Subgroups — #31913958

Order 18176.

486 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14556062	C₂	1	1	yes	maximal, characteristic
2	C₂×#17985827	C₂	1	2	yes	maximal, complemented
3	#22588369	C₂	1	1	yes	maximal, characteristic
4	#22586432	C₂	1	2	yes	maximal, complemented
5	C₂×C₂²⋊C₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄⋊Dic₇₁	—	2	1	no	—
7	C₂×#14556062	—	2	4	no	—
8	C₂×#14556062	—	2	1	no	—
9	C₂×#14556062	C₂²	1	1	yes	characteristic
10	C₂×#14556062	C₂²	1	1	yes	characteristic
11	C₂×#14556062	C₂²	1	2	yes	—
12	#17985859	—	2	1	no	—
13	#17985827	—	2	4	no	—
14	#17985762	—	2	1	no	—
15	C₂³×C₈×C₇₁	C₂²	1	1	yes	characteristic
16	C₂²×C₁₆×C₇₁	C₂²	1	2	yes	—
17	C₂²⋊C₁₆×C₇₁	C₄	1	4	yes	complemented
18	C₂³×Dic₇₁	—	4	1	no	—
19	C₂×C₄⋊Dic₇₁	—	2	4	no	—
20	C₂×C₄⋊Dic₇₁	—	2	2	no	—
21	C₂×C₄⋊Dic₇₁	—	2	1	no	—
22	C₂×C₄⋊Dic₇₁	—	2	1	no	—
23	C₂×C₄⋊Dic₇₁	—	4	2	no	—
24	C₂×C₄⋊Dic₇₁	—	4	1	no	—
25	C₂×C₂²⋊Dic₇₁	—	4	1	no	—
26	#14556062	—	2	4	no	—
27	#14556062	—	2	2	no	—
28	#14556062	—	2	2	no	—
29	#14556062	—	2	2	no	—
30	#14556062	D₄	1	4	yes	—
31	C₇₁⋊#95	—	4	1	no	—
32	C₂³×C₄×C₇₁	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₇₁	—	2	2	no	—
34	C₂²×C₈×C₇₁	—	2	2	no	—
35	C₂²×C₈×C₇₁	C₂×C₄	1	2	yes	—
36	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₇₁	Q₈	1	2	yes	—
38	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
39	C₂×C₁₆×C₇₁	—	2	2	no	—
40	C₂×C₁₆×C₇₁	—	2	2	no	—
41	C₂×C₁₆×C₇₁	C₂×C₄	1	4	yes	complemented
42	C₂²×Dic₇₁	—	4	4	no	—
43	C₂²×Dic₇₁	—	4	2	no	—
44	C₂²×Dic₇₁	—	4	1	no	—
45	C₂²×Dic₇₁	—	4	1	no	—
46	C₂²×Dic₇₁	—	8	2	no	—
47	C₂²×Dic₇₁	—	8	2	no	—
48	C₄⋊Dic₇₁	—	2	4	no	—
49	C₄⋊Dic₇₁	—	4	4	no	—
50	C₄⋊Dic₇₁	—	4	2	no	—
51	C₄⋊Dic₇₁	—	4	1	no	—
52	C₄⋊Dic₇₁	—	4	1	no	—
53	C₄⋊Dic₇₁	—	4	1	no	—
54	C₄⋊Dic₇₁	—	4	1	no	—
55	C₂²⋊Dic₇₁	—	8	2	no	—
56	C₂⁴×C₇₁	D₈	1	1	yes	characteristic
57	C₂²×C₄×C₇₁	—	2	1	no	—
58	C₂²×C₄×C₇₁	—	2	1	no	—
59	C₂²×C₄×C₇₁	—	2	1	no	—
60	C₂²×C₄×C₇₁	—	2	1	no	—
61	C₂²×C₄×C₇₁	C₄⋊C₄	1	2	yes	—
62	C₂²×C₄×C₇₁	D₈	1	1	yes	characteristic
63	C₂²×C₄×C₇₁	Q₁₆	1	2	yes	—
64	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
65	C₂×C₈×C₇₁	—	2	1	no	—
66	C₂×C₈×C₇₁	—	2	1	no	—
67	C₂×C₈×C₇₁	—	2	4	no	—
68	C₂×C₈×C₇₁	—	2	4	no	—
69	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
70	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
