Finite Group Databasebeta

← #31895056

Subgroups — #31895056

Order 18176.

400 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 107 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#22587078C211yesmaximal, characteristic
3#22587024C212yesmaximal
4#22586231C212yesmaximal
5C71×#2986C211yesmaximal, characteristic
6C2×#14556128C2211yescharacteristic
7#17986369C414yes—
8#17986351—22no—
9#17986023C2212yes—
10#17986007—22no—
11C4×#11765849C2211yescharacteristic
12#17985746—22no—
13#17985746—22no—
14#17985745—22no—
15#533×C71C2211yescharacteristic
16#446×C71—22no—
17C8×Q8×C71—22no—
18#433×C71—22no—
19#431×C71C2212yes—
20C22×Dic142—21no—
21C2×#11765849—21no—
22C2×#11765849—22no—
23#14556128—22no—
24#14556128—21no—
25#14556128C2×C412yes—
26C2×C4⋊Dic71—22no—
27C4⋊(C71⋊C8)—22no—
28C4⋊(C71⋊C8)—22no—
29#14556051—28no—
30#14556039C2×C412yes—
31C4×C71⋊C8—21no—
32C4⋊Q8×C71D412yes—
33#125×C71C2×C412yes—
34C22⋊Q8×C71—22no—
35C4×Q8×C71—22no—
36#117×C71D412yes—
37C2×C42×C71C2311yescharacteristic
38C4⋊C8×C71—24no—
39C4⋊C8×C71D412yes—
40C4⋊C8×C71D412yes—
41C22⋊C8×C71—22no—
42#97×C71—22no—
43C4×C8×C71—22no—
44C4×C8×C71—22no—
45C2×Dic142—22no—
46C2×Dic142—44no—
47C2×Dic142—41no—
48C22×Dic71—41no—
49#11765849—42no—
50#11765849—82no—
51C2×C71⋊C8—22no—
52C2×C71⋊C8—42no—
53C4⋊Dic71—24no—
54C4⋊Dic71—42no—
55C2×Q8×C71—22no—
56C22×C4×C71C2×D411yescharacteristic
57C22×C4×C71#4112yes—
58C2×C8×C71—22no—
59C2×C8×C71—22no—
60C4⋊C4×C71—24no—
61C4⋊C4×C71—22no—
62C4⋊C4×C71C22⋊C414yes—
63C22⋊C4×C71—22no—
64C42×C71—22no—
65C42×C71C22×C411yescharacteristic
66C42×C71C2×D412yes—
67C42×C71C2×D411yescharacteristic
68C23×C71#12311yescharacteristic
69Q8×C71—42no—
70Q8×C71—42no—
71C2×C4×C71—22no—
72C2×C4×C71—22no—
73C2×C4×C71—22no—
74C2×C4×C71—24no—
75C2×C4×C71C2×C22⋊C411yescharacteristic
76C2×C4×C71C4×D412yes—
77C2×C4×C71#12011yescharacteristic
78C2×C4×C71#12112yes—
79C2×Dic71—42no—
80C2×Dic71—82no—
81Dic142—44no—
82Dic142—82no—
83C8×C71—42no—
84C8×C71—42no—
85C71⋊C8—42no—
86C71⋊C8—82no—
87C22×C71—21no—
88C22×C71—22no—
89C22×C71#38611yescharacteristic
90C4×C71—21no—
91C4×C71—22no—
92C4×C71—44no—
93C4×C71#45412yes—
94C4×C71#47112yes—
95C4×C71#47312yes—
96Dic71—82no—
97#9436—711nomaximal, Sylow
98C2×C71—41no—
99C2×C71#188811yescharacteristic
100C2×C71#190812yes—
101#3139—711no—
102#2986D7111yescharacteristic
103#1561—711no—
104#1549—712no—
105#1299—712no—
106C71#943611yescharacteristic, complemented, Sylow
107#533C2×D7111yescharacteristic
108C2×C4⋊Q8—711no—
109#491—714no—
110#487—1422no—
111#446—22no—
112C8×Q8—22no—
113#433—22no—
114#431C2×D7112yes—
115#423—712no—
116#419—1422no—
117C4×M16—711no—
118#332—1422no—
119#332—1422no—
120#332—1422no—
121C22×Q8—1421no—
122C2×M16—1422no—
123C2×M16—1421no—
124C4⋊Q8—712no—
125C4⋊Q8—1422no—
126C4⋊Q8—1421no—
127C4⋊Q8D28412yes—
128#125C4×D7112yes—
129C22⋊Q8—22no—
130C4×Q8—22no—
131#117C22⋊D7112yes—
132C2×C4⋊C4—1422no—
133C2×C42C22×D7111yescharacteristic
134C4⋊C8—24no—
135C4⋊C8—1422no—
136C4⋊C8—1422no—
137C4⋊C8C22⋊D7112yes—
138C4⋊C8C22⋊D7112yes—
139#103—1428no—
140C22⋊C8—22no—
141#97—22no—
142#97—712no—
143C4×C8—22no—
144C4×C8—22no—
145C4×C8—1421no—
146C2×Q8—22no—
147C2×Q8—2844no—
148C2×Q8—2841no—
149C2×Q8—1422no—
150C22×C4—2841no—
151C22×C4C2×C22⋊D7111yescharacteristic
152C22×C4#1176586512yes—
153M16—2842no—
154M16—5682no—
155C2×C8—22no—
156C2×C8—22no—
157C2×C8—2842no—
158C2×C8—1422no—
159C4⋊C4—24no—
160C4⋊C4—22no—
161C4⋊C4—2842no—
162C4⋊C4—1424no—
163C4⋊C4(C2×D71)⋊C414yes—
164C22⋊C4—22no—
165C42—22no—
166C42C2×C4×D7111yescharacteristic
167C42C2×C22⋊D7112yes—
168C42C2×D28411yescharacteristic
169C23#1455611111yescharacteristic
170Q8—42no—
171Q8—42no—
172Q8—2844no—
173Q8—5682no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—24no—
178C2×C4—2842no—
179C2×C4—5682no—
180C2×C4C2×(C2×D71)⋊C411yescharacteristic
181C2×C4C4×C22⋊D7112yes—
182C2×C4#1455609711yescharacteristic
183C2×C4#1455610212yes—
184C8—42no—
185C8—42no—
186C8—2842no—
187C8—5682no—
188C22—21no—
189C22—22no—
190C22#1798587811yescharacteristic
191C4—21no—
192C4—22no—
193C4—44no—
194C4—5682no—
195C4#1798617212yes—
196C4#1798626212yes—
197C4#1798627412yes—
198C2—41no—
199C2#2258874311yescharacteristic
200C2#2258885512yes—
subgroupG/H
centerC22#17985878
commutatorC42×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#2986C2
socleC22×C71#386
prime radicalC23×C71#123
cosocleC2×C42×C71C23
hypercenter#2986D71
hypocenterC71#9436
LCSG ▷ C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2986

Complements

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

NH ≅ G/Ntype
106. C71#9436semidirect

Maximal subgroups

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

subgroupindex
#225987222
#225870782
#225870242
#225862312
C71×#29862
#943671