Subgroups — #30675100

Order 16768.

356 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22553443	C₂	1	2	yes	maximal, complemented
2	#22553384	C₂	1	2	yes	maximal, complemented
3	#22553384	C₂	1	2	yes	maximal, complemented
4	D₄×C₁₃₁⋊C₈	C₂	1	2	yes	maximal, complemented
5	#22553373	C₂	1	2	yes	maximal, complemented
6	#22553369	C₂	1	1	yes	maximal, characteristic, complemented
7	#22553339	C₂	1	2	yes	maximal, complemented
8	C₂×C₄⋊(C₁₃₁⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
9	#529×C₁₃₁	C₂	1	1	yes	maximal, characteristic
10	C₂×#14523497	C₂²	1	2	yes	—
11	C₂²×C₁₃₁⋊C₈	C₂²	1	1	yes	characteristic
12	C₂²×C₁₃₁⋊C₈	C₂²	1	2	yes	—
13	C₂²×C₁₃₁⋊C₈	C₂²	1	1	yes	characteristic
14	C₄⋊(C₁₃₁⋊C₈)	C₂²	1	1	yes	characteristic
15	C₄⋊(C₁₃₁⋊C₈)	C₂²	1	1	yes	characteristic
16	C₄⋊(C₁₃₁⋊C₈)	C₂²	1	2	yes	—
17	C₄⋊(C₁₃₁⋊C₈)	C₂²	1	2	yes	—
18	C₄⋊(C₁₃₁⋊C₈)	C₂²	1	2	yes	—
19	C₄⋊(C₁₃₁⋊C₈)	C₂²	1	2	yes	—
20	C₂²⋊(C₁₃₁⋊C₈)	C₂²	1	2	yes	—
21	C₂²⋊(C₁₃₁⋊C₈)	C₂²	1	2	yes	—
22	C₂²⋊(C₁₃₁⋊C₈)	C₂²	1	2	yes	—
23	#16862157	C₂²	1	2	yes	—
24	#16862157	C₂²	1	2	yes	—
25	C₄×C₁₃₁⋊C₈	C₂²	1	2	yes	—
26	#126×C₁₃₁	C₄	1	2	yes	—
27	#125×C₁₃₁	C₄	1	1	yes	characteristic
28	#124×C₁₃₁	C₄	1	1	yes	characteristic
29	#123×C₁₃₁	C₄	1	2	yes	—
30	C₂²⋊Q₈×C₁₃₁	C₄	1	1	yes	characteristic
31	#121×C₁₃₁	C₄	1	1	yes	characteristic
32	C₄×Q₈×C₁₃₁	C₂²	1	1	yes	characteristic
33	C₄×D₄×C₁₃₁	C₂²	1	2	yes	—
34	C₄×D₄×C₁₃₁	C₂²	1	1	yes	characteristic
35	#117×C₁₃₁	C₂²	1	2	yes	—
36	C₂×C₄²×C₁₃₁	C₂²	1	1	yes	characteristic
37	#14523497	—	4	2	no	—
38	C₂×C₁₃₁⋊C₈	—	4	1	no	—
39	C₂×C₁₃₁⋊C₈	—	4	1	no	—
40	C₂×C₁₃₁⋊C₈	—	4	2	no	—
41	C₂×C₁₃₁⋊C₈	C₂³	1	2	yes	—
42	C₂×C₁₃₁⋊C₈	C₂³	1	2	yes	—
43	C₂×C₁₃₁⋊C₈	C₂³	1	2	yes	—
44	C₂×C₁₃₁⋊C₈	C₂³	1	2	yes	—
45	C₂×Q₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
46	C₂×D₄×C₁₃₁	C₂×C₄	1	2	yes	—
47	C₂×D₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
51	C₂²×C₄×C₁₃₁	C₂³	1	2	yes	—
52	C₄⋊C₄×C₁₃₁	C₂×C₄	1	2	yes	—
53	C₄⋊C₄×C₁₃₁	C₂×C₄	1	2	yes	—
54	C₄⋊C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
55	C₄⋊C₄×C₁₃₁	C₂×C₄	1	2	yes	—
56	C₄⋊C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
57	C₄⋊C₄×C₁₃₁	C₂×C₄	1	2	yes	—
58	C₂²⋊C₄×C₁₃₁	C₂×C₄	1	2	yes	—
59	C₂²⋊C₄×C₁₃₁	C₂×C₄	1	2	yes	—
60	C₂²⋊C₄×C₁₃₁	C₂×C₄	1	2	yes	—
61	C₂²⋊C₄×C₁₃₁	C₂×C₄	1	2	yes	—
62	C₂²⋊C₄×C₁₃₁	C₂×C₄	1	2	yes	—
63	C₄²×C₁₃₁	C₂×C₄	1	1	yes	characteristic
64	C₄²×C₁₃₁	C₂×C₄	1	1	yes	characteristic
65	C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
66	C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
67	C₄²×C₁₃₁	C₂³	1	2	yes	—
68	C₁₃₁⋊C₈	—	2	2	no	—
