Finite Group Databasebeta

← #31884136

Subgroups — #31884136

Order 18176.

240 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 45 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#22589651C211yesmaximal, characteristic
2#22586280C211yesmaximal, characteristic, complemented
3C71×#1290C211yesmaximal, characteristic, complemented
4C22×D568C411yescharacteristic
5C2×#14556050—21no—
6#17985901—21no—
7C22⋊(C71⋊C16)—21no—
8#17985762C411yescharacteristic
9#432×C71—21no—
10C2×C4⋊C8×C71C2211yescharacteristic
11C22×D284—21no—
12C2×D568—41no—
13C2×D568—42no—
14C2×C4⋊Dic71—21no—
15C2×(C2×D71)⋊C4—41no—
16C2×(C2×D71)⋊C4—41no—
17C2×C71⋊C16—41no—
18#14556050—42no—
19C22×C8×C71C2×C411yescharacteristic
20C22×C8×C71D411yescharacteristic
21C2×C42×C71D411yescharacteristic
22C4⋊C8×C71—41no—
23C4⋊C8×C71—41no—
24C22⋊C8×C71—41no—
25#97×C71—22no—
26C23×D71—41no—
27C2×D284—22no—
28C2×D284—41no—
29C2×D284—41no—
30C2×D284—41no—
31C2×D284—41no—
32C2×D284—41no—
33C22×Dic71—41no—
34D568—42no—
35D568—42no—
36C4⋊Dic71—41no—
37C4⋊Dic71—41no—
38(C2×D71)⋊C4—82no—
39(C2×D71)⋊C4—82no—
40C71⋊C16—81no—
41C22×C4×C71C22⋊C411yescharacteristic
42C22×C4×C71D811yescharacteristic
43C22×C4×C71SD1611yescharacteristic
44C2×C8×C71—21no—
45C2×C8×C71—21no—
46C2×C8×C71—41no—
47C2×C8×C71—41no—
48C2×C8×C71—42no—
49C42×C71—21no—
50C42×C71—21no—
51C23×C71#10211yescharacteristic
52C22×D71—42no—
53C22×D71—81no—
54C22×D71—81no—
55C22×D71—81no—
56C22×D71—81no—
57C22×D71—81no—
58C22×D71—81no—
59C2×C4×C71—21no—
60C2×C4×C71—21no—
61C2×C4×C71—21no—
62C2×C4×C71—21no—
63C2×C4×C71—21no—
64C2×C4×C71—21no—
65C2×C4×C71—41no—
66C2×C4×C71#9911yescharacteristic
67C2×C4×C71#10411yescharacteristic
68C2×Dic71—81no—
69C2×Dic71—81no—
70C2×Dic71—81no—
71D284—42no—
72D284—42no—
73D284—81no—
74D284—81no—
75C8×C71—41no—
76C8×C71—41no—
77C8×C71—82no—
78C22×C71—21no—
79C22×C71—21no—
80C22×C71#32711yescharacteristic
81C22×C71#35911yescharacteristic
82C22×C71#36111yescharacteristic
83C2×D71—82no—
84C2×D71—82no—
85C2×D71—82no—
86C2×D71—161no—
87C2×D71—161no—
88C2×D71—161no—
89C2×D71—161no—
90C4×C71—21no—
91C4×C71—41no—
92C4×C71—41no—
93C4×C71AΣL(1,𝔽16)11yescharacteristic
94C4×C71#35511yescharacteristic
95Dic71—161no—
96#7080—711nomaximal, Sylow
97C2×C71—21no—
98C2×C71—21no—
99C2×C71#133911yescharacteristic
100C2×C71#134511yescharacteristic
101C2×C71#134911yescharacteristic
102D71—162no—
103#2054—711no—
104#1322—711no—
105#1290D7111yescharacteristic, complemented
106C71#708011yescharacteristic, complemented, Sylow
107C22×D8—711no—
108#432—21no—
109C2×C4⋊C8C2×D7111yescharacteristic
110C2×#102—1421no—
111#390—1421no—
112C22⋊C16—1421no—
113#340—711no—
114C22×D4—1421no—
115C2×D8—2841no—
116C2×D8—2842no—
117C22×C8C4×D7111yescharacteristic
118C22×C8D28411yescharacteristic
119C2×C4⋊C4—1421no—
120C2×C22⋊C4—2841no—
121C2×C22⋊C4—2841no—
122C2×C42C22⋊D7111yescharacteristic
123C2×C16—2841no—
124C4⋊C8—41no—
125C4⋊C8—41no—
126#102—2842no—
127C22⋊C8—41no—
128#97—22no—
129C24—2841no—
130C2×D4—2841no—
131C2×D4—2841no—
132C2×D4—2841no—
133C2×D4—2841no—
134C2×D4—2841no—
135C2×D4—1422no—
136C22×C4—2841no—
137C22×C4(C2×D71)⋊C411yescharacteristic
138C22×C4#1176585011yescharacteristic
139C22×C4#1176585211yescharacteristic
140D8—2842no—
141D8—2842no—
142C2×C8—21no—
143C2×C8—21no—
144C2×C8—41no—
145C2×C8—41no—
146C2×C8—42no—
147C4⋊C4—2841no—
148C4⋊C4—2841no—
149C22⋊C4—5682no—
150C22⋊C4—5682no—
151C42—21no—
152C42—21no—
153C16—5681no—
154C23—2842no—
155C23—5681no—
156C23—5681no—
157C23—5681no—
158C23—5681no—
159C23—5681no—
160C23—5681no—
161C23#1455604811yescharacteristic
162D4—2842no—
163D4—2842no—
164D4—5681no—
165D4—5681no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—41no—
173C2×C4—5681no—
174C2×C4—5681no—
175C2×C4—5681no—
176C2×C4#1455604211yescharacteristic
177C2×C4#1455605611yescharacteristic
178C8—41no—
179C8—41no—
180C8—82no—
181C22—21no—
182C22—21no—
183C22—5682no—
184C22—5682no—
185C22—5682no—
186C22—11361no—
187C22—11361no—
188C22—11361no—
189C22—11361no—
190C22#1798573211yescharacteristic
191C22#1798580511yescharacteristic
192C22#1798581111yescharacteristic
193C4—21no—
194C4—41no—
195C4—41no—
196C4—11361no—
197C4#1798578711yescharacteristic
198C4#1798579511yescharacteristic
199C2—21no—
200C2—21no—
201C2—11362no—
202C2#2258632411yescharacteristic
203C2#2258634211yescharacteristic
204C2#2258635411yescharacteristic
subgroupG/H
centerC22#17985732
commutatorC22×C8×C71C2×C4
FrattiniC2×C4⋊C8C2×D71
FittingC71×#1290C2
socleC22×C71#327
prime radicalC22×D568C4
cosocleC2×C4⋊C8×C71C22
hypercenter#1290D71
hypocenterC71#7080
LCSG ▷ C22×C8×C71 ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C8 ◁ #1290

Complements

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

NH ≅ G/Ntype
2. #22586280C2semidirect
3. C71×#1290C2semidirect
105. #1290D71semidirect
106. C71#7080semidirect

Maximal subgroups

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

subgroupindex
#225896512
#225862802
C71×#12902
#708071