Finite Group Databasebeta

← #31894974

Subgroups — #31894974

Order 18176.

384 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 99 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#22598722C211yesmaximal, characteristic
2#22587058C212yesmaximal
3#22586978C211yesmaximal, characteristic
4#22586211C212yesmaximal
5C71×#2987C211yesmaximal, characteristic
6C2×#14556128C2211yescharacteristic
7#17986369—22no—
8#17986351C414yes—
9#17986056C2212yes—
10#17986007—22no—
11C4×#11765849C2211yescharacteristic
12#17985746—24no—
13#17985741—22no—
14#533×C71C2211yescharacteristic
15#446×C71—24no—
16#433×C71—22no—
17#432×C71C2212yes—
18C22×Dic142—21no—
19C2×#11765849—21no—
20#14556128—22no—
21#14556128—21no—
22#14556128C2×C412yes—
23C2×C4⋊Dic71—22no—
24C4⋊(C71⋊C8)—22no—
25#14556051—28no—
26C22⋊(C71⋊C8)—22no—
27#14556039—21no—
28#14556039—22no—
29C4×C71⋊C8C2×C412yes—
30C4⋊Q8×C71C2×C412yes—
31#125×C71D412yes—
32C22⋊Q8×C71—22no—
33C4×Q8×C71—22no—
34#117×C71D412yes—
35C2×C42×C71C2311yescharacteristic
36C4⋊C8×C71—24no—
37C4⋊C8×C71D414yes—
38C22⋊C8×C71—22no—
39#97×C71—22no—
40#97×C71—22no—
41C4×C8×C71—22no—
42C2×Dic142—22no—
43C2×Dic142—44no—
44C2×Dic142—41no—
45C22×Dic71—41no—
46#11765849—42no—
47C2×C71⋊C8—22no—
48C2×C71⋊C8—42no—
49C4⋊Dic71—24no—
50C4⋊Dic71—42no—
51C2×Q8×C71—22no—
52C22×C4×C71C2×D411yescharacteristic
53C22×C4×C71#4112yes—
54C2×C8×C71—24no—
55C4⋊C4×C71—22no—
56C4⋊C4×C71—24no—
57C4⋊C4×C71C22⋊C414yes—
58C22⋊C4×C71—22no—
59C42×C71—22no—
60C42×C71C22×C411yescharacteristic
61C42×C71C2×D412yes—
62C42×C71C2×D411yescharacteristic
63C23×C71#12311yescharacteristic
64Q8×C71—82no—
65C2×C4×C71—22no—
66C2×C4×C71—24no—
67C2×C4×C71—42no—
68C2×C4×C71C2×C22⋊C411yescharacteristic
69C2×C4×C71C4×D412yes—
70C2×C4×C71#12011yescharacteristic
71C2×C4×C71#12112yes—
72C2×Dic71—42no—
73C2×Dic71—82no—
74Dic142—44no—
75Dic142—82no—
76C8×C71—44no—
77C71⋊C8—42no—
78C71⋊C8—82no—
79C22×C71—21no—
80C22×C71—22no—
81C22×C71#38611yescharacteristic
82C4×C71—21no—
83C4×C71—22no—
84C4×C71—22no—
85C4×C71—44no—
86C4×C71#45412yes—
87Dic71—82no—
88#9422—711nomaximal, Sylow
89C2×C71—41no—
90C2×C71#188811yescharacteristic
91C2×C71#190212yes—
92#3139—711no—
93#2987D7111yescharacteristic
94#1556—712no—
95#1537—711no—
96#1291—712no—
97C71#942211yescharacteristic, complemented, Sylow
98#533C2×D7111yescharacteristic
99C2×C4⋊Q8—711no—
100#491—1422no—
101#487—714no—
102#446—24no—
103#433—22no—
104#432—712no—
105#432C2×D7112yes—
106#419—1422no—
107C4×M16—711no—
108#332—1424no—
109#330—1422no—
110C22×Q8—1421no—
111C2×M16—1421no—
112C4⋊Q8—712no—
113C4⋊Q8—1422no—
114C4⋊Q8—1421no—
115C4⋊Q8C4×D7112yes—
116#125D28412yes—
117C22⋊Q8—22no—
118C4×Q8—22no—
119#117C22⋊D7112yes—
120C2×C4⋊C4—1422no—
121C2×C42C22×D7111yescharacteristic
122C4⋊C8—24no—
123C4⋊C8—1422no—
124C4⋊C8C22⋊D7114yes—
125#103—1428no—
126C22⋊C8—22no—
127C22⋊C8—1422no—
128#97—22no—
129#97—22no—
130#97—1422no—
131#97—1421no—
132C4×C8—22no—
133C4×C8—712no—
134C2×Q8—22no—
135C2×Q8—2844no—
136C2×Q8—2841no—
137C2×Q8—1422no—
138C22×C4—2841no—
139C22×C4C2×C22⋊D7111yescharacteristic
140C22×C4#1176586512yes—
141M16—2842no—
142C2×C8—24no—
143C2×C8—2842no—
144C2×C8—1422no—
145C4⋊C4—22no—
146C4⋊C4—24no—
147C4⋊C4—2842no—
148C4⋊C4—1424no—
149C4⋊C4(C2×D71)⋊C414yes—
150C22⋊C4—22no—
151C42—22no—
152C42C2×C4×D7111yescharacteristic
153C42C2×C22⋊D7112yes—
154C42C2×D28411yescharacteristic
155C23#1455611111yescharacteristic
156Q8—82no—
157Q8—2844no—
158Q8—5682no—
159C2×C4—22no—
160C2×C4—24no—
161C2×C4—42no—
162C2×C4—2842no—
163C2×C4—5682no—
164C2×C4C2×(C2×D71)⋊C411yescharacteristic
165C2×C4C4×C22⋊D7112yes—
166C2×C4#1455609711yescharacteristic
167C2×C4#1455610212yes—
168C8—44no—
169C8—2842no—
170C8—5682no—
171C22—21no—
172C22—22no—
173C22#1798587811yescharacteristic
174C4—21no—
175C4—22no—
176C4—22no—
177C4—44no—
178C4—5682no—
179C4#1798617212yes—
180C2—41no—
181C2#2258874311yescharacteristic
182C2#2258881312yes—
subgroupG/H
centerC22#17985878
commutatorC42×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#2987C2
socleC22×C71#386
prime radicalC23×C71#123
cosocleC2×C42×C71C23
hypercenter#2987D71
hypocenterC71#9422
LCSG ▷ C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2987

Complements

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

NH ≅ G/Ntype
97. C71#9422semidirect

Maximal subgroups

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

subgroupindex
#225987222
#225870582
#225869782
#225862112
C71×#29872
#942271