Subgroups — #31916065

Order 18176.

350 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22598854	C₂	1	1	yes	maximal, characteristic
2	#22588512	C₂	1	1	yes	maximal, characteristic
3	#22586436	C₂	1	4	yes	maximal
4	C₂×C₄×C₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic
5	#17986556	—	2	1	no	—
6	#17986404	C₄	1	2	yes	—
7	#17986401	—	2	1	no	—
8	#17986401	C₄	1	2	yes	—
9	#17986397	—	2	1	no	—
10	C₂×#14556064	—	2	1	no	—
11	C₂×#14556064	C₂²	1	2	yes	—
12	#17986037	—	2	1	no	—
13	#17986037	C₂²	1	2	yes	—
14	#17986023	—	2	1	no	—
15	C₂²×C₁₆×C₇₁	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
17	C₄×C₁₆×C₇₁	—	2	1	no	—
18	C₄×C₁₆×C₇₁	—	2	1	no	—
19	C₂×#11765849	—	2	2	no	—
20	#14556128	—	2	2	no	—
21	#14556120	—	2	2	no	—
22	#14556107	—	4	2	no	—
23	C₄×Dic₁₄₂	—	4	2	no	—
24	#14556087	—	2	2	no	—
25	#14556064	—	2	4	no	—
26	#14556064	—	4	2	no	—
27	#14556062	—	2	2	no	—
28	#14556062	C₂×C₄	1	4	yes	—
29	#14556060	—	2	2	no	—
30	#14556060	—	2	2	no	—
31	C₄⋊(C₇₁⋊C₈)	—	4	2	no	—
32	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
34	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
35	C₂×C₁₆×C₇₁	—	2	2	no	—
36	C₂×C₁₆×C₇₁	—	2	2	no	—
37	C₂×C₁₆×C₇₁	Q₈	1	4	yes	complemented
38	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
40	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
41	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
42	C₂×Dic₁₄₂	—	4	2	no	—
43	#11765849	—	4	2	no	—
44	#11765849	—	4	2	no	—
45	C₂×C₇₁⋊C₈	—	4	2	no	—
46	C₄⋊Dic₇₁	—	2	4	no	—
47	C₄⋊Dic₇₁	—	4	2	no	—
48	Dic₇₁⋊C₄	—	4	4	no	—
49	C₂²⋊Dic₇₁	—	4	2	no	—
50	C₄×Dic₇₁	—	4	2	no	—
51	C₂²×C₄×C₇₁	SD₁₆	1	2	yes	—
52	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
53	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₇₁	SD₁₆	1	2	yes	—
56	C₂×C₈×C₇₁	Q₁₆	1	2	yes	—
57	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₇₁	C₂×Q₈	1	2	yes	—
60	C₂×C₈×C₇₁	#41	1	2	yes	—
61	C₄²×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
62	C₄²×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
63	C₄²×C₇₁	D₈	1	2	yes	—
64	C₁₆×C₇₁	—	2	4	no	—
65	C₂³×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
66	C₂×C₄×C₇₁	—	2	4	no	—
67	C₂×C₄×C₇₁	#102	1	2	yes	—
68	C₂×C₄×C₇₁	#102	1	2	yes	—
69	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₇₁	#124	1	1	yes	characteristic
71	C₂×C₄×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
72	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
73	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
74	C₂×C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
75	C₂×Dic₇₁	—	4	4	no	—
76	Dic₁₄₂	—	8	4	no	—
77	C₈×C₇₁	#103	1	2	yes	—
78	C₈×C₇₁	#103	1	2	yes	—
79	C₈×C₇₁	C₄×Q₈	1	2	yes	—
80	C₈×C₇₁	C₂²⋊Q₈	1	2	yes	—
81	C₇₁⋊C₈	—	8	2	no	—
82	C₂²×C₇₁	—	2	2	no	—
83	C₂²×C₇₁	C₂×#102	1	1	yes	characteristic
84	C₂²×C₇₁	#487	1	1	yes	characteristic
85	C₂²×C₇₁	#498	1	1	yes	characteristic
86	C₄×C₇₁	—	2	2	no	—
87	C₄×C₇₁	#359	1	4	yes	—
88	C₄×C₇₁	#391	1	1	yes	characteristic
89	C₄×C₇₁	C₂×#103	1	1	yes	characteristic
90	C₄×C₇₁	#486	1	1	yes	characteristic
91	C₄×C₇₁	#500	1	1	yes	characteristic
92	Dic₇₁	—	8	4	no	—
93	#13285	—	71	1	no	maximal, Sylow
94	C₂×C₇₁	—	2	1	no	—
95	C₂×C₇₁	—	2	1	no	—
96	C₂×C₇₁	#1990	1	1	yes	characteristic
97	C₂×C₇₁	C₂×#359	1	2	yes	—
98	#3160	—	71	1	no	—
99	C₂×C₄×C₁₆	D₇₁	1	1	yes	characteristic
100	#1848	—	71	1	no	—
101	#1382	—	71	4	no	—
102	C₇₁	#13285	1	1	yes	characteristic, complemented, Sylow
103	#533	—	142	1	no	—
104	C₂²×C₁₆	C₂×D₇₁	1	2	yes	—
105	#500	—	71	2	no	—
106	#499	—	71	2	no	—
107	#499	—	142	1	no	—
