Subgroups — C₂×#22586302 #31907184

Order 18176.

470 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₅₆₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#17985752	C₂	1	1	yes	maximal, characteristic, complemented
3	#22586302	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₄×C₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₅₆₈	C₄	1	1	yes	characteristic
6	C₂×C₄×D₂₈₄	—	2	1	no	—
7	C₄×D₅₆₈	—	2	1	no	—
8	C₄×D₅₆₈	—	2	1	no	—
9	C₄×D₅₆₈	C₄	1	2	yes	—
10	C₄×D₅₆₈	C₂²	1	2	yes	—
11	C₂×#14556062	C₄	1	1	yes	characteristic
12	C₂×C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
13	C₂×#14556050	—	2	1	no	—
14	#17985752	—	2	1	no	—
15	#17985752	C₂²	1	2	yes	complemented
16	C₂²×C₁₆×C₇₁	—	2	1	no	—
17	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
18	C₄×C₁₆×C₇₁	—	2	1	no	—
19	C₄×C₁₆×C₇₁	C₂²	1	2	yes	complemented
20	C₂²×D₂₈₄	—	2	1	no	—
21	C₂²×C₄×D₇₁	—	4	1	no	—
22	C₂×D₅₆₈	—	2	1	no	—
23	C₂×D₅₆₈	—	2	2	no	—
24	C₂×D₅₆₈	C₈	1	4	yes	complemented
25	C₂×D₅₆₈	C₂×C₄	1	2	yes	—
26	C₂²×C₇₁⋊C₈	—	4	1	no	—
27	C₄×D₂₈₄	—	2	2	no	—
28	C₄×D₂₈₄	—	2	2	no	—
29	C₄×D₂₈₄	—	4	1	no	—
30	C₄×D₂₈₄	—	4	1	no	—
31	C₂×C₄⋊Dic₇₁	—	2	1	no	—
32	C₂×(C₂×D₇₁)⋊C₄	—	4	1	no	—
33	#14556062	—	2	1	no	—
34	#14556062	C₂×C₄	1	2	yes	—
35	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
36	C₄⋊(C₇₁⋊C₈)	—	4	1	no	—
37	#14556050	—	2	2	no	—
38	#14556050	—	2	2	no	—
39	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
42	C₂×C₁₆×C₇₁	—	2	4	no	—
43	C₂×C₁₆×C₇₁	—	2	2	no	—
44	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
46	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
47	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
48	C₂³×D₇₁	—	4	1	no	—
49	C₂×D₂₈₄	—	2	4	no	—
50	C₂×D₂₈₄	—	2	2	no	—
51	C₂×D₂₈₄	—	4	1	no	—
52	C₂×D₂₈₄	—	4	2	no	—
53	C₂²×Dic₇₁	—	4	1	no	—
54	C₂×C₄×D₇₁	—	4	2	no	—
55	C₂×C₄×D₇₁	—	4	2	no	—
56	C₂×C₄×D₇₁	—	8	2	no	—
57	C₂×C₄×D₇₁	—	8	2	no	—
58	D₅₆₈	—	2	4	no	—
59	D₅₆₈	—	2	2	no	—
60	D₅₆₈	C₂×C₈	1	4	yes	complemented
61	C₂×C₇₁⋊C₈	—	4	2	no	—
62	C₂×C₇₁⋊C₈	—	8	2	no	—
63	C₄⋊Dic₇₁	—	2	2	no	—
64	C₄⋊Dic₇₁	—	4	1	no	—
65	(C₂×D₇₁)⋊C₄	—	4	2	no	—
66	(C₂×D₇₁)⋊C₄	—	4	2	no	—
67	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₇₁	D₈	1	1	yes	characteristic
69	C₂²×C₄×C₇₁	SD₁₆	1	1	yes	characteristic
70	C₂×C₈×C₇₁	—	2	2	no	—
71	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₈×C₇₁	C₂×C₈	1	2	yes	—
