Subgroups — #31909035

Order 18176.

242 conjugacy classes of subgroups.

The subgroup lattice has 184 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	#22587088	C₂	1	1	yes	maximal, characteristic
3	#22586413	C₂	1	2	yes	maximal, complemented
4	#22586411	C₂	1	2	yes	maximal, complemented
5	C₇₁×#1564	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₇₁⋊C₁₆	C₄	1	1	yes	characteristic
7	C₂²×C₇₁⋊C₁₆	C₄	1	1	yes	characteristic
8	#17986052	—	2	1	no	—
9	C₄×C₇₁⋊C₁₆	C₄	1	1	yes	characteristic
10	C₄×C₇₁⋊C₁₆	C₄	1	1	yes	characteristic
11	C₄×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
12	C₄×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
13	#17985752	C₂²	1	2	yes	—
14	#17985750	—	2	1	no	—
15	#431×C₇₁	—	2	1	no	—
16	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
17	#335×C₇₁	C₂²	1	2	yes	—
18	#334×C₇₁	—	2	1	no	—
19	C₂×C₇₁⋊C₁₆	—	2	2	no	—
20	C₂×C₇₁⋊C₁₆	C₈	1	4	yes	complemented
21	C₂×C₇₁⋊C₁₆	C₂×C₄	1	1	yes	characteristic
22	C₂×C₇₁⋊C₁₆	C₂×C₄	1	1	yes	characteristic
23	C₂×C₇₁⋊C₁₆	C₂×C₄	1	1	yes	characteristic
24	C₂×C₇₁⋊C₁₆	C₂×C₄	1	1	yes	characteristic
25	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
26	C₂²⋊(C₇₁⋊C₈)	—	4	1	no	—
27	C₄×C₇₁⋊C₈	—	4	1	no	—
28	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₇₁	Q₈	1	1	yes	characteristic
30	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
31	C₄⋊C₈×C₇₁	—	2	2	no	—
32	C₂²⋊C₈×C₇₁	—	4	1	no	—
33	C₄×C₈×C₇₁	—	4	1	no	—
34	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
35	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
36	C₄×C₈×C₇₁	Q₈	1	1	yes	characteristic
37	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
38	C₂×C₇₁⋊C₈	—	4	2	no	—
39	C₇₁⋊C₁₆	—	2	1	no	—
40	C₇₁⋊C₁₆	—	2	1	no	—
41	C₇₁⋊C₁₆	C₂×C₈	1	2	yes	—
42	C₇₁⋊C₁₆	C₂×C₈	1	2	yes	—
43	C₂²×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
44	C₂²×C₄×C₇₁	D₈	1	1	yes	characteristic
45	C₂²×C₄×C₇₁	Q₁₆	1	1	yes	characteristic
46	C₂×C₈×C₇₁	—	2	2	no	—
47	C₂×C₈×C₇₁	—	4	2	no	—
48	C₂×C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₈×C₇₁	C₂×C₈	1	2	yes	—
51	C₂×C₈×C₇₁	M₁₆	1	2	yes	—
52	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
54	C₄²×C₇₁	C₄⋊C₄	1	1	yes	characteristic
55	C₄²×C₇₁	D₈	1	1	yes	characteristic
56	C₄²×C₇₁	Q₁₆	1	1	yes	characteristic
57	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
58	C₂³×C₇₁	#107	1	1	yes	characteristic
59	C₂×C₄×C₇₁	—	2	2	no	—
60	C₂×C₄×C₇₁	—	2	2	no	—
61	C₂×C₄×C₇₁	C₄⋊C₈	1	2	yes	—
62	C₂×C₄×C₇₁	#107	1	1	yes	characteristic
63	C₂×C₄×C₇₁	#107	1	1	yes	characteristic
64	C₂×C₄×C₇₁	#107	1	1	yes	characteristic
65	C₂×C₄×C₇₁	#108	1	2	yes	—
66	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
68	C₂×C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
69	C₈×C₇₁	—	2	1	no	—
70	C₈×C₇₁	—	2	1	no	—
71	C₈×C₇₁	—	8	2	no	—
72	C₈×C₇₁	C₄⋊C₈	1	2	yes	—
73	C₈×C₇₁	C₂²×C₈	1	1	yes	characteristic
74	C₈×C₇₁	C₂×M₁₆	1	1	yes	characteristic
75	C₇₁⋊C₈	—	8	2	no	—
76	C₂²×C₇₁	—	2	1	no	—
77	C₂²×C₇₁	—	2	1	no	—
78	C₂²×C₇₁	#335	1	2	yes	—
79	C₂²×C₇₁	C₂×#107	1	1	yes	characteristic
80	C₄×C₇₁	—	2	1	no	—
81	C₄×C₇₁	—	2	1	no	—
82	C₄×C₇₁	#335	1	2	yes	—
83	C₄×C₇₁	#368	1	2	yes	—
84	C₄×C₇₁	C₂×C₄⋊C₈	1	1	yes	characteristic
85	C₄×C₇₁	C₂×#108	1	1	yes	characteristic
86	C₄×C₇₁	#508	1	2	yes	—
87	#12008	—	71	1	no	maximal, Sylow
88	C₂×C₇₁	—	2	2	no	—
89	C₂×C₇₁	C₂×#335	1	1	yes	characteristic
90	C₂×C₇₁	#2158	1	1	yes	characteristic
91	C₂×C₇₁	C₂×#368	1	1	yes	characteristic
92	C₂×C₄×C₁₆	—	71	1	no	—
93	#1564	—	71	1	no	—
94	#1564	D₇₁	1	1	yes	characteristic
