Subgroups — C₂×#21454216 #29886110

Order 15616.

3774 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14519776	C₂	1	6	yes	maximal
2	#21454216	C₂	1	16	yes	maximal, direct product factor
3	C₂×#16839632	C₂	1	1	yes	maximal, characteristic
4	C₂×#16839631	C₂	1	6	yes	maximal
5	C₂×C₄²×Dic₆₁	C₂	1	1	yes	maximal, characteristic
6	C₂×C₆₁×#384	C₂	1	1	yes	maximal, characteristic
7	C₂³×Dic₁₂₂	C₂²	1	4	yes	—
8	C₂×#14519776	—	2	48	no	—
9	C₂×#14519776	C₂²	1	48	yes	—
10	C₂²×C₄⋊Dic₆₁	C₂²	1	6	yes	—
11	C₂²×#11758196	C₂²	1	12	yes	—
12	C₂²×C₄×Dic₆₁	C₂²	1	3	yes	—
13	C₂²×C₄×Dic₆₁	C₂²	1	3	yes	—
14	#16839632	C₂²	1	8	yes	—
15	#16839631	C₂²	1	48	yes	—
16	C₄²×Dic₆₁	C₂²	1	8	yes	—
17	C₂²×C₄⋊C₄×C₆₁	C₂²	1	6	yes	—
18	C₂²×C₄²×C₆₁	C₂²	1	1	yes	characteristic
19	C₆₁×#384	C₂²	1	8	yes	—
20	C₂²×Dic₁₂₂	—	2	48	no	—
21	C₂²×Dic₁₂₂	C₂³	1	16	yes	—
22	C₂³×Dic₆₁	C₂³	1	4	yes	—
23	C₂³×Dic₆₁	C₂³	1	1	yes	characteristic
24	C₂³×Dic₆₁	C₂³	1	3	yes	—
25	#14519776	—	2	192	no	—
26	C₂×C₄⋊Dic₆₁	—	2	24	no	—
27	C₂×C₄⋊Dic₆₁	C₂³	1	24	yes	—
28	C₂×#11758196	—	2	48	no	—
29	C₂×#11758196	C₂³	1	48	yes	—
30	C₂×C₄×Dic₆₁	D₄	1	24	yes	—
31	C₂×C₄×Dic₆₁	Q₈	1	24	yes	complemented
32	C₂×C₄×Dic₆₁	C₂³	1	12	yes	—
33	C₂×C₄×Dic₆₁	C₂³	1	12	yes	—
34	C₂³×C₄×C₆₁	C₂³	1	4	yes	—
35	C₂³×C₄×C₆₁	C₂³	1	3	yes	—
36	C₂×C₄⋊C₄×C₆₁	—	2	24	no	—
37	C₂×C₄⋊C₄×C₆₁	C₂³	1	24	yes	—
38	C₂×C₄²×C₆₁	Q₈	1	8	yes	complemented
39	C₂×C₄²×C₆₁	C₂³	1	4	yes	—
40	C₂⁴×C₆₁	C₂⁴	1	1	yes	characteristic
41	C₂²×C₄×C₆₁	—	2	24	no	—
42	C₂²×C₄×C₆₁	C₂×D₄	1	12	yes	—
43	C₂²×C₄×C₆₁	C₂×Q₈	1	12	yes	—
44	C₂²×C₄×C₆₁	C₂×Q₈	1	12	yes	—
45	C₂²×C₄×C₆₁	C₂⁴	1	8	yes	—
46	C₂²×C₄×C₆₁	C₂⁴	1	6	yes	—
47	C₂²×Dic₆₁	—	2	24	no	—
48	C₂²×Dic₆₁	C₂×D₄	1	12	yes	—
49	C₂²×Dic₆₁	C₂×D₄	1	12	yes	—
50	C₂²×Dic₆₁	C₂×Q₈	1	24	yes	—
51	C₂²×Dic₆₁	C₂⁴	1	8	yes	—
52	C₂²×Dic₆₁	C₂⁴	1	2	yes	—
53	C₂²×Dic₆₁	C₂⁴	1	6	yes	—
54	C₂×Dic₁₂₂	—	2	96	no	—
55	C₂×Dic₁₂₂	—	4	64	no	—
56	C₄⋊C₄×C₆₁	—	2	48	no	—
57	C₄²×C₆₁	C₂×Q₈	1	16	yes	complemented
58	C₄⋊Dic₆₁	—	2	48	no	—
59	#11758196	—	2	96	no	—
60	C₄×Dic₆₁	C₂×D₄	1	48	yes	—
61	C₄×Dic₆₁	C₂×Q₈	1	48	yes	complemented
62	C₂³×C₆₁	C₂²×D₄	1	6	yes	—
63	C₂³×C₆₁	C₂²×Q₈	1	6	yes	—
64	C₂³×C₆₁	C₂²×Q₈	1	2	yes	—
65	C₂³×C₆₁	C₂⁵	1	1	yes	characteristic
66	C₂×C₄×C₆₁	—	2	24	no	—
67	C₂×C₄×C₆₁	—	4	16	no	—
68	C₂×C₄×C₆₁	C₄⋊Q₈	1	48	yes	complemented
69	C₂×C₄×C₆₁	C₂²×D₄	1	12	yes	—
70	C₂×C₄×C₆₁	C₂²×Q₈	1	12	yes	—
71	C₂×C₄×C₆₁	C₂²×Q₈	1	12	yes	—
72	C₂×Dic₆₁	—	2	24	no	—
73	C₂×Dic₆₁	—	4	16	no	—
74	C₂×Dic₆₁	#127	1	16	yes	—
75	C₂×Dic₆₁	C₄⋊Q₈	1	48	yes	complemented
76	C₂×Dic₆₁	C₂²×D₄	1	12	yes	—
77	C₂×Dic₆₁	C₂²×D₄	1	12	yes	—
78	C₂×Dic₆₁	C₂²×Q₈	1	24	yes	—
79	Dic₁₂₂	—	4	64	no	—
80	C₂×#2868	—	61	1	no	maximal, Sylow
81	C₂²×C₆₁	C₂×#127	1	4	yes	—
82	C₂²×C₆₁	C₂×C₄⋊Q₈	1	12	yes	—
83	C₂²×C₆₁	C₂×C₄⋊Q₈	1	12	yes	—
84	C₂²×C₆₁	C₂³×D₄	1	3	yes	—
85	C₂²×C₆₁	C₂³×Q₈	1	3	yes	—
86	C₂²×C₆₁	C₂³×Q₈	1	1	yes	characteristic
87	C₄×C₆₁	—	4	8	no	—
88	C₄×C₆₁	C₂×C₄⋊Q₈	1	24	yes	complemented
