Subgroups — C₂×#22554077 #30680669

Order 16768.

1460 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#41×D₁₃₁	C₂	1	1	yes	maximal, characteristic
2	#22554077	C₂	1	16	yes	maximal, direct product factor
3	C₂×#16862254	C₂	1	3	yes	maximal, complemented
4	C₂×#16862253	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16862252	C₂	1	3	yes	maximal, complemented
6	C₂²×#14523496	C₂	1	3	yes	maximal, complemented
7	C₂×M₁₆×D₁₃₁	C₂	1	3	yes	maximal, complemented
8	C₂×#131×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14523514	C₄	1	1	yes	characteristic
10	C₂×#14523513	C₂²	1	3	yes	—
11	C₂×#14523512	C₄	1	3	yes	—
12	#41×D₁₃₁	C₄	1	8	yes	—
13	#41×D₁₃₁	C₂²	1	8	yes	—
14	C₂×Q₈×D₁₃₁	C₄	1	1	yes	characteristic
15	C₂×D₄×D₁₃₁	C₄	1	3	yes	—
16	C₂²×C₄×D₁₃₁	C₂²	1	3	yes	—
17	#16862254	C₂²	1	24	yes	complemented
18	#16862253	C₂²	1	8	yes	complemented
19	#16862252	C₂²	1	24	yes	complemented
20	C₂×#14523497	C₂²	1	3	yes	complemented
21	C₂×#14523496	C₂²	1	3	yes	—
22	C₂×#14523496	C₂²	1	1	yes	characteristic
23	C₂×#14523496	C₂²	1	3	yes	complemented
24	C₂×#14523496	C₂²	1	3	yes	complemented
25	C₂×#14523496	C₂²	1	24	yes	complemented
26	M₁₆×D₁₃₁	C₂²	1	24	yes	complemented
27	C₂²×C₁₃₁⋊C₈	C₂²	1	3	yes	complemented
28	C₂×C₈×D₁₃₁	C₂²	1	6	yes	complemented
29	C₂×#41×C₁₃₁	C₂²	1	1	yes	characteristic
30	#131×C₁₃₁	C₂²	1	8	yes	complemented
31	C₂×M₁₆×C₁₃₁	C₂²	1	3	yes	complemented
32	C₂²×C₈×C₁₃₁	C₂²	1	3	yes	complemented
33	C₂³×D₁₃₁	C₂×C₄	1	3	yes	—
34	#14523514	C₂×C₄	1	8	yes	—
35	#14523513	C₂×C₄	1	12	yes	—
36	#14523513	C₂³	1	12	yes	—
37	#14523512	C₂×C₄	1	24	yes	—
38	C₂×Dic₂₆₂	C₂×C₄	1	3	yes	—
39	Q₈×D₁₃₁	C₂×C₄	1	8	yes	—
40	C₂×D₅₂₄	C₂×C₄	1	3	yes	—
41	C₂×C₂²⋊D₁₃₁	C₂×C₄	1	6	yes	—
42	D₄×D₁₃₁	C₂×C₄	1	24	yes	—
43	C₂²×Dic₁₃₁	C₂×C₄	1	3	yes	—
44	C₂×C₄×D₁₃₁	C₂×C₄	1	3	yes	—
45	C₂×C₄×D₁₃₁	C₂×C₄	1	12	yes	—
46	C₂×C₄×D₁₃₁	C₂×C₄	1	3	yes	—
47	C₂×C₄×D₁₃₁	C₂³	1	3	yes	—
48	C₂×C₄×D₁₃₁	C₂³	1	12	yes	—
49	C₂×C₄×D₁₃₁	C₂³	1	1	yes	characteristic
50	#14523497	C₂³	1	12	yes	complemented
51	#14523496	C₂³	1	12	yes	—
52	#14523496	C₂³	1	4	yes	—
53	#14523496	C₂³	1	12	yes	complemented
54	#14523496	C₂³	1	12	yes	complemented
55	C₂×C₁₃₁⋊C₈	C₂³	1	3	yes	—
56	C₂×C₁₃₁⋊C₈	C₂³	1	1	yes	characteristic
57	C₂×C₁₃₁⋊C₈	C₂³	1	12	yes	complemented
58	C₈×D₁₃₁	C₂³	1	24	yes	complemented
59	#41×C₁₃₁	C₂×C₄	1	4	yes	—
60	#41×C₁₃₁	C₂³	1	4	yes	—
61	C₂×Q₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
62	C₂×D₄×C₁₃₁	C₂×C₄	1	3	yes	—
63	C₂²×C₄×C₁₃₁	C₂³	1	3	yes	—