72	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
73	C₂×C₈×C₇₁	#41	1	2	yes	—
74	C₁₆×C₇₁	—	2	4	no	—
75	C₂³×C₇₁	—	2	1	no	—
76	C₂³×C₇₁	—	2	1	no	—
77	C₂³×C₇₁	—	2	1	no	—
78	C₂³×C₇₁	—	2	1	no	—
79	C₂³×C₇₁	#107	1	2	yes	—
80	C₂³×C₇₁	SD₃₂	1	2	yes	—
81	C₂³×C₇₁	SD₃₂	1	2	yes	—
82	C₂³×C₇₁	C₂×D₈	1	1	yes	characteristic
83	C₂×C₄×C₇₁	—	2	1	no	—
84	C₂×C₄×C₇₁	—	2	1	no	—
85	C₂×C₄×C₇₁	—	2	2	no	—
86	C₂×C₄×C₇₁	—	2	2	no	—
87	C₂×C₄×C₇₁	—	2	2	no	—
88	C₂×C₄×C₇₁	—	2	2	no	—
89	C₂×C₄×C₇₁	#107	1	2	yes	—
90	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₇₁	#120	1	1	yes	characteristic
92	C₂×C₄×C₇₁	#123	1	1	yes	characteristic
93	C₂×C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
94	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
95	C₂×C₄×C₇₁	#137	1	1	yes	characteristic
96	C₂×Dic₇₁	—	4	4	no	—
97	C₂×Dic₇₁	—	8	8	no	—
98	C₂×Dic₇₁	—	8	2	no	—
99	C₂×Dic₇₁	—	8	2	no	—
100	C₂×Dic₇₁	—	8	1	no	—
101	C₂×Dic₇₁	—	8	1	no	—
102	C₂×Dic₇₁	—	8	1	no	—
103	C₂×Dic₇₁	—	8	1	no	—
104	C₂×Dic₇₁	—	8	1	no	—
105	C₂×Dic₇₁	—	8	1	no	—
106	C₈×C₇₁	—	2	2	no	—
107	C₈×C₇₁	C₄×D₄	1	2	yes	—
108	C₈×C₇₁	C₂²⋊Q₈	1	2	yes	—
109	C₂²×C₇₁	—	2	2	no	—
110	C₂²×C₇₁	—	2	2	no	—
111	C₂²×C₇₁	—	2	2	no	—
112	C₂²×C₇₁	—	2	2	no	—
113	C₂²×C₇₁	—	2	2	no	—
114	C₂²×C₇₁	—	2	2	no	—
115	C₂²×C₇₁	#367	1	4	yes	complemented
116	C₂²×C₇₁	C₂×#107	1	1	yes	characteristic
117	C₂²×C₇₁	#447	1	1	yes	characteristic
118	C₂²×C₇₁	#480	1	1	yes	characteristic
119	C₂²×C₇₁	C₂×SD₃₂	1	1	yes	characteristic
120	C₂²×C₇₁	C₂×SD₃₂	1	1	yes	characteristic
121	C₂²×C₇₁	#509	1	1	yes	characteristic
122	C₂²×C₇₁	#510	1	1	yes	characteristic
123	C₄×C₇₁	—	2	2	no	—
124	C₄×C₇₁	#385	1	1	yes	characteristic
125	C₄×C₇₁	#428	1	1	yes	characteristic
126	C₄×C₇₁	#451	1	1	yes	characteristic
127	C₄×C₇₁	#484	1	1	yes	characteristic
128	Dic₇₁	—	8	4	no	—
129	Dic₇₁	—	16	1	no	—
130	Dic₇₁	—	16	1	no	—
131	#12920	—	71	1	no	maximal, Sylow
132	C₂×C₇₁	—	2	4	no	—
133	C₂×C₇₁	#1852	1	1	yes	characteristic
134	C₂×C₇₁	C₂×#367	1	1	yes	characteristic
135	C₂×C₇₁	#2159	1	1	yes	characteristic
136	C₂×C₇₁	#2185	1	1	yes	characteristic
137	C₂×C₇₁	#2187	1	1	yes	characteristic
138	C₂×C₇₁	#2233	1	1	yes	characteristic
139	C₂×C₇₁	#2235	1	1	yes	characteristic
140	C₂²×#107	—	71	1	no	—
141	C₂×#367	—	71	2	no	—
142	C₂×C₂²⋊C₁₆	D₇₁	1	1	yes	characteristic
143	#1808	—	71	1	no	—
144	#1380	—	71	2	no	—
