Subgroups — C₈×#14519712 #29571306

Order 15616.

374 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14519712	C₂	1	1	yes	maximal, characteristic
2	#21444565	C₂	1	1	yes	maximal, characteristic
3	C₈×#11758194	C₂	1	1	yes	maximal, characteristic
4	C₂×C₈×#92418	C₂	1	1	yes	maximal, characteristic
5	#21443657	C₂	1	1	yes	maximal, characteristic
6	#21443655	C₂	1	1	yes	maximal, characteristic
7	C₈×M₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic
8	#16839815	C₂²	1	1	yes	characteristic
9	C₂×#14519712	C₄	1	2	yes	—
10	C₄×#11758194	C₂²	1	1	yes	characteristic
11	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
12	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
13	#16839526	C₄	1	2	yes	—
14	#16839514	C₄	1	2	yes	—
15	#16839512	C₄	1	2	yes	—
16	#16839484	—	2	1	no	—
17	C₈×#92418	—	2	1	no	—
18	C₈×#92418	C₄	1	2	yes	—
19	C₈×#92418	C₄	1	2	yes	—
20	C₆₁×#431	C₂²	1	1	yes	characteristic
21	C₄×M₁₆×C₆₁	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
23	C₆₁×#322	—	2	1	no	—
24	C₈²×C₆₁	—	2	1	no	—
25	C₂×#11758194	C₂×C₄	1	2	yes	—
26	C₂²×#92418	—	2	1	no	—
27	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
28	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
29	#14519712	C₈	1	8	yes	direct product factor
30	C₄⋊#92418	C₂×C₄	1	2	yes	—
31	C₂²⋊#92418	—	2	1	no	—
32	#14519688	—	2	1	no	—
33	C₄×#92418	—	2	1	no	—
34	C₄×#92418	—	2	1	no	—
35	C₄×#92418	—	2	1	no	—
36	C₄×#92418	C₂×C₄	1	2	yes	—
37	C₄×#92418	C₂×C₄	1	1	yes	characteristic
38	C₄×#92418	C₂×C₄	1	1	yes	characteristic
39	C₄×#92418	C₂×C₄	1	1	yes	characteristic
40	C₄×#92418	C₂×C₄	1	1	yes	characteristic
41	C₂×M₁₆×C₆₁	C₂×C₄	1	2	yes	—
42	C₂²×C₈×C₆₁	Q₈	1	2	yes	—
43	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
44	C₄⋊C₈×C₆₁	C₂×C₄	1	2	yes	—
45	C₂²⋊C₈×C₆₁	—	2	1	no	—
46	#97×C₆₁	—	2	1	no	—
47	C₄×C₈×C₆₁	—	2	1	no	—
48	C₄×C₈×C₆₁	—	2	1	no	—
49	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
50	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
51	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
52	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
54	C₂²×C₄×C₆₁	#41	1	1	yes	characteristic
55	M₁₆×C₆₁	—	2	2	no	—
56	M₁₆×C₆₁	C₂×C₈	1	4	yes	—
57	C₂×C₈×C₆₁	—	2	2	no	—
58	C₂×C₈×C₆₁	—	2	4	no	—
59	C₂×C₈×C₆₁	—	2	2	no	—
60	C₂×C₈×C₆₁	C₄⋊C₄	1	2	yes	—
61	C₂×C₈×C₆₁	C₄⋊C₄	1	2	yes	—
62	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
63	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
64	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
65	C₄²×C₆₁	#41	1	1	yes	characteristic
66	#11758194	—	2	2	no	—
67	#11758194	C₂×C₈	1	4	yes	—
68	C₂×#92418	—	2	4	no	—
69	C₂×#92418	—	2	2	no	—
70	C₂×#92418	—	2	2	no	—
71	C₂×#92418	—	2	2	no	—
72	C₂×#92418	—	2	2	no	—
73	C₂×#92418	C₄²	1	1	yes	characteristic
74	C₂×#92418	C₄²	1	1	yes	characteristic
