Subgroups — C₈×#16864653 #46845319

Order 34688.

456 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#16864653	C₂	1	1	yes	maximal, characteristic
2	C₈×Dic₅₄₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#31774021	C₂	1	1	yes	maximal, characteristic, complemented
4	D₄×C₂₇₁⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈×C₂²⋊D₂₇₁	C₂	1	2	yes	maximal, complemented
6	(C₂²⋊D₂₇₁)⋊C₈	C₂	1	2	yes	maximal, complemented
7	#31773949	C₂	1	1	yes	maximal, characteristic, complemented
8	#31773947	C₂	1	2	yes	maximal, complemented
9	C₂×C₈×Dic₂₇₁	C₂	1	2	yes	maximal, complemented
10	C₄×C₈×D₂₇₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×D₄×C₂₇₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#16864653	C₄	1	1	yes	characteristic
13	C₂²×C₂₇₁⋊C₈	C₂²	1	2	yes	—
14	C₂×C₈×D₂₇₁	C₂²	1	1	yes	characteristic
15	C₂×C₈×D₂₇₁	C₂²	1	1	yes	characteristic
16	C₄×Dic₅₄₂	C₂²	1	1	yes	characteristic
17	#22576125	C₄	1	1	yes	characteristic
18	D₄×Dic₂₇₁	C₄	1	1	yes	characteristic
19	C₄×C₂²⋊D₂₇₁	C₂²	1	2	yes	—
20	(C₂²⋊D₂₇₁)⋊C₄	C₄	1	2	yes	—
21	#22576118	C₄	1	1	yes	characteristic
22	#22576116	C₄	1	2	yes	—
23	C₂×C₄×Dic₂₇₁	C₂²	1	2	yes	—
24	C₄²×D₂₇₁	C₂²	1	1	yes	characteristic
25	C₄⋊(C₂₇₁⋊C₈)	C₂²	1	1	yes	characteristic
26	Dic₂₇₁⋊C₈	C₂²	1	2	yes	—
27	Dic₂₇₁⋊C₈	C₂²	1	2	yes	—
28	C₂²⋊(C₂₇₁⋊C₈)	C₂²	1	2	yes	—
29	(C₂×D₂₇₁)⋊C₈	C₂²	1	2	yes	—
30	C₄×C₂₇₁⋊C₈	C₂²	1	1	yes	characteristic
31	C₈×Dic₂₇₁	C₂²	1	2	yes	—
32	C₈×Dic₂₇₁	C₂²	1	2	yes	—
33	C₈×Dic₂₇₁	C₂²	1	1	yes	characteristic
34	C₈×Dic₂₇₁	C₂²	1	1	yes	characteristic
35	C₈×Dic₂₇₁	C₂²	1	2	yes	—
36	C₂²×C₈×C₂₇₁	C₂²	1	2	yes	—
37	C₄×D₄×C₂₇₁	C₂²	1	1	yes	characteristic
38	C₄⋊C₈×C₂₇₁	C₂²	1	1	yes	characteristic
39	C₂²⋊C₈×C₂₇₁	C₂²	1	2	yes	—
40	C₄×C₈×C₂₇₁	C₂²	1	1	yes	characteristic
41	#16864653	C₈	1	8	yes	direct product factor
42	C₂×Dic₅₄₂	C₂×C₄	1	1	yes	characteristic
43	C₂×C₂²⋊D₂₇₁	C₂×C₄	1	2	yes	—
44	C₂²×Dic₂₇₁	C₂×C₄	1	2	yes	—
45	C₂²×Dic₂₇₁	C₂×C₄	1	2	yes	—
46	C₂×C₄×D₂₇₁	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄×D₂₇₁	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄×D₂₇₁	C₂³	1	1	yes	characteristic
49	C₂×C₂₇₁⋊C₈	—	2	4	no	—
50	C₂×C₂₇₁⋊C₈	C₂³	1	1	yes	characteristic
51	C₂×C₂₇₁⋊C₈	C₂³	1	2	yes	—
52	C₂×C₂₇₁⋊C₈	C₂³	1	1	yes	characteristic
53	C₈×D₂₇₁	—	2	2	no	—
54	C₈×D₂₇₁	—	2	2	no	—
55	C₄⋊Dic₂₇₁	C₂×C₄	1	1	yes	characteristic
56	Dic₂₇₁⋊C₄	C₂×C₄	1	2	yes	—
57	Dic₂₇₁⋊C₄	C₂×C₄	1	2	yes	—
58	C₂²⋊Dic₂₇₁	C₂×C₄	1	2	yes	—
59	(C₂×D₂₇₁)⋊C₄	C₂×C₄	1	2	yes	—
60	C₄×Dic₂₇₁	C₂×C₄	1	1	yes	characteristic
61	C₄×Dic₂₇₁	C₂×C₄	1	1	yes	characteristic
62	C₄×Dic₂₇₁	C₂×C₄	1	2	yes	—
63	C₄×Dic₂₇₁	C₂×C₄	1	2	yes	—
64	C₄×Dic₂₇₁	C₂³	1	1	yes	characteristic
65	C₄×Dic₂₇₁	C₂³	1	2	yes	—
66	C₂×D₄×C₂₇₁	C₂×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₂₇₁	C₂³	1	2	yes	—
68	C₂×C₈×C₂₇₁	—	2	4	no	—
