Subgroups — C₈×#14519865 #30609686

Order 15616.

922 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	#21462625	C₂	1	1	yes	maximal, characteristic
2	C₄×#14519865	C₂	1	1	yes	maximal, characteristic
3	#21461025	C₂	1	3	yes	maximal, complemented
4	#21461021	C₂	1	3	yes	maximal, complemented
5	C₈×#11758213	C₂	1	3	yes	maximal, complemented
6	C₈×D₆₁⋊C₈	C₂	1	3	yes	maximal, complemented
7	C₈×#11758226	C₂	1	1	yes	maximal, characteristic
8	C₂×#14519865	C₄	1	2	yes	—
9	Q₈×#92420	C₄	1	2	yes	—
10	#16840790	C₄	1	6	yes	—
11	#16840781	C₂²	1	3	yes	—
12	#16840727	C₄	1	6	yes	—
13	C₄×#11758213	C₂²	1	3	yes	—
14	C₄×D₆₁⋊C₈	C₂²	1	3	yes	—
15	C₄×D₆₁⋊C₈	C₂²	1	3	yes	—
16	#92420⋊C₈	C₂²	1	6	yes	—
17	C₈×#92420	C₂²	1	1	yes	characteristic
18	C₈×#92420	C₂²	1	3	yes	—
19	C₈×#92420	C₂²	1	3	yes	—
20	C₈×#92420	C₂²	1	3	yes	—
21	C₄×#11758226	C₂²	1	1	yes	characteristic
22	Q₈×#92418	C₄	1	1	yes	characteristic
23	C₈×D₂₄₄	C₄	1	3	yes	—
24	#16839827	C₄	1	3	yes	—
25	#16839814	C₂²	1	3	yes	—
26	C₄×C₈×D₆₁	C₂²	1	3	yes	—
27	C₈×Q₈×C₆₁	C₄	1	1	yes	characteristic
28	#14519865	C₈	1	16	yes	direct product factor
29	C₂×#11758213	C₂×C₄	1	6	yes	—
30	C₂×D₆₁⋊C₈	C₂×C₄	1	6	yes	—
31	C₂×D₆₁⋊C₈	C₂×C₄	1	6	yes	—
32	C₄⋊#92420	C₂×C₄	1	6	yes	—
33	#14519834	C₂×C₄	1	6	yes	—
34	#14519832	C₂×C₄	1	6	yes	—
35	C₄×#92420	C₂×C₄	1	6	yes	—
36	C₄×#92420	C₂×C₄	1	6	yes	—
37	C₄×#92420	C₂³	1	1	yes	characteristic
38	C₄×#92420	C₂³	1	1	yes	characteristic
39	C₄×#92420	C₂³	1	3	yes	—
40	C₄×#92420	C₂³	1	3	yes	—
41	C₂×#11758226	C₂×C₄	1	1	yes	characteristic
42	C₂×C₈×D₆₁	C₂×C₄	1	3	yes	—
43	C₂×C₈×D₆₁	C₂×C₄	1	3	yes	—
44	Q₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
45	C₄×D₂₄₄	C₂×C₄	1	3	yes	—
46	#14519739	C₂×C₄	1	3	yes	—
47	#14519735	C₂×C₄	1	3	yes	—
48	C₄²×D₆₁	C₂³	1	3	yes	—
49	C₄⋊#92418	C₂×C₄	1	3	yes	—
50	#14519689	C₂×C₄	1	6	yes	—
51	C₄×#92418	C₂×C₄	1	3	yes	—
52	C₈×Dic₆₁	C₂³	1	1	yes	characteristic
53	C₈×Dic₆₁	C₂³	1	3	yes	—
54	C₄×Q₈×C₆₁	C₂×C₄	1	1	yes	characteristic
55	C₄⋊C₈×C₆₁	C₂×C₄	1	3	yes	—
56	C₄×C₈×C₆₁	C₂×C₄	1	3	yes	—
