Subgroups — C₂×#21447600 #29833039

Order 15616.

694 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₆×D₆₁	C₂	1	1	yes	maximal, characteristic
2	C₂×#16839799	C₂	1	2	yes	maximal
3	#21447600	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16839589	C₂	1	2	yes	maximal, complemented
5	C₂×#16839588	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₆₁×#366	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₈×D₆₁	C₂²	1	1	yes	characteristic
8	C₂×#14519735	—	2	2	no	—
9	C₂²×#11758185	C₂²	1	1	yes	characteristic
10	C₂×C₁₆×D₆₁	C₄	1	8	yes	complemented
11	C₂×C₁₆×D₆₁	C₂²	1	4	yes	—
12	#16839799	—	2	4	no	—
13	#16839799	C₂²	1	8	yes	—
14	C₂×#14519711	C₂²	1	2	yes	—
15	C₂×#14519708	—	2	2	no	—
16	#16839589	C₂²	1	8	yes	complemented
17	#16839588	C₂²	1	4	yes	complemented
18	C₂²×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
19	C₂×#107×C₆₁	C₂²	1	2	yes	—
20	C₆₁×#366	C₂²	1	4	yes	complemented
21	C₂²×C₄×D₆₁	D₄	1	1	yes	characteristic
22	C₂²×#92418	D₄	1	1	yes	characteristic
23	C₂×C₈×D₆₁	C₂×C₄	1	4	yes	—
24	C₂×C₈×D₆₁	D₄	1	2	yes	—
25	C₂×C₈×D₆₁	Q₈	1	4	yes	—
26	C₂×C₈×D₆₁	C₂³	1	2	yes	—
27	#14519735	—	2	8	no	—
28	#14519735	—	4	4	no	—
29	C₂×C₄⋊Dic₆₁	—	2	2	no	—
30	C₂×#11758198	—	4	2	no	—
31	C₂×C₄×Dic₆₁	—	4	2	no	—
32	C₂×#11758185	C₂×C₄	1	4	yes	—
33	C₂×#11758185	C₂³	1	2	yes	—
34	C₁₆×D₆₁	C₂×C₄	1	16	yes	complemented
35	#14519711	—	2	2	no	—
36	#14519711	C₂³	1	4	yes	—
37	#14519708	—	2	8	no	—
38	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
40	C₂×C₁₆×C₆₁	C₂×C₄	1	4	yes	—
41	C₂×C₁₆×C₆₁	C₂³	1	2	yes	—
42	#107×C₆₁	—	2	2	no	—
43	#107×C₆₁	C₂³	1	4	yes	—
44	C₂³×D₆₁	D₈	1	1	yes	characteristic
45	C₂²×C₄×C₆₁	—	4	2	no	—
46	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
47	C₂²×Dic₆₁	—	4	2	no	—
48	C₂²×Dic₆₁	D₈	1	1	yes	characteristic
49	C₂×C₄×D₆₁	C₄⋊C₄	1	4	yes	—
50	C₂×C₄×D₆₁	D₈	1	2	yes	—
51	C₂×C₄×D₆₁	Q₁₆	1	4	yes	—
52	C₂×C₄×D₆₁	C₂×D₄	1	2	yes	—
53	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
54	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
55	C₂×C₈×C₆₁	C₂×Q₈	1	2	yes	—
56	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
57	C₄⋊C₄×C₆₁	—	2	4	no	—
58	C₄⋊C₄×C₆₁	—	4	2	no	—
59	#11758198	—	4	8	no	—
60	C₄⋊Dic₆₁	—	2	4	no	—
61	C₄⋊Dic₆₁	—	4	2	no	—
62	C₄×Dic₆₁	—	4	8	no	—
63	C₂×#92418	C₄⋊C₄	1	4	yes	—
64	C₂×#92418	C₂×D₄	1	2	yes	—
65	C₈×D₆₁	C₄⋊C₄	1	8	yes	—
66	C₈×D₆₁	C₂²×C₄	1	4	yes	—
67	C₈×D₆₁	C₂×Q₈	1	4	yes	—
68	C₁₆×C₆₁	C₂²×C₄	1	4	yes	—
69	#11758185	C₂²×C₄	1	4	yes	—
70	C₂³×C₆₁	C₂×D₈	1	1	yes	characteristic
71	C₂²×D₆₁	—	2	2	no	—
72	C₂²×D₆₁	—	2	2	no	—
73	C₂²×D₆₁	#107	1	4	yes	—
74	C₂²×D₆₁	C₂×D₈	1	2	yes	—
75	C₂×C₄×C₆₁	—	4	4	no	—
76	C₂×C₄×C₆₁	—	8	4	no	—
77	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