89	Dic₆₁	—	4	8	no	—
90	Dic₆₁	C₂×#127	1	8	yes	—
91	Dic₆₁	C₂×C₄⋊Q₈	1	24	yes	complemented
92	C₂²×C₄⋊Q₈	—	61	6	no	—
93	#2868	—	61	16	no	—
94	C₂×#384	—	61	1	no	—
95	C₂×#384	—	61	6	no	—
96	C₂×#384	D₆₁	1	1	yes	characteristic
97	C₂×C₄³	—	61	1	no	—
98	C₂×C₆₁	#2868	1	8	yes	complemented
99	C₂×C₆₁	C₂²×#127	1	1	yes	characteristic
100	C₂×C₆₁	C₂²×C₄⋊Q₈	1	3	yes	—
101	C₂×C₆₁	C₂²×C₄⋊Q₈	1	3	yes	—
102	C₂³×Q₈	—	61	4	no	—
103	C₂×C₄⋊Q₈	—	61	48	no	—
104	C₂×C₄⋊Q₈	—	122	48	no	—
105	C₂²×C₄⋊C₄	—	61	12	no	—
106	C₂²×C₄⋊C₄	—	61	6	no	—
107	C₂²×C₄⋊C₄	C₂×D₆₁	1	6	yes	—
108	C₂²×C₄²	—	61	3	no	—
109	C₂²×C₄²	—	61	3	no	—
110	C₂²×C₄²	C₂×D₆₁	1	1	yes	characteristic
111	#384	—	61	48	no	—
112	#384	—	61	8	no	—
113	#384	C₂×D₆₁	1	8	yes	—
114	C₄³	—	61	8	no	—
115	C₆₁	C₂×#2868	1	1	yes	characteristic, complemented, Sylow
116	C₂²×Q₈	—	61	16	no	—
117	C₂²×Q₈	—	122	48	no	—
118	C₂³×C₄	—	61	3	no	—
119	C₂³×C₄	—	61	1	no	—
120	C₂³×C₄	—	61	4	no	—
121	C₂³×C₄	C₂²×D₆₁	1	4	yes	—
122	C₂³×C₄	C₂²×D₆₁	1	3	yes	—
123	C₄⋊Q₈	—	122	192	no	—
124	C₂×C₄⋊C₄	—	2	24	no	—
125	C₂×C₄⋊C₄	—	61	48	no	—
126	C₂×C₄⋊C₄	—	61	24	no	—
127	C₂×C₄⋊C₄	—	122	48	no	—
128	C₂×C₄⋊C₄	—	122	24	no	—
129	C₂×C₄⋊C₄	C₂²×D₆₁	1	24	yes	—
130	C₂×C₄²	—	61	24	no	—
131	C₂×C₄²	—	61	24	no	—
132	C₂×C₄²	—	61	12	no	—
133	C₂×C₄²	—	61	12	no	—
134	C₂×C₄²	Dic₁₂₂	1	8	yes	complemented
135	C₂×C₄²	C₂²×D₆₁	1	4	yes	—
136	C₂⁴	C₂³×D₆₁	1	1	yes	characteristic
137	C₂×Q₈	—	122	96	no	—
138	C₂×Q₈	—	244	64	no	—
139	C₂²×C₄	—	2	24	no	—
140	C₂²×C₄	—	61	12	no	—
141	C₂²×C₄	—	61	12	no	—
142	C₂²×C₄	—	61	24	no	—
143	C₂²×C₄	—	61	6	no	—
144	C₂²×C₄	—	61	2	no	—
145	C₂²×C₄	—	61	8	no	—
146	C₂²×C₄	—	122	24	no	—
147	C₂²×C₄	C₂×Dic₁₂₂	1	12	yes	—
148	C₂²×C₄	D₄×D₆₁	1	12	yes	—
149	C₂²×C₄	Q₈×D₆₁	1	12	yes	—
150	C₂²×C₄	C₂³×D₆₁	1	8	yes	—
151	C₂²×C₄	C₂³×D₆₁	1	6	yes	—
152	C₄⋊C₄	—	2	48	no	—
153	C₄⋊C₄	—	122	96	no	—
154	C₄⋊C₄	—	122	48	no	—
155	C₄²	—	61	48	no	—
156	C₄²	—	61	48	no	—
157	C₄²	C₂×Dic₁₂₂	1	16	yes	complemented
158	C₂³	C₂×D₄×D₆₁	1	6	yes	—
159	C₂³	C₂×Q₈×D₆₁	1	6	yes	—
160	C₂³	C₂²×Dic₁₂₂	1	2	yes	—
161	C₂³	C₂⁴×D₆₁	1	1	yes	characteristic
162	Q₈	—	244	64	no	—
163	C₂×C₄	—	2	24	no	—
164	C₂×C₄	—	4	16	no	—
165	C₂×C₄	—	61	16	no	—
166	C₂×C₄	—	61	48	no	—
167	C₂×C₄	—	61	12	no	—
168	C₂×C₄	—	61	12	no	—
169	C₂×C₄	—	61	24	no	—
170	C₂×C₄	—	122	24	no	—
171	C₂×C₄	—	244	16	no	—
172	C₂×C₄	#14519776	1	48	yes	complemented
173	C₂×C₄	C₂×D₄×D₆₁	1	12	yes	—
174	C₂×C₄	C₂×Q₈×D₆₁	1	12	yes	—
175	C₂×C₄	C₂²×Dic₁₂₂	1	12	yes	—
176	C₂²	#127×D₆₁	1	4	yes	—
177	C₂²	C₄⋊Q₈×D₆₁	1	12	yes	—
178	C₂²	C₂×#14519776	1	12	yes	—
179	C₂²	C₂²×D₄×D₆₁	1	3	yes	—
180	C₂²	C₂²×Q₈×D₆₁	1	3	yes	—
181	C₂²	C₂³×Dic₁₂₂	1	1	yes	characteristic
182	C₄	—	4	8	no	—
183	C₄	—	61	8	no	—
184	C₄	—	61	24	no	—
185	C₄	—	244	8	no	—
186	C₄	C₂×#14519776	1	24	yes	complemented
187	C₂	#21454216	1	8	yes	direct product factor
188	C₂	C₂×#127×D₆₁	1	1	yes	characteristic
189	C₂	C₂×C₄⋊Q₈×D₆₁	1	3	yes	—
190	C₂	C₂²×#14519776	1	3	yes	—