57	C₂×Q₈×C₆₁	C₄²	1	1	yes	characteristic
58	#11758226	C₂×C₈	1	8	yes	—
59	C₂×D₂₄₄	C₄²	1	3	yes	—
60	C₂×C₄×D₆₁	C₂²×C₄	1	3	yes	—
61	C₂×C₄×D₆₁	C₂²×C₄	1	3	yes	—
62	C₂×C₄×D₆₁	C₂²×C₄	1	3	yes	—
63	C₂×#92420	C₂²×C₄	1	6	yes	—
64	C₂×#92420	C₂²×C₄	1	2	yes	—
65	C₂×#92420	C₂²×C₄	1	6	yes	—
66	C₂×#92420	C₂²×C₄	1	2	yes	—
67	#11758213	C₂×C₈	1	24	yes	complemented
68	D₆₁⋊C₈	—	2	12	no	—
69	D₆₁⋊C₈	C₂×C₈	1	24	yes	complemented
70	C₂×C₈×C₆₁	C₂²×C₄	1	3	yes	—
71	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
72	C₄⋊C₄×C₆₁	C₄²	1	3	yes	—
73	C₄²×C₆₁	C₂²×C₄	1	3	yes	—
74	#11758198	C₄²	1	6	yes	—
75	C₄⋊Dic₆₁	C₄²	1	3	yes	—
76	C₄×Dic₆₁	C₂²×C₄	1	3	yes	—
77	C₄×Dic₆₁	C₂²×C₄	1	3	yes	—
78	C₄×Dic₆₁	C₂⁴	1	1	yes	characteristic
79	C₂×#92418	C₂²×C₄	1	3	yes	—
80	C₂×#92418	C₂²×C₄	1	1	yes	characteristic
81	C₈×D₆₁	—	2	6	no	—
82	C₈×D₆₁	—	2	6	no	—
83	C₂²×D₆₁	C₂×C₄²	1	3	yes	—
84	Q₈×C₆₁	C₄×C₈	1	4	yes	—
85	C₂×C₄×C₆₁	C₂×C₄²	1	3	yes	—
86	C₂×C₄×C₆₁	C₂×C₄²	1	3	yes	—
87	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
88	C₂×Dic₆₁	C₂×C₄²	1	3	yes	—
89	C₂×Dic₆₁	C₂³×C₄	1	1	yes	characteristic
90	C₂×Dic₆₁	C₂³×C₄	1	1	yes	characteristic
91	D₂₄₄	C₄×C₈	1	12	yes	—
92	C₄×D₆₁	—	2	3	no	—
93	C₄×D₆₁	—	2	3	no	—
94	C₄×D₆₁	—	2	6	no	—
95	C₄×D₆₁	C₂²×C₈	1	12	yes	—
96	#92420	—	2	6	no	—
97	#92420	C₂²×C₈	1	4	yes	—
98	#92420	C₂²×C₈	1	12	yes	—
99	#92420	#131	1	4	yes	complemented
100	C₈×C₆₁	—	2	3	no	—
101	C₈×C₆₁	#131	1	2	yes	complemented
102	#92418	—	2	3	no	—
103	#92418	#131	1	2	yes	complemented
104	C₈×#131	—	61	1	no	maximal, Sylow
105	C₂²×C₆₁	C₂²×C₄²	1	1	yes	characteristic
106	C₂×D₆₁	—	2	3	no	—
107	C₂×D₆₁	—	2	3	no	—
108	C₂×D₆₁	C₂×C₄×C₈	1	6	yes	—
109	C₄×C₆₁	—	2	3	no	—
110	C₄×C₆₁	C₂×C₄×C₈	1	6	yes	—
111	C₄×C₆₁	C₂×#131	1	1	yes	characteristic
112	C₄×C₆₁	C₂×#131	1	1	yes	characteristic
113	Dic₆₁	—	2	3	no	—
114	Dic₆₁	C₂³×C₈	1	2	yes	—
115	Dic₆₁	C₂×#131	1	2	yes	—
116	C₈×#41	—	61	1	no	—
117	C₈×#41	—	61	1	no	—
118	C₄×#131	—	61	1	no	—
119	#1450	—	61	3	no	—
120	C₈×M₁₆	—	61	3	no	—
121	C₈×M₁₆	—	61	3	no	—
122	C₂×C₈²	—	61	3	no	—
123	C₂×C₆₁	C₂²×C₄×C₈	1	1	yes	characteristic
