Finite Group Databasebeta

← #22599790

Subgroups — #22599790

Order 9088.

322 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 117 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#17986611C211yesmaximal, characteristic
2#17986306C212yesmaximal, complemented
3#17986303C212yesmaximal, complemented
4#17986255C212yesmaximal
5#17986238C212yesmaximal
6#17986209C212yesmaximal, complemented
7#17986206C212yesmaximal, complemented
8#17986026C211yesmaximal, characteristic
9#541×C71C211yesmaximal, characteristic
10C2×#11765855C2212yes—
11C2×#11765855C2212yes—
12C2×#11765849C2211yescharacteristic
13C22×C71⋊C8C2211yescharacteristic
14#14556128C2211yescharacteristic
15#14556121—22no—
16#14556120C2211yescharacteristic
17#14556118—22no—
18#14556107C2212yes—
19C4×Dic142C2212yes—
20#14556061C2212yes—
21#14556059C2212yes—
22C4⋊(C71⋊C8)C2212yes—
23C4⋊(C71⋊C8)C2212yes—
24#14556052C2212yes—
25#14556052C2212yes—
26#14556051C2212yes—
27#14556051C2212yes—
28C4⋊Q8×C71C2211yescharacteristic
29#126×C71—22no—
30#125×C71—22no—
31#125×C71C2211yescharacteristic
32C22⋊Q8×C71C2212yes—
33C4×Q8×C71C2212yes—
34#117×C71C2211yescharacteristic
35C2×Dic142C2312yes—
36#11765855—42no—
37#11765855—42no—
38#11765849—41no—
39C2×C71⋊C8—41no—
40C2×C71⋊C8C2312yes—
41C2×C71⋊C8C2312yes—
42C4⋊Dic71—22no—
43C4⋊Dic71C2312yes—
44Dic71⋊C4—22no—
45Dic71⋊C4—22no—
46Dic71⋊C4—22no—
47C22⋊Dic71—22no—
48C4×Dic71—22no—
49C2×Q8×C71C2312yes—
50C22×C4×C71C2311yescharacteristic
51C4⋊C4×C71—22no—
52C4⋊C4×C71—22no—
53C4⋊C4×C71—22no—
54C4⋊C4×C71—22no—
55C4⋊C4×C71D412yes—
56C4⋊C4×C71C2312yes—
57C22⋊C4×C71—22no—
58C22⋊C4×C71D412yes—
59C42×C71—22no—
60C42×C71C2311yescharacteristic
61C42×C71C2311yescharacteristic
62C23×C71C2×D411yescharacteristic
63Q8×C71—22no—
64Q8×C71—22no—
65C2×C4×C71—21no—
66C2×C4×C71—21no—
67C2×C4×C71—22no—
68C2×C4×C71—22no—
69C2×C4×C71C2×D412yes—
70C2×C4×C71C2×D412yes—
71C2×C4×C71C2×D411yescharacteristic
72C2×C4×C71C2411yescharacteristic
73C2×Dic71—22no—
74C2×Dic71—22no—
75Dic142—22no—
76Dic142—22no—
77C71⋊C8—22no—
78C71⋊C8—22no—
79C22×C71—21no—
80C22×C71—22no—
81C22×C71C22×D411yescharacteristic
82C4×C71—21no—
83C4×C71—22no—
84C4×C71—22no—
85C4×C71—42no—
86C4×C71—42no—
87C4×C71H(5,𝔽2)11yescharacteristic
88C4×C712−1+411yescharacteristic
89Dic71—42no—
90Dic71—42no—
91C2×C71—41no—
92C2×C71#53711yescharacteristic
93C2×C71#57812yes—
94#3261—711nomaximal, Sylow
95C71#326111yescharacteristic, complemented, Sylow
96#541—711no—
97#541D7111yescharacteristic
98#479—712no—
99#479—712no—
100#470—712no—
101#467—712no—
102#462—712no—
103#462—712no—
104#424—711no—
105C2×Q16—712no—
106C2×Q16—712no—
107C2×M16—711no—
108C22×C8—711no—
109C4⋊Q8—711no—
110C4⋊Q8C2×D7111yescharacteristic
111#126—22no—
112#126—1422no—
113#125—22no—
114#125—711no—
115#125—1422no—
116#125C2×D7111yescharacteristic
117C22⋊Q8—712no—
118C22⋊Q8C2×D7112yes—
119C4×Q8—712no—
120C4×Q8C2×D7112yes—
121#117C2×D7111yescharacteristic
122#107—712no—
123#106—712no—
124C4⋊C8—712no—
125C4⋊C8—712no—
126#103—712no—
127#103—712no—
128#103—712no—
129#103—712no—
130C2×Q8—712no—
131C2×Q8C22×D7112yes—
132C22×C4C22×D7111yescharacteristic
133Q16—2842no—
134Q16—2842no—
135M16—2841no—
136C2×C8—2841no—
137C2×C8—712no—
138C2×C8—712no—
139C4⋊C4—22no—
140C4⋊C4—22no—
141C4⋊C4—22no—
142C4⋊C4—22no—
143C4⋊C4—712no—
144C4⋊C4—1422no—
145C4⋊C4—1422no—
146C4⋊C4—1422no—
147C4⋊C4—1422no—
148C4⋊C4C22⋊D7112yes—
149C4⋊C4C22×D7112yes—
150C22⋊C4—22no—
151C22⋊C4—1422no—
152C22⋊C4C22⋊D7112yes—
153C42—22no—
154C42—1422no—
155C42C22×D7111yescharacteristic
156C42C22×D7111yescharacteristic
157C23C2×C22⋊D7111yescharacteristic
158Q8—22no—
159Q8—22no—
160Q8—1422no—
161Q8—1422no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—1422no—
167C2×C4—1422no—
168C2×C4C2×C22⋊D7112yes—
169C2×C4C2×C22⋊D7112yes—
170C2×C4C2×C22⋊D7111yescharacteristic
171C2×C4C23×D7111yescharacteristic
172C8—1422no—
173C8—1422no—
174C22—21no—
175C22—22no—
176C22C22×C22⋊D7111yescharacteristic
177C4—21no—
178C4—22no—
179C4—22no—
180C4—42no—
181C4—42no—
182C4—2842no—
183C4—2842no—
184C4#1455617411yescharacteristic
185C4#1455617511yescharacteristic
186C2—41no—
187C2#1798657711yescharacteristic
188C2#1798681412yes—
subgroupG/H
centerC22C22×C22⋊D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#541×C71C2
socleC22×C71C22×D4
hypercenter#541D71
hypocenterC71#3261
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ #117 ◁ #541

Complements

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

NH ≅ G/Ntype
2. #17986306C2semidirect
3. #17986303C2semidirect
6. #17986209C2semidirect
7. #17986206C2semidirect
95. C71#3261semidirect