74	C₂×C₈×C₇₁	M₁₆	1	2	yes	—
75	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
77	C₄²×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
78	C₄²×C₇₁	D₈	1	1	yes	characteristic
79	C₄²×C₇₁	SD₁₆	1	1	yes	characteristic
80	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
81	C₁₆×C₇₁	—	2	4	no	—
82	C₂³×C₇₁	#102	1	1	yes	characteristic
83	C₂²×D₇₁	—	4	4	no	—
84	C₂²×D₇₁	—	4	2	no	—
85	C₂²×D₇₁	—	8	2	no	—
86	C₂²×D₇₁	—	8	2	no	—
87	C₂×C₄×C₇₁	—	2	2	no	—
88	C₂×C₄×C₇₁	C₂²⋊C₈	1	2	yes	—
89	C₂×C₄×C₇₁	#102	1	1	yes	characteristic
90	C₂×C₄×C₇₁	#102	1	1	yes	characteristic
91	C₂×C₄×C₇₁	#102	1	1	yes	characteristic
92	C₂×C₄×C₇₁	#104	1	2	yes	—
93	C₂×C₄×C₇₁	D₁₆	1	2	yes	—
94	C₂×C₄×C₇₁	SD₃₂	1	2	yes	—
95	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
97	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
98	C₂×Dic₇₁	—	4	2	no	—
99	C₂×Dic₇₁	—	8	2	no	—
100	D₂₈₄	—	2	4	no	—
101	D₂₈₄	—	4	4	no	—
102	D₂₈₄	—	4	2	no	—
103	C₄×D₇₁	—	8	4	no	—
104	C₄×D₇₁	—	8	2	no	—
105	C₄×D₇₁	—	8	2	no	—
106	C₈×C₇₁	—	2	1	no	—
107	C₈×C₇₁	—	2	1	no	—
108	C₈×C₇₁	C₂²⋊C₈	1	2	yes	—
109	C₈×C₇₁	C₂²×C₈	1	1	yes	characteristic
110	C₈×C₇₁	C₂×M₁₆	1	1	yes	characteristic
111	C₇₁⋊C₈	—	8	2	no	—
112	C₂²×C₇₁	D₄⋊C₈	1	2	yes	—
113	C₂²×C₇₁	#357	1	2	yes	—
114	C₂²×C₇₁	#359	1	2	yes	—
115	C₂²×C₇₁	C₂×#102	1	1	yes	characteristic
116	C₂×D₇₁	—	4	4	no	—
117	C₂×D₇₁	—	8	8	no	—
118	C₂×D₇₁	—	8	2	no	—
119	C₂×D₇₁	—	8	2	no	—
120	C₄×C₇₁	—	2	1	no	—
121	C₄×C₇₁	—	2	1	no	—
122	C₄×C₇₁	D₄⋊C₈	1	2	yes	—
123	C₄×C₇₁	#357	1	2	yes	—
124	C₄×C₇₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
125	C₄×C₇₁	C₂×#104	1	1	yes	characteristic
126	C₄×C₇₁	C₂×D₁₆	1	1	yes	characteristic
127	C₄×C₇₁	C₂×SD₃₂	1	1	yes	characteristic
128	Dic₇₁	—	8	2	no	—
129	C₂×#1331	—	71	1	no	maximal, Sylow
130	C₂×C₇₁	#1331	1	4	yes	complemented
131	C₂×C₇₁	C₂×D₄⋊C₈	1	1	yes	characteristic
132	C₂×C₇₁	C₂×#357	1	1	yes	characteristic
133	C₂×C₇₁	C₂×#359	1	1	yes	characteristic
134	D₇₁	—	8	4	no	—
135	C₂×C₄×D₈	—	71	1	no	—
136	C₂×C₄×C₁₆	D₇₁	1	1	yes	characteristic, complemented
137	C₂×#335	—	71	1	no	—
138	#1331	—	71	4	no	—
139	C₇₁	C₂×#1331	1	1	yes	characteristic, complemented, Sylow
140	C₂²×D₈	—	71	1	no	—
141	C₂×C₄×D₄	—	142	1	no	—
142	C₂²×C₁₆	—	2	1	no	—
143	C₄×D₈	—	71	2	no	—
144	C₄×D₈	—	71	2	no	—
145	C₄×D₈	—	142	1	no	—
146	C₄×D₈	—	142	1	no	—
147	C₂×#107	—	71	1	no	—
148	C₂×C₄⋊C₈	—	142	1	no	—
149	C₂×#102	—	142	1	no	—