124	C₂×C₆₁	C₄×#131	1	1	yes	characteristic
125	C₂×C₆₁	C₄×#131	1	1	yes	characteristic
126	D₆₁	—	2	6	no	—
127	C₂×#131	—	61	2	no	—
128	C₄×#41	—	61	1	no	—
129	C₈×Q₈	—	61	2	no	—
130	C₈×Q₈	—	61	1	no	—
131	C₈×Q₈	#6923	1	1	yes	characteristic
132	C₈×D₄	—	61	6	no	—
133	C₈×D₄	—	61	3	no	—
134	C₈×D₄	—	61	3	no	—
135	#431	—	61	3	no	—
136	#431	—	61	3	no	—
137	#405	—	61	6	no	—
138	C₄×M₁₆	—	61	3	no	—
139	C₂×C₄×C₈	—	61	3	no	—
140	C₂×C₄×C₈	—	61	3	no	—
141	C₂×C₄×C₈	—	61	3	no	—
142	#322	—	61	6	no	—
143	C₈²	—	61	1	no	—
144	C₈²	—	61	3	no	—
145	C₈²	—	61	3	no	—
146	C₈²	—	61	3	no	—
147	C₆₁	C₈×#131	1	1	yes	characteristic, complemented, Sylow
148	C₂×#41	—	61	1	no	—
149	#131	—	61	16	no	—
150	C₂×M₁₆	—	61	6	no	—
151	C₂²×C₈	—	61	6	no	—
152	C₂²×C₈	—	61	6	no	—
153	C₂²×C₈	—	61	3	no	—
154	C₂²×C₈	—	61	3	no	—
155	C₄×Q₈	—	61	1	no	—
156	C₄×Q₈	C₂×#6923	1	1	yes	characteristic
157	C₄×D₄	—	61	3	no	—
158	C₄×D₄	—	61	3	no	—
159	#117	—	61	3	no	—
160	C₂×C₄²	—	61	3	no	—
161	C₄⋊C₈	—	61	6	no	—
162	C₄⋊C₈	—	61	3	no	—
163	C₄⋊C₈	C₂×#6923	1	3	yes	—
164	C₂²⋊C₈	—	61	6	no	—
165	C₂²⋊C₈	—	61	6	no	—
166	#97	—	61	6	no	—
167	C₄×C₈	—	61	6	no	—
168	C₄×C₈	—	61	6	no	—
169	C₄×C₈	—	61	1	no	—
170	C₄×C₈	—	61	1	no	—
171	C₄×C₈	—	61	3	no	—
172	C₄×C₈	—	61	3	no	—
173	C₄×C₈	—	61	3	no	—
174	C₄×C₈	—	61	1	no	—
175	C₄×C₈	—	61	3	no	—
176	C₄×C₈	C₂×#6923	1	3	yes	—
177	#41	—	61	8	no	—
178	C₂×Q₈	C₄×#6923	1	1	yes	characteristic
179	C₂×D₄	—	61	3	no	—
180	C₂²×C₄	—	61	3	no	—
181	C₂²×C₄	—	61	3	no	—
182	C₂²×C₄	—	61	3	no	—
183	M₁₆	—	61	24	no	—
184	C₂×C₈	—	61	2	no	—
185	C₂×C₈	—	61	2	no	—
186	C₂×C₈	—	61	6	no	—
187	C₂×C₈	—	61	24	no	—
188	C₂×C₈	—	61	1	no	—
189	C₂×C₈	—	61	6	no	—
190	C₂×C₈	—	61	3	no	—
191	C₂×C₈	—	122	6	no	—
192	C₂×C₈	—	122	12	no	—
193	C₂×C₈	—	122	6	no	—
194	C₂×C₈	C₂²×#6923	1	3	yes	—
195	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
196	C₄⋊C₄	—	61	3	no	—
197	C₄⋊C₄	C₄×#6923	1	3	yes	—
198	C₂²⋊C₄	—	61	6	no	—
199	C₄²	—	61	3	no	—
200	C₄²	—	61	3	no	—
201	C₄²	—	61	1	no	—
202	C₄²	C₂²×#6923	1	3	yes	—
203	C₂³	—	61	3	no	—
