Finite Group Databasebeta

← #22599681

Subgroups — #22599681

Order 9088.

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

normal subgroups only

This group has 348 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#17986551C212yesmaximal
2#17986293C212yesmaximal, complemented
3#17986292C212yesmaximal, complemented
4#17986240C211yesmaximal, characteristic
5#17986238C212yesmaximal
6#17986208C212yesmaximal, complemented
7#17986207C212yesmaximal, complemented
8C2×Dic71⋊C8C211yesmaximal, characteristic
9#467×C71C211yesmaximal, characteristic
10C2×Dic284C2211yescharacteristic
11C2×#11765855C2212yes—
12C22×C71⋊C8C2211yescharacteristic
13#14556127C2212yes—
14#14556126C2212yes—
15#14556118—22no—
16#14556107C2212yes—
17#14556105—22no—
18Q8×Dic71C2212yes—
19#14556093—22no—
20#14556093C2212yes—
21#14556084—22no—
22C2×C4×Dic71C2211yescharacteristic
23#14556062C2211yescharacteristic
24#14556061C2212yes—
25Dic71⋊C8C2212yes—
26Dic71⋊C8C2212yes—
27#14556053C2212yes—
28#14556052C2212yes—
29#14556051C2212yes—
30C2×Q16×C71C2211yescharacteristic
31C22×C8×C71C2211yescharacteristic
32C22⋊Q8×C71C2212yes—
33#107×C71C2211yescharacteristic
34#103×C71C2212yes—
35C2×Dic142—22no—
36C2×Dic142C2312yes—
37C22×Dic71D411yescharacteristic
38C22×Dic71D411yescharacteristic
39Dic284—41no—
40#11765855—42no—
41C2×C71⋊C8—41no—
42C2×C71⋊C8C2312yes—
43C4⋊Dic71—22no—
44C4⋊Dic71C2312yes—
45Dic71⋊C4—22no—
46Dic71⋊C4—22no—
47Dic71⋊C4—22no—
48Dic71⋊C4—22no—
49Dic71⋊C4—22no—
50C22⋊Dic71—22no—
51C4×Dic71—22no—
52C4×Dic71—22no—
53C4×Dic71D412yes—
54C4×Dic71C2312yes—
55C2×Q8×C71C2312yes—
56C22×C4×C71C2311yescharacteristic
57Q16×C71—41no—
58C2×C8×C71—41no—
59C2×C8×C71C2311yescharacteristic
60C2×C8×C71C2311yescharacteristic
61C4⋊C4×C71—22no—
62C4⋊C4×C71C2312yes—
63C22⋊C4×C71—22no—
64C23×C71C2×D411yescharacteristic
65Q8×C71—22no—
66Q8×C71—22no—
67C2×C4×C71—21no—
68C2×C4×C71—21no—
69C2×C4×C71—22no—
70C2×C4×C71—22no—
71C2×C4×C71C2×D411yescharacteristic
72C2×C4×C71C2411yescharacteristic
73C2×Dic71—22no—
74C2×Dic71—22no—
75C2×Dic71—41no—
76C2×Dic71Q1614yescomplemented
77C2×Dic71C2×D412yes—
78C2×Dic71C2×D412yes—
79Dic142—22no—
80Dic142—22no—
81Dic142—44no—
82C8×C71—21no—
83C8×C71—21no—
84C71⋊C8—22no—
85C22×C71—21no—
86C22×C71—21no—
87C22×C71C2×Q1612yes—
88C22×C71C22×D411yescharacteristic
89C4×C71—21no—
90C4×C71—42no—
91C4×C71—42no—
92C4×C71H(5,𝔽2)11yescharacteristic
93C4×C712−1+411yescharacteristic
94Dic71—22no—
95Dic71—42no—
96Dic71—42no—
97Dic71C2×Q1614yes—
98C2×C71—22no—
99C2×C71#53711yescharacteristic
100C2×C71C22×Q1611yescharacteristic
101C2×C71#57511yescharacteristic
102#3249—711nomaximal, Sylow
103C71#324911yescharacteristic, complemented, Sylow
104#533—712no—
105#477—712no—
106#477—712no—
107#467—711no—
108#467—712no—
109#467D7111yescharacteristic
110#462—712no—
111#462—712no—
112C2×C4⋊C8—711no—
113C2×Q16—712no—
114C2×Q16—711no—
115C2×Q16C2×D7111yescharacteristic
116C22×C8—711no—
117C22×C8C2×D7111yescharacteristic
118C4⋊Q8—712no—
119C4⋊Q8—712no—
120#125—1422no—
121C22⋊Q8—712no—
122C22⋊Q8—1422no—
123C22⋊Q8C2×D7112yes—
124C4×Q8—712no—
125C4×Q8—712no—
126C4×Q8—1422no—
127#117—1422no—
128C2×C42—711no—
129#107—711no—
130#107—712no—
131#107C2×D7111yescharacteristic
132C4⋊C8—712no—
133C4⋊C8—712no—
134#103—712no—
135#103—712no—
136#103—712no—
137#103C2×D7112yes—
138C2×Q8—712no—
139C2×Q8—1422no—
140C2×Q8C22×D7112yes—
141C22×C4—711no—
142C22×C4—711no—
143C22×C4C22×D7111yescharacteristic
144Q16—41no—
145Q16—2841no—
146Q16—2842no—
147C2×C8—41no—
148C2×C8—2841no—
149C2×C8—712no—
150C2×C8C22×D7111yescharacteristic
151C2×C8C22×D7111yescharacteristic
152C4⋊C4—22no—
153C4⋊C4—712no—
154C4⋊C4—1422no—
155C4⋊C4—1422no—
156C4⋊C4—1422no—
157C4⋊C4—1422no—
158C4⋊C4—1422no—
159C4⋊C4—1422no—
160C4⋊C4C22×D7112yes—
161C22⋊C4—22no—
162C22⋊C4—1422no—
163C42—712no—
164C42—712no—
165C42—1422no—
166C42—1422no—
167C23D4×D7111yescharacteristic
168Q8—22no—
169Q8—22no—
170Q8—2844no—
171Q8—1422no—
172Q8—1422no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—2841no—
178C2×C4—712no—
179C2×C4—714no—
180C2×C4—712no—
181C2×C4—1422no—
182C2×C4—1422no—
183C2×C4D4×D7111yescharacteristic
184C2×C4C23×D7111yescharacteristic
185C8—21no—
186C8—21no—
187C8—1422no—
188C22—21no—
189C22—21no—
190C22Q16×D7112yes—
191C22C2×D4×D7111yescharacteristic
192C4—21no—
193C4—42no—
194C4—42no—
195C4—2842no—
196C4—2842no—
197C4—714no—
198C4—1422no—
199C4#1455617311yescharacteristic
200C4#1455617511yescharacteristic
201C2—22no—
202C2#1798657611yescharacteristic
203C2C2×Q16×D7111yescharacteristic
204C2#1798679611yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#467×C71C2
socleC22×C71C22×D4
hypercenter#467D71
hypocenterC71#3249
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

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

NH ≅ G/Ntype
2. #17986293C2semidirect
3. #17986292C2semidirect
6. #17986208C2semidirect
7. #17986207C2semidirect
76. C2×Dic71Q16semidirect
103. C71#3249semidirect