Subgroups — C₂×#22553384 #30674641

Order 16768.

564 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14523497	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×C₁₃₁⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#22553384	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₄⋊(C₁₃₁⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₂²⋊(C₁₃₁⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₂²⋊(C₁₃₁⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#16862157	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₄×C₁₃₁	C₂	1	1	yes	maximal, characteristic
9	C₂×#14523497	—	2	4	no	—
10	C₂×#14523497	C₂²	1	4	yes	complemented
11	C₂²×C₁₃₁⋊C₈	—	2	4	no	—
12	C₂²×C₁₃₁⋊C₈	C₂²	1	1	yes	characteristic
13	C₂²×C₁₃₁⋊C₈	C₂²	1	1	yes	characteristic
14	C₂²×C₁₃₁⋊C₈	C₂²	1	1	yes	characteristic
15	C₂²×C₁₃₁⋊C₈	C₂²	1	1	yes	characteristic
16	C₂²×C₁₃₁⋊C₈	C₂²	1	4	yes	complemented
17	C₄⋊(C₁₃₁⋊C₈)	C₂²	1	4	yes	complemented
18	C₂²⋊(C₁₃₁⋊C₈)	C₂²	1	4	yes	complemented
19	C₂²⋊(C₁₃₁⋊C₈)	C₂²	1	4	yes	complemented
20	#16862157	C₂²	1	4	yes	complemented
21	C₂²×D₄×C₁₃₁	C₄	1	1	yes	characteristic
22	C₂³×C₄×C₁₃₁	C₂²	1	1	yes	characteristic
23	C₂³×C₄×C₁₃₁	C₂²	1	1	yes	characteristic
24	C₄×D₄×C₁₃₁	C₄	1	4	yes	—
25	C₄×D₄×C₁₃₁	C₂²	1	4	yes	—
26	C₂×C₄⋊C₄×C₁₃₁	C₄	1	1	yes	characteristic
27	C₂×C₂²⋊C₄×C₁₃₁	C₄	1	1	yes	characteristic
28	C₂×C₂²⋊C₄×C₁₃₁	C₄	1	1	yes	characteristic
29	C₂×C₄²×C₁₃₁	C₂²	1	1	yes	characteristic
30	#14523497	—	2	8	no	—
31	C₂×C₁₃₁⋊C₈	—	2	8	no	—
32	C₂×C₁₃₁⋊C₈	—	2	2	no	—
33	C₂×C₁₃₁⋊C₈	—	2	2	no	—
34	C₂×C₁₃₁⋊C₈	—	2	2	no	—
35	C₂×C₁₃₁⋊C₈	D₄	1	4	yes	complemented
36	C₂×C₁₃₁⋊C₈	C₂³	1	2	yes	—
37	C₂×C₁₃₁⋊C₈	C₂³	1	2	yes	—
38	C₂×C₁₃₁⋊C₈	C₂³	1	2	yes	—
39	C₂×C₁₃₁⋊C₈	C₂³	1	2	yes	—
40	C₂⁴×C₁₃₁	C₂×C₄	1	1	yes	characteristic
41	C₂⁴×C₁₃₁	C₂×C₄	1	1	yes	characteristic
42	C₂×D₄×C₁₃₁	—	2	4	no	—
43	C₂×D₄×C₁₃₁	C₂×C₄	1	4	yes	—
44	C₂²×C₄×C₁₃₁	—	2	2	no	—
45	C₂²×C₄×C₁₃₁	—	2	2	no	—
46	C₂²×C₄×C₁₃₁	—	2	2	no	—
47	C₂²×C₄×C₁₃₁	—	2	2	no	—
48	C₂²×C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₃₁	C₂×C₄	1	2	yes	—
53	C₂²×C₄×C₁₃₁	C₂×C₄	1	2	yes	—
54	C₂²×C₄×C₁₃₁	C₂³	1	2	yes	—
55	C₂²×C₄×C₁₃₁	C₂³	1	2	yes	—
56	C₂²×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
