Finite Group Databasebeta

← #22601141

Subgroups — #22601141

Order 9088.

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

normal subgroups only

This group has 326 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#17986728C211yesmaximal, characteristic
2#17986727C211yesmaximal, characteristic
3#17986366C212yesmaximal
4#17986350C211yesmaximal, characteristic
5#17986305C212yesmaximal
6#17986284C211yesmaximal, characteristic
7#17986209C211yesmaximal, characteristic
8#17986207C212yesmaximal
9#17986129C211yesmaximal, characteristic
10#17986103C212yesmaximal
11C4×Q16×C71C211yesmaximal, characteristic
12C2×#11765855C2212yes—
13#14556128—21no—
14#14556128C2211yescharacteristic
15#14556128C2211yescharacteristic
16#14556126C2212yes—
17#14556120—21no—
18#14556119—22no—
19#14556119C2212yes—
20#14556118—22no—
21Q8×Dic71C2212yes—
22C4×Dic142—21no—
23C4×Dic142—21no—
24C4×Dic142C2211yescharacteristic
25#14556093C2212yes—
26#14556059C2212yes—
27C4⋊(C71⋊C8)C2211yescharacteristic
28Dic71⋊C8C2212yes—
29#14556053C2212yes—
30#14556053C2212yes—
31#14556052C2212yes—
32#14556051C2212yes—
33#14556038C2212yes—
34C2×Q16×C71C2211yescharacteristic
35C4×Q8×C71C2211yescharacteristic
36C4×Q8×C71C2211yescharacteristic
37#107×C71C2211yescharacteristic
38#103×C71C2211yescharacteristic
39#103×C71C2211yescharacteristic
40C4×C8×C71C2211yescharacteristic
41C2×Dic142—21no—
42C2×Dic142D411yescharacteristic
43C2×Dic142C2312yes—
44#11765855—24no—
45C2×C71⋊C8C2312yes—
46C4⋊Dic71—21no—
47C4⋊Dic71—21no—
48C4⋊Dic71—21no—
49C4⋊Dic71—22no—
50C4⋊Dic71—22no—
51C4⋊Dic71D411yescharacteristic
52C4⋊Dic71C2312yes—
53Dic71⋊C4—22no—
54Dic71⋊C4—22no—
55Dic71⋊C4—22no—
56Dic71⋊C4—22no—
57Dic71⋊C4D412yes—
58C4×Dic71—22no—
59C4×Dic71—22no—
60C4×Dic71C2312yes—
61C2×Q8×C71C2311yescharacteristic
62C2×Q8×C71C2311yescharacteristic
63Q16×C71—22no—
64C2×C8×C71C2311yescharacteristic
65C2×C8×C71C2311yescharacteristic
66C4⋊C4×C71—21no—
67C4⋊C4×C71—21no—
68C4⋊C4×C71C2311yescharacteristic
69C4⋊C4×C71C2311yescharacteristic
70C42×C71—21no—
71C42×C71—21no—
72C42×C71C2311yescharacteristic
73Q8×C71—21no—
74Q8×C71—21no—
75Q8×C71—21no—
76Q8×C71#4112yes—
77C2×C4×C71—21no—
78C2×C4×C71—21no—
79C2×C4×C71—21no—
80C2×C4×C71—21no—
81C2×C4×C71C2×D411yescharacteristic
82C2×C4×C71C2×D411yescharacteristic
83C2×C4×C71C2411yescharacteristic
84C2×Dic71—22no—
85C2×Dic71—22no—
86C2×Dic71C2×D412yes—
87C2×Dic71C2×D412yes—
88Dic142—22no—
89Dic142—22no—
90Dic142—22no—
91Dic142—42no—
92C8×C71—21no—
93C8×C71—21no—
94C71⋊C8—22no—
95C22×C71C22×D411yescharacteristic
96C4×C71—22no—
97C4×C71—21no—
98C4×C71—41no—
99C4×C71—41no—
100C4×C71—41no—
101C4×C71#13712yes—
102C4×C71C2×#4111yescharacteristic
103C4×C712−1+411yescharacteristic
