Subgroups — #31890731

Order 18176.

346 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22588478	C₂	1	1	yes	maximal, characteristic
2	C₂×#17985752	C₂	1	1	yes	maximal, characteristic
3	#22587030	C₂	1	2	yes	maximal
4	#22586214	C₂	1	2	yes	maximal
5	C₇₁×#2960	C₂	1	1	yes	maximal, characteristic
6	#17986023	C₂²	1	2	yes	—
7	C₂×C₄⋊(C₇₁⋊C₈)	C₂²	1	2	yes	—
8	#17985764	—	2	1	no	—
9	#17985764	—	2	1	no	—
10	#17985752	—	2	1	no	—
11	#17985752	—	2	1	no	—
12	#17985748	—	2	2	no	—
13	#17985745	—	2	2	no	—
14	#533×C₇₁	C₂²	1	1	yes	characteristic
15	#446×C₇₁	—	2	1	no	—
16	#446×C₇₁	C₄	1	2	yes	—
17	C₈×Q₈×C₇₁	—	2	1	no	—
18	C₈×Q₈×C₇₁	C₄	1	2	yes	—
19	#424×C₇₁	—	2	1	no	—
20	#423×C₇₁	C₂²	1	1	yes	characteristic
21	#405×C₇₁	—	2	1	no	—
22	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
23	C₂×#11765849	—	2	2	no	—
24	C₂²×C₇₁⋊C₈	—	2	2	no	—
25	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
26	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
27	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
28	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
29	C₂×M₁₆×C₇₁	—	2	1	no	—
30	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
32	C₄⋊Q₈×C₇₁	D₄	1	1	yes	characteristic
33	C₄⋊Q₈×C₇₁	D₄	1	1	yes	characteristic
34	#125×C₇₁	—	2	1	no	—
35	C₂²⋊Q₈×C₇₁	—	2	1	no	—
36	C₂²⋊Q₈×C₇₁	Q₈	1	2	yes	—
37	C₄×Q₈×C₇₁	—	2	1	no	—
38	C₄×Q₈×C₇₁	C₂×C₄	1	2	yes	—
39	#117×C₇₁	—	2	1	no	—
40	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
41	C₄⋊C₈×C₇₁	—	2	2	no	—
42	C₄⋊C₈×C₇₁	—	2	2	no	—
43	C₄⋊C₈×C₇₁	—	2	1	no	—
44	C₄⋊C₈×C₇₁	C₂×C₄	1	2	yes	—
45	#97×C₇₁	—	2	2	no	—
46	C₄×C₈×C₇₁	—	2	2	no	—
47	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
48	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
50	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
51	#11765849	—	4	2	no	—
52	#11765849	—	4	2	no	—
53	C₂×C₇₁⋊C₈	—	4	2	no	—
54	C₂×C₇₁⋊C₈	—	4	2	no	—
55	C₂×C₇₁⋊C₈	—	8	2	no	—
56	C₂×Q₈×C₇₁	—	2	1	no	—
57	C₂×Q₈×C₇₁	C₄⋊C₄	1	2	yes	—
58	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
60	C₂²×C₄×C₇₁	#41	1	1	yes	characteristic
61	M₁₆×C₇₁	—	4	2	no	—
62	C₂×C₈×C₇₁	—	2	2	no	—
63	C₂×C₈×C₇₁	—	2	2	no	—
64	C₂×C₈×C₇₁	—	2	2	no	—
65	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
69	C₄⋊C₄×C₇₁	—	2	2	no	—
70	C₄⋊C₄×C₇₁	—	2	1	no	—
71	C₄⋊C₄×C₇₁	—	2	1	no	—
72	C₄⋊C₄×C₇₁	—	2	1	no	—
73	C₄⋊C₄×C₇₁	—	2	2	no	—
74	C₄⋊C₄×C₇₁	C₄⋊C₄	1	2	yes	—
75	C₂²⋊C₄×C₇₁	—	2	2	no	—
76	C₄²×C₇₁	—	2	2	no	—
77	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
78	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
79	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₇₁	#41	1	1	yes	characteristic
81	C₂³×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
82	Q₈×C₇₁	—	4	2	no	—
83	Q₈×C₇₁	—	4	1	no	—
