Finite Group Databasebeta

← #22589307

Subgroups — #22589307

Order 9088.

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

normal subgroups only

This group has 260 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
1C2×#14556119C211yesmaximal, characteristic
2C2×#14556038C211yesmaximal, characteristic
3#17985868C211yesmaximal, characteristic
4#17985762C211yesmaximal, characteristic
5#17985761C211yesmaximal, characteristic
6#17985761C211yesmaximal, characteristic
7#340×C71C211yesmaximal, characteristic
8C22×C71⋊C8C2211yescharacteristic
9#14556119—22no—
10#14556119C414yescomplemented
11C2×C4⋊Dic71—21no—
12C2×C4⋊Dic71C2211yescharacteristic
13C2×C4⋊Dic71C2211yescharacteristic
14C2×Dic71⋊C4—21no—
15C2×Dic71⋊C4—21no—
16C2×Dic71⋊C4—21no—
17C2×C4×Dic71C2211yescharacteristic
18#14556038—22no—
19C22×C8×C71C2211yescharacteristic
20C2×C4⋊C4×C71C2211yescharacteristic
21C2×C4⋊C4×C71C2211yescharacteristic
22C22×Dic71—21no—
23C22×Dic71—21no—
24C22×Dic71D411yescharacteristic
25C22×Dic71D411yescharacteristic
26C2×C71⋊C8—21no—
27C2×C71⋊C8—21no—
28C2×C71⋊C8Q812yes—
29C4⋊Dic71—21no—
30C4⋊Dic71—24no—
31C4⋊Dic71—21no—
32C4⋊Dic71—21no—
33C4⋊Dic71C2×C412yes—
34Dic71⋊C4—24no—
35Dic71⋊C4—42no—
36Dic71⋊C4—42no—
37C4×Dic71C2×C412yes—
38C4×Dic71D412yes—
39C22×C4×C71—21no—
40C22×C4×C71—21no—
41C22×C4×C71C2311yescharacteristic
42C2×C8×C71—21no—
43C2×C8×C71—21no—
44C2×C8×C71Q812yes—
45C4⋊C4×C71—21no—
46C4⋊C4×C71—21no—
47C4⋊C4×C71—21no—
48C4⋊C4×C71C2×C412yes—
49C23×C71C2×D411yescharacteristic
50C2×C4×C71—22no—
51C2×C4×C71—41no—
52C2×C4×C71—41no—
53C2×C4×C71—41no—
54C2×C4×C71—41no—
55C2×C4×C71—41no—
56C2×C4×C71C22×C411yescharacteristic
57C2×C4×C71C2×D411yescharacteristic
58C2×C4×C71C2×Q811yescharacteristic
59C2×C4×C71C2×Q811yescharacteristic
60C2×C4×C71#4111yescharacteristic
61C2×C4×C71#4111yescharacteristic
62C2×Dic71—21no—
63C2×Dic71—21no—
64C2×Dic71—21no—
65C2×Dic71—21no—
66C2×Dic71—22no—
67C2×Dic71—41no—
68C2×Dic71—41no—
69C2×Dic71—41no—
70C2×Dic71—41no—
71C2×Dic71—41no—
72C2×Dic71C22⋊C412yes—
73C2×Dic71C22⋊C412yes—
74C8×C71—22no—
75C71⋊C8—22no—
76C22×C71C2×C22⋊C411yescharacteristic
77C22×C71#12411yescharacteristic
78C22×C71C4⋊Q811yescharacteristic
79C22×C71AGL(1,ℤ/8)11yescharacteristic
80C22×C71AGL(1,ℤ/8)11yescharacteristic
81C22×C71#13711yescharacteristic
82C22×C71#13711yescharacteristic
83C4×C71—42no—
84C4×C71—81no—
85C4×C71C4×Q811yescharacteristic
86C4×C71C4×Q811yescharacteristic
87C4×C71C22⋊Q811yescharacteristic
88C4×C71C22⋊Q811yescharacteristic
89Dic71—22no—
90Dic71—22no—
91Dic71—42no—
92Dic71—81no—
93C2×C71#39111yescharacteristic
94C2×C71#41711yescharacteristic
95C2×C71#41711yescharacteristic
96C2×C71#49011yescharacteristic
