Subgroups — C₂×#21447579 #29832997

Order 15616.

734 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#11758185	C₂	1	1	yes	maximal, characteristic
2	C₂×#16839717	C₂	1	1	yes	maximal, characteristic
3	#21447579	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16839580	C₂	1	4	yes	maximal, complemented
5	C₂²×M₃₂×C₆₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11758194	C₄	1	1	yes	characteristic
7	C₂×#14519736	C₄	1	1	yes	characteristic
8	C₂²×#11758185	—	2	4	no	—
9	C₂²×#11758185	C₂²	1	2	yes	—
10	C₂²×#11758185	C₂²	1	4	yes	—
11	#16839717	C₄	1	4	yes	—
12	#16839717	C₂²	1	4	yes	—
13	C₂×#14519687	C₄	1	2	yes	—
14	C₂×C₈×Dic₆₁	C₂²	1	2	yes	—
15	#16839580	C₂²	1	16	yes	complemented
16	C₂³×C₈×C₆₁	C₂²	1	1	yes	characteristic
17	C₂×M₃₂×C₆₁	—	2	4	no	—
18	C₂×M₃₂×C₆₁	C₂²	1	4	yes	—
19	C₂²×C₁₆×C₆₁	C₂²	1	2	yes	—
20	C₂×#11758194	—	2	4	no	—
21	C₂×#11758194	C₂×C₄	1	4	yes	—
22	C₂²×#92418	C₂×C₄	1	2	yes	—
23	#14519736	C₈	1	4	yes	—
24	#14519736	C₂×C₄	1	4	yes	—
25	C₂×C₄⋊Dic₆₁	C₈	1	2	yes	—
26	C₂×C₂²⋊Dic₆₁	C₈	1	2	yes	—
27	C₂×C₄×Dic₆₁	C₂×C₄	1	2	yes	—
28	C₂×#11758185	—	2	4	no	—
29	C₂×#11758185	—	2	8	no	—
30	C₂×#11758185	C₂³	1	4	yes	—
31	#14519687	C₂×C₄	1	8	yes	—
32	C₈×Dic₆₁	C₂×C₄	1	4	yes	—
33	C₈×Dic₆₁	C₂³	1	4	yes	—
34	C₂³×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
37	C₂²×C₈×C₆₁	D₄	1	4	yes	—
38	C₂²×C₈×C₆₁	Q₈	1	4	yes	—
39	C₂²×C₈×C₆₁	C₂³	1	2	yes	—
40	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
41	M₃₂×C₆₁	—	2	8	no	—
42	C₂×C₁₆×C₆₁	—	2	4	no	—
43	C₂×C₁₆×C₆₁	C₂³	1	4	yes	—
44	C₂⁴×C₆₁	C₂×C₈	1	1	yes	characteristic
45	C₂²×C₄×C₆₁	C₄⋊C₄	1	4	yes	—
46	C₂²×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
47	C₂²×C₄×C₆₁	C₂×C₈	1	2	yes	—
48	C₂²×C₄×C₆₁	M₁₆	1	4	yes	—
49	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₆₁	C₂²×C₄	1	2	yes	—
51	C₂²×Dic₆₁	C₂×C₈	1	4	yes	—
52	C₂×C₈×C₆₁	C₄⋊C₄	1	4	yes	—
53	C₂×C₈×C₆₁	C₄⋊C₄	1	8	yes	—
54	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
56	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
57	C₂×C₈×C₆₁	C₂×D₄	1	4	yes	—
58	C₂×C₈×C₆₁	C₂×Q₈	1	4	yes	—
59	C₂×C₈×C₆₁	C₂×Q₈	1	2	yes	—
60	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
61	C₂²⋊Dic₆₁	C₂×C₈	1	8	yes	—
62	C₄⋊Dic₆₁	C₂×C₈	1	8	yes	—
63	C₄×Dic₆₁	C₂×C₈	1	4	yes	—
64	C₄×Dic₆₁	C₂²×C₄	1	4	yes	—
65	#11758194	—	2	8	no	—
66	C₂×#92418	—	2	4	no	—
67	C₂×#92418	C₂²×C₄	1	4	yes	—
68	C₁₆×C₆₁	—	2	4	no	—
69	#11758185	—	2	4	no	—
70	C₂³×C₆₁	—	2	4	no	—
71	C₂³×C₆₁	C₄⋊C₈	1	4	yes	—
72	C₂³×C₆₁	C₂²×C₈	1	2	yes	—
73	C₂³×C₆₁	C₂²×C₈	1	1	yes	characteristic
74	C₂×C₄×C₆₁	C₄⋊C₈	1	8	yes	—
75	C₂×C₄×C₆₁	C₄⋊C₈	1	4	yes	—