145	C₇₁	#12920	1	1	yes	characteristic, complemented, Sylow
146	C₂³×C₈	C₂×D₇₁	1	1	yes	characteristic
147	C₂²×C₄⋊C₄	—	142	1	no	—
148	C₂²×C₁₆	C₂×D₇₁	1	2	yes	—
149	C₂×#107	—	71	2	no	—
150	C₂×#107	—	71	1	no	—
151	C₂×#107	—	71	1	no	—
152	C₂×#107	—	142	4	no	—
153	C₂×#107	—	142	1	no	—
154	#380	—	142	1	no	—
155	#367	—	142	4	no	—
156	C₂²⋊C₁₆	Dic₇₁	1	4	yes	complemented
157	#340	—	142	1	no	—
158	C₂³×C₄	—	284	1	no	—
159	C₂³×C₄	D₂₈₄	1	1	yes	characteristic
160	C₂²×C₈	—	2	2	no	—
161	C₂²×C₈	—	2	2	no	—
162	C₂²×C₈	Dic₁₄₂	1	2	yes	—
163	C₂²×C₈	D₂₈₄	1	1	yes	characteristic
164	C₂²×C₈	C₂×Dic₇₁	1	2	yes	—
165	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
166	C₂×C₄⋊C₄	—	284	2	no	—
167	C₂×C₄⋊C₄	—	284	1	no	—
168	C₂×C₄⋊C₄	—	142	1	no	—
169	C₂×C₄⋊C₄	—	142	1	no	—
170	C₂×C₄⋊C₄	—	142	4	no	—
171	C₂×C₄⋊C₄	—	142	2	no	—
172	C₂×C₂²⋊C₄	—	284	1	no	—
173	C₂×C₁₆	—	2	2	no	—
174	C₂×C₁₆	—	2	2	no	—
175	C₂×C₁₆	C₂×Dic₇₁	1	4	yes	complemented
176	#107	—	71	4	no	—
177	#107	—	142	2	no	—
178	#107	—	142	2	no	—
179	#107	—	142	4	no	—
180	#107	—	142	2	no	—
181	#95	—	284	1	no	—
182	C₂⁴	D₅₆₈	1	1	yes	characteristic
183	C₂²×C₄	—	2	1	no	—
184	C₂²×C₄	—	2	1	no	—
185	C₂²×C₄	—	2	1	no	—
186	C₂²×C₄	—	2	1	no	—
187	C₂²×C₄	—	284	4	no	—
188	C₂²×C₄	—	284	2	no	—
189	C₂²×C₄	—	284	1	no	—
190	C₂²×C₄	—	284	1	no	—
191	C₂²×C₄	—	568	2	no	—
192	C₂²×C₄	—	568	2	no	—
193	C₂²×C₄	C₄⋊Dic₇₁	1	2	yes	—
194	C₂²×C₄	D₅₆₈	1	1	yes	characteristic
195	C₂²×C₄	Dic₂₈₄	1	2	yes	—
196	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
197	C₂×C₈	—	2	1	no	—
198	C₂×C₈	—	2	1	no	—
199	C₂×C₈	—	2	4	no	—
200	C₂×C₈	—	2	4	no	—
201	C₂×C₈	C₄⋊Dic₇₁	1	2	yes	—
202	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
203	C₂×C₈	D₄×D₇₁	1	2	yes	—
204	C₂×C₈	C₂×Dic₁₄₂	1	1	yes	characteristic
205	C₂×C₈	#11765864	1	2	yes	—
206	C₄⋊C₄	—	284	4	no	—
207	C₄⋊C₄	—	284	2	no	—
208	C₄⋊C₄	—	284	1	no	—
209	C₄⋊C₄	—	284	1	no	—
210	C₄⋊C₄	—	284	1	no	—
211	C₄⋊C₄	—	284	1	no	—
212	C₄⋊C₄	—	142	4	no	—
213	C₂²⋊C₄	—	568	2	no	—
214	C₁₆	—	2	4	no	—
215	C₂³	—	2	1	no	—
216	C₂³	—	2	1	no	—
217	C₂³	—	2	1	no	—
218	C₂³	—	2	1	no	—
219	C₂³	#14556062	1	2	yes	—
220	C₂³	#14556073	1	2	yes	—
221	C₂³	#14556073	1	2	yes	—
222	C₂³	C₂×D₅₆₈	1	1	yes	characteristic
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	284	4	no	—
230	C₂×C₄	—	568	8	no	—
231	C₂×C₄	—	568	2	no	—
232	C₂×C₄	—	568	2	no	—