69	C₁₃₁⋊C₈	—	2	2	no	—
70	C₁₃₁⋊C₈	—	2	2	no	—
71	C₁₃₁⋊C₈	—	2	2	no	—
72	C₂³×C₁₃₁	C₂²×C₄	1	2	yes	—
73	C₂³×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
74	Q₈×C₁₃₁	—	4	1	no	—
75	D₄×C₁₃₁	—	2	4	no	—
76	D₄×C₁₃₁	—	4	1	no	—
77	C₂×C₄×C₁₃₁	—	2	2	no	—
78	C₂×C₄×C₁₃₁	—	2	1	no	—
79	C₂×C₄×C₁₃₁	—	2	2	no	—
80	C₂×C₄×C₁₃₁	—	2	1	no	—
81	C₂×C₄×C₁₃₁	—	2	2	no	—
82	C₂×C₄×C₁₃₁	—	4	1	no	—
83	C₂×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₄×C₁₃₁	C₂²×C₄	1	2	yes	—
89	C₂×C₄×C₁₃₁	C₂²×C₄	1	2	yes	—
90	C₂×C₄×C₁₃₁	C₂²×C₄	1	2	yes	—
91	C₂×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
92	C₂²×C₁₃₁	—	2	1	no	—
93	C₂²×C₁₃₁	—	2	1	no	—
94	C₂²×C₁₃₁	—	2	2	no	—
95	C₂²×C₁₃₁	—	2	2	no	—
96	C₂²×C₁₃₁	—	2	2	no	—
97	C₂²×C₁₃₁	#131	1	2	yes	—
98	C₂²×C₁₃₁	C₂³×C₄	1	1	yes	characteristic
99	C₄×C₁₃₁	—	2	1	no	—
100	C₄×C₁₃₁	—	2	2	no	—
101	C₄×C₁₃₁	—	2	1	no	—
102	C₄×C₁₃₁	—	2	1	no	—
103	C₄×C₁₃₁	—	2	2	no	—
104	C₄×C₁₃₁	—	2	1	no	—
105	C₄×C₁₃₁	—	2	1	no	—
106	C₄×C₁₃₁	—	2	2	no	—
107	C₄×C₁₃₁	H(5,𝔽₂)	1	1	yes	characteristic
108	C₄×C₁₃₁	2₋¹⁺⁴	1	1	yes	characteristic
109	C₂×C₁₃₁	—	2	2	no	—
110	C₂×C₁₃₁	—	4	2	no	—
111	C₂×C₁₃₁	#520	1	1	yes	characteristic
112	C₂×C₁₃₁	C₂×#131	1	1	yes	characteristic
113	C₂×C₁₃₁	#568	1	1	yes	characteristic
114	C₁₃₁	#2992	1	1	yes	characteristic, complemented, Sylow
115	#2992	—	131	1	no	maximal, Sylow
116	#529	D₁₃₁	1	1	yes	characteristic
117	#446	—	131	2	no	—
118	#435	—	131	2	no	—
119	#435	—	131	2	no	—
120	C₈×D₄	—	131	2	no	—
121	#433	—	131	2	no	—
122	#432	—	131	1	no	—
123	#424	—	131	2	no	—
124	C₂×C₄⋊C₈	—	131	1	no	—
125	C₂×M₁₆	—	131	2	no	—
126	C₂²×C₈	—	131	1	no	—
127	C₂²×C₈	—	131	1	no	—
128	C₂²×C₈	—	131	2	no	—
129	#126	Dic₁₃₁	1	2	yes	—
130	#125	Dic₁₃₁	1	1	yes	characteristic
131	#124	Dic₁₃₁	1	1	yes	characteristic
132	#123	Dic₁₃₁	1	2	yes	—
133	C₂²⋊Q₈	Dic₁₃₁	1	1	yes	characteristic
134	#121	Dic₁₃₁	1	1	yes	characteristic
135	C₄×Q₈	C₂×D₁₃₁	1	1	yes	characteristic
136	C₄×D₄	C₂×D₁₃₁	1	2	yes	—
137	C₄×D₄	C₂×D₁₃₁	1	1	yes	characteristic
138	#117	C₂×D₁₃₁	1	2	yes	—
139	C₂×C₄²	C₂×D₁₃₁	1	1	yes	characteristic
140	C₄⋊C₈	—	131	2	no	—
141	C₄⋊C₈	—	131	2	no	—
142	C₄⋊C₈	—	131	1	no	—
143	C₄⋊C₈	—	131	2	no	—
144	C₄⋊C₈	—	131	2	no	—
145	C₄⋊C₈	—	131	1	no	—
146	C₂²⋊C₈	—	131	2	no	—
147	C₂²⋊C₈	—	131	2	no	—
148	C₂²⋊C₈	—	131	2	no	—
149	#97	—	131	2	no	—
150	#97	—	131	2	no	—
151	C₄×C₈	—	131	2	no	—
152	C₂×Q₈	C₂×Dic₁₃₁	1	1	yes	characteristic
153	C₂×D₄	C₂×Dic₁₃₁	1	2	yes	—
154	C₂×D₄	C₂×Dic₁₃₁	1	1	yes	characteristic
155	C₂²×C₄	C₂×Dic₁₃₁	1	1	yes	characteristic
156	C₂²×C₄	C₂×Dic₁₃₁	1	1	yes	characteristic