76	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	4	yes	—
77	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
78	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
79	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
80	C₂×C₄×C₆₁	C₂×M₁₆	1	2	yes	—
81	C₂×C₄×C₆₁	C₂×M₁₆	1	4	yes	—
82	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
83	C₂×Dic₆₁	—	2	8	no	—
84	C₂×Dic₆₁	C₂²×C₈	1	8	yes	—
85	C₈×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
86	C₈×C₆₁	C₂×C₄⋊C₄	1	4	yes	—
87	C₈×C₆₁	C₂²×D₄	1	1	yes	characteristic
88	C₈×C₆₁	C₂²×Q₈	1	1	yes	characteristic
89	#92418	—	2	4	no	—
90	C₂×#2151	—	61	1	no	maximal, Sylow
91	C₂²×C₆₁	—	2	4	no	—
92	C₂²×C₆₁	—	2	8	no	—
93	C₂²×C₆₁	C₂×C₄⋊C₈	1	4	yes	—
94	C₂²×C₆₁	C₂×C₄⋊C₈	1	2	yes	—
95	C₂²×C₆₁	#504	1	4	yes	—
96	C₂²×C₆₁	C₂³×C₈	1	1	yes	characteristic
97	C₄×C₆₁	C₂×C₄⋊C₈	1	2	yes	—
98	C₄×C₆₁	C₂×C₄⋊C₈	1	4	yes	—
99	C₄×C₆₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
100	C₄×C₆₁	C₂²×M₁₆	1	1	yes	characteristic
101	Dic₆₁	—	2	8	no	—
102	C₂²×M₃₂	D₆₁	1	1	yes	characteristic
103	C₂³×C₁₆	—	61	1	no	—
104	C₂×#405	—	61	1	no	—
105	#2151	—	61	8	no	—
106	C₂×C₄⋊C₁₆	—	61	4	no	—
107	C₂×C₆₁	—	2	4	no	—
108	C₂×C₆₁	#2151	1	4	yes	complemented
109	C₂×C₆₁	C₂²×C₄⋊C₈	1	1	yes	characteristic
110	C₂×C₆₁	C₂×#504	1	2	yes	—
111	C₂²×M₁₆	—	61	1	no	—
112	C₂³×C₈	C₂×D₆₁	1	1	yes	characteristic
113	C₂×#117	—	61	1	no	—
114	C₂×M₃₂	—	2	4	no	—
115	C₂×M₃₂	C₂×D₆₁	1	4	yes	—
116	C₂²×C₁₆	—	61	4	no	—
117	C₂²×C₁₆	—	61	2	no	—
118	C₂²×C₁₆	—	122	4	no	—
119	C₂²×C₁₆	C₂×D₆₁	1	2	yes	—
120	#405	—	61	4	no	—
121	#405	—	61	4	no	—
122	C₂×#97	—	61	2	no	—
123	C₂×C₄×C₈	—	61	2	no	—
124	C₄⋊C₁₆	—	61	16	no	—
125	C₆₁	C₂×#2151	1	1	yes	characteristic, complemented, Sylow
126	C₂³×C₄	C₄×D₆₁	1	1	yes	characteristic
127	C₂×M₁₆	—	61	4	no	—
128	C₂×M₁₆	—	122	4	no	—
129	C₂²×C₈	—	61	2	no	—
130	C₂²×C₈	Dic₁₂₂	1	4	yes	—
131	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
132	C₂²×C₈	C₄×D₆₁	1	2	yes	—
133	C₂²×C₈	C₂²⋊D₆₁	1	4	yes	—
134	C₂²×C₈	C₂²×D₆₁	1	2	yes	—
135	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
136	#117	—	61	4	no	—
137	#117	—	61	4	no	—
138	C₂×C₄⋊C₄	—	61	2	no	—
139	C₂×C₂²⋊C₄	—	61	2	no	—
140	C₂×C₄²	—	61	2	no	—
141	M₃₂	—	2	8	no	—
142	C₂×C₁₆	—	2	4	no	—
143	C₂×C₁₆	—	61	4	no	—
144	C₂×C₁₆	—	122	4	no	—
145	C₂×C₁₆	—	122	8	no	—
146	C₂×C₁₆	C₂²×D₆₁	1	4	yes	—
147	#97	—	61	8	no	—
148	C₄×C₈	—	61	4	no	—
149	C₄×C₈	—	61	4	no	—
150	C₂⁴	C₈×D₆₁	1	1	yes	characteristic
151	C₂²×C₄	—	61	4	no	—
152	C₂²×C₄	C₈×D₆₁	1	1	yes	characteristic
153	C₂²×C₄	C₈×D₆₁	1	2	yes	—
154	C₂²×C₄	#11758189	1	4	yes	—
155	C₂²×C₄	#11758196	1	4	yes	—