64	M₁₆×C₁₃₁	C₂³	1	12	yes	complemented
65	C₂×C₈×C₁₃₁	C₂³	1	3	yes	—
66	C₂×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
67	C₂×C₈×C₁₃₁	C₂³	1	12	yes	complemented
68	C₂²×D₁₃₁	—	4	6	no	—
69	C₂²×D₁₃₁	C₂²×C₄	1	1	yes	characteristic
70	C₂²×D₁₃₁	C₂²×C₄	1	3	yes	—
71	C₂²×D₁₃₁	C₂²×C₄	1	12	yes	—
72	Dic₂₆₂	C₂²×C₄	1	12	yes	—
73	D₅₂₄	C₂²×C₄	1	12	yes	—
74	C₂²⋊D₁₃₁	C₂²×C₄	1	24	yes	—
75	C₂×Dic₁₃₁	C₂²×C₄	1	1	yes	characteristic
76	C₂×Dic₁₃₁	C₂²×C₄	1	3	yes	—
77	C₂×Dic₁₃₁	C₂²×C₄	1	12	yes	—
78	C₄×D₁₃₁	C₂²×C₄	1	2	yes	—
79	C₄×D₁₃₁	C₂²×C₄	1	12	yes	—
80	C₄×D₁₃₁	C₂²×C₄	1	12	yes	—
81	C₄×D₁₃₁	C₂²×C₄	1	6	yes	—
82	C₄×D₁₃₁	C₂⁴	1	6	yes	—
83	C₄×D₁₃₁	C₂⁴	1	2	yes	—
84	C₁₃₁⋊C₈	C₂⁴	1	6	yes	—
85	C₁₃₁⋊C₈	C₂⁴	1	2	yes	—
86	C₂³×C₁₃₁	C₂²×C₄	1	3	yes	—
87	Q₈×C₁₃₁	C₂²×C₄	1	4	yes	—
88	D₄×C₁₃₁	C₂²×C₄	1	12	yes	—
89	C₂×C₄×C₁₃₁	C₂²×C₄	1	3	yes	—
90	C₂×C₄×C₁₃₁	C₂²×C₄	1	6	yes	—
91	C₂×C₄×C₁₃₁	C₂⁴	1	6	yes	—
92	C₂×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
93	C₈×C₁₃₁	C₂⁴	1	2	yes	—
94	C₈×C₁₃₁	C₂⁴	1	6	yes	—
95	C₂²×C₁₃₁	—	2	6	no	—
96	C₂²×C₁₃₁	C₂³×C₄	1	6	yes	—
97	C₂²×C₁₃₁	C₂³×C₄	1	1	yes	characteristic
98	C₂×D₁₃₁	—	2	2	no	—
99	C₂×D₁₃₁	—	2	6	no	—
100	C₂×D₁₃₁	—	4	12	no	—
101	C₂×D₁₃₁	C₂³×C₄	1	2	yes	—
102	C₂×D₁₃₁	C₂³×C₄	1	6	yes	—
103	C₄×C₁₃₁	C₂³×C₄	1	1	yes	characteristic
104	C₄×C₁₃₁	C₂³×C₄	1	6	yes	—
105	C₄×C₁₃₁	C₂⁵	1	1	yes	characteristic
106	Dic₁₃₁	C₂³×C₄	1	2	yes	—
107	Dic₁₃₁	C₂³×C₄	1	6	yes	—
108	C₂×C₁₃₁	—	2	6	no	—
109	C₂×C₁₃₁	#568	1	2	yes	complemented
110	C₂×C₁₃₁	C₂⁴×C₄	1	1	yes	characteristic
111	D₁₃₁	—	2	2	no	—
112	D₁₃₁	—	2	6	no	—
113	C₁₃₁	C₂×#568	1	1	yes	characteristic, complemented, Sylow
114	C₂×#568	—	131	1	no	maximal, Sylow
115	C₂²×#41	—	131	1	no	—
116	#568	—	131	16	no	—
117	C₂×#131	—	131	1	no	—
118	C₂×#131	—	131	3	no	—
119	C₂×#131	—	131	3	no	—
120	C₂×#131	D₁₃₁	1	1	yes	characteristic, complemented
121	C₂²×M₁₆	—	131	3	no	—
122	C₂²×M₁₆	—	131	3	no	—
123	C₂×#41	—	131	8	no	—
124	C₂×#41	—	131	8	no	—
125	C₂×#41	—	131	1	no	—
126	C₂×#41	—	131	3	no	—
127	C₂×#41	—	131	3	no	—
128	C₂×#41	C₂×D₁₃₁	1	1	yes	characteristic
129	C₂²×Q₈	—	131	1	no	—
130	C₂²×D₄	—	131	3	no	—
131	C₂³×C₄	—	131	3	no	—
132	#131	—	131	8	no	—
133	#131	—	131	24	no	—
134	#131	—	131	24	no	—
135	#131	C₂×D₁₃₁	1	8	yes	complemented
136	C₂×M₁₆	—	131	3	no	—
137	C₂×M₁₆	—	131	24	no	—
138	C₂×M₁₆	—	131	3	no	—
139	C₂×M₁₆	—	131	1	no	—
140	C₂×M₁₆	—	131	3	no	—
141	C₂×M₁₆	—	131	3	no	—
142	C₂×M₁₆	—	131	24	no	—
143	C₂×M₁₆	C₂×D₁₃₁	1	3	yes	complemented
144	C₂²×C₈	—	131	3	no	—
145	C₂²×C₈	—	131	6	no	—