57	C₄⋊C₄×C₁₃₁	C₂×C₄	1	4	yes	—
58	C₂²⋊C₄×C₁₃₁	C₂×C₄	1	4	yes	—
59	C₂²⋊C₄×C₁₃₁	C₂×C₄	1	4	yes	—
60	C₄²×C₁₃₁	C₂×C₄	1	2	yes	—
61	C₄²×C₁₃₁	C₂³	1	2	yes	—
62	C₁₃₁⋊C₈	—	2	2	no	—
63	C₁₃₁⋊C₈	—	2	2	no	—
64	C₁₃₁⋊C₈	—	2	2	no	—
65	C₁₃₁⋊C₈	C₂×D₄	1	4	yes	complemented
66	C₂³×C₁₃₁	—	2	2	no	—
67	C₂³×C₁₃₁	—	2	2	no	—
68	C₂³×C₁₃₁	—	2	2	no	—
69	C₂³×C₁₃₁	—	2	2	no	—
70	C₂³×C₁₃₁	—	2	2	no	—
71	C₂³×C₁₃₁	M₁₆	1	4	yes	complemented
72	C₂³×C₁₃₁	C₂²×C₄	1	2	yes	—
73	C₂³×C₁₃₁	C₂²×C₄	1	2	yes	—
74	C₂³×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
75	D₄×C₁₃₁	—	2	8	no	—
76	C₂×C₄×C₁₃₁	—	2	2	no	—
77	C₂×C₄×C₁₃₁	—	2	2	no	—
78	C₂×C₄×C₁₃₁	—	2	2	no	—
79	C₂×C₄×C₁₃₁	—	2	2	no	—
80	C₂×C₄×C₁₃₁	—	2	4	no	—
81	C₂×C₄×C₁₃₁	—	2	4	no	—
82	C₂×C₄×C₁₃₁	—	2	2	no	—
83	C₂×C₄×C₁₃₁	—	2	2	no	—
84	C₂×C₄×C₁₃₁	—	2	2	no	—
85	C₂×C₄×C₁₃₁	—	2	2	no	—
86	C₂×C₄×C₁₃₁	C₂²×C₄	1	2	yes	—
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	2	yes	—
90	C₂×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₄×C₁₃₁	C₂×D₄	1	2	yes	—
92	C₂×C₄×C₁₃₁	#41	1	2	yes	—
93	C₂×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
94	C₂²×C₁₃₁	—	2	2	no	—
95	C₂²×C₁₃₁	—	2	2	no	—
96	C₂²×C₁₃₁	—	2	8	no	—
97	C₂²×C₁₃₁	—	2	2	no	—
98	C₂²×C₁₃₁	—	2	2	no	—
99	C₂²×C₁₃₁	—	2	2	no	—
100	C₂²×C₁₃₁	—	4	4	no	—
101	C₂²×C₁₃₁	C₄×D₄	1	2	yes	—
102	C₂²×C₁₃₁	C₂×M₁₆	1	2	yes	—
103	C₂²×C₁₃₁	C₂×M₁₆	1	4	yes	complemented
104	C₂²×C₁₃₁	#131	1	2	yes	—
105	C₂²×C₁₃₁	C₂³×C₄	1	1	yes	characteristic
106	C₄×C₁₃₁	—	2	2	no	—
107	C₄×C₁₃₁	—	2	2	no	—
108	C₄×C₁₃₁	—	2	2	no	—
109	C₄×C₁₃₁	—	2	2	no	—
110	C₄×C₁₃₁	C₄×D₄	1	2	yes	—
111	C₄×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
112	C₄×C₁₃₁	C₂×#41	1	1	yes	characteristic
113	C₂×C₁₃₁	—	2	4	no	—
114	C₂×C₁₃₁	—	4	2	no	—
115	C₂×C₁₃₁	#435	1	4	yes	complemented
116	C₂×C₁₃₁	C₂×C₄×D₄	1	1	yes	characteristic
117	C₂×C₁₃₁	C₂²×M₁₆	1	1	yes	characteristic
118	C₂×C₁₃₁	C₂×#131	1	1	yes	characteristic
119	C₁₃₁	C₂×#435	1	1	yes	characteristic, complemented, Sylow
120	C₂×#435	—	131	1	no	maximal, Sylow
121	C₂²×M₁₆	—	131	1	no	—
122	C₂³×C₈	—	131	1	no	—
123	C₂×C₄×D₄	D₁₃₁	1	1	yes	characteristic
124	#435	—	131	8	no	—
125	C₂×C₄⋊C₈	—	131	1	no	—
126	C₂×C₂²⋊C₈	—	131	1	no	—
127	C₂×C₂²⋊C₈	—	131	1	no	—
