	1	no	—
88	C₄×D₁₃₁	—	2	4	no	—
89	C₄×D₁₃₁	—	2	2	no	—
90	C₁₃₁⋊C₈	—	2	1	no	—
91	C₁₃₁⋊C₈	—	2	2	no	—
92	C₁₃₁⋊C₈	—	2	1	no	—
93	Q₈×C₁₃₁	—	2	2	no	—
94	C₂×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
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	C₈×C₁₃₁	—	2	1	no	—
100	C₈×C₁₃₁	—	2	2	no	—
101	C₈×C₁₃₁	—	2	1	no	—
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	1	no	—
106	C₂×D₁₃₁	—	2	1	no	—
107	C₂×D₁₃₁	—	2	1	no	—
108	C₂×D₁₃₁	C₂×M₁₆	1	4	yes	—
109	C₄×C₁₃₁	—	2	2	no	—
110	C₄×C₁₃₁	—	2	1	no	—
111	C₄×C₁₃₁	—	2	2	no	—
112	C₄×C₁₃₁	C₂×M₁₆	1	2	yes	—
113	C₄×C₁₃₁	H(5,𝔽₂)	1	1	yes	characteristic
114	C₄×C₁₃₁	2₋¹⁺⁴	1	1	yes	characteristic
115	Dic₁₃₁	—	2	2	no	—
116	Dic₁₃₁	—	2	1	no	—
117	Dic₁₃₁	—	2	1	no	—
118	Dic₁₃₁	#131	1	2	yes	—
119	C₂×C₁₃₁	#520	1	1	yes	characteristic
120	C₂×C₁₃₁	C₂²×M₁₆	1	1	yes	characteristic
121	C₂×C₁₃₁	C₂×#131	1	1	yes	characteristic
122	D₁₃₁	—	2	4	no	—
123	D₁₃₁	—	4	1	no	—
124	C₁₃₁	#2990	1	1	yes	characteristic, complemented, Sylow
125	#2990	—	131	1	no	maximal, Sylow
126	C₄×#41	—	131	1	no	—
127	#446	—	131	1	no	—
128	#446	D₁₃₁	1	1	yes	characteristic, complemented
129	#435	—	131	2	no	—
130	#435	—	131	2	no	—
131	C₈×D₄	—	131	1	no	—
132	C₈×D₄	—	131	1	no	—
133	#432	—	131	2	no	—
134	#431	—	131	1	no	—
135	#423	—	131	1	no	—
136	C₂×C₄⋊C₈	—	131	2	no	—
137	C₂×#41	—	131	1	no	—
138	C₂×M₁₆	—	131	2	no	—
139	C₂²×C₈	—	131	2	no	—
140	C₂²×C₈	—	131	2	no	—
141	C₄×Q₈	—	131	1	no	—
142	C₄×Q₈	C₂×D₁₃₁	1	1	yes	characteristic
143	C₄×D₄	—	131	2	no	—
144	C₄×D₄	—	131	1	no	—
145	C₄×D₄	—	131	1	no	—
146	C₄×D₄	—	131	2	no	—
147	#117	—	131	1	no	—
148	#117	—	131	2	no	—
149	C₂×C₄²	—	131	2	no	—
150	C₂×C₄²	—	131	1	no	—
151	C₄⋊C₈	—	131	1	no	—
152	C₄⋊C₈	—	131	2	no	—
153	C₄⋊C₈	—	131	1	no	—
154	C₄⋊C₈	—	131	2	no	—
155	C₄⋊C₈	—	131	1	no	—
156	C₄⋊C₈	C₂×D₁₃₁	1	2	yes	—
157	C₄⋊C₈	C₂×D₁₃₁	1	1	yes	characteristic
158	C₂²⋊C₈	—	131	1	no	—
159	C₂²⋊C₈	—	131	1	no	—
160	C₂²⋊C₈	—	131	2	no	—
161	C₂²⋊C₈	—	131	2	no	—
162	#97	—	131	2	no	—
163	#97	C₂×D₁₃₁	1	2	yes	—
164	C₄×C₈	—	131	1	no	—
165	C₄×C₈	C₂×D₁₃₁	1	1	yes	characteristic
166	#41	—	262	4	no	—
167	C₂×Q₈	C₄×D₁₃₁	1	1	yes	characteristic
168	C₂×D₄	—	131	2	no	—
169	C₂×D₄	—	131	1	no	—
170	C₂²×C₄	—	131	2	no	—
171	C₂²×C₄	—	131	1	no	—
172	C₂²×C₄	—	131	1	no	—
173	C₂²×C₄	—	131	2	no	—
174	C₂²×C₄	—	131	2	no	—
175	C₂²×C₄	—	131	1	no	—
176	M₁₆	—	524	2	no	—
177	C₂×C₈	—	524	2	no	—
178	C₂×C₈	—	524	2	no	—
179	C₂×C₈	—	131	1	no	—
180	C₂×C₈	—	131	1	no	—
181	C₂×C₈	—	131	2	no	—
182	C₂×C₈	C₂²×D₁₃₁	1	1	yes	characteristic
183	C₂×C₈	C₂²×D₁₃₁	1	1	yes	characteristic
184	C₂×C₈	C₂²×D₁₃₁	1	2	yes	—
