Subgroups — #31902167

Order 18176.

366 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	#22588500	C₂	1	1	yes	maximal, characteristic
2	#22588345	C₂	1	2	yes	maximal
3	#22586224	C₂	1	2	yes	maximal
4	#22586175	C₂	1	1	yes	maximal, characteristic
5	C₇₁×#1984	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556062	—	2	1	no	—
7	C₂×#14556062	—	2	1	no	—
8	C₂×C₄⋊(C₇₁⋊C₈)	C₂²	1	2	yes	—
9	C₂×#14556052	—	2	2	no	—
10	#17985869	C₂²	1	2	yes	—
11	#17985761	—	2	2	no	—
12	C₂×C₄×Q₈×C₇₁	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
15	#336×C₇₁	—	2	1	no	—
16	#336×C₇₁	—	2	1	no	—
17	C₂²×C₇₁⋊C₈	—	2	2	no	—
18	C₂×C₄⋊Dic₇₁	—	2	2	no	—
19	C₂×C₄⋊Dic₇₁	—	2	2	no	—
20	C₂×C₄⋊Dic₇₁	—	2	2	no	—
21	#14556062	—	4	2	no	—
22	#14556062	—	4	2	no	—
23	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
24	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
25	#14556052	—	4	4	no	—
26	C₂²×Q₈×C₇₁	D₄	1	1	yes	characteristic
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	C₄×Q₈×C₇₁	—	2	2	no	—
31	C₄×Q₈×C₇₁	—	2	2	no	—
32	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
33	C₂×C₄⋊C₄×C₇₁	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₇₁	—	2	1	no	—
35	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
36	C₄⋊C₈×C₇₁	—	2	2	no	—
37	C₄×C₈×C₇₁	—	2	2	no	—
38	C₂²×Dic₇₁	—	4	2	no	—
39	C₂×C₇₁⋊C₈	—	4	2	no	—
40	C₂×C₇₁⋊C₈	—	4	2	no	—
41	C₂×C₇₁⋊C₈	—	4	2	no	—
42	C₄⋊Dic₇₁	—	4	2	no	—
43	C₄⋊Dic₇₁	—	4	2	no	—
44	C₄⋊Dic₇₁	—	4	2	no	—
45	C₄⋊Dic₇₁	—	4	2	no	—
46	C₄⋊Dic₇₁	—	8	2	no	—
47	C₂×Q₈×C₇₁	—	2	1	no	—
48	C₂×Q₈×C₇₁	—	2	2	no	—
49	C₂×Q₈×C₇₁	—	2	2	no	—
50	C₂×Q₈×C₇₁	Q₁₆	1	2	yes	—
51	C₂²×C₄×C₇₁	—	2	1	no	—
52	C₂²×C₄×C₇₁	—	2	1	no	—
53	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₇₁	#41	1	1	yes	characteristic
56	C₂×C₈×C₇₁	—	2	1	no	—
57	C₂×C₈×C₇₁	—	2	1	no	—
58	C₂×C₈×C₇₁	—	2	1	no	—
59	C₂×C₈×C₇₁	—	2	1	no	—
60	C₂×C₈×C₇₁	—	4	1	no	—
61	C₂×C₈×C₇₁	—	4	1	no	—
62	C₂×C₈×C₇₁	—	4	1	no	—
63	C₂×C₈×C₇₁	SD₁₆	1	2	yes	—
64	C₂×C₈×C₇₁	Q₁₆	1	2	yes	—
65	C₄⋊C₄×C₇₁	—	2	2	no	—
66	C₄⋊C₄×C₇₁	—	4	2	no	—
67	C₄²×C₇₁	—	4	1	no	—
68	C₄²×C₇₁	—	4	1	no	—
69	C₄²×C₇₁	#41	1	2	yes	—
70	C₄²×C₇₁	#41	1	2	yes	—
71	C₂³×C₇₁	#121	1	1	yes	characteristic
72	Q₈×C₇₁	—	2	4	no	—
73	Q₈×C₇₁	—	4	2	no	—
74	C₂×C₄×C₇₁	—	2	1	no	—
75	C₂×C₄×C₇₁	—	2	1	no	—
76	C₂×C₄×C₇₁	—	2	2	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	—	4	1	no	—
81	C₂×C₄×C₇₁	—	4	1	no	—
82	C₂×C₄×C₇₁	#123	1	1	yes	characteristic
83	C₂×C₄×C₇₁	#123	1	2	yes	—
84	C₂×C₄×C₇₁	#124	1	1	yes	characteristic
85	C₂×C₄×C₇₁	#126	1	2	yes	—
86	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
87	C₂×C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
88	C₂×C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
89	C₂×C₄×C₇₁	#135	1	1	yes	characteristic
