Subgroups — #31913841

Order 18176.

282 conjugacy classes of subgroups.

The subgroup lattice has 168 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	#22588512	C₂	1	1	yes	maximal, characteristic
3	#22586433	C₂	1	4	yes	maximal
4	C₇₁×#1848	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₇₁⋊C₁₆	C₂²	1	2	yes	—
6	C₂×#14556064	—	2	1	no	—
7	C₂×#14556064	C₂²	1	2	yes	—
8	#17986023	—	2	1	no	—
9	C₄×C₇₁⋊C₁₆	C₄	1	2	yes	—
10	C₄×C₇₁⋊C₁₆	C₄	1	2	yes	—
11	C₂×#108×C₇₁	—	2	1	no	—
12	C₂×#108×C₇₁	C₂²	1	2	yes	—
13	#423×C₇₁	—	2	1	no	—
14	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
15	C₂×#11765849	—	2	2	no	—
16	C₂×C₇₁⋊C₁₆	—	2	2	no	—
17	C₂×C₇₁⋊C₁₆	—	2	2	no	—
18	C₂×C₇₁⋊C₁₆	C₂×C₄	1	4	yes	—
19	#14556064	—	2	4	no	—
20	#14556064	—	4	2	no	—
21	C₄⋊(C₇₁⋊C₈)	—	4	2	no	—
22	C₂×M₁₆×C₇₁	—	2	2	no	—
23	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
25	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
26	#108×C₇₁	—	2	4	no	—
27	#108×C₇₁	—	4	2	no	—
28	C₄⋊C₈×C₇₁	—	4	2	no	—
29	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₇₁	Q₈	1	1	yes	characteristic
32	C₄×C₈×C₇₁	Q₈	1	1	yes	characteristic
33	#11765849	—	4	2	no	—
34	#11765849	—	4	2	no	—
35	C₂×C₇₁⋊C₈	—	4	2	no	—
36	C₇₁⋊C₁₆	—	2	4	no	—
37	C₂²×C₄×C₇₁	Q₁₆	1	2	yes	—
38	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
39	M₁₆×C₇₁	—	4	2	no	—
40	M₁₆×C₇₁	—	4	2	no	—
41	C₂×C₈×C₇₁	—	4	2	no	—
42	C₂×C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₇₁	SD₁₆	1	2	yes	—
45	C₂×C₈×C₇₁	SD₁₆	1	2	yes	—
46	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₈×C₇₁	#41	1	2	yes	—
49	C₂×C₈×C₇₁	#41	1	2	yes	—
50	C₄²×C₇₁	C₄⋊C₄	1	1	yes	characteristic
51	C₄²×C₇₁	C₄⋊C₄	1	1	yes	characteristic
52	C₄²×C₇₁	D₈	1	2	yes	—
53	C₂³×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
54	C₂×C₄×C₇₁	—	2	4	no	—
55	C₂×C₄×C₇₁	#107	1	2	yes	—
56	C₂×C₄×C₇₁	#107	1	2	yes	—
57	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₇₁	#124	1	1	yes	characteristic
59	C₂×C₄×C₇₁	#124	1	1	yes	characteristic
60	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
61	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
62	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
63	C₈×C₇₁	—	8	2	no	—
64	C₈×C₇₁	#106	1	2	yes	—
65	C₈×C₇₁	#106	1	2	yes	—
66	C₈×C₇₁	#117	1	2	yes	—
67	C₈×C₇₁	#125	1	2	yes	—
68	C₇₁⋊C₈	—	8	2	no	—
69	C₂²×C₇₁	—	2	2	no	—
70	C₂²×C₇₁	C₂×#107	1	1	yes	characteristic
71	C₂²×C₇₁	#487	1	1	yes	characteristic
72	C₂²×C₇₁	#487	1	1	yes	characteristic
73	C₄×C₇₁	—	2	2	no	—
74	C₄×C₇₁	#368	1	4	yes	—
75	C₄×C₇₁	#382	1	1	yes	characteristic
76	C₄×C₇₁	C₂×#106	1	1	yes	characteristic
77	C₄×C₇₁	#486	1	1	yes	characteristic
78	C₄×C₇₁	#486	1	1	yes	characteristic
79	#12891	—	71	1	no	maximal, Sylow
80	C₂×C₇₁	—	2	1	no	—
81	C₂×C₇₁	—	2	1	no	—
82	C₂×C₇₁	#1834	1	1	yes	characteristic
83	C₂×C₇₁	C₂×#368	1	2	yes	—
84	C₂×C₄×C₁₆	—	71	1	no	—
85	#1848	—	71	1	no	—
86	#1848	D₇₁	1	1	yes	characteristic
87	#1381	—	71	4	no	—
88	C₇₁	#12891	1	1	yes	characteristic, complemented, Sylow
89	C₂²×C₁₆	—	71	2	no	—
90	C₂×#108	—	2	1	no	—
91	C₂×#108	—	71	2	no	—
92	C₂×#108	—	142	1	no	—
