Subgroups — #31913900

Order 18176.

314 conjugacy classes of subgroups.

The subgroup lattice has 170 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17985828	C₂	1	4	yes	maximal
2	C₂×C₄×C₇₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	#22588512	C₂	1	1	yes	maximal, characteristic
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	#17985828	—	2	8	no	—
10	C₄×C₇₁⋊C₁₆	C₄	1	2	yes	—
11	C₄×C₇₁⋊C₁₆	C₄	1	2	yes	—
12	C₂×#108×C₇₁	—	2	1	no	—
13	C₂×#108×C₇₁	C₂²	1	2	yes	—
14	#423×C₇₁	—	2	1	no	—
15	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
16	C₂×#11765849	—	2	2	no	—
17	C₂×C₇₁⋊C₁₆	C₂×C₄	1	4	yes	—
18	C₂×C₇₁⋊C₁₆	D₄	1	4	yes	—
19	C₂×C₇₁⋊C₁₆	Q₈	1	4	yes	—
20	#14556064	—	2	4	no	—
21	#14556064	—	4	2	no	—
22	C₄⋊(C₇₁⋊C₈)	—	4	2	no	—
23	C₂×M₁₆×C₇₁	—	2	2	no	—
24	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
26	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
27	#108×C₇₁	—	2	4	no	—
28	#108×C₇₁	—	4	2	no	—
29	C₄⋊C₈×C₇₁	—	4	2	no	—
30	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₇₁	Q₈	1	1	yes	characteristic
33	C₄×C₈×C₇₁	Q₈	1	1	yes	characteristic
34	#11765849	—	4	2	no	—
35	#11765849	—	4	2	no	—
36	C₂×C₇₁⋊C₈	—	4	2	no	—
37	C₇₁⋊C₁₆	C₄⋊C₄	1	8	yes	—
38	C₂²×C₄×C₇₁	Q₁₆	1	2	yes	—
39	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
40	M₁₆×C₇₁	—	4	2	no	—
41	M₁₆×C₇₁	—	4	2	no	—
42	C₂×C₈×C₇₁	—	4	2	no	—
43	C₂×C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₇₁	D₈	1	2	yes	—
46	C₂×C₈×C₇₁	Q₁₆	1	2	yes	—
47	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
49	C₂×C₈×C₇₁	C₂×Q₈	1	2	yes	—
50	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
51	C₄²×C₇₁	C₄⋊C₄	1	1	yes	characteristic
52	C₄²×C₇₁	C₄⋊C₄	1	1	yes	characteristic
53	C₄²×C₇₁	D₈	1	2	yes	—
54	C₂³×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
55	C₂×C₄×C₇₁	—	2	4	no	—
56	C₂×C₄×C₇₁	#107	1	2	yes	—
57	C₂×C₄×C₇₁	#107	1	2	yes	—
58	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₄×C₇₁	#127	1	1	yes	characteristic
60	C₂×C₄×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
61	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
62	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
63	C₂×C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
64	C₈×C₇₁	—	8	2	no	—
65	C₈×C₇₁	#107	1	2	yes	—
66	C₈×C₇₁	#107	1	2	yes	—
67	C₈×C₇₁	C₂×C₄⋊C₄	1	2	yes	—
68	C₈×C₇₁	C₄⋊Q₈	1	2	yes	—
69	C₇₁⋊C₈	—	8	2	no	—
70	C₂²×C₇₁	—	2	2	no	—
71	C₂²×C₇₁	C₂×#107	1	1	yes	characteristic
72	C₂²×C₇₁	#494	1	1	yes	characteristic
73	C₂²×C₇₁	#500	1	1	yes	characteristic
74	C₄×C₇₁	—	2	2	no	—
75	C₄×C₇₁	#368	1	4	yes	complemented
76	C₄×C₇₁	#384	1	1	yes	characteristic
77	C₄×C₇₁	C₂×#107	1	1	yes	characteristic
78	C₄×C₇₁	#493	1	1	yes	characteristic
79	C₄×C₇₁	#500	1	1	yes	characteristic
80	#12909	—	71	1	no	maximal, Sylow
81	C₂×C₇₁	—	2	1	no	—
82	C₂×C₇₁	—	2	1	no	—
83	C₂×C₇₁	#1845	1	1	yes	characteristic
84	C₂×C₇₁	C₂×#368	1	2	yes	—
85	C₂×#368	—	71	4	no	—
86	C₂×C₄×C₁₆	—	71	1	no	—
87	#1848	—	71	1	no	—
88	#1848	D₇₁	1	1	yes	characteristic