84	Q₈×C₇₁	—	4	1	no	—
85	C₂×C₄×C₇₁	—	2	1	no	—
86	C₂×C₄×C₇₁	—	2	1	no	—
87	C₂×C₄×C₇₁	—	2	2	no	—
88	C₂×C₄×C₇₁	—	2	2	no	—
89	C₂×C₄×C₇₁	—	4	1	no	—
90	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₇₁	#117	1	1	yes	characteristic
93	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
94	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
95	C₂×C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
96	C₂×C₄×C₇₁	#135	1	2	yes	—
97	C₂×C₄×C₇₁	#137	1	2	yes	—
98	C₈×C₇₁	—	2	1	no	—
99	C₈×C₇₁	—	2	1	no	—
100	C₈×C₇₁	—	2	1	no	—
101	C₈×C₇₁	—	2	1	no	—
102	C₈×C₇₁	—	4	2	no	—
103	C₇₁⋊C₈	—	8	2	no	—
104	C₇₁⋊C₈	—	8	2	no	—
105	C₂²×C₇₁	—	2	1	no	—
106	C₂²×C₇₁	—	2	1	no	—
107	C₂²×C₇₁	#380	1	1	yes	characteristic
108	C₂²×C₇₁	#479	1	2	yes	—
109	C₄×C₇₁	—	2	1	no	—
110	C₄×C₇₁	—	2	1	no	—
111	C₄×C₇₁	—	2	1	no	—
112	C₄×C₇₁	—	4	2	no	—
113	C₄×C₇₁	—	4	2	no	—
114	C₄×C₇₁	#416	1	1	yes	characteristic
115	C₄×C₇₁	#419	1	1	yes	characteristic
116	C₄×C₇₁	#464	1	2	yes	—
117	C₄×C₇₁	#472	1	1	yes	characteristic
118	C₄×C₇₁	#473	1	1	yes	characteristic
119	#8689	—	71	1	no	maximal, Sylow
120	C₂×C₇₁	—	2	2	no	—
121	C₂×C₇₁	#1805	1	1	yes	characteristic
122	C₂×C₇₁	#1813	1	1	yes	characteristic
123	C₂×C₇₁	#1816	1	1	yes	characteristic
124	#2960	D₇₁	1	1	yes	characteristic
125	#1837	—	71	1	no	—
126	C₂×#335	—	71	1	no	—
127	#1550	—	71	2	no	—
128	#1292	—	71	2	no	—
129	C₇₁	#8689	1	1	yes	characteristic, complemented, Sylow
130	#533	C₂×D₇₁	1	1	yes	characteristic
131	#446	—	2	1	no	—
132	#446	Dic₇₁	1	2	yes	—
133	C₈×Q₈	—	2	1	no	—
134	C₈×Q₈	Dic₇₁	1	2	yes	—
135	#424	—	2	1	no	—
136	#423	—	71	2	no	—
137	#423	C₂×D₇₁	1	1	yes	characteristic
138	C₂×C₄⋊C₈	—	71	2	no	—
139	#405	—	2	1	no	—
140	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
141	#341	—	142	1	no	—
142	#341	—	142	1	no	—
143	#335	—	142	1	no	—
144	#335	—	142	1	no	—
145	#333	—	142	2	no	—
146	#332	—	142	2	no	—
147	C₂×M₁₆	—	2	1	no	—
148	C₂×M₁₆	—	142	2	no	—
149	C₂²×C₈	—	142	2	no	—
150	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
151	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
152	C₄⋊Q₈	D₂₈₄	1	1	yes	characteristic
153	C₄⋊Q₈	D₂₈₄	1	1	yes	characteristic
154	#125	—	2	1	no	—
155	C₂²⋊Q₈	—	2	1	no	—
156	C₂²⋊Q₈	Dic₁₄₂	1	2	yes	—
157	C₄×Q₈	—	2	1	no	—
158	C₄×Q₈	C₂×Dic₇₁	1	2	yes	—
159	#117	—	2	1	no	—
160	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
161	C₄⋊C₈	—	2	2	no	—
162	C₄⋊C₈	—	2	2	no	—
163	C₄⋊C₈	—	2	1	no	—
164	C₄⋊C₈	—	142	2	no	—
165	C₄⋊C₈	—	142	2	no	—
166	C₄⋊C₈	—	142	2	no	—
167	C₄⋊C₈	—	142	2	no	—
168	C₄⋊C₈	C₂×Dic₇₁	1	2	yes	—
169	#97	—	2	2	no	—
170	C₄×C₈	—	2	2	no	—
171	C₄×C₈	C₂×Dic₇₁	1	1	yes	characteristic
172	C₄×C₈	C₂×Dic₇₁	1	1	yes	characteristic
173	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