78	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
79	C₂×C₄×C₆₁	C₂×Q₁₆	1	2	yes	—
80	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
81	C₂×Dic₆₁	—	4	4	no	—
82	C₂×Dic₆₁	—	8	4	no	—
83	C₂×Dic₆₁	#107	1	4	yes	—
84	C₂×Dic₆₁	C₂×D₈	1	2	yes	—
85	C₄×D₆₁	#107	1	8	yes	—
86	C₄×D₆₁	C₂×C₄⋊C₄	1	4	yes	—
87	C₄×D₆₁	C₂×Q₁₆	1	4	yes	—
88	C₈×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
89	C₈×C₆₁	C₂³×C₄	1	1	yes	characteristic
90	C₈×C₆₁	C₂²×Q₈	1	1	yes	characteristic
91	#92418	C₂×C₄⋊C₄	1	4	yes	—
92	C₂×#2158	—	61	1	no	maximal, Sylow
93	C₂²×C₆₁	C₂×#107	1	2	yes	—
94	C₂²×C₆₁	#508	1	2	yes	—
95	C₂²×C₆₁	#508	1	2	yes	—
96	C₂²×C₆₁	C₂²×D₈	1	1	yes	characteristic
97	C₂×D₆₁	—	2	2	no	—
98	C₂×D₆₁	—	2	2	no	—
99	C₂×D₆₁	—	2	8	no	—
100	C₂×D₆₁	C₂×#107	1	4	yes	—
101	C₄×C₆₁	—	8	4	no	—
102	C₄×C₆₁	C₂×#107	1	2	yes	—
103	C₄×C₆₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
104	C₄×C₆₁	C₂²×Q₁₆	1	1	yes	characteristic
105	Dic₆₁	—	8	4	no	—
106	Dic₆₁	C₂×#107	1	4	yes	—
107	C₂³×C₁₆	—	61	1	no	—
108	C₂×#427	—	61	2	no	—
109	#2158	—	61	8	no	—
110	C₂×#367	—	61	2	no	—
111	C₂×#366	—	61	1	no	—
112	C₂×#366	D₆₁	1	1	yes	characteristic, complemented
113	C₂×C₆₁	#2158	1	4	yes	complemented
114	C₂×C₆₁	C₂²×#107	1	1	yes	characteristic
115	C₂×C₆₁	C₂×#508	1	1	yes	characteristic
116	C₂×C₆₁	C₂×#508	1	1	yes	characteristic
117	D₆₁	—	2	4	no	—
118	C₂³×C₈	—	61	1	no	—
119	C₂×#117	—	122	2	no	—
120	C₂²×C₁₆	—	61	1	no	—
121	C₂²×C₁₆	—	61	8	no	—
122	C₂²×C₁₆	—	61	4	no	—
123	C₂²×C₁₆	C₂×D₆₁	1	1	yes	characteristic
124	#427	—	61	8	no	—
125	#427	—	122	4	no	—
126	C₂×#107	—	61	2	no	—
127	C₂×#107	C₂×D₆₁	1	2	yes	—
128	C₂×#106	—	122	2	no	—
129	#367	—	61	8	no	—
130	#366	—	61	4	no	—
131	#366	C₂×D₆₁	1	4	yes	complemented
132	C₆₁	C₂×#2158	1	1	yes	characteristic, complemented, Sylow
133	C₂³×C₄	—	61	1	no	—
134	C₂²×C₈	—	61	1	no	—
135	C₂²×C₈	—	61	4	no	—
136	C₂²×C₈	—	61	2	no	—
137	C₂²×C₈	—	61	4	no	—
138	C₂²×C₈	—	61	2	no	—
139	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
140	#117	—	122	8	no	—
141	#117	—	244	4	no	—
142	C₂×C₄⋊C₄	—	2	2	no	—
143	C₂×C₄⋊C₄	—	122	2	no	—
144	C₂×C₂²⋊C₄	—	244	2	no	—
145	C₂×C₄²	—	244	2	no	—
146	C₂×C₁₆	—	61	4	no	—
147	C₂×C₁₆	—	61	2	no	—
148	C₂×C₁₆	—	61	16	no	—
149	C₂×C₁₆	C₄×D₆₁	1	4	yes	—
150	C₂×C₁₆	C₂²×D₆₁	1	2	yes	—
151	#107	—	2	2	no	—
152	#107	—	61	4	no	—
153	#107	—	122	2	no	—
154	#107	C₂²×D₆₁	1	4	yes	—
155	#106	—	122	8	no	—
156	C₂⁴	—	61	1	no	—
157	C₂²×C₄	—	4	2	no	—
158	C₂²×C₄	—	61	1	no	—
159	C₂²×C₄	—	61	2	no	—
160	C₂²×C₄	—	61	4	no	—
161	C₂²×C₄	—	61	2	no	—
162	C₂²×C₄	—	61	4	no	—
163	C₂²×C₄	—	244	2	no	—
164	C₂²×C₄	D₄×D₆₁	1	1	yes	characteristic
165	C₂×C₈	—	61	2	no	—
166	C₂×C₈	—	61	4	no	—
