Subgroups — #31891222

Order 18176.

454 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22588499	C₂	1	2	yes	maximal
2	#22588496	C₂	1	2	yes	maximal
3	#22588468	C₂	1	1	yes	maximal, characteristic
4	C₂×C₈×C₇₁⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₇₁×#1834	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556061	—	2	4	no	—
7	C₂×#14556059	—	2	4	no	—
8	C₂×C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
10	#17985869	C₂²	1	2	yes	—
11	#17985762	—	2	2	no	—
12	C₈×C₇₁⋊C₈	C₄	1	4	yes	complemented
13	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
14	#384×C₇₁	C₂²	1	2	yes	—
15	#340×C₇₁	—	2	2	no	—
16	C₂²×C₇₁⋊C₈	D₄	1	2	yes	—
17	C₂²×C₇₁⋊C₈	D₄	1	2	yes	—
18	C₂×C₄⋊Dic₇₁	—	2	2	no	—
19	C₂×C₄⋊Dic₇₁	—	2	2	no	—
20	C₂×C₄⋊Dic₇₁	—	2	2	no	—
21	#14556061	—	4	8	no	—
22	#14556059	—	4	8	no	—
23	C₄×C₇₁⋊C₈	C₂×C₄	1	2	yes	—
24	C₄×C₇₁⋊C₈	C₂×C₄	1	2	yes	—
25	C₄×C₇₁⋊C₈	Q₈	1	2	yes	—
26	C₄×C₇₁⋊C₈	Q₈	1	2	yes	—
27	C₂²×C₈×C₇₁	D₄	1	2	yes	—
28	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
29	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
30	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
31	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
32	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
33	C₄×C₈×C₇₁	Q₈	1	2	yes	—
34	C₂²×Dic₇₁	—	4	2	no	—
35	C₂×C₇₁⋊C₈	C₄⋊C₄	1	4	yes	—
36	C₂×C₇₁⋊C₈	C₄⋊C₄	1	4	yes	—
37	C₂×C₇₁⋊C₈	D₈	1	4	yes	—
38	C₂×C₇₁⋊C₈	SD₁₆	1	8	yes	—
39	C₂×C₇₁⋊C₈	Q₁₆	1	4	yes	—
40	C₄⋊Dic₇₁	—	4	4	no	—
41	C₄⋊Dic₇₁	—	4	4	no	—
42	C₄⋊Dic₇₁	—	4	4	no	—
43	C₂²×C₄×C₇₁	—	4	2	no	—
44	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
47	C₂×C₈×C₇₁	—	2	4	no	—
48	C₂×C₈×C₇₁	C₄⋊C₄	1	4	yes	—
49	C₄⋊C₄×C₇₁	—	4	4	no	—
50	C₄⋊C₄×C₇₁	—	4	4	no	—
51	C₄⋊C₄×C₇₁	—	4	4	no	—
52	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
53	C₄²×C₇₁	C₂×Q₈	1	1	yes	characteristic
54	C₄²×C₇₁	C₂×Q₈	1	1	yes	characteristic
55	C₄²×C₇₁	C₂×Q₈	1	1	yes	characteristic
56	C₂³×C₇₁	#127	1	1	yes	characteristic
57	C₂×C₄×C₇₁	—	8	2	no	—
58	C₂×C₄×C₇₁	—	8	2	no	—
59	C₂×C₄×C₇₁	—	8	2	no	—
60	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
64	C₂×C₄×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
65	C₂×C₄×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
66	C₂×C₄×C₇₁	C₂×D₈	1	2	yes	—
67	C₂×C₄×C₇₁	C₂×SD₁₆	1	4	yes	—
68	C₂×C₄×C₇₁	C₂×Q₁₆	1	2	yes	—
69	C₂×C₄×C₇₁	#135	1	4	yes	—
70	C₂×Dic₇₁	—	8	2	no	—
71	C₂×Dic₇₁	—	8	2	no	—
72	C₂×Dic₇₁	—	8	2	no	—
73	C₈×C₇₁	—	2	4	no	—
74	C₇₁⋊C₈	#106	1	8	yes	complemented
75	C₇₁⋊C₈	#107	1	8	yes	complemented
76	C₂²×C₇₁	#384	1	1	yes	characteristic
77	C₂²×C₇₁	#492	1	2	yes	—
78	C₂²×C₇₁	#493	1	1	yes	characteristic
79	C₂²×C₇₁	#494	1	1	yes	characteristic
80	C₂²×C₇₁	#495	1	2	yes	—
81	C₄×C₇₁	—	16	2	no	—
82	C₄×C₇₁	C₂×#106	1	2	yes	—
83	C₄×C₇₁	C₂×#107	1	2	yes	—
84	C₄×C₇₁	#427	1	2	yes	—
85	C₄×C₇₁	#498	1	2	yes	—
86	C₄×C₇₁	#499	1	2	yes	—
87	C₄×C₇₁	#500	1	2	yes	—
88	Dic₇₁	—	16	2	no	—
89	#8814	—	71	1	no	maximal, Sylow
90	C₂×C₇₁	#1706	1	2	yes	—
91	C₂×C₇₁	#1707	1	2	yes	—
92	C₂×C₇₁	#1844	1	1	yes	characteristic
93	C₂×C₇₁	#1845	1	1	yes	characteristic
94	C₂×C₇₁	#1846	1	1	yes	characteristic
95	#1845	—	71	2	no	—
96	#1844	—	71	2	no	—
97	#1834	—	71	1	no	—
98	#1834	D₇₁	1	1	yes	characteristic
99	C₂×C₈²	—	71	1	no	—
100	C₇₁	#8814	1	1	yes	characteristic, complemented, Sylow