233	C₂×C₄	—	568	1	no	—
234	C₂×C₄	—	568	1	no	—
235	C₂×C₄	—	568	1	no	—
236	C₂×C₄	—	568	1	no	—
237	C₂×C₄	—	568	1	no	—
238	C₂×C₄	—	568	1	no	—
239	C₂×C₄	#14556062	1	2	yes	—
240	C₂×C₄	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
241	C₂×C₄	#14556098	1	1	yes	characteristic
242	C₂×C₄	#14556112	1	1	yes	characteristic
243	C₂×C₄	C₂×Dic₂₈₄	1	1	yes	characteristic
244	C₂×C₄	#14556156	1	1	yes	characteristic
245	C₂×C₄	#14556161	1	1	yes	characteristic
246	C₈	—	2	2	no	—
247	C₈	D₄×Dic₇₁	1	2	yes	—
248	C₈	#14556105	1	2	yes	—
249	C₂²	—	2	2	no	—
250	C₂²	—	2	2	no	—
251	C₂²	—	2	2	no	—
252	C₂²	—	2	2	no	—
253	C₂²	—	2	2	no	—
254	C₂²	—	2	2	no	—
255	C₂²	#17985827	1	4	yes	complemented
256	C₂²	C₂×#14556062	1	1	yes	characteristic
257	C₂²	#17986135	1	1	yes	characteristic
258	C₂²	#17986313	1	1	yes	characteristic
259	C₂²	C₂×#14556073	1	1	yes	characteristic
260	C₂²	C₂×#14556073	1	1	yes	characteristic
261	C₂²	#17986437	1	1	yes	characteristic
262	C₂²	#17986442	1	1	yes	characteristic
263	C₄	—	2	2	no	—
264	C₄	—	568	4	no	—
265	C₄	—	1136	1	no	—
266	C₄	—	1136	1	no	—
267	C₄	#17985874	1	1	yes	characteristic
268	C₄	#17986041	1	1	yes	characteristic
269	C₄	#17986157	1	1	yes	characteristic
270	C₄	#17986335	1	1	yes	characteristic
271	C₂	—	2	4	no	—
272	C₂	#22588530	1	1	yes	characteristic
273	C₂	C₂×#17985827	1	1	yes	characteristic
274	C₂	#22590128	1	1	yes	characteristic
275	C₂	#22590259	1	1	yes	characteristic
276	C₂	#22590271	1	1	yes	characteristic
277	C₂	#22590526	1	1	yes	characteristic
278	C₂	#22590538	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₅₆₈
commutator	C₂×C₈×C₇₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₂×C₂²⋊C₁₆×C₇₁	C₂
socle	C₂³×C₇₁	C₂×D₈
prime radical	C₂⁴×C₇₁	D₈
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	C₂×C₂²⋊C₁₆	D₇₁
hypocenter	C₇₁	#12920
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×C₂²⋊C₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×#17985827	C₂	semidirect
4. #22586432	C₂	semidirect
17. C₂²⋊C₁₆×C₇₁	C₄	semidirect
41. C₂×C₁₆×C₇₁	C₂×C₄	semidirect
115. C₂²×C₇₁	#367	semidirect
145. C₇₁	#12920	semidirect
156. C₂²⋊C₁₆	Dic₇₁	semidirect
175. C₂×C₁₆	C₂×Dic₇₁	semidirect
255. C₂²	#17985827	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#14556062	2
C₂×#17985827	2
#22588369	2
#22586432	2
C₂×C₂²⋊C₁₆×C₇₁	2
#12920	71