128	C₂×#97	—	131	1	no	—
129	C₂²×D₄	Dic₁₃₁	1	1	yes	characteristic
130	C₂³×C₄	C₂×D₁₃₁	1	1	yes	characteristic
131	C₂³×C₄	C₂×D₁₃₁	1	1	yes	characteristic
132	C₂×M₁₆	—	262	4	no	—
133	C₂×M₁₆	—	131	4	no	—
134	C₂²×C₈	—	262	4	no	—
135	C₂²×C₈	—	131	4	no	—
136	C₂²×C₈	—	131	1	no	—
137	C₂²×C₈	—	131	1	no	—
138	C₂²×C₈	—	131	1	no	—
139	C₂²×C₈	—	131	1	no	—
140	C₄×D₄	Dic₁₃₁	1	4	yes	—
141	C₄×D₄	C₂×D₁₃₁	1	4	yes	—
142	C₂×C₄⋊C₄	Dic₁₃₁	1	1	yes	characteristic
143	C₂×C₂²⋊C₄	Dic₁₃₁	1	1	yes	characteristic
144	C₂×C₂²⋊C₄	Dic₁₃₁	1	1	yes	characteristic
145	C₂×C₄²	C₂×D₁₃₁	1	1	yes	characteristic
146	C₄⋊C₈	—	131	4	no	—
147	C₂²⋊C₈	—	131	4	no	—
148	C₂²⋊C₈	—	131	4	no	—
149	#97	—	131	4	no	—
150	C₂⁴	C₂×Dic₁₃₁	1	1	yes	characteristic
151	C₂⁴	C₂×Dic₁₃₁	1	1	yes	characteristic
152	C₂×D₄	—	2	4	no	—
153	C₂×D₄	C₂×Dic₁₃₁	1	4	yes	—
154	C₂²×C₄	—	2	2	no	—
155	C₂²×C₄	—	2	2	no	—
156	C₂²×C₄	—	2	2	no	—
157	C₂²×C₄	—	2	2	no	—
158	C₂²×C₄	C₂×Dic₁₃₁	1	1	yes	characteristic
159	C₂²×C₄	C₂×Dic₁₃₁	1	1	yes	characteristic
160	C₂²×C₄	C₂×Dic₁₃₁	1	1	yes	characteristic
161	C₂²×C₄	C₂×Dic₁₃₁	1	1	yes	characteristic
162	C₂²×C₄	C₂×Dic₁₃₁	1	2	yes	—
163	C₂²×C₄	C₂×Dic₁₃₁	1	2	yes	—
164	C₂²×C₄	C₂²×D₁₃₁	1	2	yes	—
165	C₂²×C₄	C₂²×D₁₃₁	1	2	yes	—
166	C₂²×C₄	C₂²×D₁₃₁	1	1	yes	characteristic
167	M₁₆	—	262	8	no	—
168	C₂×C₈	—	262	2	no	—
169	C₂×C₈	—	262	2	no	—
170	C₂×C₈	—	262	2	no	—
171	C₂×C₈	—	262	8	no	—
172	C₂×C₈	—	131	2	no	—
173	C₂×C₈	—	131	2	no	—
174	C₂×C₈	—	131	2	no	—
175	C₂×C₈	—	131	4	no	—
176	C₂×C₈	—	131	2	no	—
177	C₄⋊C₄	C₂×Dic₁₃₁	1	4	yes	—
178	C₂²⋊C₄	C₂×Dic₁₃₁	1	4	yes	—
179	C₂²⋊C₄	C₂×Dic₁₃₁	1	4	yes	—
180	C₄²	C₂×Dic₁₃₁	1	2	yes	—
181	C₄²	C₂²×D₁₃₁	1	2	yes	—
182	C₂³	—	2	2	no	—
183	C₂³	—	2	2	no	—
184	C₂³	—	2	2	no	—
185	C₂³	—	2	2	no	—
186	C₂³	—	2	2	no	—
187	C₂³	#14523497	1	4	yes	complemented
188	C₂³	C₂²×Dic₁₃₁	1	2	yes	—
189	C₂³	C₂²×Dic₁₃₁	1	2	yes	—
190	C₂³	C₂²×Dic₁₃₁	1	1	yes	characteristic
191	D₄	—	2	8	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	4	no	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	C₂²×Dic₁₃₁	1	2	yes	—
203	C₂×C₄	C₂²×Dic₁₃₁	1	2	yes	—
204	C₂×C₄	C₂²×Dic₁₃₁	1	2	yes	—
205	C₂×C₄	C₂²×Dic₁₃₁	1	2	yes	—
206	C₂×C₄	C₂²×Dic₁₃₁	1	1	yes	characteristic
207	C₂×C₄	D₄×D₁₃₁	1	2	yes	—