156	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
157	C₂²×C₄	C₂×C₄×D₆₁	1	2	yes	—
158	M₁₆	—	122	8	no	—
159	C₂×C₈	—	61	4	no	—
160	C₂×C₈	—	122	4	no	—
161	C₂×C₈	#11758196	1	4	yes	—
162	C₂×C₈	#11758196	1	8	yes	—
163	C₂×C₈	C₂×Dic₁₂₂	1	4	yes	—
164	C₂×C₈	C₂×Dic₁₂₂	1	2	yes	—
165	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
166	C₂×C₈	C₂×C₄×D₆₁	1	2	yes	—
167	C₂×C₈	C₂×C₂²⋊D₆₁	1	2	yes	—
168	C₂×C₈	C₂×C₂²⋊D₆₁	1	4	yes	—
169	C₂×C₈	C₂³×D₆₁	1	1	yes	characteristic
170	C₄⋊C₄	—	61	8	no	—
171	C₂²⋊C₄	—	61	8	no	—
172	C₄²	—	61	4	no	—
173	C₄²	—	61	4	no	—
174	C₁₆	—	2	4	no	—
175	C₁₆	—	122	4	no	—
176	C₂³	—	2	4	no	—
177	C₂³	#14519706	1	4	yes	—
178	C₂³	C₂×C₈×D₆₁	1	2	yes	—
179	C₂³	C₂×C₈×D₆₁	1	1	yes	characteristic
180	C₂×C₄	—	61	8	no	—
181	C₂×C₄	—	122	8	no	—
182	C₂×C₄	#14519706	1	8	yes	—
183	C₂×C₄	#14519706	1	4	yes	—
184	C₂×C₄	C₂×#11758196	1	4	yes	—
185	C₂×C₄	C₂×#11758196	1	2	yes	—
186	C₂×C₄	C₂×C₈×D₆₁	1	1	yes	characteristic
187	C₂×C₄	C₂×C₈×D₆₁	1	2	yes	—
188	C₂×C₄	C₂×#11758189	1	2	yes	—
189	C₂×C₄	C₂×#11758189	1	4	yes	—
190	C₂×C₄	C₂²×C₄×D₆₁	1	1	yes	characteristic
191	C₈	—	122	4	no	—
192	C₈	C₂×#11758196	1	4	yes	—
193	C₈	C₂×#11758196	1	2	yes	—
194	C₈	C₂²×C₂²⋊D₆₁	1	1	yes	characteristic
195	C₈	C₂²×Dic₁₂₂	1	1	yes	characteristic
196	C₂²	—	2	8	no	—
197	C₂²	—	2	4	no	—
198	C₂²	C₂×#14519706	1	4	yes	—
199	C₂²	C₂×#14519706	1	2	yes	—
200	C₂²	#16840180	1	4	yes	—
201	C₂²	C₂²×C₈×D₆₁	1	1	yes	characteristic
202	C₄	—	122	8	no	—
203	C₄	C₂×#14519706	1	2	yes	—
204	C₄	C₂×#14519706	1	4	yes	—
205	C₄	C₂²×#11758196	1	1	yes	characteristic
206	C₄	C₂²×#11758189	1	1	yes	characteristic
207	C₂	—	2	4	no	—
208	C₂	#21447579	1	4	yes	direct product factor
209	C₂	C₂²×#14519706	1	1	yes	characteristic
210	C₂	C₂×#16840180	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈	C₂²×D₆₁
commutator	C₂²×C₆₁	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₆₁
Fitting	C₂²×M₃₂×C₆₁	C₂
socle	C₂³×C₆₁	C₂²×C₈
prime radical	C₂⁴×C₆₁	C₂×C₈
cosocle	C₂×C₈×C₆₁	C₂⁴
hypercenter	C₂²×M₃₂	D₆₁
hypocenter	C₆₁	C₂×#2151
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₂²×M₃₂

Complements

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

N	H ≅ G/N	type
3. #21447579	C₂	direct (N × H)
4. C₂×#16839580	C₂	semidirect
15. #16839580	C₂²	semidirect
108. C₂×C₆₁	#2151	semidirect
125. C₆₁	C₂×#2151	semidirect
208. C₂	#21447579	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#11758185	2
C₂×#16839717	2
#21447579	2
C₂×#16839580	2
C₂²×M₃₂×C₆₁	2
C₂×#2151	61

Subgroups — C2×#21447579 #29832997

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21447579 #29832997