150	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
151	C₄×C₁₆	—	2	1	no	—
152	C₄×C₁₆	C₂×D₇₁	1	2	yes	complemented
153	#335	—	71	2	no	—
154	#335	—	142	1	no	—
155	C₂²×D₄	—	142	1	no	—
156	C₂³×C₄	—	284	1	no	—
157	C₂×D₈	—	71	4	no	—
158	C₂×D₈	—	71	2	no	—
159	C₂×D₈	—	142	1	no	—
160	C₂×D₈	—	142	2	no	—
161	C₂²×C₈	—	284	1	no	—
162	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
163	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
164	C₄×D₄	—	284	1	no	—
165	C₄×D₄	—	284	1	no	—
166	C₄×D₄	—	142	2	no	—
167	C₄×D₄	—	142	2	no	—
168	C₂×C₄⋊C₄	—	142	1	no	—
169	C₂×C₂²⋊C₄	—	284	1	no	—
170	C₂×C₄²	D₂₈₄	1	1	yes	characteristic
171	C₂×C₁₆	—	2	4	no	—
172	C₂×C₁₆	—	2	2	no	—
173	#107	—	71	2	no	—
174	#107	—	142	1	no	—
175	C₄⋊C₈	—	284	1	no	—
176	C₄⋊C₈	—	142	2	no	—
177	#102	—	142	2	no	—
178	#102	—	142	2	no	—
179	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
180	C₄×C₈	D₂₈₄	1	1	yes	characteristic
181	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
182	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
183	C₂⁴	—	284	1	no	—
184	C₂×D₄	—	284	1	no	—
185	C₂×D₄	—	284	2	no	—
186	C₂×D₄	—	142	4	no	—
187	C₂×D₄	—	142	2	no	—
188	C₂²×C₄	—	284	1	no	—
189	C₂²×C₄	—	284	2	no	—
190	C₂²×C₄	—	284	2	no	—
191	C₂²×C₄	—	568	2	no	—
192	C₂²×C₄	—	568	2	no	—
193	C₂²×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
194	C₂²×C₄	D₅₆₈	1	1	yes	characteristic
195	C₂²×C₄	#11765854	1	1	yes	characteristic
196	D₈	—	71	4	no	—
197	D₈	—	142	4	no	—
198	D₈	—	142	2	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	284	2	no	—
201	C₂×C₈	—	568	2	no	—
202	C₂×C₈	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
203	C₂×C₈	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
204	C₂×C₈	C₈×D₇₁	1	2	yes	—
205	C₂×C₈	#11765848	1	2	yes	—
206	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
207	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
208	C₄⋊C₄	—	284	1	no	—
209	C₄⋊C₄	—	142	2	no	—
210	C₂²⋊C₄	—	284	2	no	—
211	C₂²⋊C₄	—	284	2	no	—
212	C₄²	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
213	C₄²	D₅₆₈	1	1	yes	characteristic
214	C₄²	#11765854	1	1	yes	characteristic
215	C₄²	C₂×D₂₈₄	1	1	yes	characteristic
216	C₁₆	—	2	4	no	—
217	C₂³	—	284	4	no	—
218	C₂³	—	284	2	no	—
219	C₂³	—	568	2	no	—
220	C₂³	—	568	2	no	—
221	C₂³	#14556050	1	1	yes	characteristic
222	D₄	—	284	4	no	—
223	D₄	—	284	2	no	—
