Subgroups — #31915466

Order 18176.

290 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17985819	C₂	1	1	yes	maximal, characteristic
2	#22590100	C₂	1	1	yes	maximal, characteristic
3	#22590099	C₂	1	1	yes	maximal, characteristic
4	C₂×#17985774	C₂	1	1	yes	maximal, characteristic
5	#22588220	C₂	1	1	yes	maximal, characteristic
6	#22586428	C₂	1	1	yes	maximal, characteristic
7	C₇₁×#1379	C₂	1	1	yes	maximal, characteristic
8	C₂×#14556068	C₂²	1	1	yes	characteristic
9	C₂²×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
10	C₂×#14556063	C₄	1	1	yes	characteristic
11	#17986035	C₄	1	1	yes	characteristic
12	#17985954	C₂²	1	1	yes	characteristic
13	C₂×C₈×Dic₇₁	C₂²	1	1	yes	characteristic
14	#17985819	—	2	2	no	—
15	C₄⋊(C₇₁⋊C₁₆)	—	2	2	no	—
16	Dic₇₁⋊C₁₆	—	2	2	no	—
17	#17985774	—	2	2	no	—
18	#17985760	C₄	1	1	yes	characteristic
19	#17985758	C₄	1	1	yes	characteristic
20	C₂×M₃₂×C₇₁	C₂²	1	1	yes	characteristic
21	C₂²×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
22	#405×C₇₁	C₂²	1	1	yes	characteristic
23	C₂×#11765849	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
25	#14556087	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×Dic₇₁	C₂×C₄	1	1	yes	characteristic
27	#14556068	—	2	1	no	—
28	#14556068	—	2	1	no	—
29	C₂×C₇₁⋊C₁₆	—	2	1	no	—
30	C₂×C₇₁⋊C₁₆	—	2	1	no	—
31	C₂×C₇₁⋊C₁₆	—	2	1	no	—
32	C₂×C₇₁⋊C₁₆	D₄	1	2	yes	—
33	#14556063	—	2	2	no	—
34	#14556061	C₈	1	2	yes	—
35	#14556059	C₈	1	2	yes	—
36	#14556038	—	2	1	no	—
37	C₈×Dic₇₁	—	2	1	no	—
38	C₈×Dic₇₁	D₄	1	2	yes	—
39	C₈×Dic₇₁	Q₈	1	2	yes	—
40	C₂×M₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
42	#117×C₇₁	C₂×C₄	1	1	yes	characteristic
43	M₃₂×C₇₁	—	2	1	no	—
44	M₃₂×C₇₁	—	2	1	no	—
45	C₂×C₁₆×C₇₁	—	2	1	no	—
46	C₂×C₁₆×C₇₁	—	2	1	no	—
47	C₂×C₁₆×C₇₁	—	2	1	no	—
48	C₂×C₁₆×C₇₁	Q₈	1	2	yes	—
49	#97×C₇₁	—	2	1	no	—
50	C₄×C₈×C₇₁	—	2	1	no	—
51	C₂²×Dic₇₁	—	2	1	no	—
52	#11765849	—	2	1	no	—
53	#11765849	—	2	1	no	—
54	C₂×C₇₁⋊C₈	—	2	1	no	—
55	C₂×C₇₁⋊C₈	C₄⋊C₄	1	2	yes	—
56	C₂×C₇₁⋊C₈	C₂×C₈	1	1	yes	characteristic
57	C₂×C₇₁⋊C₈	C₂×C₈	1	1	yes	characteristic
58	C₂×C₇₁⋊C₈	M₁₆	1	2	yes	—
59	C₄⋊Dic₇₁	C₂×C₈	1	2	yes	—
60	C₂²⋊Dic₇₁	—	2	1	no	—
61	C₄×Dic₇₁	—	2	1	no	—
62	C₄×Dic₇₁	—	2	1	no	—
63	C₄×Dic₇₁	C₄⋊C₄	1	2	yes	—
64	C₇₁⋊C₁₆	—	2	2	no	—
65	C₇₁⋊C₁₆	—	4	1	no	—
66	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
67	M₁₆×C₇₁	—	2	1	no	—
68	M₁₆×C₇₁	—	2	1	no	—
69	C₂×C₈×C₇₁	—	2	1	no	—
70	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
72	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
73	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
74	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
75	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