90	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
91	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	2	yes	—
92	C₂×C₄×C₇₁	#137	1	1	yes	characteristic
93	C₂×Dic₇₁	—	8	2	no	—
94	C₂×Dic₇₁	—	8	2	no	—
95	C₂×Dic₇₁	—	8	2	no	—
96	C₈×C₇₁	—	2	2	no	—
97	C₈×C₇₁	—	2	2	no	—
98	C₈×C₇₁	—	8	1	no	—
99	C₇₁⋊C₈	—	8	2	no	—
100	C₂²×C₇₁	#397	1	1	yes	characteristic
101	C₂²×C₇₁	#460	1	1	yes	characteristic
102	C₂²×C₇₁	#462	1	1	yes	characteristic
103	C₂²×C₇₁	#467	1	1	yes	characteristic
104	C₂²×C₇₁	#469	1	1	yes	characteristic
105	C₂²×C₇₁	#472	1	1	yes	characteristic
106	C₂²×C₇₁	#473	1	1	yes	characteristic
107	C₄×C₇₁	—	2	2	no	—
108	C₄×C₇₁	—	4	2	no	—
109	C₄×C₇₁	—	8	1	no	—
110	C₄×C₇₁	#482	1	2	yes	—
111	C₄×C₇₁	#483	1	1	yes	characteristic
112	C₄×C₇₁	#484	1	1	yes	characteristic
113	C₄×C₇₁	#484	1	2	yes	—
114	C₄×C₇₁	#487	1	1	yes	characteristic
115	C₄×C₇₁	#489	1	1	yes	characteristic
116	Dic₇₁	—	16	2	no	—
117	#10618	—	71	1	no	maximal, Sylow
118	C₂×C₇₁	#1670	1	1	yes	characteristic
119	C₂×C₇₁	#1676	1	1	yes	characteristic
120	C₂×C₇₁	#1688	1	1	yes	characteristic
121	C₂×C₇₁	#1694	1	1	yes	characteristic
122	C₂×C₇₁	#2072	1	1	yes	characteristic
123	C₂×C₇₁	#2076	1	1	yes	characteristic
124	C₂×C₇₁	#2078	1	1	yes	characteristic
125	#1984	D₇₁	1	1	yes	characteristic
126	#1845	—	71	1	no	—
127	#1803	—	71	2	no	—
128	#1296	—	71	2	no	—
129	#1277	—	71	1	no	—
130	C₇₁	#10618	1	1	yes	characteristic, complemented, Sylow
131	C₂×C₄×Q₈	C₂×D₇₁	1	1	yes	characteristic
132	C₂×#107	—	142	1	no	—
133	C₂×#107	—	142	1	no	—
134	C₂×C₄⋊C₈	—	71	2	no	—
135	C₂×C₄⋊C₈	C₂×D₇₁	1	1	yes	characteristic
136	C₂×#103	—	142	2	no	—
137	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
138	#384	—	71	2	no	—
139	#340	—	142	2	no	—
140	#336	—	2	1	no	—
141	#336	—	2	1	no	—
142	C₂²×Q₈	D₂₈₄	1	1	yes	characteristic
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	C₄×Q₈	—	2	2	no	—
148	C₄×Q₈	—	2	2	no	—
149	C₂×C₄⋊C₄	—	2	1	no	—
150	C₂×C₄⋊C₄	—	142	2	no	—
151	C₂×C₄⋊C₄	—	142	2	no	—
152	C₂×C₄⋊C₄	—	142	2	no	—
153	C₂×C₄⋊C₄	D₂₈₄	1	1	yes	characteristic
154	C₂×C₄²	—	2	1	no	—
155	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
156	#107	—	284	2	no	—
157	#107	—	284	2	no	—
158	C₄⋊C₈	—	2	2	no	—
159	C₄⋊C₈	—	142	2	no	—
160	C₄⋊C₈	—	142	2	no	—
161	#103	—	284	4	no	—
162	C₄×C₈	—	2	2	no	—
163	C₂×Q₈	—	2	1	no	—
164	C₂×Q₈	—	2	2	no	—
165	C₂×Q₈	—	2	2	no	—
166	C₂×Q₈	Dic₂₈₄	1	2	yes	—
167	C₂²×C₄	—	2	1	no	—
168	C₂²×C₄	—	2	1	no	—
169	C₂²×C₄	—	284	2	no	—
170	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
171	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
172	C₂²×C₄	#11765865	1	1	yes	characteristic
173	C₂×C₈	—	2	1	no	—
174	C₂×C₈	—	2	1	no	—
175	C₂×C₈	—	2	1	no	—
176	C₂×C₈	—	2	1	no	—
