Subgroups — #31913831

Order 18176.

342 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₇₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	#22588500	C₂	1	1	yes	maximal, characteristic
3	#22586431	C₂	1	4	yes	maximal
4	C₇₁×#1845	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₇₁⋊C₁₆	C₂²	1	2	yes	—
6	C₂×#14556062	—	2	1	no	—
7	C₂×#14556062	C₂²	1	2	yes	—
8	#17985869	—	2	1	no	—
9	C₄×C₇₁⋊C₁₆	C₄	1	4	yes	complemented
10	C₂×#107×C₇₁	—	2	1	no	—
11	C₂×#107×C₇₁	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
13	#384×C₇₁	—	2	1	no	—
14	C₂×C₄⋊Dic₇₁	—	2	2	no	—
15	C₂×C₄⋊Dic₇₁	—	4	1	no	—
16	C₂×C₄⋊Dic₇₁	—	4	1	no	—
17	C₂×C₇₁⋊C₁₆	—	2	2	no	—
18	C₂×C₇₁⋊C₁₆	—	2	2	no	—
19	C₂×C₇₁⋊C₁₆	C₂×C₄	1	4	yes	—
20	#14556062	—	2	4	no	—
21	#14556062	—	4	2	no	—
22	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
24	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
25	C₂×C₄⋊C₄×C₇₁	—	4	1	no	—
26	C₂×C₄⋊C₄×C₇₁	—	4	1	no	—
27	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
28	#107×C₇₁	—	2	4	no	—
29	#107×C₇₁	—	4	2	no	—
30	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
31	C₄×C₈×C₇₁	Q₈	1	2	yes	—
32	C₂²×Dic₇₁	—	4	2	no	—
33	C₄⋊Dic₇₁	—	4	2	no	—
34	C₄⋊Dic₇₁	—	4	2	no	—
35	C₄⋊Dic₇₁	—	8	2	no	—
36	C₄⋊Dic₇₁	—	8	2	no	—
37	C₇₁⋊C₁₆	—	2	4	no	—
38	C₂²×C₄×C₇₁	—	4	2	no	—
39	C₂²×C₄×C₇₁	D₈	1	1	yes	characteristic
40	C₂²×C₄×C₇₁	D₈	1	1	yes	characteristic
41	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
42	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
43	C₂×C₈×C₇₁	SD₁₆	1	2	yes	—
44	C₂×C₈×C₇₁	SD₁₆	1	2	yes	—
45	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₈×C₇₁	#41	1	2	yes	—
48	C₂×C₈×C₇₁	#41	1	2	yes	—
49	C₄⋊C₄×C₇₁	—	4	2	no	—
50	C₄⋊C₄×C₇₁	—	4	2	no	—
51	C₄⋊C₄×C₇₁	—	8	2	no	—
52	C₄⋊C₄×C₇₁	—	8	2	no	—
53	C₄²×C₇₁	C₄⋊C₄	1	2	yes	—
54	C₄²×C₇₁	Q₁₆	1	2	yes	—
55	C₂³×C₇₁	C₂×D₈	1	1	yes	characteristic
56	C₂×C₄×C₇₁	—	8	2	no	—
57	C₂×C₄×C₇₁	—	8	2	no	—
58	C₂×C₄×C₇₁	—	8	2	no	—
59	C₂×C₄×C₇₁	#107	1	2	yes	—
60	C₂×C₄×C₇₁	#107	1	2	yes	—
61	C₂×C₄×C₇₁	D₁₆	1	2	yes	—
62	C₂×C₄×C₇₁	SD₃₂	1	2	yes	—
63	C₂×C₄×C₇₁	SD₃₂	1	2	yes	—
64	C₂×C₄×C₇₁	Q₃₂	1	2	yes	—
65	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₇₁	#124	1	1	yes	characteristic
67	C₂×C₄×C₇₁	#124	1	1	yes	characteristic
68	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
69	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
70	C₂×C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