208	C₂×C₄	#14523512	1	2	yes	—
209	C₂×C₄	C₂³×D₁₃₁	1	1	yes	characteristic
210	C₈	—	262	2	no	—
211	C₈	—	262	2	no	—
212	C₈	—	262	2	no	—
213	C₈	—	131	4	no	—
214	C₂²	—	2	8	no	—
215	C₂²	—	2	2	no	—
216	C₂²	—	2	2	no	—
217	C₂²	—	2	2	no	—
218	C₂²	—	2	2	no	—
219	C₂²	—	2	2	no	—
220	C₂²	—	4	4	no	—
221	C₂²	D₄×Dic₁₃₁	1	2	yes	—
222	C₂²	C₂×#14523497	1	2	yes	—
223	C₂²	C₂×#14523497	1	4	yes	complemented
224	C₂²	#16862253	1	2	yes	—
225	C₂²	C₂³×Dic₁₃₁	1	1	yes	characteristic
226	C₄	—	2	2	no	—
227	C₄	—	2	2	no	—
228	C₄	—	2	2	no	—
229	C₄	—	2	2	no	—
230	C₄	D₄×Dic₁₃₁	1	2	yes	—
231	C₄	C₂×D₄×D₁₃₁	1	1	yes	characteristic
232	C₄	C₂×#14523512	1	1	yes	characteristic
233	C₂	—	2	4	no	—
234	C₂	—	4	2	no	—
235	C₂	#22553384	1	4	yes	direct product factor
236	C₂	C₂×D₄×Dic₁₃₁	1	1	yes	characteristic
237	C₂	C₂²×#14523497	1	1	yes	characteristic
238	C₂	C₂×#16862253	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₁₃₁
commutator	C₂²×C₁₃₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₃₁
Fitting	C₂×C₄×D₄×C₁₃₁	C₂
socle	C₂³×C₁₃₁	C₂²×C₄
prime radical	C₂²×D₄×C₁₃₁	C₄
cosocle	C₂×C₄×C₁₃₁	C₂⁴
hypercenter	C₂×C₄×D₄	D₁₃₁
hypocenter	C₁₃₁	C₂×#435
LCS	G ▷ C₂²×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
1. C₂²×#14523497	C₂	semidirect
2. C₂³×C₁₃₁⋊C₈	C₂	semidirect
3. #22553384	C₂	direct (N × H)
4. C₂×C₄⋊(C₁₃₁⋊C₈)	C₂	semidirect
5. C₂×C₂²⋊(C₁₃₁⋊C₈)	C₂	semidirect
6. C₂×C₂²⋊(C₁₃₁⋊C₈)	C₂	semidirect
7. C₂×#16862157	C₂	semidirect
10. C₂×#14523497	C₂²	semidirect
16. C₂²×C₁₃₁⋊C₈	C₂²	semidirect
17. C₄⋊(C₁₃₁⋊C₈)	C₂²	semidirect
18. C₂²⋊(C₁₃₁⋊C₈)	C₂²	semidirect
19. C₂²⋊(C₁₃₁⋊C₈)	C₂²	semidirect
20. #16862157	C₂²	semidirect
35. C₂×C₁₃₁⋊C₈	D₄	semidirect
65. C₁₃₁⋊C₈	C₂×D₄	semidirect
71. C₂³×C₁₃₁	M₁₆	semidirect
103. C₂²×C₁₃₁	C₂×M₁₆	semidirect
115. C₂×C₁₃₁	#435	semidirect
119. C₁₃₁	C₂×#435	semidirect
187. C₂³	#14523497	semidirect
223. C₂²	C₂×#14523497	semidirect
235. C₂	#22553384	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#14523497	2
C₂³×C₁₃₁⋊C₈	2
#22553384	2
C₂×C₄⋊(C₁₃₁⋊C₈)	2
C₂×C₂²⋊(C₁₃₁⋊C₈)	2
C₂×#16862157	2
C₂×C₄×D₄×C₁₃₁	2
C₂×#435	131

Subgroups — C2×#22553384 #30674641

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22553384 #30674641