177	C₂×C₈	—	4	1	no	—
178	C₂×C₈	—	4	1	no	—
179	C₂×C₈	—	4	1	no	—
180	C₂×C₈	—	284	2	no	—
181	C₂×C₈	—	284	2	no	—
182	C₂×C₈	—	284	2	no	—
183	C₂×C₈	#11765852	1	2	yes	—
184	C₂×C₈	#11765855	1	2	yes	—
185	C₄⋊C₄	—	2	2	no	—
186	C₄⋊C₄	—	4	2	no	—
187	C₄⋊C₄	—	284	2	no	—
188	C₄⋊C₄	—	284	2	no	—
189	C₄⋊C₄	—	284	2	no	—
190	C₄⋊C₄	—	284	2	no	—
191	C₄⋊C₄	—	568	2	no	—
192	C₄²	—	4	1	no	—
193	C₄²	—	4	1	no	—
194	C₄²	#11765864	1	2	yes	—
195	C₄²	#11765864	1	2	yes	—
196	C₂³	#14556102	1	1	yes	characteristic
197	Q₈	—	2	4	no	—
198	Q₈	—	4	2	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	4	1	no	—
203	C₂×C₄	—	4	1	no	—
204	C₂×C₄	—	4	1	no	—
205	C₂×C₄	—	4	1	no	—
206	C₂×C₄	—	4	1	no	—
207	C₂×C₄	—	568	2	no	—
208	C₂×C₄	—	568	2	no	—
209	C₂×C₄	—	568	2	no	—
210	C₂×C₄	#14556109	1	1	yes	characteristic
211	C₂×C₄	#14556112	1	2	yes	—
212	C₂×C₄	#14556115	1	1	yes	characteristic
213	C₂×C₄	#14556121	1	2	yes	—
214	C₂×C₄	C₂×#11765852	1	1	yes	characteristic
215	C₂×C₄	C₂×#11765855	1	1	yes	characteristic
216	C₂×C₄	C₂×Dic₂₈₄	1	1	yes	characteristic
217	C₂×C₄	#14556150	1	1	yes	characteristic
218	C₂×C₄	#14556153	1	1	yes	characteristic
219	C₂×C₄	#14556156	1	2	yes	—
220	C₂×C₄	#14556158	1	1	yes	characteristic
221	C₈	—	2	2	no	—
222	C₈	—	2	2	no	—
223	C₈	—	8	1	no	—
224	C₈	—	568	2	no	—
225	C₂²	#17985924	1	1	yes	characteristic
226	C₂²	#17986199	1	1	yes	characteristic
227	C₂²	#17986209	1	1	yes	characteristic
228	C₂²	#17986240	1	1	yes	characteristic
229	C₂²	#17986252	1	1	yes	characteristic
230	C₂²	#17986269	1	1	yes	characteristic
231	C₂²	#17986274	1	1	yes	characteristic
232	C₄	—	2	2	no	—
233	C₄	—	4	2	no	—
234	C₄	—	8	1	no	—
235	C₄	—	1136	2	no	—
236	C₄	#17986324	1	2	yes	—
237	C₄	#17986327	1	1	yes	characteristic
238	C₄	#17986332	1	1	yes	characteristic
239	C₄	#17986335	1	2	yes	—
240	C₄	#17986349	1	1	yes	characteristic
241	C₄	#17986358	1	1	yes	characteristic
242	C₂	#22587710	1	1	yes	characteristic
243	C₂	#22587744	1	1	yes	characteristic
244	C₂	#22587824	1	1	yes	characteristic
245	C₂	#22587858	1	1	yes	characteristic
246	C₂	#22589748	1	1	yes	characteristic
247	C₂	#22589768	1	1	yes	characteristic
248	C₂	#22589780	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14556102
commutator, cosocle	C₂×C₄²×C₇₁	C₂³
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₇₁×#1984	C₂
socle, prime radical	C₂³×C₇₁	#121
hypercenter	#1984	D₇₁
hypocenter	C₇₁	#10618
LCS	G ▷ C₂×C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1984

Complements

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

N	H ≅ G/N	type
130. C₇₁	#10618	semidirect

Maximal subgroups

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

subgroup	index
#22588500	2
#22588345	2
#22586224	2
#22586175	2
C₇₁×#1984	2
#10618	71