95	#1372	—	71	2	no	—
96	#1371	—	71	2	no	—
97	C₇₁	#12008	1	1	yes	characteristic, complemented, Sylow
98	C₂²×C₁₆	—	71	1	no	—
99	C₂²×C₁₆	—	71	1	no	—
100	#431	—	2	1	no	—
101	#431	—	142	1	no	—
102	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
103	C₄×C₁₆	—	71	1	no	—
104	C₄×C₁₆	—	71	1	no	—
105	C₄×C₁₆	—	71	1	no	—
106	C₄×C₁₆	—	71	1	no	—
107	#335	—	71	2	no	—
108	#335	C₂×D₇₁	1	2	yes	—
109	#334	—	2	1	no	—
110	#334	—	142	1	no	—
111	C₂²×C₈	Dic₁₄₂	1	1	yes	characteristic
112	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
113	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
114	C₂×C₁₆	—	71	4	no	—
115	C₂×C₁₆	—	71	1	no	—
116	C₂×C₁₆	—	71	1	no	—
117	C₂×C₁₆	—	71	1	no	—
118	C₂×C₁₆	—	71	1	no	—
119	C₂×C₁₆	—	142	2	no	—
120	C₄⋊C₈	—	2	2	no	—
121	C₄⋊C₈	—	142	2	no	—
122	C₂²⋊C₈	—	4	1	no	—
123	C₂²⋊C₈	—	284	1	no	—
124	C₄×C₈	—	4	1	no	—
125	C₄×C₈	—	284	1	no	—
126	C₄×C₈	Dic₁₄₂	1	1	yes	characteristic
127	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
128	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
129	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
130	C₂²×C₄	Dic₇₁⋊C₄	1	1	yes	characteristic
131	C₂²×C₄	#11765850	1	1	yes	characteristic
132	C₂²×C₄	#11765855	1	1	yes	characteristic
133	C₂×C₈	—	2	2	no	—
134	C₂×C₈	—	4	2	no	—
135	C₂×C₈	—	284	2	no	—
136	C₂×C₈	Dic₇₁⋊C₄	1	1	yes	characteristic
137	C₂×C₈	Dic₇₁⋊C₄	1	1	yes	characteristic
138	C₂×C₈	C₈×D₇₁	1	2	yes	—
139	C₂×C₈	#11765848	1	2	yes	—
140	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
141	C₂×C₈	C₂×Dic₁₄₂	1	1	yes	characteristic
142	C₄²	Dic₇₁⋊C₄	1	1	yes	characteristic
143	C₄²	#11765850	1	1	yes	characteristic
144	C₄²	#11765855	1	1	yes	characteristic
145	C₄²	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
146	C₁₆	—	71	2	no	—
147	C₁₆	—	71	2	no	—
148	C₁₆	—	142	1	no	—
149	C₁₆	—	142	1	no	—
150	C₂³	#14556061	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	Dic₇₁⋊C₈	1	2	yes	—
154	C₂×C₄	#14556061	1	1	yes	characteristic
155	C₂×C₄	#14556061	1	1	yes	characteristic
156	C₂×C₄	#14556061	1	1	yes	characteristic
157	C₂×C₄	#14556063	1	2	yes	—
158	C₂×C₄	C₂×Dic₇₁⋊C₄	1	1	yes	characteristic
159	C₂×C₄	C₂×#11765850	1	1	yes	characteristic
160	C₂×C₄	C₂×#11765855	1	1	yes	characteristic
161	C₈	—	2	1	no	—
162	C₈	—	2	1	no	—
163	C₈	—	8	2	no	—
164	C₈	—	568	2	no	—
165	C₈	Dic₇₁⋊C₈	1	2	yes	—
166	C₈	C₂×C₈×D₇₁	1	1	yes	characteristic
167	C₈	C₂×#11765848	1	1	yes	characteristic
168	C₂²	—	2	1	no	—
169	C₂²	—	2	1	no	—
170	C₂²	#17985751	1	2	yes	—
171	C₂²	C₂×#14556061	1	1	yes	characteristic
172	C₄	—	2	1	no	—
173	C₄	—	2	1	no	—
174	C₄	#17985751	1	2	yes	—
175	C₄	#17985828	1	2	yes	—
176	C₄	C₂×Dic₇₁⋊C₈	1	1	yes	characteristic
177	C₄	C₂×#14556063	1	1	yes	characteristic
178	C₄	#17986429	1	2	yes	—
179	C₂	—	2	2	no	—
180	C₂	C₂×#17985751	1	1	yes	characteristic
181	C₂	#22590122	1	1	yes	characteristic
182	C₂	C₂×#17985828	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	#11765850
commutator	C₈×C₇₁	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₇₁
Fitting	C₇₁×#1564	C₂
socle	C₂²×C₇₁	C₂×#107
prime radical	C₂³×C₇₁	#107
cosocle	C₄×C₈×C₇₁	C₂³
hypercenter	#1564	D₇₁
hypocenter	C₇₁	#12008
LCS	G ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1564

Complements

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

N	H ≅ G/N	type
3. #22586413	C₂	semidirect
4. #22586411	C₂	semidirect
20. C₂×C₇₁⋊C₁₆	C₈	semidirect
97. C₇₁	#12008	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
#22587088	2
#22586413	2
#22586411	2
C₇₁×#1564	2
#12008	71