76	C₄⋊C₄×C₇₁	C₂×C₈	1	2	yes	—
77	C₂²⋊C₄×C₇₁	—	2	1	no	—
78	C₄²×C₇₁	—	2	1	no	—
79	C₁₆×C₇₁	—	2	2	no	—
80	C₁₆×C₇₁	—	4	1	no	—
81	C₂³×C₇₁	#131	1	1	yes	characteristic
82	C₂×C₄×C₇₁	—	2	2	no	—
83	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
85	C₂×C₄×C₇₁	C₄×Q₈	1	1	yes	characteristic
86	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
87	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
88	C₂×C₄×C₇₁	#131	1	1	yes	characteristic
89	C₂×Dic₇₁	—	2	2	no	—
90	C₂×Dic₇₁	—	2	2	no	—
91	C₂×Dic₇₁	—	4	2	no	—
92	C₈×C₇₁	—	4	1	no	—
93	C₈×C₇₁	#121	1	1	yes	characteristic
94	C₈×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
95	C₈×C₇₁	#125	1	1	yes	characteristic
96	C₈×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
97	C₇₁⋊C₈	—	4	1	no	—
98	C₇₁⋊C₈	C₄⋊C₈	1	2	yes	—
99	C₇₁⋊C₈	C₄⋊C₈	1	2	yes	—
100	C₂²×C₇₁	—	2	2	no	—
101	C₂²×C₇₁	#424	1	1	yes	characteristic
102	C₂²×C₇₁	C₈×D₄	1	1	yes	characteristic
103	C₂²×C₇₁	#446	1	1	yes	characteristic
104	C₄×C₇₁	—	4	2	no	—
105	C₄×C₇₁	#389	1	1	yes	characteristic
106	C₄×C₇₁	C₂×C₄⋊C₈	1	1	yes	characteristic
107	C₄×C₇₁	#436	1	1	yes	characteristic
108	C₄×C₇₁	C₈×Q₈	1	1	yes	characteristic
109	Dic₇₁	—	4	2	no	—
110	Dic₇₁	—	4	2	no	—
111	#13162	—	71	1	no	maximal, Sylow
112	C₂×C₇₁	—	2	1	no	—
113	C₂×C₇₁	—	2	1	no	—
114	C₂×C₇₁	#1939	1	1	yes	characteristic
115	C₂×C₇₁	#2171	1	2	yes	—
116	C₂×#364	—	71	1	no	—
117	#2151	—	71	1	no	—
118	#2151	—	71	1	no	—
119	C₂×#346	—	71	1	no	—
120	#1772	—	71	1	no	—
121	#1379	—	71	1	no	—
122	#1379	D₇₁	1	1	yes	characteristic
123	C₇₁	#13162	1	1	yes	characteristic, complemented, Sylow
124	C₂×M₃₂	—	71	1	no	—
125	C₂×M₃₂	C₂×D₇₁	1	1	yes	characteristic
126	C₂²×C₁₆	—	71	1	no	—
127	C₂²×C₁₆	C₂×D₇₁	1	1	yes	characteristic
128	C₂×#108	—	71	1	no	—
129	#427	—	71	1	no	—
130	#405	—	71	1	no	—
131	#405	C₂×D₇₁	1	1	yes	characteristic
132	C₂×C₄×C₈	—	71	1	no	—
133	#364	—	142	2	no	—
134	C₄⋊C₁₆	—	142	2	no	—
135	C₄⋊C₁₆	—	142	2	no	—
136	#346	—	142	2	no	—
137	#339	—	71	1	no	—
138	#339	—	71	1	no	—
139	C₂×M₁₆	—	71	1	no	—
140	C₂×M₁₆	C₄×D₇₁	1	1	yes	characteristic
141	C₂²×C₈	—	71	1	no	—
142	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
143	#117	—	71	1	no	—
144	#117	C₄×D₇₁	1	1	yes	characteristic
145	C₂×C₄²	—	71	1	no	—
146	M₃₂	—	2	1	no	—
147	M₃₂	—	2	1	no	—
148	M₃₂	—	142	1	no	—
149	M₃₂	—	142	1	no	—
150	C₂×C₁₆	—	2	1	no	—
151	C₂×C₁₆	—	2	1	no	—
152	C₂×C₁₆	—	2	1	no	—
153	C₂×C₁₆	—	71	2	no	—
154	C₂×C₁₆	—	142	1	no	—
155	C₂×C₁₆	—	142	1	no	—
156	C₂×C₁₆	—	142	1	no	—
157	C₂×C₁₆	Dic₁₄₂	1	2	yes	—
158	#108	—	142	2	no	—
159	#107	—	71	2	no	—
160	#106	—	71	2	no	—
161	#97	—	2	1	no	—
162	#97	—	142	1	no	—
163	C₄×C₈	—	2	1	no	—