167	C₂×C₈	—	61	4	no	—
168	C₂×C₈	—	61	4	no	—
169	C₂×C₈	—	61	8	no	—
170	C₂×C₈	C₂×C₄×D₆₁	1	2	yes	—
171	C₂×C₈	D₄×D₆₁	1	1	yes	characteristic
172	C₂×C₈	Q₈×D₆₁	1	2	yes	—
173	C₂×C₈	C₂³×D₆₁	1	1	yes	characteristic
174	C₄⋊C₄	—	2	4	no	—
175	C₄⋊C₄	—	4	2	no	—
176	C₄⋊C₄	—	122	4	no	—
177	C₄⋊C₄	—	244	2	no	—
178	C₂²⋊C₄	—	244	8	no	—
179	C₄²	—	244	8	no	—
180	C₁₆	—	61	4	no	—
181	C₁₆	C₂×C₄×D₆₁	1	4	yes	—
182	C₂³	—	61	2	no	—
183	C₂³	—	61	4	no	—
184	C₂³	—	122	2	no	—
185	C₂³	—	122	2	no	—
186	C₂³	D₈×D₆₁	1	1	yes	characteristic
187	C₂×C₄	—	4	4	no	—
188	C₂×C₄	—	8	4	no	—
189	C₂×C₄	—	61	2	no	—
190	C₂×C₄	—	61	4	no	—
191	C₂×C₄	—	61	4	no	—
192	C₂×C₄	—	61	4	no	—
193	C₂×C₄	—	61	8	no	—
194	C₂×C₄	—	488	4	no	—
195	C₂×C₄	—	244	4	no	—
196	C₂×C₄	C₄⋊C₄×D₆₁	1	2	yes	—
197	C₂×C₄	D₈×D₆₁	1	1	yes	characteristic
198	C₂×C₄	Q₁₆×D₆₁	1	2	yes	—
199	C₂×C₄	C₂×D₄×D₆₁	1	1	yes	characteristic
200	C₈	—	61	4	no	—
201	C₈	C₄⋊C₄×D₆₁	1	2	yes	—
202	C₈	C₂²×C₄×D₆₁	1	1	yes	characteristic
203	C₈	C₂×Q₈×D₆₁	1	1	yes	characteristic
204	C₂²	—	61	4	no	—
205	C₂²	—	122	8	no	—
206	C₂²	—	122	2	no	—
207	C₂²	—	122	2	no	—
208	C₂²	#107×D₆₁	1	2	yes	—
209	C₂²	#16840191	1	2	yes	—
210	C₂²	#16840195	1	2	yes	—
211	C₂²	C₂×D₈×D₆₁	1	1	yes	characteristic
212	C₄	—	8	4	no	—
213	C₄	—	61	4	no	—
214	C₄	—	488	4	no	—
215	C₄	#107×D₆₁	1	2	yes	—
216	C₄	C₂×C₄⋊C₄×D₆₁	1	1	yes	characteristic
217	C₄	C₂×Q₁₆×D₆₁	1	1	yes	characteristic
218	C₂	—	122	4	no	—
219	C₂	#21447600	1	4	yes	direct product factor
220	C₂	C₂×#107×D₆₁	1	1	yes	characteristic
221	C₂	C₂×#16840191	1	1	yes	characteristic
222	C₂	C₂×#16840195	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₈×D₆₁
commutator	C₈×C₆₁	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₆₁
Fitting	C₂×C₆₁×#366	C₂
socle	C₂³×C₆₁	C₂×D₈
prime radical	C₂³×D₆₁	D₈
cosocle	C₂×C₈×C₆₁	C₂⁴
hypercenter	C₂×#366	D₆₁
hypocenter	C₆₁	C₂×#2158
LCS	G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#366

Complements

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

N	H ≅ G/N	type
3. #21447600	C₂	direct (N × H)
4. C₂×#16839589	C₂	semidirect
5. C₂×#16839588	C₂	semidirect
6. C₂×C₆₁×#366	C₂	semidirect
10. C₂×C₁₆×D₆₁	C₄	semidirect
16. #16839589	C₂²	semidirect
17. #16839588	C₂²	semidirect
20. C₆₁×#366	C₂²	semidirect
34. C₁₆×D₆₁	C₂×C₄	semidirect
112. C₂×#366	D₆₁	semidirect
113. C₂×C₆₁	#2158	semidirect
131. #366	C₂×D₆₁	semidirect
132. C₆₁	C₂×#2158	semidirect
219. C₂	#21447600	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×C₁₆×D₆₁	2
C₂×#16839799	2
#21447600	2
C₂×#16839589	2
C₂×#16839588	2
C₂×C₆₁×#366	2
C₂×#2158	61

Subgroups — C2×#21447600 #29833039

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21447600 #29833039