Subgroups — #31909101

Order 18176.

226 conjugacy classes of subgroups.

The subgroup lattice has 166 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22587103	C₂	1	2	yes	maximal
2	#22586414	C₂	1	2	yes	maximal, complemented
3	#22586412	C₂	1	2	yes	maximal, complemented
4	C₂×C₄×C₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic
5	#17986056	—	2	2	no	—
6	#17985752	C₂²	1	4	yes	—
7	#17985750	—	2	2	no	—
8	C₂²×C₁₆×C₇₁	C₄	1	1	yes	characteristic
9	C₂²×C₁₆×C₇₁	C₄	1	1	yes	characteristic
10	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
11	C₄×C₁₆×C₇₁	C₄	1	1	yes	characteristic
12	C₄×C₁₆×C₇₁	C₄	1	1	yes	characteristic
13	C₄×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
14	C₄×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
15	C₄⋊(C₇₁⋊C₈)	—	2	4	no	—
16	C₂²⋊(C₇₁⋊C₈)	—	4	2	no	—
17	#14556039	—	4	2	no	—
18	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₇₁	Q₈	1	1	yes	characteristic
20	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
21	C₂×C₁₆×C₇₁	—	2	2	no	—
22	C₂×C₁₆×C₇₁	—	2	2	no	—
23	C₂×C₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
24	C₂×C₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
25	C₂×C₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
26	C₂×C₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
29	C₄×C₈×C₇₁	Q₈	1	1	yes	characteristic
30	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
31	C₂×C₇₁⋊C₈	—	4	4	no	—
32	C₂²×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
33	C₂²×C₄×C₇₁	D₈	1	1	yes	characteristic
34	C₂²×C₄×C₇₁	Q₁₆	1	1	yes	characteristic
35	C₂×C₈×C₇₁	—	2	2	no	—
36	C₂×C₈×C₇₁	—	2	1	no	—
37	C₂×C₈×C₇₁	—	2	1	no	—
38	C₂×C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
42	C₄²×C₇₁	C₄⋊C₄	1	1	yes	characteristic
43	C₄²×C₇₁	D₈	1	1	yes	characteristic
44	C₄²×C₇₁	Q₁₆	1	1	yes	characteristic
45	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
46	C₁₆×C₇₁	—	2	1	no	—
47	C₁₆×C₇₁	—	2	1	no	—
48	C₁₆×C₇₁	M₁₆	1	2	yes	—
49	C₁₆×C₇₁	M₁₆	1	2	yes	—
50	C₂³×C₇₁	#107	1	1	yes	characteristic
51	C₂×C₄×C₇₁	—	2	2	no	—
52	C₂×C₄×C₇₁	—	2	1	no	—
53	C₂×C₄×C₇₁	—	2	1	no	—
54	C₂×C₄×C₇₁	—	2	2	no	—
55	C₂×C₄×C₇₁	#107	1	1	yes	characteristic
56	C₂×C₄×C₇₁	#107	1	1	yes	characteristic
57	C₂×C₄×C₇₁	#107	1	1	yes	characteristic
58	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
60	C₂×C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
61	C₈×C₇₁	—	2	1	no	—
62	C₈×C₇₁	—	2	1	no	—
63	C₈×C₇₁	#108	1	2	yes	—
64	C₈×C₇₁	C₂×M₁₆	1	1	yes	characteristic
65	C₈×C₇₁	C₂×M₁₆	1	1	yes	characteristic
66	C₇₁⋊C₈	—	8	4	no	—
67	C₂²×C₇₁	—	2	1	no	—
68	C₂²×C₇₁	—	2	1	no	—
69	C₂²×C₇₁	#368	1	2	yes	—
70	C₂²×C₇₁	C₂×#107	1	1	yes	characteristic
71	C₄×C₇₁	—	2	1	no	—
72	C₄×C₇₁	—	2	1	no	—
73	C₄×C₇₁	—	2	1	no	—
74	C₄×C₇₁	—	2	1	no	—
75	C₄×C₇₁	#423	1	1	yes	characteristic
76	C₄×C₇₁	C₂×#108	1	1	yes	characteristic
77	C₄×C₇₁	#508	1	2	yes	—
78	#12024	—	71	1	no	maximal, Sylow
79	C₂×C₇₁	—	2	2	no	—
80	C₂×C₇₁	#1567	1	1	yes	characteristic
81	C₂×C₇₁	#2158	1	1	yes	characteristic
82	C₂×C₇₁	C₂×#368	1	1	yes	characteristic
83	C₂×C₄×C₁₆	D₇₁	1	1	yes	characteristic
84	#1568	—	71	2	no	—
85	#1372	—	71	2	no	—
86	#1371	—	71	2	no	—
87	C₇₁	#12024	1	1	yes	characteristic, complemented, Sylow