69	C₂×C₈×C₂₇₁	C₂³	1	1	yes	characteristic
70	C₂×C₈×C₂₇₁	C₂³	1	2	yes	—
71	C₂×C₈×C₂₇₁	C₂³	1	1	yes	characteristic
72	C₄⋊C₄×C₂₇₁	C₂×C₄	1	1	yes	characteristic
73	C₂²⋊C₄×C₂₇₁	C₂×C₄	1	2	yes	—
74	C₄²×C₂₇₁	C₂³	1	1	yes	characteristic
75	C₂²×D₂₇₁	C₂²×C₄	1	1	yes	characteristic
76	Dic₅₄₂	C₂×C₈	1	4	yes	complemented
77	C₂²⋊D₂₇₁	C₂×C₈	1	8	yes	complemented
78	C₂×Dic₂₇₁	—	2	4	no	—
79	C₂×Dic₂₇₁	C₂×C₈	1	8	yes	complemented
80	C₂×Dic₂₇₁	C₂²×C₄	1	1	yes	characteristic
81	C₂×Dic₂₇₁	C₂²×C₄	1	2	yes	—
82	C₂×Dic₂₇₁	C₂²×C₄	1	1	yes	characteristic
83	C₂×Dic₂₇₁	C₂²×C₄	1	2	yes	—
84	C₂×Dic₂₇₁	C₂²×C₄	1	1	yes	characteristic
85	C₄×D₂₇₁	—	2	1	no	—
86	C₄×D₂₇₁	—	2	1	no	—
87	C₄×D₂₇₁	—	2	2	no	—
88	C₄×D₂₇₁	C₂×C₈	1	4	yes	complemented
89	C₂₇₁⋊C₈	—	2	1	no	—
90	C₂₇₁⋊C₈	—	2	2	no	—
91	C₂₇₁⋊C₈	#41	1	2	yes	complemented
92	C₂³×C₂₇₁	C₂²×C₄	1	2	yes	—
93	D₄×C₂₇₁	C₂×C₈	1	4	yes	complemented
94	C₂×C₄×C₂₇₁	—	2	2	no	—
95	C₂×C₄×C₂₇₁	—	2	2	no	—
96	C₂×C₄×C₂₇₁	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₄×C₂₇₁	C₂²×C₄	1	2	yes	—
98	C₂×C₄×C₂₇₁	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₄×C₂₇₁	C₂⁴	1	1	yes	characteristic
100	C₈×C₂₇₁	—	2	2	no	—
101	C₈×C₂₇₁	—	2	1	no	—
102	C₈×C₂₇₁	#41	1	2	yes	complemented
103	C₂²×C₂₇₁	—	2	2	no	—
104	C₂²×C₂₇₁	—	2	2	no	—
105	C₂²×C₂₇₁	C₂²×C₈	1	4	yes	—
106	C₂²×C₂₇₁	C₂³×C₄	1	1	yes	characteristic
107	C₂×D₂₇₁	—	2	1	no	—
108	C₂×D₂₇₁	—	2	1	no	—
109	C₂×D₂₇₁	C₂²×C₈	1	2	yes	—
110	C₄×C₂₇₁	—	2	2	no	—
111	C₄×C₂₇₁	—	2	1	no	—
112	C₄×C₂₇₁	C₂²×C₈	1	2	yes	—
113	C₄×C₂₇₁	C₂×#41	1	1	yes	characteristic
114	C₄×C₂₇₁	C₂×#41	1	1	yes	characteristic
115	Dic₂₇₁	—	2	1	no	—
116	Dic₂₇₁	—	2	2	no	—
117	Dic₂₇₁	C₂²×C₈	1	2	yes	—
118	Dic₂₇₁	C₂²×C₈	1	4	yes	—
119	Dic₂₇₁	#131	1	2	yes	—
120	C₂×C₂₇₁	—	2	4	no	—
121	C₂×C₂₇₁	C₄×#41	1	1	yes	characteristic
122	C₂×C₂₇₁	C₂³×C₈	1	1	yes	characteristic
123	C₂×C₂₇₁	C₂×#131	1	1	yes	characteristic
124	D₂₇₁	—	2	2	no	—
125	C₂₇₁	C₈×#41	1	1	yes	characteristic, complemented, Sylow
126	C₈×#41	—	271	1	no	maximal, Sylow
127	C₄×#41	—	271	1	no	—
128	C₈×Q₈	—	271	1	no	—
129	C₈×Q₈	—	271	1	no	—
130	C₈×D₄	—	271	2	no	—
131	C₈×D₄	—	271	1	no	—
132	C₈×D₄	—	271	2	no	—
133	C₈×D₄	D₂₇₁	1	1	yes	characteristic, complemented
134	#431	—	271	1	no	—
135	#431	—	271	2	no	—
136	C₂×C₄×C₈	—	271	2	no	—
137	C₂×C₄×C₈	—	271	1	no	—
138	C₂×#41	—	271	1	no	—
139	C₂²×C₈	—	271	2	no	—
140	C₂²×C₈	—	271	1	no	—
141	C₂²×C₈	—	271	1	no	—
142	C₂²×C₈	C₂×D₂₇₁	1	2	yes	—
143	C₄×Q₈	—	271	1	no	—
144	C₄×Q₈	—	271	1	no	—
145	C₄×D₄	—	271	1	no	—
146	C₄×D₄	—	271	2	no	—
147	C₄×D₄	—	271	2	no	—
148	C₄×D₄	C₂×D₂₇₁	1	1	yes	characteristic
149	#117	—	271	1	no	—
150	#117	—	271	2	no	—
151	C₂×C₄²	—	271	2	no	—
152	C₂×C₄²	—	271	1	no	—
153	C₄⋊C₈	—	271	2	no	—
154	C₄⋊C₈	—	271	2	no	—
155	C₄⋊C₈	—	271	1	no	—
156	C₄⋊C₈	C₂×D₂₇₁	1	1	yes	characteristic
157	C₂²⋊C₈	—	271	2	no	—