75	C₂×#92418	C₄²	1	1	yes	characteristic
76	C₂×#92418	C₄²	1	1	yes	characteristic
77	C₂×#92418	C₂×C₈	1	4	yes	—
78	C₂³×C₆₁	C₄×Q₈	1	1	yes	characteristic
79	C₂×C₄×C₆₁	—	2	2	no	—
80	C₂×C₄×C₆₁	—	2	2	no	—
81	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
82	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₆₁	#117	1	1	yes	characteristic
84	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
85	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
86	C₂×C₄×C₆₁	C₄×Q₈	1	1	yes	characteristic
87	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
88	C₂×C₄×C₆₁	#131	1	2	yes	—
89	C₈×C₆₁	—	2	2	no	—
90	C₈×C₆₁	—	2	4	no	—
91	C₈×C₆₁	—	4	2	no	—
92	C₈×C₆₁	#108	1	4	yes	complemented
93	#92418	—	2	4	no	—
94	#92418	—	2	1	no	—
95	#92418	—	2	1	no	—
96	#92418	—	2	4	no	—
97	#92418	—	4	2	no	—
98	#92418	C₄×C₈	1	2	yes	—
99	#92418	C₄×C₈	1	2	yes	—
100	C₈×#108	—	61	1	no	maximal, Sylow
101	C₂²×C₆₁	—	2	1	no	—
102	C₂²×C₆₁	—	2	1	no	—
103	C₂²×C₆₁	C₄×C₄⋊C₄	1	1	yes	characteristic
104	C₂²×C₆₁	C₈×Q₈	1	2	yes	—
105	C₄×C₆₁	—	2	1	no	—
106	C₄×C₆₁	—	2	1	no	—
107	C₄×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
108	C₄×C₆₁	#405	1	1	yes	characteristic
109	C₄×C₆₁	C₂×#108	1	2	yes	—
110	C₄×C₆₁	C₈×D₄	1	2	yes	—
111	C₄×C₆₁	#443	1	2	yes	—
112	C₄×#108	—	61	1	no	—
113	#1564	—	61	1	no	—
114	C₈×M₁₆	—	61	1	no	—
115	C₈×M₁₆	D₆₁	1	1	yes	characteristic
116	C₂×C₈²	—	61	1	no	—
117	#1278	—	61	1	no	—
118	#1278	—	61	1	no	—
119	C₂×C₆₁	—	2	2	no	—
120	C₂×C₆₁	C₈×C₄⋊C₄	1	1	yes	characteristic
121	C₂×C₆₁	#1772	1	1	yes	characteristic
122	C₂×C₆₁	C₄×#108	1	1	yes	characteristic
123	#431	—	61	1	no	—
124	#431	C₂×D₆₁	1	1	yes	characteristic
125	C₂×#108	—	61	2	no	—
126	C₄×M₁₆	—	61	1	no	—
127	C₄×M₁₆	C₂×D₆₁	1	1	yes	characteristic
128	C₂×C₄×C₈	—	61	1	no	—
129	C₂×C₄×C₈	—	61	1	no	—
130	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
131	#341	—	61	2	no	—
132	#335	—	61	2	no	—
133	#334	—	61	2	no	—
134	#322	—	2	1	no	—
135	#322	—	122	1	no	—
136	C₈²	—	2	1	no	—
137	C₈²	—	61	2	no	—
138	C₈²	—	61	2	no	—
139	C₈²	—	122	1	no	—
140	C₆₁	C₈×#108	1	1	yes	characteristic, complemented, Sylow
141	C₂×M₁₆	—	61	2	no	—
142	C₂×M₁₆	C₄×D₆₁	1	2	yes	—
143	C₂²×C₈	—	61	1	no	—
144	C₂²×C₈	—	61	1	no	—
145	C₂²×C₈	—	122	1	no	—
146	C₂²×C₈	Dic₁₂₂	1	2	yes	—
147	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
148	#108	—	61	8	no	—
149	C₄⋊C₈	—	61	2	no	—
150	C₄⋊C₈	C₄×D₆₁	1	2	yes	—
151	C₂²⋊C₈	—	2	1	no	—
152	C₂²⋊C₈	—	122	1	no	—
153	#97	—	2	1	no	—
154	#97	—	122	1	no	—
155	C₄×C₈	—	2	1	no	—
156	C₄×C₈	—	2	1	no	—
157	C₄×C₈	—	61	1	no	—
158	C₄×C₈	—	61	1	no	—
159	C₄×C₈	—	61	1	no	—
160	C₄×C₈	—	61	1	no	—
161	C₄×C₈	—	61	2	no	—
162	C₄×C₈	—	122	1	no	—
163	C₄×C₈	—	122	1	no	—
164	C₄×C₈	—	122	1	no	—
165	C₄×C₈	C₄×D₆₁	1	2	yes	—
166	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