146	C₂²×C₈	C₂×D₁₃₁	1	3	yes	complemented
147	C₂⁴	—	131	3	no	—
148	#41	—	131	8	no	—
149	#41	—	131	24	no	—
150	#41	—	131	12	no	—
151	#41	—	131	12	no	—
152	#41	C₄×D₁₃₁	1	4	yes	—
153	#41	C₂²×D₁₃₁	1	4	yes	—
154	C₂×Q₈	—	131	8	no	—
155	C₂×Q₈	—	131	3	no	—
156	C₂×Q₈	C₄×D₁₃₁	1	1	yes	characteristic
157	C₂×D₄	—	131	6	no	—
158	C₂×D₄	—	131	24	no	—
159	C₂×D₄	—	131	3	no	—
160	C₂×D₄	C₄×D₁₃₁	1	3	yes	—
161	C₂²×C₄	—	131	3	no	—
162	C₂²×C₄	—	131	3	no	—
163	C₂²×C₄	—	131	3	no	—
164	C₂²×C₄	—	131	12	no	—
165	C₂²×C₄	—	131	12	no	—
166	C₂²×C₄	—	131	3	no	—
167	C₂²×C₄	—	131	1	no	—
168	C₂²×C₄	C₂²×D₁₃₁	1	3	yes	—
169	M₁₆	—	131	12	no	—
170	M₁₆	—	131	12	no	—
171	M₁₆	—	131	4	no	—
172	M₁₆	—	131	12	no	—
173	M₁₆	—	131	12	no	—
174	M₁₆	C₂²×D₁₃₁	1	12	yes	complemented
175	C₂×C₈	—	131	12	no	—
176	C₂×C₈	—	131	3	no	—
177	C₂×C₈	—	131	1	no	—
178	C₂×C₈	—	131	24	no	—
179	C₂×C₈	C₂²×D₁₃₁	1	3	yes	—
180	C₂×C₈	C₂²×D₁₃₁	1	1	yes	characteristic
181	C₂×C₈	C₂²×D₁₃₁	1	12	yes	complemented
182	C₂³	—	524	6	no	—
183	C₂³	—	131	1	no	—
184	C₂³	—	131	3	no	—
185	C₂³	—	131	12	no	—
186	C₂³	C₂×C₄×D₁₃₁	1	3	yes	—
187	Q₈	—	131	12	no	—
188	Q₈	C₂×C₄×D₁₃₁	1	4	yes	—
189	D₄	—	131	24	no	—
190	D₄	—	131	12	no	—
191	D₄	C₂×C₄×D₁₃₁	1	12	yes	—
192	C₂×C₄	—	131	1	no	—
193	C₂×C₄	—	131	3	no	—
194	C₂×C₄	—	131	12	no	—
195	C₂×C₄	—	131	2	no	—
196	C₂×C₄	—	131	12	no	—
197	C₂×C₄	—	131	12	no	—
198	C₂×C₄	—	131	6	no	—
199	C₂×C₄	—	131	6	no	—
200	C₂×C₄	—	131	2	no	—
201	C₂×C₄	C₂×C₄×D₁₃₁	1	3	yes	—
202	C₂×C₄	C₂×C₄×D₁₃₁	1	6	yes	—
203	C₂×C₄	C₂³×D₁₃₁	1	6	yes	—
204	C₂×C₄	C₂³×D₁₃₁	1	1	yes	characteristic
205	C₈	—	131	2	no	—
206	C₈	—	131	6	no	—
207	C₈	C₂³×D₁₃₁	1	6	yes	—
208	C₈	C₂³×D₁₃₁	1	2	yes	—
209	C₂²	—	2	6	no	—
210	C₂²	—	262	2	no	—
211	C₂²	—	262	6	no	—
212	C₂²	—	524	12	no	—
213	C₂²	—	131	2	no	—
214	C₂²	—	131	6	no	—
215	C₂²	C₂²×C₄×D₁₃₁	1	6	yes	—
216	C₂²	C₂²×C₄×D₁₃₁	1	1	yes	characteristic
217	C₄	—	131	2	no	—
218	C₄	—	131	6	no	—
219	C₄	C₂²×C₄×D₁₃₁	1	1	yes	characteristic
220	C₄	C₂²×C₄×D₁₃₁	1	6	yes	—
221	C₄	C₂⁴×D₁₃₁	1	1	yes	characteristic
222	C₂	—	2	6	no	—
223	C₂	—	262	2	no	—
224	C₂	—	262	6	no	—
225	C₂	#22554077	1	2	yes	direct product factor
226	C₂	C₂³×C₄×D₁₃₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂³×D₁₃₁
commutator	C₂×C₁₃₁	C₂⁴×C₄
Frattini	C₄	C₂⁴×D₁₃₁
Fitting	C₂×#131×C₁₃₁	C₂
socle	C₂²×C₁₃₁	C₂³×C₄
prime radical	C₂×#41×D₁₃₁	C₂
cosocle	C₄×C₁₃₁	C₂⁵
hypercenter	C₂×#131	D₁₃₁
hypocenter	C₁₃₁	C₂×#568
LCS	G ▷ C₂×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#131