71	C₂×Dic₇₁	—	8	2	no	—
72	C₂×Dic₇₁	—	8	2	no	—
73	C₂×Dic₇₁	—	8	2	no	—
74	C₈×C₇₁	#106	1	4	yes	—
75	C₈×C₇₁	#117	1	2	yes	—
76	C₈×C₇₁	#125	1	2	yes	—
77	C₂²×C₇₁	C₂×#107	1	1	yes	characteristic
78	C₂²×C₇₁	#486	1	1	yes	characteristic
79	C₂²×C₇₁	#486	1	1	yes	characteristic
80	C₂²×C₇₁	C₂×D₁₆	1	1	yes	characteristic
81	C₂²×C₇₁	C₂×SD₃₂	1	1	yes	characteristic
82	C₂²×C₇₁	C₂×SD₃₂	1	1	yes	characteristic
83	C₂²×C₇₁	C₂×Q₃₂	1	1	yes	characteristic
84	C₄×C₇₁	—	16	2	no	—
85	C₄×C₇₁	#366	1	4	yes	—
86	C₄×C₇₁	#367	1	4	yes	—
87	C₄×C₇₁	#382	1	1	yes	characteristic
88	C₄×C₇₁	C₂×#106	1	1	yes	characteristic
89	C₄×C₇₁	#487	1	1	yes	characteristic
90	C₄×C₇₁	#487	1	1	yes	characteristic
91	Dic₇₁	—	16	2	no	—
92	#12888	—	71	1	no	maximal, Sylow
93	C₂×C₇₁	#1834	1	1	yes	characteristic
94	C₂×C₇₁	C₂×#366	1	1	yes	characteristic
95	C₂×C₇₁	C₂×#367	1	1	yes	characteristic
96	C₂×C₇₁	#2240	1	1	yes	characteristic
97	C₂×C₇₁	#2240	1	1	yes	characteristic
98	C₂×C₇₁	#2241	1	1	yes	characteristic
99	C₂×C₇₁	#2241	1	1	yes	characteristic
100	C₂×C₄×C₁₆	—	71	1	no	—
101	#1845	—	71	1	no	—
102	#1845	D₇₁	1	1	yes	characteristic
103	#1380	—	71	4	no	—
104	C₇₁	#12888	1	1	yes	characteristic, complemented, Sylow
105	C₂²×C₁₆	—	71	2	no	—
106	C₂×#107	—	2	1	no	—
107	C₂×#107	—	71	2	no	—
108	C₂×#107	—	142	1	no	—
109	C₂×#107	C₂×D₇₁	1	2	yes	—
110	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
111	#384	—	2	1	no	—
112	#384	—	142	1	no	—
113	C₄×C₁₆	—	71	4	no	—
114	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
115	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
116	C₂×C₄⋊C₄	—	2	2	no	—
117	C₂×C₄⋊C₄	—	4	1	no	—
118	C₂×C₄⋊C₄	—	4	1	no	—
119	C₂×C₄⋊C₄	—	284	1	no	—
120	C₂×C₄⋊C₄	—	284	1	no	—
121	C₂×C₄⋊C₄	—	142	2	no	—
122	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
123	C₂×C₁₆	—	71	4	no	—
124	C₂×C₁₆	—	142	2	no	—
125	C₂×C₁₆	—	142	2	no	—
126	#107	—	2	4	no	—
127	#107	—	4	2	no	—
128	#107	—	284	2	no	—
129	#107	—	142	4	no	—
130	C₄×C₈	Dic₁₄₂	1	2	yes	—
131	C₄×C₈	C₄×D₇₁	1	2	yes	—
132	C₂²×C₄	—	4	2	no	—
133	C₂²×C₄	—	284	2	no	—
134	C₂²×C₄	#11765850	1	1	yes	characteristic
135	C₂²×C₄	#11765850	1	1	yes	characteristic
136	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
137	C₂×C₈	Dic₇₁⋊C₄	1	2	yes	—
138	C₂×C₈	#11765852	1	2	yes	—
139	C₂×C₈	#11765853	1	2	yes	—
140	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
141	C₂×C₈	C₂×Dic₁₄₂	1	1	yes	characteristic
142	C₂×C₈	#11765864	1	2	yes	—