101	C₂×#107	—	142	4	no	—
102	C₂×#106	—	142	4	no	—
103	C₂×C₄×C₈	—	71	1	no	—
104	C₂×C₄×C₈	—	71	1	no	—
105	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
106	#384	—	71	2	no	—
107	#384	C₂×D₇₁	1	2	yes	—
108	#340	—	2	2	no	—
109	#340	—	142	2	no	—
110	C₈²	—	71	4	no	—
111	C₂²×C₈	—	71	2	no	—
112	C₂²×C₈	—	71	2	no	—
113	C₂²×C₈	C₂²⋊D₇₁	1	2	yes	—
114	C₂×C₄⋊C₄	—	2	2	no	—
115	C₂×C₄⋊C₄	—	2	2	no	—
116	C₂×C₄⋊C₄	—	2	2	no	—
117	C₂×C₄⋊C₄	—	142	2	no	—
118	C₂×C₄⋊C₄	—	142	2	no	—
119	C₂×C₄⋊C₄	—	142	2	no	—
120	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
121	#107	—	284	8	no	—
122	#106	—	284	8	no	—
123	C₄×C₈	—	71	2	no	—
124	C₄×C₈	—	71	2	no	—
125	C₄×C₈	—	71	2	no	—
126	C₄×C₈	—	71	2	no	—
127	C₄×C₈	Dic₁₄₂	1	2	yes	—
128	C₄×C₈	C₄×D₇₁	1	2	yes	—
129	C₂²×C₄	—	4	2	no	—
130	C₂²×C₄	—	284	2	no	—
131	C₂²×C₄	D₄×D₇₁	1	1	yes	characteristic
132	C₂²×C₄	D₄×D₇₁	1	1	yes	characteristic
133	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
134	C₂×C₈	—	2	4	no	—
135	C₂×C₈	—	71	4	no	—
136	C₂×C₈	—	71	4	no	—
137	C₂×C₈	—	71	4	no	—
138	C₂×C₈	—	71	8	no	—
139	C₂×C₈	—	71	4	no	—
140	C₂×C₈	Dic₇₁⋊C₄	1	4	yes	—
141	C₄⋊C₄	—	4	4	no	—
142	C₄⋊C₄	—	4	4	no	—
143	C₄⋊C₄	—	4	4	no	—
144	C₄⋊C₄	—	284	4	no	—
145	C₄⋊C₄	—	284	4	no	—
146	C₄⋊C₄	—	284	4	no	—
147	C₄²	C₂×C₄×D₇₁	1	1	yes	characteristic
148	C₄²	Q₈×D₇₁	1	1	yes	characteristic
149	C₄²	Q₈×D₇₁	1	1	yes	characteristic
150	C₄²	C₂×Dic₁₄₂	1	1	yes	characteristic
151	C₂³	#14556124	1	1	yes	characteristic
152	C₂×C₄	—	8	2	no	—
153	C₂×C₄	—	8	2	no	—
154	C₂×C₄	—	8	2	no	—
155	C₂×C₄	—	568	2	no	—
156	C₂×C₄	—	568	2	no	—
157	C₂×C₄	—	568	2	no	—
158	C₂×C₄	C₄⋊C₄×D₇₁	1	1	yes	characteristic
159	C₂×C₄	C₄⋊C₄×D₇₁	1	1	yes	characteristic
160	C₂×C₄	C₂×Dic₇₁⋊C₄	1	1	yes	characteristic
161	C₂×C₄	#14556126	1	1	yes	characteristic
162	C₂×C₄	#14556127	1	1	yes	characteristic
163	C₂×C₄	#14556127	1	1	yes	characteristic
164	C₂×C₄	D₈×D₇₁	1	2	yes	—
165	C₂×C₄	SD₁₆×D₇₁	1	4	yes	—
166	C₂×C₄	Q₁₆×D₇₁	1	2	yes	—
167	C₂×C₄	#14556148	1	4	yes	—
168	C₈	—	2	4	no	—
169	C₈	—	71	8	no	—
170	C₈	—	71	8	no	—
171	C₂²	#17985868	1	1	yes	characteristic
172	C₂²	#17986370	1	2	yes	—
173	C₂²	#17986374	1	1	yes	characteristic
174	C₂²	#17986377	1	1	yes	characteristic
175	C₂²	#17986381	1	2	yes	—
176	C₄	—	16	2	no	—
177	C₄	—	1136	2	no	—
178	C₄	#106×D₇₁	1	2	yes	—
179	C₄	#107×D₇₁	1	2	yes	—
180	C₄	#17986035	1	2	yes	—
181	C₄	#17986396	1	2	yes	—
182	C₄	#17986398	1	2	yes	—
183	C₄	#17986403	1	2	yes	—
184	C₂	#22587931	1	2	yes	—
185	C₂	#22587936	1	2	yes	—
186	C₂	#22588495	1	1	yes	characteristic
187	C₂	#22588498	1	1	yes	characteristic
188	C₂	#22588502	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14556124
commutator	C₄²×C₇₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₇₁×#1834	C₂
socle, prime radical	C₂³×C₇₁	#127
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	#1834	D₇₁
hypocenter	C₇₁	#8814
LCS	G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1834

Complements

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

N	H ≅ G/N	type
12. C₈×C₇₁⋊C₈	C₄	semidirect
74. C₇₁⋊C₈	#106	semidirect
75. C₇₁⋊C₈	#107	semidirect
100. C₇₁	#8814	semidirect

Maximal subgroups

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

subgroup	index
#22588499	2
#22588496	2
#22588468	2
C₂×C₈×C₇₁⋊C₈	2
C₇₁×#1834	2
#8814	71