108	#498	—	142	1	no	—
109	C₂×#108	—	71	2	no	—
110	C₂×#108	—	142	1	no	—
111	#427	—	71	2	no	—
112	#427	—	142	1	no	—
113	#423	—	142	1	no	—
114	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
115	C₄×C₁₆	—	2	1	no	—
116	C₄×C₁₆	—	2	1	no	—
117	C₂×M₁₆	—	142	2	no	—
118	C₂²×C₈	D₂₈₄	1	1	yes	characteristic
119	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
120	C₄⋊Q₈	—	142	2	no	—
121	#125	—	142	2	no	—
122	C₂²⋊Q₈	—	284	2	no	—
123	C₄×Q₈	—	284	2	no	—
124	#117	—	142	2	no	—
125	C₂×C₄²	D₂₈₄	1	1	yes	characteristic
126	C₂×C₁₆	—	2	2	no	—
127	C₂×C₁₆	—	2	2	no	—
128	C₂×C₁₆	Dic₁₄₂	1	4	yes	complemented
129	#108	—	284	2	no	—
130	#108	—	142	4	no	—
131	#107	—	71	4	no	—
132	#107	—	142	2	no	—
133	#106	—	142	2	no	—
134	#106	—	142	2	no	—
135	C₄⋊C₈	—	284	2	no	—
136	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
137	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
138	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
139	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
140	C₂×Q₈	—	284	2	no	—
141	C₂²×C₄	#11765854	1	2	yes	—
142	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
143	M₁₆	—	284	2	no	—
144	M₁₆	—	284	2	no	—
145	C₂×C₈	—	284	2	no	—
146	C₂×C₈	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
147	C₂×C₈	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
148	C₂×C₈	#11765854	1	2	yes	—
149	C₂×C₈	Dic₂₈₄	1	2	yes	—
150	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
151	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
152	C₂×C₈	C₂×Dic₁₄₂	1	2	yes	—
153	C₂×C₈	#11765865	1	2	yes	—
154	C₄⋊C₄	—	284	4	no	—
155	C₄⋊C₄	—	284	2	no	—
156	C₄⋊C₄	—	142	4	no	—
157	C₂²⋊C₄	—	284	2	no	—
158	C₄²	—	284	2	no	—
159	C₄²	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
160	C₄²	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
161	C₄²	D₅₆₈	1	2	yes	—
162	C₁₆	—	2	4	no	—
163	C₂³	C₂×#11765854	1	1	yes	characteristic
164	Q₈	—	568	4	no	—
165	C₂×C₄	—	2	4	no	—
166	C₂×C₄	—	284	4	no	—
167	C₂×C₄	#14556050	1	2	yes	—
168	C₂×C₄	#14556050	1	2	yes	—
169	C₂×C₄	C₂×(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
170	C₂×C₄	#14556116	1	1	yes	characteristic
171	C₂×C₄	#14556128	1	1	yes	characteristic
172	C₂×C₄	C₂×D₅₆₈	1	1	yes	characteristic
173	C₂×C₄	C₂×#11765854	1	1	yes	characteristic
174	C₂×C₄	C₂×Dic₂₈₄	1	1	yes	characteristic
175	C₈	—	568	2	no	—
176	C₈	#14556053	1	2	yes	—
177	C₈	#14556053	1	2	yes	—
178	C₈	C₄×Dic₁₄₂	1	2	yes	—
179	C₈	#14556107	1	2	yes	—
180	C₂²	—	2	2	no	—
181	C₂²	C₂×#14556050	1	1	yes	characteristic
182	C₂²	#17986350	1	1	yes	characteristic
183	C₂²	#17986397	1	1	yes	characteristic
184	C₄	—	2	2	no	—
185	C₄	—	568	4	no	—
186	C₄	#17985807	1	4	yes	—
187	C₄	#17985905	1	1	yes	characteristic
188	C₄	C₂×#14556053	1	1	yes	characteristic
189	C₄	#17986346	1	1	yes	characteristic
190	C₄	#17986404	1	1	yes	characteristic
191	C₂	—	2	1	no	—
192	C₂	—	2	1	no	—
193	C₂	#22589301	1	1	yes	characteristic
194	C₂	C₂×#17985807	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₅₆₈
commutator	C₂×C₈×C₇₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₂×C₄×C₁₆×C₇₁	C₂
socle	C₂²×C₇₁	C₂×#102
prime radical	C₂³×C₇₁	C₂×SD₁₆
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	C₂×C₄×C₁₆	D₇₁
hypocenter	C₇₁	#13285
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
37. C₂×C₁₆×C₇₁	Q₈	semidirect
102. C₇₁	#13285	semidirect
128. C₂×C₁₆	Dic₁₄₂	semidirect

Maximal subgroups

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

subgroup	index
#22598854	2
#22588512	2
#22586436	2
C₂×C₄×C₁₆×C₇₁	2
#13285	71