157	C₂²×C₄	C₂²×D₁₃₁	1	1	yes	characteristic
158	C₂²×C₄	C₂²×D₁₃₁	1	2	yes	—
159	M₁₆	—	524	2	no	—
160	C₂×C₈	—	524	1	no	—
161	C₂×C₈	—	524	2	no	—
162	C₂×C₈	—	524	1	no	—
163	C₂×C₈	—	131	2	no	—
164	C₂×C₈	—	131	2	no	—
165	C₂×C₈	—	131	2	no	—
166	C₂×C₈	—	131	2	no	—
167	C₄⋊C₄	C₂×Dic₁₃₁	1	2	yes	—
168	C₄⋊C₄	C₂×Dic₁₃₁	1	2	yes	—
169	C₄⋊C₄	C₂×Dic₁₃₁	1	1	yes	characteristic
170	C₄⋊C₄	C₂×Dic₁₃₁	1	2	yes	—
171	C₄⋊C₄	C₂×Dic₁₃₁	1	1	yes	characteristic
172	C₄⋊C₄	C₂×Dic₁₃₁	1	2	yes	—
173	C₂²⋊C₄	C₂×Dic₁₃₁	1	2	yes	—
174	C₂²⋊C₄	C₂×Dic₁₃₁	1	2	yes	—
175	C₂²⋊C₄	C₂×Dic₁₃₁	1	2	yes	—
176	C₂²⋊C₄	C₂×Dic₁₃₁	1	2	yes	—
177	C₂²⋊C₄	C₂×Dic₁₃₁	1	2	yes	—
178	C₄²	C₂×Dic₁₃₁	1	1	yes	characteristic
179	C₄²	C₂×Dic₁₃₁	1	1	yes	characteristic
180	C₄²	C₂²×D₁₃₁	1	2	yes	—
181	C₄²	C₂²×D₁₃₁	1	1	yes	characteristic
182	C₄²	C₂²×D₁₃₁	1	1	yes	characteristic
183	C₂³	C₂²×Dic₁₃₁	1	2	yes	—
184	C₂³	C₂²×Dic₁₃₁	1	1	yes	characteristic
185	Q₈	—	4	1	no	—
186	D₄	—	2	4	no	—
187	D₄	—	4	1	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	4	1	no	—
194	C₂×C₄	C₂²×Dic₁₃₁	1	1	yes	characteristic
195	C₂×C₄	C₂²×Dic₁₃₁	1	1	yes	characteristic
196	C₂×C₄	C₂²×Dic₁₃₁	1	1	yes	characteristic
197	C₂×C₄	C₂²×Dic₁₃₁	1	1	yes	characteristic
198	C₂×C₄	C₂²×Dic₁₃₁	1	1	yes	characteristic
199	C₂×C₄	C₂²×Dic₁₃₁	1	2	yes	—
200	C₂×C₄	C₂²×Dic₁₃₁	1	2	yes	—
201	C₂×C₄	C₂²×Dic₁₃₁	1	2	yes	—
202	C₂×C₄	C₂³×D₁₃₁	1	1	yes	characteristic
203	C₈	—	262	2	no	—
204	C₈	—	262	2	no	—
205	C₈	—	262	2	no	—
206	C₈	—	262	2	no	—
207	C₂²	—	2	1	no	—
208	C₂²	—	2	1	no	—
209	C₂²	—	2	2	no	—
210	C₂²	—	2	2	no	—
211	C₂²	—	2	2	no	—
212	C₂²	#16862253	1	2	yes	—
213	C₂²	C₂³×Dic₁₃₁	1	1	yes	characteristic
214	C₄	—	2	1	no	—
215	C₄	—	2	2	no	—
216	C₄	—	2	1	no	—
217	C₄	—	2	1	no	—
218	C₄	—	2	2	no	—
219	C₄	—	2	1	no	—
220	C₄	—	2	1	no	—
221	C₄	—	2	2	no	—
222	C₄	#16862292	1	1	yes	characteristic
223	C₄	#16862293	1	1	yes	characteristic
224	C₂	—	2	2	no	—
225	C₂	—	4	2	no	—
226	C₂	#22553795	1	1	yes	characteristic
227	C₂	C₂×#16862253	1	1	yes	characteristic
228	C₂	#22554079	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	#2992
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. #22553384	C₂	semidirect
3. #22553384	C₂	semidirect
4. D₄×C₁₃₁⋊C₈	C₂	semidirect
5. #22553373	C₂	semidirect
6. #22553369	C₂	semidirect
7. #22553339	C₂	semidirect
8. C₂×C₄⋊(C₁₃₁⋊C₈)	C₂	semidirect
114. C₁₃₁	#2992	semidirect

Maximal subgroups

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

subgroup	index
#22553443	2
#22553384	2
D₄×C₁₃₁⋊C₈	2
#22553373	2
#22553369	2
#22553339	2
C₂×C₄⋊(C₁₃₁⋊C₈)	2
#529×C₁₃₁	2
#2992	131