97C2×C71#49111yescharacteristic
98C2×C71#50111yescharacteristic
99C2×C71#50111yescharacteristic
100#1992—711nomaximal, Sylow
101C71#199211yescharacteristic, complemented, Sylow
102C2×#125—711no—
103C2×#97—711no—
104#384—711no—
105#340—711no—
106#340—711no—
107#340—711no—
108#340D7111yescharacteristic
109C22×C8—711no—
110C22×C8C2×D7111yescharacteristic
111#125—714no—
112#125—1422no—
113C2×C4⋊C4—711no—
114C2×C4⋊C4—711no—
115C2×C4⋊C4—1421no—
116C2×C4⋊C4—1421no—
117C2×C4⋊C4—1421no—
118C2×C4⋊C4—1421no—
119C2×C4⋊C4C2×D7111yescharacteristic
120C2×C4⋊C4C2×D7111yescharacteristic
121C2×C42—711no—
122#97—1422no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—711no—
126C22×C4—711no—
127C22×C4—1421no—
128C22×C4—1421no—
129C22×C4C22×D7111yescharacteristic
130C2×C8—21no—
131C2×C8—21no—
132C2×C8—712no—
133C2×C8—1421no—
134C2×C8—1421no—
135C2×C8Dic14212yes—
136C4⋊C4—21no—
137C4⋊C4—21no—
138C4⋊C4—21no—
139C4⋊C4—2842no—
140C4⋊C4—2842no—
141C4⋊C4—712no—
142C4⋊C4—1424no—
143C4⋊C4—1424no—
144C4⋊C4—1421no—
145C4⋊C4—1421no—
146C4⋊C4—1421no—
147C4⋊C4C4×D7112yes—
148C42—712no—
149C42—712no—
150C23D4×D7111yescharacteristic
151C2×C4—22no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—2841no—
158C2×C4—2841no—
159C2×C4—2841no—
160C2×C4—2841no—
161C2×C4—2841no—
162C2×C4—712no—
163C2×C4—712no—
164C2×C4—1422no—
165C2×C4—1421no—
166C2×C4—1421no—
167C2×C4—1421no—
168C2×C4—1421no—
169C2×C4C2×C4×D7111yescharacteristic
170C2×C4D4×D7111yescharacteristic
171C2×C4Q8×D7111yescharacteristic
172C2×C4C2×Dic14211yescharacteristic
173C2×C4#1176586411yescharacteristic
174C2×C4#1176586511yescharacteristic
175C8—22no—
176C8—1422no—
177C22C22⋊C4×D7111yescharacteristic
178C22#1455611411yescharacteristic
179C22#1455612711yescharacteristic
180C22#1455615211yescharacteristic
181C22#1455615511yescharacteristic
182C22#1455615711yescharacteristic
183C22#1455616011yescharacteristic
184C4—42no—
185C4—81no—
186C4—2842no—
187C4—5681no—
188C4—1422no—
189C4—1422no—
190C4#1455609311yescharacteristic
191C4C4×Dic14211yescharacteristic
192C4#1455610511yescharacteristic
193C4#1455610611yescharacteristic
194C2#1798590311yescharacteristic
195C2#1798599711yescharacteristic
196C2#1798599911yescharacteristic
197C2#1798636211yescharacteristic
198C2#1798636611yescharacteristic
199C2#1798640611yescharacteristic
200C2#1798640711yescharacteristic
subgroupG/H
centerC23D4×D71
commutatorC2×C4×C71C22×C4
FrattiniC22×C4C22×D71
Fitting#340×C71C2
socleC23×C71C2×D4
hypercenter#340D71
hypocenterC71#1992
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C22×C4 ◁ #340

Complements

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

NH ≅ G/Ntype
10. #14556119C4semidirect
101. C71#1992semidirect