Finite Group Databasebeta

← #22598718

Subgroups — #22598718

Order 9088.

380 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 129 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#17986552C211yesmaximal, characteristic
2C2×#14556126C211yesmaximal, characteristic
3#17986366C214yesmaximal, complemented
4#17986348C214yesmaximal, complemented
5#17986009C211yesmaximal, characteristic
6#17986008C211yesmaximal, characteristic
7#17986007C211yesmaximal, characteristic
8M16×Dic71C211yesmaximal, characteristic
9#419×C71C211yesmaximal, characteristic
10C22×Dic142C2211yescharacteristic
11C2×#11765849C2211yescharacteristic
12#14556127C2212yes—
13#14556126—22no—
14#14556126C2212yes—
15#14556126C2212yes—
16#14556119C2212yes—
17#14556105—22no—
18C4×Dic142—22no—
19#14556087C2211yescharacteristic
20C2×Dic71⋊C4—22no—
21C2×C4×Dic71C2211yescharacteristic
22#14556053C2214yes—
23#14556052C2214yes—
24#14556051C2214yes—
25#14556038C2212yes—
26C8×Dic71C2212yes—
27C22×Q8×C71C2211yescharacteristic
28C2×M16×C71C2211yescharacteristic
29#117×C71C2211yescharacteristic
30#103×C71C2214yes—
31C2×Dic142—22no—
32C2×Dic142—22no—
33C2×Dic142—21no—
34C2×Dic142—22no—
35C2×Dic142C2312yes—
36C22×Dic71—21no—
37C22×Dic71D412yes—
38#11765849—22no—
39C2×C71⋊C8C2312yes—
40C4⋊Dic71—22no—
41C4⋊Dic71C2312yes—
42Dic71⋊C4—24no—
43Dic71⋊C4—24no—
44Dic71⋊C4—22no—
45Dic71⋊C4—22no—
46C22⋊Dic71—21no—
47C4×Dic71—21no—
48C4×Dic71D412yes—
49C4×Dic71C2311yescharacteristic
50C4×Dic71C2311yescharacteristic
51C2×Q8×C71—21no—
52C2×Q8×C71—22no—
53C2×Q8×C71—22no—
54C2×Q8×C71C2312yes—
55C22×C4×C71—21no—
56C22×C4×C71C2311yescharacteristic
57M16×C71—22no—
58C2×C8×C71C2312yes—
59C4⋊C4×C71C2312yes—
60C22⋊C4×C71—21no—
61C42×C71—21no—
62C23×C71C2×D411yescharacteristic
63Q8×C71—22no—
64Q8×C71—22no—
65Q8×C71—42no—
66C2×C4×C71—22no—
67C2×C4×C71—22no—
68C2×C4×C71—42no—
69C2×C4×C71C2×D411yescharacteristic
70C2×C4×C71#4112yes—
71C2×C4×C71#4112yes—
72C2×C4×C71C2411yescharacteristic
73C2×Dic71—24no—
74C2×Dic71—22no—
75C2×Dic71—22no—
76C2×Dic71—42no—
77C2×Dic71C2×D412yes—
78C2×Dic71C2×D412yes—
79Dic142—22no—
80Dic142—22no—
81Dic142—42no—
82Dic142—44no—
83C8×C71—22no—
84C71⋊C8—22no—
85C22×C71—22no—
86C22×C71#12412yes—
87C22×C71C22×D411yescharacteristic
88C4×C71—42no—
89C4×C71—42no—
90C4×C71#12412yes—
91C4×C71C2×#4111yescharacteristic
92C4×C71C2×#4111yescharacteristic
93Dic71—22no—
94Dic71—42no—
95Dic71—42no—
96Dic71#13714yes—
97C2×C71—22no—
98C2×C71C2×#12411yescharacteristic
99C2×C71C2×#13712yes—
100#3139—711nomaximal, Sylow
101C71#313911yescharacteristic, complemented, Sylow
102#533—711no—
103C2×C4⋊Q8—711no—
104#491—714no—
105#487—714no—
106#419—711no—
107#419—711no—
108#419—711no—
109#419D7111yescharacteristic
110C4×M16—711no—
111C22×Q8—711no—
112C22×Q8C2×D7111yescharacteristic
113C2×M16—711no—
114C2×M16C2×D7111yescharacteristic
115C4⋊Q8—712no—
116C4⋊Q8—712no—
117C4⋊Q8—712no—
118C4⋊Q8—1422no—
119#125—712no—
120C22⋊Q8—1422no—
121C4×Q8—1422no—
122#117—711no—
123#117C2×D7111yescharacteristic
124C2×C4⋊C4—1422no—
125C2×C42—711no—
126#103—714no—
127#103—714no—
128#103—714no—
129#103C2×D7114yes—
130#97—712no—
131C4×C8—712no—
132C2×Q8—21no—
133C2×Q8—22no—
134C2×Q8—22no—
135C2×Q8—712no—
136C2×Q8—1422no—
137C2×Q8—1422no—
138C2×Q8—1422no—
139C2×Q8—1421no—
140C2×Q8C22×D7112yes—
141C22×C4—21no—
142C22×C4—712no—
143C22×C4—1421no—
144C22×C4C22×D7111yescharacteristic
145M16—22no—
146M16—1422no—
147C2×C8—712no—
148C2×C8C22×D7112yes—
149C4⋊C4—712no—
150C4⋊C4—1424no—
151C4⋊C4—1424no—
152C4⋊C4—1422no—
153C4⋊C4—1422no—
154C4⋊C4—1422no—
155C4⋊C4C22×D7112yes—
156C22⋊C4—21no—
157C22⋊C4—1421no—
158C42—21no—
159C42—712no—
160C42—711no—
161C42—711no—
162C42—1421no—
163C23D4×D7111yescharacteristic
164Q8—22no—
165Q8—22no—
166Q8—42no—
167Q8—2844no—
168Q8—2842no—
169Q8—1422no—
170Q8—1422no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—42no—
174C2×C4—2842no—
175C2×C4—712no—
176C2×C4—712no—
177C2×C4—1422no—
178C2×C4—1422no—
179C2×C4—1424no—
180C2×C4D4×D7111yescharacteristic
181C2×C4#1176586412yes—
182C2×C4#1176586512yes—
183C2×C4C23×D7111yescharacteristic
184C8—22no—
185C8—1422no—
186C22—22no—
187C22#1455611412yes—
188C22C2×D4×D7111yescharacteristic
189C4—42no—
190C4—42no—
191C4—2842no—
192C4—2842no—
193C4—714no—
194C4—1422no—
195C4#1455611412yes—
196C4C2×#1176586411yescharacteristic
197C4C2×#1176586511yescharacteristic
198C2—22no—
199C2C2×#1455611411yescharacteristic
200C2#137×D7112yes—
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#419×C71C2
socleC22×C71C22×D4
hypercenter#419D71
hypocenterC71#3139
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
3. #17986366C2semidirect
4. #17986348C2semidirect
101. C71#3139semidirect