	subgroup	G/H
center	C₂⁴	C₂³×D₆₁
commutator, cosocle	C₂³×C₆₁	C₂⁵
Frattini	C₂³	C₂⁴×D₆₁
Fitting	C₂×C₆₁×#384	C₂
socle, prime radical	C₂⁴×C₆₁	C₂⁴
hypercenter	C₂×#384	D₆₁
hypocenter	C₆₁	C₂×#2868
LCS	G ▷ C₂³×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×#384

Complements

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

N	H ≅ G/N	type
2. #21454216	C₂	direct (N × H)
31. C₂×C₄×Dic₆₁	Q₈	semidirect
38. C₂×C₄²×C₆₁	Q₈	semidirect
57. C₄²×C₆₁	C₂×Q₈	semidirect
61. C₄×Dic₆₁	C₂×Q₈	semidirect
68. C₂×C₄×C₆₁	C₄⋊Q₈	semidirect
75. C₂×Dic₆₁	C₄⋊Q₈	semidirect
88. C₄×C₆₁	C₂×C₄⋊Q₈	semidirect
91. Dic₆₁	C₂×C₄⋊Q₈	semidirect
98. C₂×C₆₁	#2868	semidirect
115. C₆₁	C₂×#2868	semidirect
134. C₂×C₄²	Dic₁₂₂	semidirect
157. C₄²	C₂×Dic₁₂₂	semidirect
172. C₂×C₄	#14519776	semidirect
186. C₄	C₂×#14519776	semidirect
187. C₂	#21454216	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×#14519776	2
#21454216	2
C₂×#16839632	2
C₂×#16839631	2
C₂×C₄²×Dic₆₁	2
C₂×C₆₁×#384	2
C₂×#2868	61

Subgroups — C2×#21454216 #29886110

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21454216 #29886110