164	C₄×C₈	—	71	2	no	—
165	C₄×C₈	—	71	2	no	—
166	C₄×C₈	—	142	1	no	—
167	C₂²×C₄	—	142	1	no	—
168	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
169	M₁₆	—	2	1	no	—
170	M₁₆	—	2	1	no	—
171	M₁₆	—	142	1	no	—
172	M₁₆	—	142	1	no	—
173	C₂×C₈	—	2	1	no	—
174	C₂×C₈	—	71	2	no	—
175	C₂×C₈	—	71	1	no	—
176	C₂×C₈	—	71	1	no	—
177	C₂×C₈	—	71	2	no	—
178	C₂×C₈	—	142	1	no	—
179	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
180	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
181	C₂×C₈	Q₈×D₇₁	1	1	yes	characteristic
182	C₂×C₈	C₂×Dic₁₄₂	1	1	yes	characteristic
183	C₂×C₈	#11765864	1	1	yes	characteristic
184	C₂×C₈	#11765865	1	1	yes	characteristic
185	C₄⋊C₄	—	71	2	no	—
186	C₄⋊C₄	C₈×D₇₁	1	2	yes	—
187	C₂²⋊C₄	—	2	1	no	—
188	C₂²⋊C₄	—	142	1	no	—
189	C₄²	—	2	1	no	—
190	C₄²	—	71	2	no	—
191	C₄²	—	142	1	no	—
192	C₄²	—	142	1	no	—
193	C₁₆	—	2	2	no	—
194	C₁₆	—	4	1	no	—
195	C₁₆	—	284	1	no	—
196	C₁₆	—	142	2	no	—
197	C₂³	#14556136	1	1	yes	characteristic
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	284	2	no	—
200	C₂×C₄	—	142	2	no	—
201	C₂×C₄	—	142	2	no	—
202	C₂×C₄	C₄⋊C₄×D₇₁	1	1	yes	characteristic
203	C₂×C₄	(C₂²⋊D₇₁)⋊C₄	1	1	yes	characteristic
204	C₂×C₄	C₄×Dic₁₄₂	1	1	yes	characteristic
205	C₂×C₄	C₂×C₈×D₇₁	1	1	yes	characteristic
206	C₂×C₄	M₁₆×D₇₁	1	1	yes	characteristic
207	C₂×C₄	#14556134	1	1	yes	characteristic
208	C₈	—	4	1	no	—
209	C₈	—	284	1	no	—
210	C₈	—	71	2	no	—
211	C₈	—	71	2	no	—
212	C₈	#14556101	1	1	yes	characteristic
213	C₈	#14556105	1	1	yes	characteristic
214	C₈	#14556118	1	1	yes	characteristic
215	C₈	#14556127	1	1	yes	characteristic
216	C₂²	—	2	2	no	—
217	C₂²	#17986024	1	1	yes	characteristic
218	C₂²	(C₂²⋊D₇₁)⋊C₈	1	1	yes	characteristic
219	C₂²	#17986127	1	1	yes	characteristic
220	C₄	—	4	2	no	—
221	C₄	—	284	2	no	—
222	C₄	—	284	2	no	—
223	C₄	#17985896	1	1	yes	characteristic
224	C₄	C₄⋊C₈×D₇₁	1	1	yes	characteristic
225	C₄	#17986074	1	1	yes	characteristic
226	C₄	C₈×Dic₁₄₂	1	1	yes	characteristic
227	C₂	—	2	1	no	—
228	C₂	—	2	1	no	—
229	C₂	#22589057	1	1	yes	characteristic
230	C₂	#22590179	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	D₄×D₇₁
commutator	C₂×C₄×C₇₁	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₇₁×#1379	C₂
socle	C₂²×C₇₁	C₈×D₄
prime radical	C₂³×C₇₁	#131
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	#1379	D₇₁
hypocenter	C₇₁	#13162
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #1379

Complements

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

N	H ≅ G/N	type
123. C₇₁	#13162	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17985819	2
#22590100	2
#22590099	2
C₂×#17985774	2
#22588220	2
#22586428	2
C₇₁×#1379	2
#13162	71