174	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
175	C₂×Q₈	—	2	1	no	—
176	C₂×Q₈	C₄⋊Dic₇₁	1	2	yes	—
177	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
178	C₂²×C₄	C₂×Dic₁₄₂	1	1	yes	characteristic
179	C₂²×C₄	#11765865	1	1	yes	characteristic
180	M₁₆	—	4	2	no	—
181	M₁₆	—	284	2	no	—
182	M₁₆	—	284	2	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	—	284	2	no	—
187	C₂×C₈	—	284	2	no	—
188	C₂×C₈	—	568	2	no	—
189	C₂×C₈	C₂²⋊Dic₇₁	1	1	yes	characteristic
190	C₂×C₈	C₂²⋊Dic₇₁	1	1	yes	characteristic
191	C₂×C₈	C₂²⋊Dic₇₁	1	1	yes	characteristic
192	C₂×C₈	C₂²⋊Dic₇₁	1	1	yes	characteristic
193	C₄⋊C₄	—	2	2	no	—
194	C₄⋊C₄	—	2	1	no	—
195	C₄⋊C₄	—	2	1	no	—
196	C₄⋊C₄	—	2	1	no	—
197	C₄⋊C₄	—	2	2	no	—
198	C₄⋊C₄	C₄⋊Dic₇₁	1	2	yes	—
199	C₂²⋊C₄	—	2	2	no	—
200	C₄²	—	2	2	no	—
201	C₄²	C₂²×Dic₇₁	1	1	yes	characteristic
202	C₄²	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
203	C₄²	C₂×D₂₈₄	1	1	yes	characteristic
204	C₄²	#11765865	1	1	yes	characteristic
205	C₂³	#14556107	1	1	yes	characteristic
206	Q₈	—	4	2	no	—
207	Q₈	—	4	1	no	—
208	Q₈	—	4	1	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	4	1	no	—
214	C₂×C₄	C₂×C₂²⋊Dic₇₁	1	1	yes	characteristic
215	C₂×C₄	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
216	C₂×C₄	#14556087	1	1	yes	characteristic
217	C₂×C₄	#14556102	1	1	yes	characteristic
218	C₂×C₄	#14556102	1	1	yes	characteristic
219	C₂×C₄	#14556107	1	1	yes	characteristic
220	C₂×C₄	#14556148	1	2	yes	—
221	C₂×C₄	#14556159	1	2	yes	—
222	C₈	—	2	1	no	—
223	C₈	—	2	1	no	—
224	C₈	—	2	1	no	—
225	C₈	—	2	1	no	—
226	C₈	—	4	2	no	—
227	C₈	—	568	2	no	—
228	C₈	—	568	2	no	—
229	C₂²	—	2	1	no	—
230	C₂²	—	2	1	no	—
231	C₂²	#17985859	1	1	yes	characteristic
232	C₂²	#17986306	1	2	yes	—
233	C₄	—	2	1	no	—
234	C₄	—	2	1	no	—
235	C₄	—	2	1	no	—
236	C₄	—	4	2	no	—
237	C₄	—	4	2	no	—
238	C₄	#17985995	1	1	yes	characteristic
239	C₄	#17986008	1	1	yes	characteristic
240	C₄	#17986221	1	2	yes	—
241	C₄	#17986269	1	1	yes	characteristic
242	C₄	#17986274	1	1	yes	characteristic
243	C₂	—	2	2	no	—
244	C₂	#22588355	1	1	yes	characteristic
245	C₂	#22588390	1	1	yes	characteristic
246	C₂	#22588404	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#17985859
commutator	C₄²×C₇₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₇₁×#2960	C₂
socle	C₂²×C₇₁	#380
prime radical	C₂³×C₇₁	C₂²⋊Q₈
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	#2960	D₇₁
hypocenter	C₇₁	#8689
LCS	G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960

Complements

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

N	H ≅ G/N	type
129. C₇₁	#8689	semidirect

Maximal subgroups

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

subgroup	index
#22588478	2
C₂×#17985752	2
#22587030	2
#22586214	2
C₇₁×#2960	2
#8689	71