143	C₂×C₈	#11765866	1	2	yes	—
144	C₄⋊C₄	—	4	2	no	—
145	C₄⋊C₄	—	4	2	no	—
146	C₄⋊C₄	—	8	2	no	—
147	C₄⋊C₄	—	8	2	no	—
148	C₄⋊C₄	—	284	2	no	—
149	C₄⋊C₄	—	284	2	no	—
150	C₄⋊C₄	—	568	2	no	—
151	C₄⋊C₄	—	568	2	no	—
152	C₄²	Dic₇₁⋊C₄	1	2	yes	—
153	C₄²	#11765855	1	2	yes	—
154	C₁₆	—	142	4	no	—
155	C₂³	C₂×#11765850	1	1	yes	characteristic
156	C₂×C₄	—	8	2	no	—
157	C₂×C₄	—	8	2	no	—
158	C₂×C₄	—	8	2	no	—
159	C₂×C₄	—	568	2	no	—
160	C₂×C₄	—	568	2	no	—
161	C₂×C₄	—	568	2	no	—
162	C₂×C₄	#14556061	1	2	yes	—
163	C₂×C₄	#14556061	1	2	yes	—
164	C₂×C₄	#14556069	1	2	yes	—
165	C₂×C₄	#14556071	1	2	yes	—
166	C₂×C₄	#14556072	1	2	yes	—
167	C₂×C₄	#14556074	1	2	yes	—
168	C₂×C₄	C₂×Dic₇₁⋊C₄	1	1	yes	characteristic
169	C₂×C₄	#14556115	1	1	yes	characteristic
170	C₂×C₄	#14556117	1	1	yes	characteristic
171	C₂×C₄	C₂×#11765852	1	1	yes	characteristic
172	C₂×C₄	C₂×#11765853	1	1	yes	characteristic
173	C₂×C₄	C₂×#11765855	1	1	yes	characteristic
174	C₈	#14556059	1	4	yes	—
175	C₈	#14556086	1	2	yes	—
176	C₈	#14556119	1	2	yes	—
177	C₂²	C₂×#14556061	1	1	yes	characteristic
178	C₂²	#17986345	1	1	yes	characteristic
179	C₂²	#17986347	1	1	yes	characteristic
180	C₂²	C₂×#14556069	1	1	yes	characteristic
181	C₂²	C₂×#14556071	1	1	yes	characteristic
182	C₂²	C₂×#14556072	1	1	yes	characteristic
183	C₂²	C₂×#14556074	1	1	yes	characteristic
184	C₄	—	16	2	no	—
185	C₄	—	1136	2	no	—
186	C₄	#17985824	1	4	yes	—
187	C₄	#17985826	1	4	yes	—
188	C₄	#17985864	1	1	yes	characteristic
189	C₄	C₂×#14556059	1	1	yes	characteristic
190	C₄	#17986349	1	1	yes	characteristic
191	C₄	#17986351	1	1	yes	characteristic
192	C₂	#22588467	1	1	yes	characteristic
193	C₂	C₂×#17985824	1	1	yes	characteristic
194	C₂	C₂×#17985826	1	1	yes	characteristic
195	C₂	#22590563	1	1	yes	characteristic
196	C₂	#22590565	1	1	yes	characteristic
197	C₂	#22590567	1	1	yes	characteristic
198	C₂	#22590569	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#11765850
commutator	C₂×C₈×C₇₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₇₁×#1845	C₂
socle, prime radical	C₂³×C₇₁	C₂×D₈
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	#1845	D₇₁
hypocenter	C₇₁	#12888
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1845

Complements

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

N	H ≅ G/N	type
9. C₄×C₇₁⋊C₁₆	C₄	semidirect
104. C₇₁	#12888	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₇₁⋊C₁₆	2
#22588500	2
#22586431	2
C₇₁×#1845	2
#12888	71