Complements

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

N	H ≅ G/N	type
2. #22554077	C₂	direct (N × H)
3. C₂×#16862254	C₂	semidirect
4. C₂×#16862253	C₂	semidirect
5. C₂×#16862252	C₂	semidirect
6. C₂²×#14523496	C₂	semidirect
7. C₂×M₁₆×D₁₃₁	C₂	semidirect
8. C₂×#131×C₁₃₁	C₂	semidirect
17. #16862254	C₂²	semidirect
18. #16862253	C₂²	semidirect
19. #16862252	C₂²	semidirect
20. C₂×#14523497	C₂²	semidirect
23. C₂×#14523496	C₂²	semidirect
24. C₂×#14523496	C₂²	semidirect
25. C₂×#14523496	C₂²	semidirect
26. M₁₆×D₁₃₁	C₂²	semidirect
27. C₂²×C₁₃₁⋊C₈	C₂²	semidirect
28. C₂×C₈×D₁₃₁	C₂²	semidirect
30. #131×C₁₃₁	C₂²	semidirect
31. C₂×M₁₆×C₁₃₁	C₂²	semidirect
32. C₂²×C₈×C₁₃₁	C₂²	semidirect
50. #14523497	C₂³	semidirect
53. #14523496	C₂³	semidirect
54. #14523496	C₂³	semidirect
57. C₂×C₁₃₁⋊C₈	C₂³	semidirect
58. C₈×D₁₃₁	C₂³	semidirect
64. M₁₆×C₁₃₁	C₂³	semidirect
67. C₂×C₈×C₁₃₁	C₂³	semidirect
109. C₂×C₁₃₁	#568	semidirect
113. C₁₃₁	C₂×#568	semidirect
120. C₂×#131	D₁₃₁	semidirect
135. #131	C₂×D₁₃₁	semidirect
143. C₂×M₁₆	C₂×D₁₃₁	semidirect
146. C₂²×C₈	C₂×D₁₃₁	semidirect
174. M₁₆	C₂²×D₁₃₁	semidirect
181. C₂×C₈	C₂²×D₁₃₁	semidirect
225. C₂	#22554077	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#41×D₁₃₁	2
#22554077	2
C₂×#16862254	2
C₂×#16862253	2
C₂×#16862252	2
C₂²×#14523496	2
C₂×M₁₆×D₁₃₁	2
C₂×#131×C₁₃₁	2
C₂×#568	131

Subgroups — C2×#22554077 #30680669

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22554077 #30680669