Subgroups — #31896134

Order 18176.

370 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22589098	C₂	1	2	yes	maximal
2	#22588500	C₂	1	1	yes	maximal, characteristic
3	C₂×#17985750	C₂	1	1	yes	maximal, characteristic
4	#22586224	C₂	1	2	yes	maximal
5	C₇₁×#1770	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556062	—	2	1	no	—
7	C₂×#14556062	—	2	1	no	—
8	C₂×#14556061	—	2	2	no	—
9	C₂×C₄⋊(C₇₁⋊C₈)	C₂²	1	2	yes	—
10	#17985869	C₂²	1	2	yes	—
11	#17985761	—	2	2	no	—
12	#17985750	C₄	1	4	yes	complemented
13	C₂×C₄⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
14	C₂×#97×C₇₁	—	2	1	no	—
15	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
16	C₄×C₄⋊C₄×C₇₁	C₂²	1	1	yes	characteristic
17	#336×C₇₁	—	2	1	no	—
18	C₂²×C₇₁⋊C₈	—	2	2	no	—
19	C₂×C₄⋊Dic₇₁	—	2	2	no	—
20	C₂×C₄⋊Dic₇₁	—	2	2	no	—
21	C₂×C₄⋊Dic₇₁	—	2	2	no	—
22	#14556062	—	4	2	no	—
23	#14556062	—	4	2	no	—
24	#14556061	—	4	4	no	—
25	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
26	C₄⋊(C₇₁⋊C₈)	C₂×C₄	1	4	yes	—
27	C₂²×C₈×C₇₁	—	2	1	no	—
28	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
30	C₂×C₄⋊C₄×C₇₁	D₄	1	1	yes	characteristic
31	C₂×C₄⋊C₄×C₇₁	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₇₁	—	2	1	no	—
33	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
34	C₄⋊C₈×C₇₁	—	2	1	no	—
35	C₄⋊C₈×C₇₁	Q₈	1	2	yes	—
36	#97×C₇₁	—	4	2	no	—
37	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
38	C₄×C₈×C₇₁	Q₈	1	2	yes	—
39	#95×C₇₁	—	2	1	no	—
40	C₂²×Dic₇₁	—	4	2	no	—
41	C₂×C₇₁⋊C₈	—	2	4	no	—
42	C₂×C₇₁⋊C₈	—	4	2	no	—
43	C₂×C₇₁⋊C₈	—	4	2	no	—
44	C₄⋊Dic₇₁	—	4	2	no	—
45	C₄⋊Dic₇₁	—	4	2	no	—
46	C₄⋊Dic₇₁	—	4	2	no	—
47	C₄⋊Dic₇₁	—	4	2	no	—
48	C₄⋊Dic₇₁	—	8	2	no	—
49	C₂²×C₄×C₇₁	—	2	1	no	—
50	C₂²×C₄×C₇₁	—	2	1	no	—
51	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₇₁	#41	1	1	yes	characteristic
54	C₂×C₈×C₇₁	—	2	1	no	—
55	C₂×C₈×C₇₁	—	2	1	no	—
56	C₂×C₈×C₇₁	—	2	2	no	—
57	C₂×C₈×C₇₁	—	4	1	no	—
58	C₂×C₈×C₇₁	—	4	1	no	—
59	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
60	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
61	C₂×C₈×C₇₁	D₈	1	2	yes	—
62	C₂×C₈×C₇₁	Q₁₆	1	2	yes	—
63	C₄⋊C₄×C₇₁	—	2	2	no	—
64	C₄⋊C₄×C₇₁	—	2	2	no	—
65	C₄²×C₇₁	—	4	1	no	—
66	C₄²×C₇₁	—	4	1	no	—
67	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
68	C₄²×C₇₁	C₂×Q₈	1	1	yes	characteristic
69	C₄²×C₇₁	C₂×Q₈	1	1	yes	characteristic
70	C₄²×C₇₁	#41	1	1	yes	characteristic
71	C₂³×C₇₁	#121	1	1	yes	characteristic
72	C₂×C₄×C₇₁	—	2	1	no	—
73	C₂×C₄×C₇₁	—	2	1	no	—
74	C₂×C₄×C₇₁	—	4	1	no	—
75	C₂×C₄×C₇₁	—	4	1	no	—
76	C₂×C₄×C₇₁	—	4	1	no	—
77	C₂×C₄×C₇₁	—	4	1	no	—
78	C₂×C₄×C₇₁	—	4	1	no	—
79	C₂×C₄×C₇₁	—	4	1	no	—
80	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
82	C₂×C₄×C₇₁	C₄×Q₈	1	1	yes	characteristic
83	C₂×C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
84	C₂×C₄×C₇₁	#125	1	1	yes	characteristic
85	C₂×C₄×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
86	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