158	C₂²⋊C₈	—	271	2	no	—
159	C₂²⋊C₈	C₂×D₂₇₁	1	2	yes	—
160	C₄×C₈	—	271	2	no	—
161	C₄×C₈	—	271	1	no	—
162	C₄×C₈	—	271	2	no	—
163	C₄×C₈	—	271	1	no	—
164	C₄×C₈	—	271	2	no	—
165	C₄×C₈	—	271	1	no	—
166	C₄×C₈	C₂×D₂₇₁	1	1	yes	characteristic
167	#41	—	271	8	no	—
168	C₂×Q₈	—	271	1	no	—
169	C₂×D₄	—	271	2	no	—
170	C₂×D₄	C₄×D₂₇₁	1	1	yes	characteristic
171	C₂²×C₄	—	271	2	no	—
172	C₂²×C₄	—	271	1	no	—
173	C₂²×C₄	—	271	2	no	—
174	C₂²×C₄	—	271	1	no	—
175	C₂²×C₄	—	271	1	no	—
176	C₂²×C₄	C₂²×D₂₇₁	1	2	yes	—
177	C₂×C₈	—	2	4	no	—
178	C₂×C₈	—	271	1	no	—
179	C₂×C₈	—	271	2	no	—
180	C₂×C₈	—	271	1	no	—
181	C₂×C₈	—	542	2	no	—
182	C₂×C₈	—	542	4	no	—
183	C₂×C₈	—	542	2	no	—
184	C₂×C₈	C₂²×D₂₇₁	1	2	yes	—
185	C₂×C₈	C₂²×D₂₇₁	1	1	yes	characteristic
186	C₂×C₈	C₂²×D₂₇₁	1	1	yes	characteristic
187	C₄⋊C₄	—	271	2	no	—
188	C₄⋊C₄	—	271	2	no	—
189	C₄⋊C₄	—	271	1	no	—
190	C₄⋊C₄	C₄×D₂₇₁	1	1	yes	characteristic
191	C₂²⋊C₄	—	271	2	no	—
192	C₂²⋊C₄	—	271	2	no	—
193	C₂²⋊C₄	C₄×D₂₇₁	1	2	yes	—
194	C₄²	—	271	2	no	—
195	C₄²	—	271	1	no	—
196	C₄²	—	271	2	no	—
197	C₄²	—	271	1	no	—
198	C₄²	—	271	1	no	—
199	C₄²	—	271	2	no	—
200	C₄²	C₂²×D₂₇₁	1	1	yes	characteristic
201	C₂³	—	271	1	no	—
202	C₂³	C₂×C₄×D₂₇₁	1	2	yes	—
203	Q₈	—	271	4	no	—
204	D₄	—	271	8	no	—
205	D₄	C₈×D₂₇₁	1	4	yes	complemented
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	271	8	no	—
209	C₂×C₄	—	271	1	no	—
210	C₂×C₄	—	271	1	no	—
211	C₂×C₄	—	271	2	no	—
212	C₂×C₄	—	271	4	no	—
213	C₂×C₄	—	271	1	no	—
214	C₂×C₄	—	271	2	no	—
215	C₂×C₄	—	542	2	no	—
216	C₂×C₄	—	542	4	no	—
217	C₂×C₄	—	542	1	no	—
218	C₂×C₄	—	542	1	no	—
219	C₂×C₄	C₂×C₄×D₂₇₁	1	2	yes	—
220	C₂×C₄	C₂×C₄×D₂₇₁	1	1	yes	characteristic
221	C₂×C₄	C₂×C₄×D₂₇₁	1	1	yes	characteristic
222	C₂×C₄	C₂³×D₂₇₁	1	1	yes	characteristic
223	C₈	—	2	1	no	—
224	C₈	—	2	2	no	—
225	C₈	—	271	2	no	—
226	C₈	—	542	2	no	—
227	C₈	—	542	1	no	—
228	C₈	#16864653	1	2	yes	direct product factor
229	C₂²	—	2	2	no	—
230	C₂²	—	2	2	no	—
231	C₂²	—	271	2	no	—
232	C₂²	—	542	1	no	—
233	C₂²	—	542	1	no	—
234	C₂²	C₂×C₈×D₂₇₁	1	4	yes	—
235	C₂²	C₂²×C₄×D₂₇₁	1	1	yes	characteristic
236	C₄	—	2	2	no	—
237	C₄	—	2	1	no	—
238	C₄	—	271	2	no	—
239	C₄	—	271	4	no	—
240	C₄	—	271	2	no	—
241	C₄	—	542	1	no	—
242	C₄	—	542	2	no	—
243	C₄	C₂×C₈×D₂₇₁	1	2	yes	—
244	C₄	#41×D₂₇₁	1	1	yes	characteristic
245	C₄	C₂×#16864653	1	1	yes	characteristic
246	C₂	—	2	4	no	—
247	C₂	—	542	2	no	—
248	C₂	C₄×#16864653	1	1	yes	characteristic
249	C₂	C₂²×C₈×D₂₇₁	1	1	yes	characteristic
250	C₂	#131×D₂₇₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²×D₂₇₁
commutator	C₂×C₂₇₁	C₂³×C₈
Frattini	C₂×C₄	C₂³×D₂₇₁
Fitting	C₈×D₄×C₂₇₁	C₂
socle	C₂²×C₂₇₁	C₂³×C₄
prime radical	C₂×#16864653	C₄
cosocle	C₂×C₄×C₂₇₁	C₂⁴
hypercenter	C₈×D₄	D₂₇₁
hypocenter	C₂₇₁	C₈×#41
LCS	G ▷ C₂×C₂₇₁ ▷ C₂₇₁
UCS	C₁ ◁ C₂×C₈ ◁ C₈×D₄