204	Q₈	C₈×#6923	1	4	yes	—
205	D₄	—	61	12	no	—
206	C₂×C₄	—	61	1	no	—
207	C₂×C₄	—	61	1	no	—
208	C₂×C₄	—	61	12	no	—
209	C₂×C₄	—	61	3	no	—
210	C₂×C₄	—	122	6	no	—
211	C₂×C₄	—	122	3	no	—
212	C₂×C₄	—	122	3	no	—
213	C₂×C₄	C₂×C₄×#6923	1	3	yes	—
214	C₂×C₄	C₂×C₄×#6923	1	3	yes	—
215	C₂×C₄	C₂³×#6923	1	1	yes	characteristic
216	C₈	—	2	3	no	—
217	C₈	—	61	4	no	—
218	C₈	—	61	12	no	—
219	C₈	—	61	4	no	—
220	C₈	—	61	2	no	—
221	C₈	—	122	6	no	—
222	C₈	—	122	3	no	—
223	C₈	#14519865	1	2	yes	direct product factor
224	C₂²	—	61	6	no	—
225	C₂²	—	122	3	no	—
226	C₂²	—	122	3	no	—
227	C₂²	C₂²×C₄×#6923	1	1	yes	characteristic
228	C₄	—	2	3	no	—
229	C₄	—	61	2	no	—
230	C₄	—	61	2	no	—
231	C₄	—	122	3	no	—
232	C₄	C₂×C₈×#6923	1	6	yes	—
233	C₄	D₆₁⋊#131	1	1	yes	characteristic
234	C₄	C₂×#14519865	1	1	yes	characteristic
235	C₂	—	122	6	no	—
236	C₂	C₂²×C₈×#6923	1	1	yes	characteristic
237	C₂	#131×#6923	1	1	yes	characteristic
238	C₂	C₄×#14519865	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²×#6923
commutator	C₂×C₆₁	C₂²×C₄×C₈
Frattini	C₂×C₄	C₂³×#6923
Fitting	C₈×Q₈×C₆₁	C₄
socle	C₂²×C₆₁	C₂²×C₄²
prime radical	C₂×#11758226	C₂×C₄
cosocle	C₄×Dic₆₁	C₂⁴
hypercenter	C₈×Q₈	#6923
hypocenter	C₆₁	C₈×#131
LCS	G ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₈ ◁ C₈×Q₈

Complements

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

N	H ≅ G/N	type
3. #21461025	C₂	semidirect
4. #21461021	C₂	semidirect
5. C₈×#11758213	C₂	semidirect
6. C₈×D₆₁⋊C₈	C₂	semidirect
28. #14519865	C₈	direct (N × H)
67. #11758213	C₂×C₈	semidirect
69. D₆₁⋊C₈	C₂×C₈	semidirect
99. #92420	#131	semidirect
101. C₈×C₆₁	#131	semidirect
103. #92418	#131	semidirect
147. C₆₁	C₈×#131	semidirect
223. C₈	#14519865	direct (N × H)

Maximal subgroups

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

subgroup	index
#21462625	2
C₄×#14519865	2
#21461025	2
#21461021	2
C₈×#11758213	2
C₈×D₆₁⋊C₈	2
C₈×#11758226	2
C₈×#131	61

Subgroups — C8×#14519865 #30609686

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×#14519865 #30609686