Finite Group Databasebeta

← #22598993

Subgroups — #22598993

Order 9088.

338 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 121 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#17986398C211yesmaximal, characteristic, complemented
3#17986378C211yesmaximal, characteristic, complemented
4#17986369C211yesmaximal, characteristic, complemented
5#17986367C211yesmaximal, characteristic, complemented
6#17986255C212yesmaximal
7#17986238C212yesmaximal
8#17986104C212yesmaximal, complemented
9C4×#11765855C212yesmaximal, complemented
10#17985955C211yesmaximal, characteristic
11#541×C71C211yesmaximal, characteristic
12C2×#11765855C2212yes—
13C2×#11765855C2212yes—
14C2×#11765849C2211yescharacteristic
15C22×C71⋊C8C2211yescharacteristic
16#14556128C2211yescharacteristic
17#14556121—22no—
18#14556120C2211yescharacteristic
19#14556118—22no—
20#14556107C2212yes—
21C4×Dic142C2212yes—
22#14556061C2212yes—
23#14556059C2212yes—
24#14556052C2212yes—
25#14556052C2212yes—
26#14556051C2212yes—
27#14556051C2212yes—
28#14556039C2211yescharacteristic
29#14556039C2211yescharacteristic
30C4×C71⋊C8C2211yescharacteristic
31C4×C71⋊C8C2211yescharacteristic
32C4⋊Q8×C71C2211yescharacteristic
33#126×C71—22no—
34#125×C71—22no—
35#125×C71C2211yescharacteristic
36C22⋊Q8×C71C2212yes—
37C4×Q8×C71C2212yes—
38#117×C71C2211yescharacteristic
39C2×Dic142C2312yes—
40#11765855—24no—
41#11765855—24no—
42#11765849—22no—
43C2×C71⋊C8—22no—
44C2×C71⋊C8C2311yescharacteristic
45C2×C71⋊C8C2311yescharacteristic
46C2×C71⋊C8C2311yescharacteristic
47C2×C71⋊C8C2311yescharacteristic
48C4⋊Dic71—22no—
49C4⋊Dic71C2312yes—
50Dic71⋊C4—22no—
51Dic71⋊C4—22no—
52Dic71⋊C4—22no—
53C22⋊Dic71—22no—
54C4×Dic71—22no—
55C2×Q8×C71C2312yes—
56C22×C4×C71C2311yescharacteristic
57C4⋊C4×C71—22no—
58C4⋊C4×C71—22no—
59C4⋊C4×C71—22no—
60C4⋊C4×C71—22no—
61C4⋊C4×C71D412yes—
62C4⋊C4×C71C2312yes—
63C22⋊C4×C71—22no—
64C22⋊C4×C71D412yes—
65C42×C71—22no—
66C42×C71C2311yescharacteristic
67C42×C71C2311yescharacteristic
68C23×C71C2×D411yescharacteristic
69Q8×C71—22no—
70Q8×C71—22no—
71C2×C4×C71—21no—
72C2×C4×C71—21no—
73C2×C4×C71—22no—
74C2×C4×C71—22no—
75C2×C4×C71C2×D412yes—
76C2×C4×C71C2×D412yes—
77C2×C4×C71C2×D411yescharacteristic
78C2×C4×C71C2411yescharacteristic
79C2×Dic71—22no—
80C2×Dic71—22no—
81Dic142—22no—
82Dic142—22no—
83C71⋊C8—21no—
84C71⋊C8—21no—
85C71⋊C8#4112yes—
86C71⋊C8#4112yes—
87C22×C71—21no—
88C22×C71—22no—
89C22×C71C22×D411yescharacteristic
90C4×C71—21no—
91C4×C71—22no—
92C4×C71—22no—
93C4×C71—42no—
94C4×C71—42no—
95C4×C71C2×#4111yescharacteristic
96C4×C71C2×#4111yescharacteristic
97Dic71—42no—
98Dic71—42no—
99C2×C71—41no—
100C2×C71#53211yescharacteristic
101C2×C71#57812yes—
102#3177—711nomaximal, Sylow
103C71#317711yescharacteristic, complemented, Sylow
104#541—711no—
105#541D7111yescharacteristic
106#499—711no—
107#494—711no—
108#491—711no—
109#491—711no—
110#470—712no—
111#467—712no—
112#441—712no—
113C4×Q16—712no—
114#405—711no—
115C2×Q16—712no—
116C2×Q16—712no—
117C2×M16—711no—
118C22×C8—711no—
119C4⋊Q8—711no—
120C4⋊Q8C2×D7111yescharacteristic
121#126—22no—
122#126—1422no—
123#125—22no—
124#125—711no—
125#125—1422no—
126#125C2×D7111yescharacteristic
127C22⋊Q8—712no—
128C22⋊Q8C2×D7112yes—
129C4×Q8—712no—
130C4×Q8C2×D7112yes—
131#117C2×D7111yescharacteristic
132#107—712no—
133#106—712no—
134#103—712no—
135#103—712no—
136#103—712no—
137#103—712no—
138#97—711no—
139#97—711no—
140C4×C8—711no—
141C4×C8—711no—
142C2×Q8—712no—
143C2×Q8C22×D7112yes—
144C22×C4C22×D7111yescharacteristic
145Q16—1424no—
146Q16—1424no—
147M16—1422no—
148C2×C8—711no—
149C2×C8—711no—
150C2×C8—711no—
151C2×C8—711no—
152C2×C8—1422no—
153C4⋊C4—22no—
154C4⋊C4—22no—
155C4⋊C4—22no—
156C4⋊C4—22no—
157C4⋊C4—712no—
158C4⋊C4—1422no—
159C4⋊C4—1422no—
160C4⋊C4—1422no—
161C4⋊C4—1422no—
162C4⋊C4C22⋊D7112yes—
163C4⋊C4C22×D7112yes—
164C22⋊C4—22no—
165C22⋊C4—1422no—
166C22⋊C4C22⋊D7112yes—
167C42—22no—
168C42—1422no—
169C42C22×D7111yescharacteristic
170C42C22×D7111yescharacteristic
171C23C2×C22⋊D7111yescharacteristic
172Q8—22no—
173Q8—22no—
174Q8—1422no—
175Q8—1422no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—1422no—
181C2×C4—1422no—
182C2×C4C2×C22⋊D7112yes—
183C2×C4C2×C22⋊D7112yes—
184C2×C4C2×C22⋊D7111yescharacteristic
185C2×C4C23×D7111yescharacteristic
186C8—712no—
187C8—712no—
188C8—1421no—
189C8—1421no—
190C22—21no—
191C22—22no—
192C22C22×C22⋊D7111yescharacteristic
193C4—21no—
194C4—22no—
195C4—22no—
196C4—42no—
197C4—42no—
198C4—2842no—
199C4—2842no—
200C4#41×D7111yescharacteristic
201C4#41×D7111yescharacteristic
202C2—41no—
203C2#1798654511yescharacteristic
204C2#1798681412yes—
subgroupG/H
centerC22C22×C22⋊D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#541×C71C2
socleC22×C71C22×D4
hypercenter#541D71
hypocenterC71#3177
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. #17986398C2semidirect
3. #17986378C2semidirect
4. #17986369C2semidirect
5. #17986367C2semidirect
8. #17986104C2semidirect
9. C4×#11765855C2semidirect
103. C71#3177semidirect