87	C₂×C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
88	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
89	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	2	yes	—
90	C₂×C₄×C₇₁	#137	1	2	yes	—
91	C₂×C₄×C₇₁	#137	1	1	yes	characteristic
92	C₂×Dic₇₁	—	8	2	no	—
93	C₂×Dic₇₁	—	8	2	no	—
94	C₂×Dic₇₁	—	8	2	no	—
95	C₈×C₇₁	—	2	2	no	—
96	C₈×C₇₁	—	4	2	no	—
97	C₈×C₇₁	#107	1	4	yes	complemented
98	C₇₁⋊C₈	—	4	4	no	—
99	C₂²×C₇₁	#389	1	1	yes	characteristic
100	C₂²×C₇₁	#459	1	1	yes	characteristic
101	C₂²×C₇₁	#462	1	1	yes	characteristic
102	C₂²×C₇₁	#469	1	1	yes	characteristic
103	C₂²×C₇₁	#470	1	1	yes	characteristic
104	C₂²×C₇₁	#472	1	1	yes	characteristic
105	C₂²×C₇₁	#473	1	1	yes	characteristic
106	C₄×C₇₁	—	2	1	no	—
107	C₄×C₇₁	—	2	1	no	—
108	C₄×C₇₁	—	8	1	no	—
109	C₄×C₇₁	—	8	1	no	—
110	C₄×C₇₁	C₂×#107	1	1	yes	characteristic
111	C₄×C₇₁	#428	1	1	yes	characteristic
112	C₄×C₇₁	#440	1	2	yes	—
113	C₄×C₇₁	#474	1	1	yes	characteristic
114	C₄×C₇₁	#477	1	1	yes	characteristic
115	C₄×C₇₁	#501	1	2	yes	—
116	Dic₇₁	—	16	2	no	—
117	#9620	—	71	1	no	maximal, Sylow
118	C₂×C₇₁	#1687	1	1	yes	characteristic
119	C₂×C₇₁	#1689	1	1	yes	characteristic
120	C₂×C₇₁	#1694	1	1	yes	characteristic
121	C₂×C₇₁	#1696	1	1	yes	characteristic
122	C₂×C₇₁	#1944	1	1	yes	characteristic
123	C₂×C₇₁	#1947	1	1	yes	characteristic
124	C₂×C₇₁	#1952	1	1	yes	characteristic
125	#1947	—	71	2	no	—
126	#1845	—	71	1	no	—
127	#1770	D₇₁	1	1	yes	characteristic
128	C₂×#334	—	71	1	no	—
129	#1296	—	71	2	no	—
130	C₇₁	#9620	1	1	yes	characteristic, complemented, Sylow
131	C₂×#107	—	142	1	no	—
132	C₂×#107	—	142	1	no	—
133	C₂×#107	—	142	2	no	—
134	C₂×C₄⋊C₈	—	71	2	no	—
135	C₂×C₄⋊C₈	C₂×D₇₁	1	1	yes	characteristic
136	C₂×#97	—	2	1	no	—
137	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
138	#384	—	71	2	no	—
139	C₄×C₄⋊C₄	C₂×D₇₁	1	1	yes	characteristic
140	#340	—	142	2	no	—
141	#336	—	2	1	no	—
142	#334	—	71	4	no	—
143	C₂²×C₈	—	2	1	no	—
144	C₂²×C₈	—	142	2	no	—
145	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
146	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
147	C₂×C₄⋊C₄	—	142	2	no	—
148	C₂×C₄⋊C₄	—	142	2	no	—
149	C₂×C₄⋊C₄	—	142	2	no	—
150	C₂×C₄⋊C₄	D₂₈₄	1	1	yes	characteristic
151	C₂×C₄⋊C₄	D₂₈₄	1	1	yes	characteristic
152	C₂×C₄²	—	2	1	no	—
153	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
154	#107	—	284	4	no	—
155	#107	—	284	2	no	—
156	#107	—	284	2	no	—
157	C₄⋊C₈	—	2	1	no	—
158	C₄⋊C₈	—	71	4	no	—
159	C₄⋊C₈	—	142	2	no	—
160	C₄⋊C₈	Dic₁₄₂	1	2	yes	—
161	#97	—	4	2	no	—
162	C₄×C₈	Dic₁₄₂	1	2	yes	—
163	C₄×C₈	C₄×D₇₁	1	2	yes	—
164	#95	—	2	1	no	—
165	C₂²×C₄	—	2	1	no	—
166	C₂²×C₄	—	2	1	no	—
167	C₂²×C₄	—	284	2	no	—
168	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
169	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
170	C₂²×C₄	#11765865	1	1	yes	characteristic
171	C₂×C₈	—	2	1	no	—
172	C₂×C₈	—	2	1	no	—
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	—	4	1	no	—
175	C₂×C₈	—	4	1	no	—
176	C₂×C₈	—	284	2	no	—
177	C₂×C₈	—	284	2	no	—