167	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
168	C₂²×C₄	C₂×Dic₁₂₂	1	1	yes	characteristic
169	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
170	C₂²×C₄	#11758222	1	1	yes	characteristic
171	M₁₆	—	2	2	no	—
172	M₁₆	—	61	4	no	—
173	M₁₆	—	122	2	no	—
174	M₁₆	C₈×D₆₁	1	4	yes	—
175	C₂×C₈	—	2	2	no	—
176	C₂×C₈	—	2	4	no	—
177	C₂×C₈	—	2	2	no	—
178	C₂×C₈	—	61	1	no	—
179	C₂×C₈	—	61	1	no	—
180	C₂×C₈	—	61	4	no	—
181	C₂×C₈	—	61	1	no	—
182	C₂×C₈	—	61	1	no	—
183	C₂×C₈	—	122	4	no	—
184	C₂×C₈	—	122	2	no	—
185	C₂×C₈	—	122	2	no	—
186	C₂×C₈	—	122	2	no	—
187	C₂×C₈	—	122	2	no	—
188	C₂×C₈	#11758196	1	2	yes	—
189	C₂×C₈	#11758196	1	2	yes	—
190	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
191	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
192	C₄²	#11758222	1	1	yes	characteristic
193	C₄²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
194	C₂³	C₄×Dic₁₂₂	1	1	yes	characteristic
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	C₄²×D₆₁	1	1	yes	characteristic
198	C₂×C₄	C₂×#11758196	1	1	yes	characteristic
199	C₂×C₄	#14519734	1	1	yes	characteristic
200	C₂×C₄	C₄×C₂²⋊D₆₁	1	1	yes	characteristic
201	C₂×C₄	C₄×C₂²⋊D₆₁	1	1	yes	characteristic
202	C₂×C₄	C₄×Dic₁₂₂	1	1	yes	characteristic
203	C₂×C₄	C₂×C₈×D₆₁	1	2	yes	—
204	C₂×C₄	#14519783	1	2	yes	—
205	C₈	—	2	2	no	—
206	C₈	—	2	4	no	—
207	C₈	—	4	2	no	—
208	C₈	—	61	2	no	—
209	C₈	—	61	2	no	—
210	C₈	—	122	4	no	—
211	C₈	—	122	1	no	—
212	C₈	—	122	1	no	—
213	C₈	—	122	4	no	—
214	C₈	—	244	2	no	—
215	C₈	#14519712	1	4	yes	direct product factor
216	C₂²	—	2	1	no	—
217	C₂²	—	2	1	no	—
218	C₂²	C₄×#11758196	1	1	yes	characteristic
219	C₂²	C₈×Dic₁₂₂	1	2	yes	—
220	C₄	—	2	1	no	—
221	C₄	—	2	1	no	—
222	C₄	C₄×C₈×D₆₁	1	1	yes	characteristic
223	C₄	#16839716	1	1	yes	characteristic
224	C₄	C₂×#14519712	1	2	yes	—
225	C₄	C₈×C₂²⋊D₆₁	1	2	yes	—
226	C₄	#16839877	1	2	yes	—
227	C₂	—	2	2	no	—
228	C₂	C₈×#11758196	1	1	yes	characteristic
229	C₂	#21445699	1	1	yes	characteristic
230	C₂	C₄×#14519712	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₈	C₂²⋊D₆₁
commutator	C₄×C₆₁	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₈×M₁₆×C₆₁	C₂
socle	C₂²×C₆₁	C₄×C₄⋊C₄
prime radical	C₂³×C₆₁	C₄×Q₈
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	C₈×M₁₆	D₆₁
hypocenter	C₆₁	C₈×#108
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₄×C₈ ◁ C₂×C₄×C₈ ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
29. #14519712	C₈	direct (N × H)
92. C₈×C₆₁	#108	semidirect
140. C₆₁	C₈×#108	semidirect
215. C₈	#14519712	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×#14519712	2
#21444565	2
C₈×#11758194	2
C₂×C₈×#92418	2
#21443657	2
#21443655	2
C₈×M₁₆×C₆₁	2
C₈×#108	61

Subgroups — C8×#14519712 #29571306

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×#14519712 #29571306