185	C₄⋊C₄	—	131	2	no	—
186	C₄⋊C₄	—	131	1	no	—
187	C₄⋊C₄	C₄×D₁₃₁	1	2	yes	—
188	C₄⋊C₄	C₄×D₁₃₁	1	1	yes	characteristic
189	C₂²⋊C₄	—	131	2	no	—
190	C₂²⋊C₄	—	131	2	no	—
191	C₂²⋊C₄	—	131	2	no	—
192	C₄²	—	131	2	no	—
193	C₄²	—	131	1	no	—
194	C₄²	—	131	2	no	—
195	C₄²	—	131	1	no	—
196	C₄²	—	131	1	no	—
197	C₄²	C₂²×D₁₃₁	1	1	yes	characteristic
198	C₄²	C₂²×D₁₃₁	1	2	yes	—
199	C₂³	—	131	1	no	—
200	C₂³	—	131	2	no	—
201	Q₈	—	2	2	no	—
202	D₄	—	262	4	no	—
203	D₄	—	131	4	no	—
204	C₂×C₄	—	262	4	no	—
205	C₂×C₄	—	262	4	no	—
206	C₂×C₄	—	262	2	no	—
207	C₂×C₄	—	262	2	no	—
208	C₂×C₄	—	262	2	no	—
209	C₂×C₄	—	262	1	no	—
210	C₂×C₄	—	262	2	no	—
211	C₂×C₄	—	262	1	no	—
212	C₂×C₄	—	131	1	no	—
213	C₂×C₄	—	131	2	no	—
214	C₂×C₄	—	131	1	no	—
215	C₂×C₄	—	131	1	no	—
216	C₂×C₄	C₂×C₄×D₁₃₁	1	1	yes	characteristic
217	C₂×C₄	C₂×C₄×D₁₃₁	1	2	yes	—
218	C₂×C₄	C₂×C₄×D₁₃₁	1	2	yes	—
219	C₂×C₄	C₂×C₄×D₁₃₁	1	1	yes	characteristic
220	C₂×C₄	C₂³×D₁₃₁	1	1	yes	characteristic
221	C₈	—	2	1	no	—
222	C₈	—	2	2	no	—
223	C₈	—	2	1	no	—
224	C₈	—	262	1	no	—
225	C₈	—	262	2	no	—
226	C₈	—	262	1	no	—
227	C₂²	—	262	2	no	—
228	C₂²	—	262	2	no	—
229	C₂²	—	262	1	no	—
230	C₂²	—	262	1	no	—
231	C₂²	—	262	1	no	—
232	C₂²	—	131	4	no	—
233	C₂²	C₂²×C₄×D₁₃₁	1	1	yes	characteristic
234	C₄	—	2	2	no	—
235	C₄	—	2	2	no	—
236	C₄	—	2	1	no	—
237	C₄	—	262	2	no	—
238	C₄	—	262	1	no	—
239	C₄	—	262	1	no	—
240	C₄	—	131	2	no	—
241	C₄	M₁₆×D₁₃₁	1	2	yes	—
242	C₄	#16862292	1	1	yes	characteristic
243	C₄	#16862294	1	1	yes	characteristic
244	C₂	—	262	4	no	—
245	C₂	—	524	1	no	—
246	C₂	#22553794	1	1	yes	characteristic
247	C₂	C₂×M₁₆×D₁₃₁	1	1	yes	characteristic
248	C₂	#131×D₁₃₁	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₃₁
commutator, socle	C₂²×C₁₃₁	C₂³×C₄
Fitting	#446×C₁₃₁	C₂
prime radical	C₂×#14523514	C₄
cosocle	C₂×C₄×C₁₃₁	C₂⁴
hypercenter	#446	D₁₃₁
hypocenter	C₁₃₁	#2990
LCS	G ▷ C₂²×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂×C₄ ◁ #446

Complements

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

N	H ≅ G/N	type
2. #22553443	C₂	semidirect
3. #22553385	C₂	semidirect
4. #22553382	C₂	semidirect
5. C₈×D₅₂₄	C₂	semidirect
6. #22553377	C₂	semidirect
7. #22553367	C₂	semidirect
8. #22553363	C₂	semidirect
9. #22553334	C₂	semidirect
10. C₄⋊C₈×D₁₃₁	C₂	semidirect
11. #446×C₁₃₁	C₂	semidirect
77. D₅₂₄	M₁₆	semidirect
124. C₁₃₁	#2990	semidirect
128. #446	D₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14523514	2
#22553443	2
#22553385	2
#22553382	2
C₈×D₅₂₄	2
#22553377	2
#22553367	2
#22553363	2
#22553334	2
C₄⋊C₈×D₁₃₁	2
#446×C₁₃₁	2
#2990	131