178	C₂×C₈	—	142	4	no	—
179	C₂×C₈	Dic₇₁⋊C₄	1	2	yes	—
180	C₂×C₈	Dic₇₁⋊C₄	1	2	yes	—
181	C₂×C₈	#11765850	1	2	yes	—
182	C₂×C₈	#11765855	1	2	yes	—
183	C₄⋊C₄	—	2	2	no	—
184	C₄⋊C₄	—	2	2	no	—
185	C₄⋊C₄	—	284	2	no	—
186	C₄⋊C₄	—	284	2	no	—
187	C₄⋊C₄	—	284	2	no	—
188	C₄⋊C₄	—	284	2	no	—
189	C₄⋊C₄	—	568	2	no	—
190	C₄²	—	4	1	no	—
191	C₄²	—	4	1	no	—
192	C₄²	C₂×C₄×D₇₁	1	1	yes	characteristic
193	C₄²	C₂×Dic₁₄₂	1	1	yes	characteristic
194	C₄²	C₂×Dic₁₄₂	1	1	yes	characteristic
195	C₄²	#11765865	1	1	yes	characteristic
196	C₂³	#14556102	1	1	yes	characteristic
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	4	1	no	—
200	C₂×C₄	—	4	1	no	—
201	C₂×C₄	—	4	1	no	—
202	C₂×C₄	—	4	1	no	—
203	C₂×C₄	—	4	1	no	—
204	C₂×C₄	—	4	1	no	—
205	C₂×C₄	—	568	2	no	—
206	C₂×C₄	—	568	2	no	—
207	C₂×C₄	—	568	2	no	—
208	C₂×C₄	C₂×Dic₇₁⋊C₄	1	1	yes	characteristic
209	C₂×C₄	C₄×D₂₈₄	1	1	yes	characteristic
210	C₂×C₄	C₄×Dic₁₄₂	1	1	yes	characteristic
211	C₂×C₄	#14556107	1	1	yes	characteristic
212	C₂×C₄	#14556120	1	1	yes	characteristic
213	C₂×C₄	#14556128	1	1	yes	characteristic
214	C₂×C₄	C₂×#11765850	1	1	yes	characteristic
215	C₂×C₄	C₂×#11765855	1	1	yes	characteristic
216	C₂×C₄	#14556153	1	1	yes	characteristic
217	C₂×C₄	#14556156	1	2	yes	—
218	C₂×C₄	#14556158	1	1	yes	characteristic
219	C₂×C₄	#14556161	1	2	yes	—
220	C₈	—	2	2	no	—
221	C₈	—	4	2	no	—
222	C₈	—	284	4	no	—
223	C₈	#14556061	1	4	yes	complemented
224	C₂²	#17985897	1	1	yes	characteristic
225	C₂²	#17986194	1	1	yes	characteristic
226	C₂²	#17986209	1	1	yes	characteristic
227	C₂²	#17986252	1	1	yes	characteristic
228	C₂²	#17986257	1	1	yes	characteristic
229	C₂²	#17986269	1	1	yes	characteristic
230	C₂²	#17986274	1	1	yes	characteristic
231	C₄	—	2	1	no	—
232	C₄	—	2	1	no	—
233	C₄	—	8	1	no	—
234	C₄	—	8	1	no	—
235	C₄	—	1136	2	no	—
236	C₄	C₂×#14556061	1	1	yes	characteristic
237	C₄	#17986042	1	1	yes	characteristic
238	C₄	#17986101	1	2	yes	—
239	C₄	#17986279	1	1	yes	characteristic
240	C₄	#17986294	1	1	yes	characteristic
241	C₄	#17986408	1	2	yes	—
242	C₂	#22587819	1	1	yes	characteristic
243	C₂	#22587829	1	1	yes	characteristic
244	C₂	#22587858	1	1	yes	characteristic
245	C₂	#22587868	1	1	yes	characteristic
246	C₂	#22589083	1	1	yes	characteristic
247	C₂	#22589098	1	1	yes	characteristic
248	C₂	#22589125	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14556102
commutator	C₄²×C₇₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₇₁×#1770	C₂
socle, prime radical	C₂³×C₇₁	#121
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	#1770	D₇₁
hypocenter	C₇₁	#9620
LCS	G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1770

Complements

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

N	H ≅ G/N	type
12. #17985750	C₄	semidirect
97. C₈×C₇₁	#107	semidirect
130. C₇₁	#9620	semidirect
223. C₈	#14556061	semidirect

Maximal subgroups

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

subgroup	index
#22589098	2
#22588500	2
C₂×#17985750	2
#22586224	2
C₇₁×#1770	2
#9620	71