Complements

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

N	H ≅ G/N	type
2. C₈×Dic₅₄₂	C₂	semidirect
3. #31774021	C₂	semidirect
4. D₄×C₂₇₁⋊C₈	C₂	semidirect
5. C₈×C₂²⋊D₂₇₁	C₂	semidirect
6. (C₂²⋊D₂₇₁)⋊C₈	C₂	semidirect
7. #31773949	C₂	semidirect
8. #31773947	C₂	semidirect
9. C₂×C₈×Dic₂₇₁	C₂	semidirect
10. C₄×C₈×D₂₇₁	C₂	semidirect
11. C₈×D₄×C₂₇₁	C₂	semidirect
41. #16864653	C₈	direct (N × H)
76. Dic₅₄₂	C₂×C₈	semidirect
77. C₂²⋊D₂₇₁	C₂×C₈	semidirect
79. C₂×Dic₂₇₁	C₂×C₈	semidirect
88. C₄×D₂₇₁	C₂×C₈	semidirect
91. C₂₇₁⋊C₈	#41	semidirect
93. D₄×C₂₇₁	C₂×C₈	semidirect
102. C₈×C₂₇₁	#41	semidirect
125. C₂₇₁	C₈×#41	semidirect
133. C₈×D₄	D₂₇₁	semidirect
205. D₄	C₈×D₂₇₁	semidirect
228. C₈	#16864653	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×#16864653	2
C₈×Dic₅₄₂	2
#31774021	2
D₄×C₂₇₁⋊C₈	2
C₈×C₂²⋊D₂₇₁	2
(C₂²⋊D₂₇₁)⋊C₈	2
#31773949	2
#31773947	2
C₂×C₈×Dic₂₇₁	2
C₄×C₈×D₂₇₁	2
C₈×D₄×C₂₇₁	2
C₈×#41	271

Subgroups — C8×#16864653 #46845319

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×#16864653 #46845319