224	D₄	—	142	4	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	284	2	no	—
227	C₂×C₄	—	568	2	no	—
228	C₂×C₄	—	568	4	no	—
229	C₂×C₄	—	568	2	no	—
230	C₂×C₄	—	568	2	no	—
231	C₂×C₄	(C₂×D₇₁)⋊C₈	1	2	yes	—
232	C₂×C₄	#14556050	1	1	yes	characteristic
233	C₂×C₄	#14556050	1	1	yes	characteristic
234	C₂×C₄	#14556050	1	1	yes	characteristic
235	C₂×C₄	#14556056	1	2	yes	—
236	C₂×C₄	D₁₁₃₆	1	2	yes	—
237	C₂×C₄	#14556073	1	2	yes	—
238	C₂×C₄	C₂×(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
239	C₂×C₄	C₂×D₅₆₈	1	1	yes	characteristic
240	C₂×C₄	C₂×#11765854	1	1	yes	characteristic
241	C₈	—	2	1	no	—
242	C₈	—	2	1	no	—
243	C₈	—	568	2	no	—
244	C₈	(C₂×D₇₁)⋊C₈	1	2	yes	—
245	C₈	C₂×C₈×D₇₁	1	1	yes	characteristic
246	C₈	C₂×#11765848	1	1	yes	characteristic
247	C₂²	—	284	4	no	—
248	C₂²	—	568	2	no	—
249	C₂²	—	568	2	no	—
250	C₂²	—	568	8	no	—
251	C₂²	#17985728	1	2	yes	—
252	C₂²	#17985801	1	2	yes	—
253	C₂²	#17985807	1	2	yes	—
254	C₂²	C₂×#14556050	1	1	yes	characteristic
255	C₄	—	2	1	no	—
256	C₄	—	2	1	no	—
257	C₄	—	568	2	no	—
258	C₄	#17985728	1	2	yes	—
259	C₄	#17985801	1	2	yes	—
260	C₄	C₂×(C₂×D₇₁)⋊C₈	1	1	yes	characteristic
261	C₄	C₂×#14556056	1	1	yes	characteristic
262	C₄	C₂×D₁₁₃₆	1	1	yes	characteristic
263	C₄	C₂×#14556073	1	1	yes	characteristic
264	C₂	—	568	4	no	—
265	C₂	#22586302	1	4	yes	direct product factor
266	C₂	C₂×#17985728	1	1	yes	characteristic
267	C₂	C₂×#17985801	1	1	yes	characteristic
268	C₂	C₂×#17985807	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₅₆₈
commutator	C₈×C₇₁	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₇₁
Fitting	C₂×C₄×C₁₆×C₇₁	C₂
socle	C₂³×C₇₁	#102
prime radical	C₂²×D₅₆₈	C₄
cosocle	C₄×C₈×C₇₁	C₂³
hypercenter	C₂×C₄×C₁₆	D₇₁
hypocenter	C₇₁	C₂×#1331
LCS	G ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×#17985752	C₂	semidirect
3. #22586302	C₂	direct (N × H)
4. C₂×C₄×C₁₆×C₇₁	C₂	semidirect
15. #17985752	C₂²	semidirect
19. C₄×C₁₆×C₇₁	C₂²	semidirect
24. C₂×D₅₆₈	C₈	semidirect
60. D₅₆₈	C₂×C₈	semidirect
130. C₂×C₇₁	#1331	semidirect
136. C₂×C₄×C₁₆	D₇₁	semidirect
139. C₇₁	C₂×#1331	semidirect
152. C₄×C₁₆	C₂×D₇₁	semidirect
265. C₂	#22586302	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×D₅₆₈	2
C₂×#17985752	2
#22586302	2
C₂×C₄×C₁₆×C₇₁	2
C₂×#1331	71

Subgroups — C2×#22586302 #31907184

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22586302 #31907184