104Dic71—22no—
105Dic71—22no—
106Dic71—42no—
107Dic71—42no—
108C2×C71#54811yescharacteristic
109C2×C71C2×#13711yescharacteristic
110C2×C71#57811yescharacteristic
111#3374—711nomaximal, Sylow
112C71#337411yescharacteristic, complemented, Sylow
113#557—711no—
114#557—711no—
115#491—712no—
116#487—711no—
117#479—712no—
118#475—711no—
119#462—712no—
120#462—711no—
121#446—711no—
122#441—712no—
123C4×Q16D7111yescharacteristic
124C2×Q16—712no—
125C2×Q16C2×D7111yescharacteristic
126C4⋊Q8—712no—
127C4⋊Q8—711no—
128C4⋊Q8—711no—
129C4⋊Q8—1421no—
130#125—712no—
131#125—1421no—
132#125—1422no—
133#125—1422no—
134C4×Q8—711no—
135C4×Q8—712no—
136C4×Q8—712no—
137C4×Q8—1421no—
138C4×Q8—1421no—
139C4×Q8C2×D7111yescharacteristic
140C4×Q8C2×D7111yescharacteristic
141#107C2×D7111yescharacteristic
142#106—712no—
143C4⋊C8—712no—
144C4⋊C8—711no—
145#103—712no—
146#103—712no—
147#103—712no—
148#103—712no—
149#103C2×D7111yescharacteristic
150#103C2×D7111yescharacteristic
151#97—712no—
152C4×C8C2×D7111yescharacteristic
153C2×Q8—711no—
154C2×Q8—712no—
155C2×Q8—1421no—
156C2×Q8C22×D7111yescharacteristic
157C2×Q8C22×D7111yescharacteristic
158Q16—22no—
159Q16—1424no—
160C2×C8—712no—
161C2×C8C22×D7111yescharacteristic
162C2×C8C22×D7111yescharacteristic
163C4⋊C4—21no—
164C4⋊C4—21no—
165C4⋊C4—712no—
166C4⋊C4—711no—
167C4⋊C4—712no—
168C4⋊C4—1422no—
169C4⋊C4—1422no—
170C4⋊C4—1422no—
171C4⋊C4—1422no—
172C4⋊C4—1422no—
173C4⋊C4—1422no—
174C4⋊C4—1421no—
175C4⋊C4—1421no—
176C4⋊C4—1421no—
177C4⋊C4C22×D7111yescharacteristic
178C4⋊C4C22×D7111yescharacteristic
179C42—21no—
180C42—21no—
181C42—712no—
182C42—1422no—
183C42—1422no—
184C42C22×D7111yescharacteristic
185Q8—21no—
186Q8—21no—
187Q8—21no—
188Q8—2842no—
189Q8—1422no—
190Q8—1422no—
191Q8—1422no—
192Q8#1176586512yes—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—712no—
198C2×C4—712no—
199C2×C4—1422no—
200C2×C4—1422no—
201C2×C4D4×D7111yescharacteristic
202C2×C4D4×D7111yescharacteristic
203C2×C4C23×D7111yescharacteristic
204C8—21no—
205C8—21no—
206C8—1422no—
207C22C2×D4×D7111yescharacteristic
208C4—22no—
209C4—21no—
210C4—41no—
211C4—41no—
212C4—41no—
213C4—2842no—
214C4—2842no—
215C4—1422no—
216C4—1422no—
217C4#1455616012yes—
218C4C2×#1176586511yescharacteristic
219C4#1455617511yescharacteristic
220C2#1798665811yescharacteristic
221C2C2×#1455616011yescharacteristic
222C2#1798681311yescharacteristic
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
FittingC4×Q16×C71C2
socleC22×C71C22×D4
hypercenterC4×Q16D71
hypocenterC71#3374
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ C4×Q16

Complements

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

NH ≅ G/Ntype
112. C71#3374semidirect