93	C₂×#108	C₂×D₇₁	1	2	yes	—
94	#423	—	2	1	no	—
95	#423	—	142	1	no	—
96	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
97	C₄×C₁₆	—	71	2	no	—
98	C₄×C₁₆	—	71	2	no	—
99	C₂×M₁₆	—	2	2	no	—
100	C₂×M₁₆	—	142	2	no	—
101	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
102	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
103	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
104	C₂×C₁₆	—	71	4	no	—
105	C₂×C₁₆	—	142	2	no	—
106	C₂×C₁₆	—	142	2	no	—
107	#108	—	2	4	no	—
108	#108	—	4	2	no	—
109	#108	—	284	2	no	—
110	#108	—	142	4	no	—
111	C₄⋊C₈	—	4	2	no	—
112	C₄⋊C₈	—	284	2	no	—
113	C₄×C₈	Dic₁₄₂	1	1	yes	characteristic
114	C₄×C₈	Dic₁₄₂	1	1	yes	characteristic
115	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
116	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
117	C₂²×C₄	#11765855	1	2	yes	—
118	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
119	M₁₆	—	4	2	no	—
120	M₁₆	—	4	2	no	—
121	M₁₆	—	284	2	no	—
122	M₁₆	—	284	2	no	—
123	C₂×C₈	—	4	2	no	—
124	C₂×C₈	—	284	2	no	—
125	C₂×C₈	Dic₇₁⋊C₄	1	1	yes	characteristic
126	C₂×C₈	Dic₇₁⋊C₄	1	1	yes	characteristic
127	C₂×C₈	#11765852	1	2	yes	—
128	C₂×C₈	#11765853	1	2	yes	—
129	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
130	C₂×C₈	C₂×Dic₁₄₂	1	1	yes	characteristic
131	C₂×C₈	#11765864	1	2	yes	—
132	C₂×C₈	#11765866	1	2	yes	—
133	C₄²	Dic₇₁⋊C₄	1	1	yes	characteristic
134	C₄²	Dic₇₁⋊C₄	1	1	yes	characteristic
135	C₄²	#11765850	1	2	yes	—
136	C₁₆	—	142	4	no	—
137	C₂³	C₂×#11765855	1	1	yes	characteristic
138	C₂×C₄	—	2	4	no	—
139	C₂×C₄	#14556061	1	2	yes	—
140	C₂×C₄	#14556061	1	2	yes	—
141	C₂×C₄	C₂×Dic₇₁⋊C₄	1	1	yes	characteristic
142	C₂×C₄	#14556115	1	1	yes	characteristic
143	C₂×C₄	#14556117	1	1	yes	characteristic
144	C₂×C₄	C₂×#11765850	1	1	yes	characteristic
145	C₂×C₄	C₂×#11765852	1	1	yes	characteristic
146	C₂×C₄	C₂×#11765853	1	1	yes	characteristic
147	C₈	—	8	2	no	—
148	C₈	—	568	2	no	—
149	C₈	#14556059	1	2	yes	—
150	C₈	#14556059	1	2	yes	—
151	C₈	#14556086	1	2	yes	—
152	C₈	#14556119	1	2	yes	—
153	C₂²	—	2	2	no	—
154	C₂²	C₂×#14556061	1	1	yes	characteristic
155	C₂²	#17986349	1	1	yes	characteristic
156	C₂²	#17986351	1	1	yes	characteristic
157	C₄	—	2	2	no	—
158	C₄	#17985828	1	4	yes	—
159	C₄	#17985864	1	1	yes	characteristic
160	C₄	C₂×#14556059	1	1	yes	characteristic
161	C₄	#17986345	1	1	yes	characteristic
162	C₄	#17986347	1	1	yes	characteristic
163	C₂	—	2	1	no	—
164	C₂	—	2	1	no	—
165	C₂	#22588467	1	1	yes	characteristic
166	C₂	C₂×#17985828	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×#11765850
commutator	C₂×C₈×C₇₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₇₁×#1848	C₂
socle	C₂²×C₇₁	C₂×#107
prime radical	C₂³×C₇₁	C₂×Q₁₆
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	#1848	D₇₁
hypocenter	C₇₁	#12891
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1848

Complements

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

N	H ≅ G/N	type
88. C₇₁	#12891	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
#22588512	2
#22586433	2
C₇₁×#1848	2
#12891	71