Finite Group Databasebeta

← #31911713

Subgroups — #31911713

Order 18176.

216 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 65 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22590592C211yesmaximal, characteristic
2#22590242C211yesmaximal, characteristic
3#22587711C211yesmaximal, characteristic
4#22587707C211yesmaximal, characteristic
5#22586411C211yesmaximal, characteristic
6#22586390C211yesmaximal, characteristic
7C71×#1362C211yesmaximal, characteristic
8C2×#14556074—21no—
9C2×#14556074—21no—
10#17986379C2211yescharacteristic
11#17986285—21no—
12#17986281—21no—
13C8×Dic142C2211yescharacteristic
14Dic71⋊C16—21no—
15C4×C71⋊C16C2211yescharacteristic
16C16×Dic71—21no—
17#17985752C2211yescharacteristic
18#17985745—21no—
19#17985744—21no—
20#494×C71C2211yescharacteristic
21C4⋊C16×C71C2211yescharacteristic
22#335×C71C2211yescharacteristic
23C2×Dic284—21no—
24C2×Dic284—21no—
25#14556128—21no—
26C4×Dic142D411yescharacteristic
27#14556074—42no—
28#14556074—42no—
29C2×C71⋊C16D411yescharacteristic
30C2×C71⋊C16D411yescharacteristic
31#14556060—41no—
32#14556059—41no—
33C4⋊(C71⋊C8)—21no—
34C4⋊(C71⋊C8)D411yescharacteristic
35Dic71⋊C8—21no—
36#14556053—41no—
37#14556051—41no—
38C8×Dic71—21no—
39C2×Q16×C71—21no—
40C2×Q16×C71—21no—
41C4⋊Q8×C71—21no—
42C2×C16×C71—21no—
43C4⋊C8×C71—21no—
44C4×C8×C71C2311yescharacteristic
45C2×Dic142—41no—
46C2×Dic142D811yescharacteristic
47Dic284—41no—
48Dic284—41no—
49Dic284—81no—
50C2×C71⋊C8—21no—
51C2×C71⋊C8—41no—
52C4⋊Dic71—41no—
53C4⋊Dic71—41no—
54C4⋊Dic71D811yescharacteristic
55Dic71⋊C4—21no—
56C4×Dic71—21no—
57C71⋊C16—21no—
58C71⋊C16SD1612yes—
59C2×Q8×C71—41no—
60Q16×C71—41no—
61Q16×C71—41no—
62Q16×C71—81no—
63C2×C8×C71—41no—
64C2×C8×C71C2×D411yescharacteristic
65C2×C8×C71#4111yescharacteristic
66C4⋊C4×C71—41no—
67C4⋊C4×C71—41no—
68C42×C71C2×D411yescharacteristic
69C16×C71—41no—
70Q8×C71—81no—
71Q8×C71—81no—
72C2×C4×C71—81no—
73C2×C4×C71#12111yescharacteristic
74C2×C4×C71C2×D811yescharacteristic
75C2×C4×C71#13511yescharacteristic
76C2×Dic71—21no—
77C2×Dic71—21no—
78C2×Dic71—81no—
79Dic142—21no—
80Dic142—81no—
81Dic142—81no—
82Dic142Q3212yescomplemented
83C8×C71—21no—
84C8×C71—81no—
85C8×C71C2×SD1611yescharacteristic
86C8×C71AGL(1,ℤ/8)11yescharacteristic
87C71⋊C8—41no—
88C71⋊C8—81no—
89C22×C71#46611yescharacteristic
90C4×C71—21no—
91C4×C71—161no—
92C4×C71#46011yescharacteristic
93C4×C71#47211yescharacteristic
94C4×C71C2×Q3211yescharacteristic
95C4×C71#50811yescharacteristic
96Dic71—22no—
97Dic71—41no—
98Dic71—161no—
99#12440—711nomaximal, Sylow
100C2×C71#166811yescharacteristic
101C2×C71#221611yescharacteristic
102C2×C71#222211yescharacteristic
103#2247—711no—
104Q8×C16—711no—
105#1670—711no—
106#1670—711no—
107#1371—711no—
108#1362—711no—
109#1362D7111yescharacteristic
110C71#1244011yescharacteristic, complemented, Sylow
111C2×Q32—1421no—
112C2×Q32—1421no—
113#494—711no—
114#494C2×D7111yescharacteristic
115#475—1421no—
116#475—1421no—
117C8×Q8—711no—
118C4⋊C16—1421no—
119C4⋊C16C2×D7111yescharacteristic
120C4×C16—711no—
121C4×C16—1421no—
122#335—711no—
123#335C2×D7111yescharacteristic
124#332—1421no—
125#332—1421no—
126C2×Q16—21no—
127C2×Q16—21no—
128C2×Q16—1421no—
129C2×Q16—1421no—
130C4⋊Q8—21no—
131C4⋊Q8—1421no—
132C4×Q8—711no—
133Q32—2842no—
134Q32—2842no—
135C2×C16—21no—
136C2×C16—711no—
137C2×C16—711no—
138#106—2841no—
139#106—2841no—
140C4⋊C8—21no—
141C4⋊C8—711no—
142C4⋊C8—1421no—
143C4⋊C8—1421no—
144#103—2841no—
145#103—2841no—
146C4×C8—1421no—
147C4×C8C22×D7111yescharacteristic
148C2×Q8—41no—
149C2×Q8—2841no—
150C2×Q8—711no—
151Q16—41no—
152Q16—41no—
153Q16—81no—
154Q16—2841no—
155Q16—2841no—
156Q16—5681no—
157C2×C8—41no—
158C2×C8—2841no—
159C2×C8—1421no—
160C2×C8D4×D7111yescharacteristic
161C2×C8#1176586511yescharacteristic
162C4⋊C4—41no—
163C4⋊C4—41no—
164C4⋊C4—2841no—
165C4⋊C4—2841no—
166C4⋊C4—711no—
167C4⋊C4—1421no—
168C42—1421no—
169C42D4×D7111yescharacteristic
170C16—41no—
171C16—712no—
172C16—1421no—
173Q8—81no—
174Q8—81no—
175Q8—5681no—
176Q8—5681no—
177Q8—712no—
178Q8—1421no—
179C2×C4—81no—
180C2×C4—5681no—
181C2×C4—1421no—
182C2×C4—1421no—
183C2×C4#1455610111yescharacteristic
184C2×C4D8×D7111yescharacteristic
185C2×C4#1455614911yescharacteristic
186C8—21no—
187C8—81no—
188C8—2841no—
189C8—5681no—
190C8SD16×D7111yescharacteristic
191C8#1455615211yescharacteristic
192C22#1798623411yescharacteristic
193C4—21no—
194C4—161no—
195C4—2841no—
196C4—11361no—
197C4—1422no—
198C4#1798619811yescharacteristic
199C4#1798626811yescharacteristic
200C4Q32×D7111yescharacteristic
201C4#1798643011yescharacteristic
202C2#2258769911yescharacteristic
203C2#2259043011yescharacteristic
204C2#2259046411yescharacteristic
subgroupG/H
centerC22#17986234
commutator, cosocleC4×C8×C71C23
FrattiniC4×C8C22×D71
FittingC71×#1362C2
socle, prime radicalC22×C71#466
hypercenter#1362D71
hypocenterC71#12440
LCSG ▷ C4×C8×C71 ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1362

Complements

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

NH ≅ G/Ntype
82. Dic142Q32semidirect
110. C71#12440semidirect

Maximal subgroups

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

subgroupindex
#225905922
#225902422
#225877112
#225877072
#225864112
#225863902
C71×#13622
#1244071