88	C₂²×C₁₆	Dic₇₁	1	1	yes	characteristic
89	C₂²×C₁₆	Dic₇₁	1	1	yes	characteristic
90	#432	—	142	2	no	—
91	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
92	C₄×C₁₆	Dic₇₁	1	1	yes	characteristic
93	C₄×C₁₆	Dic₇₁	1	1	yes	characteristic
94	C₄×C₁₆	C₂×D₇₁	1	1	yes	characteristic
95	C₄×C₁₆	C₂×D₇₁	1	1	yes	characteristic
96	#335	—	71	4	no	—
97	#334	—	142	2	no	—
98	C₂²×C₈	Dic₁₄₂	1	1	yes	characteristic
99	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
100	C₂×C₄²	D₂₈₄	1	1	yes	characteristic
101	C₂×C₁₆	—	2	2	no	—
102	C₂×C₁₆	—	2	2	no	—
103	C₂×C₁₆	C₂×Dic₇₁	1	1	yes	characteristic
104	C₂×C₁₆	C₂×Dic₇₁	1	1	yes	characteristic
105	C₂×C₁₆	C₂×Dic₇₁	1	1	yes	characteristic
106	C₂×C₁₆	C₂×Dic₇₁	1	1	yes	characteristic
107	C₄⋊C₈	—	142	4	no	—
108	C₂²⋊C₈	—	284	2	no	—
109	#97	—	284	2	no	—
110	C₄×C₈	Dic₁₄₂	1	1	yes	characteristic
111	C₄×C₈	D₂₈₄	1	1	yes	characteristic
112	C₄×C₈	C₂×Dic₇₁	1	1	yes	characteristic
113	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
114	C₂²×C₄	C₄⋊Dic₇₁	1	1	yes	characteristic
115	C₂²×C₄	D₅₆₈	1	1	yes	characteristic
116	C₂²×C₄	Dic₂₈₄	1	1	yes	characteristic
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	2	1	no	—
119	C₂×C₈	—	2	1	no	—
120	C₂×C₈	—	284	4	no	—
121	C₂×C₈	C₄⋊Dic₇₁	1	1	yes	characteristic
122	C₂×C₈	C₄⋊Dic₇₁	1	1	yes	characteristic
123	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
124	C₂×C₈	C₂×Dic₁₄₂	1	1	yes	characteristic
125	C₄²	C₄⋊Dic₇₁	1	1	yes	characteristic
126	C₄²	D₅₆₈	1	1	yes	characteristic
127	C₄²	Dic₂₈₄	1	1	yes	characteristic
128	C₄²	C₂×D₂₈₄	1	1	yes	characteristic
129	C₁₆	—	2	1	no	—
130	C₁₆	—	2	1	no	—
131	C₁₆	#11765849	1	2	yes	—
132	C₁₆	#11765849	1	2	yes	—
133	C₂³	#14556062	1	1	yes	characteristic
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	2	1	no	—
138	C₂×C₄	#14556062	1	1	yes	characteristic
139	C₂×C₄	#14556062	1	1	yes	characteristic
140	C₂×C₄	#14556062	1	1	yes	characteristic
141	C₂×C₄	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
142	C₂×C₄	C₂×D₅₆₈	1	1	yes	characteristic
143	C₂×C₄	C₂×Dic₂₈₄	1	1	yes	characteristic
144	C₈	—	2	1	no	—
145	C₈	—	2	1	no	—
146	C₈	—	568	4	no	—
147	C₈	#14556064	1	2	yes	—
148	C₈	C₂×#11765849	1	1	yes	characteristic
149	C₈	C₂×#11765849	1	1	yes	characteristic
150	C₂²	—	2	1	no	—
151	C₂²	—	2	1	no	—
152	C₂²	#17985829	1	2	yes	—
153	C₂²	C₂×#14556062	1	1	yes	characteristic
154	C₄	—	2	1	no	—
155	C₄	—	2	1	no	—
156	C₄	—	2	1	no	—
157	C₄	—	2	1	no	—
158	C₄	#17986023	1	1	yes	characteristic
159	C₄	C₂×#14556064	1	1	yes	characteristic
160	C₄	#17986431	1	2	yes	—
161	C₂	—	2	2	no	—
162	C₂	#22587099	1	1	yes	characteristic
163	C₂	#22590124	1	1	yes	characteristic
164	C₂	C₂×#17985829	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₅₆₈
commutator	C₂×C₈×C₇₁	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₇₁
Fitting	C₂×C₄×C₁₆×C₇₁	C₂
socle	C₂²×C₇₁	C₂×#107
prime radical	C₂³×C₇₁	#107
cosocle	C₄×C₈×C₇₁	C₂³
hypercenter	C₂×C₄×C₁₆	D₇₁
hypocenter	C₇₁	#12024
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
2. #22586414	C₂	semidirect
3. #22586412	C₂	semidirect
87. C₇₁	#12024	semidirect

Maximal subgroups

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

subgroup	index
#22587103	2
#22586414	2
#22586412	2
C₂×C₄×C₁₆×C₇₁	2
#12024	71