89	C₇₁	#12909	1	1	yes	characteristic, complemented, Sylow
90	C₂²×C₁₆	—	71	2	no	—
91	C₂×#108	—	2	1	no	—
92	C₂×#108	—	71	2	no	—
93	C₂×#108	—	142	1	no	—
94	C₂×#108	C₂×D₇₁	1	2	yes	—
95	#423	—	2	1	no	—
96	#423	—	142	1	no	—
97	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
98	#368	—	142	8	no	—
99	C₄×C₁₆	—	71	2	no	—
100	C₄×C₁₆	—	71	2	no	—
101	C₂×M₁₆	—	2	2	no	—
102	C₂×M₁₆	—	142	2	no	—
103	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
104	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
105	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
106	C₂×C₁₆	—	71	4	no	—
107	C₂×C₁₆	—	71	4	no	—
108	C₂×C₁₆	—	71	4	no	—
109	#108	—	2	4	no	—
110	#108	—	4	2	no	—
111	#108	—	284	2	no	—
112	#108	—	142	4	no	—
113	C₄⋊C₈	—	4	2	no	—
114	C₄⋊C₈	—	284	2	no	—
115	C₄×C₈	Dic₁₄₂	1	1	yes	characteristic
116	C₄×C₈	Dic₁₄₂	1	1	yes	characteristic
117	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
118	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
119	C₂²×C₄	#11765855	1	2	yes	—
120	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
121	M₁₆	—	4	2	no	—
122	M₁₆	—	4	2	no	—
123	M₁₆	—	284	2	no	—
124	M₁₆	—	284	2	no	—
125	C₂×C₈	—	4	2	no	—
126	C₂×C₈	—	284	2	no	—
127	C₂×C₈	Dic₇₁⋊C₄	1	1	yes	characteristic
128	C₂×C₈	Dic₇₁⋊C₄	1	1	yes	characteristic
129	C₂×C₈	#11765850	1	2	yes	—
130	C₂×C₈	#11765855	1	2	yes	—
131	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
132	C₂×C₈	D₄×D₇₁	1	2	yes	—
133	C₂×C₈	Q₈×D₇₁	1	2	yes	—
134	C₂×C₈	C₂×Dic₁₄₂	1	1	yes	characteristic
135	C₄²	Dic₇₁⋊C₄	1	1	yes	characteristic
136	C₄²	Dic₇₁⋊C₄	1	1	yes	characteristic
137	C₄²	#11765850	1	2	yes	—
138	C₁₆	—	71	8	no	—
139	C₂³	C₂×#11765855	1	1	yes	characteristic
140	C₂×C₄	—	2	4	no	—
141	C₂×C₄	#14556061	1	2	yes	—
142	C₂×C₄	#14556061	1	2	yes	—
143	C₂×C₄	C₂×Dic₇₁⋊C₄	1	1	yes	characteristic
144	C₂×C₄	#14556124	1	1	yes	characteristic
145	C₂×C₄	#14556126	1	1	yes	characteristic
146	C₂×C₄	C₂×#11765850	1	1	yes	characteristic
147	C₂×C₄	C₂×#11765850	1	1	yes	characteristic
148	C₂×C₄	C₂×#11765855	1	1	yes	characteristic
149	C₈	—	8	2	no	—
150	C₈	—	568	2	no	—
151	C₈	#14556061	1	2	yes	—
152	C₈	#14556061	1	2	yes	—
153	C₈	C₄⋊C₄×D₇₁	1	2	yes	—
154	C₈	#14556127	1	2	yes	—
155	C₂²	—	2	2	no	—
156	C₂²	C₂×#14556061	1	1	yes	characteristic
157	C₂²	#17986378	1	1	yes	characteristic
158	C₂²	#17986402	1	1	yes	characteristic
159	C₄	—	2	2	no	—
160	C₄	#17985828	1	4	yes	complemented
161	C₄	#17985868	1	1	yes	characteristic
162	C₄	C₂×#14556061	1	1	yes	characteristic
163	C₄	#17986375	1	1	yes	characteristic
164	C₄	#17986402	1	1	yes	characteristic
165	C₂	—	2	1	no	—
166	C₂	—	2	1	no	—
167	C₂	#22588499	1	1	yes	characteristic
168	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	#12909
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
75. C₄×C₇₁	#368	semidirect
89. C₇₁	#12909	semidirect
160. C₄	#17985828	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17985828	2
C₂×C₄×C₇₁⋊C₁₆	2
#22588512	2
C₇₁×#1848	2
#12909	71