Finite Group Databasebeta

← #22601215

Subgroups — #22601215

Order 9088.

338 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 133 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#17986727C211yesmaximal, characteristic, complemented
2#17986398C212yesmaximal, complemented
3#17986395C211yesmaximal, characteristic, complemented
4#17986303C212yesmaximal, complemented
5#17986299C212yesmaximal
6#17986291C211yesmaximal, characteristic, complemented
7#17986279C211yesmaximal, characteristic
8Q8×C71⋊C8C211yesmaximal, characteristic, complemented
9C4×#11765853C211yesmaximal, characteristic
10C4×#11765853C212yesmaximal
11#554×C71C211yesmaximal, characteristic
12C2×#11765853C2211yescharacteristic
13#14556128C2211yescharacteristic
14#14556126—21no—
15#14556120C2212yes—
16#14556118—22no—
17C4×Dic142C2211yescharacteristic
18C4×Dic142C2212yes—
19#14556093—21no—
20#14556061C2212yes—
21#14556061C2212yes—
22#14556059C2212yes—
23#14556059C2211yescharacteristic
24#14556059C2212yes—
25C4⋊(C71⋊C8)C2211yescharacteristic
26C4⋊(C71⋊C8)C2212yes—
27#14556051C2212yes—
28#14556051C2211yescharacteristic
29#14556049C2212yes—
30#14556049C2211yescharacteristic
31C4×C71⋊C8C2211yescharacteristic
32C4×C71⋊C8C2212yes—
33C4⋊Q8×C71C2211yescharacteristic
34#125×C71C2212yes—
35C22⋊Q8×C71—21no—
36C22⋊Q8×C71—22no—
37C4×Q8×C71C2211yescharacteristic
38C4×D4×C71C2211yescharacteristic
39C4×D4×C71C2212yes—
40C2×C4⋊C4×C71—21no—
41C2×Dic142—21no—
42C2×Dic142C2311yescharacteristic
43#11765853Q814yescomplemented
44C2×C71⋊C8C2311yescharacteristic
45C2×C71⋊C8C2312yes—
46C2×C71⋊C8C2311yescharacteristic
47C4⋊Dic71—21no—
48C4⋊Dic71C2312yes—
49C4⋊Dic71C2311yescharacteristic
50Dic71⋊C4—22no—
51Dic71⋊C4—21no—
52Dic71⋊C4—22no—
53Dic71⋊C4—22no—
54C4×Dic71—22no—
55C4×Dic71—21no—
56C2×Q8×C71—21no—
57C2×Q8×C71D411yescharacteristic
58C2×D4×C71C2311yescharacteristic
59C22×C4×C71—21no—
60C22×C4×C71—22no—
61C4⋊C4×C71—22no—
62C4⋊C4×C71—21no—
63C4⋊C4×C71—22no—
64C4⋊C4×C71D411yescharacteristic
65C4⋊C4×C71D412yes—
66C4⋊C4×C71C2311yescharacteristic
67C4⋊C4×C71C2312yes—
68C22⋊C4×C71—21no—
69C22⋊C4×C71—22no—
70C42×C71C2312yes—
71C42×C71C2311yescharacteristic
72C23×C71—21no—
73Q8×C71—22no—
74Q8×C71—81no—
75D4×C71—21no—
76D4×C71C2×Q812yes—
77C2×C4×C71—21no—
78C2×C4×C71—22no—
79C2×C4×C71—42no—
80C2×C4×C71—82no—
81C2×C4×C71C2×D412yes—
82C2×C4×C71C2×D411yescharacteristic
83C2×C4×C71C2×D412yes—
84C2×C4×C71C2×D411yescharacteristic
85C2×C4×C71C2411yescharacteristic
86C2×Dic71—21no—
87C2×Dic71—21no—
88C2×Dic71—22no—
89Dic142—21no—
90Dic142—42no—
91Dic142C2×Q812yes—
92C71⋊C8—21no—
93C71⋊C8—22no—
94C71⋊C8C2×Q812yes—
95C22×C71—22no—
96C22×C71—41no—
97C22×C71—41no—
98C22×C71C22×D411yescharacteristic
99C4×C71—22no—
100C4×C71—22no—
101C4×C71—21no—
102C4×C71—41no—
103C4×C71—42no—
104C4×C71#13512yes—
105C4×C71C22×Q811yescharacteristic
106C4×C712−1+411yescharacteristic
107Dic71—22no—
108Dic71—42no—
109Dic71—41no—
110C2×C71—42no—
111C2×C71D4×Q811yescharacteristic
112C2×C71C2×#13511yescharacteristic
113C2×C71#57711yescharacteristic
114#3381—711nomaximal, Sylow
115C71#338111yescharacteristic, complemented, Sylow
116#557—711no—
117#554D7111yescharacteristic
118#499—712no—
119#498—711no—
120#479—712no—
121#478—712no—
122#477—711no—
123#474—711no—
124C8×Q8—711no—
125C4×SD16—711no—
126C4×SD16—712no—
127C2×SD16—711no—
128C4⋊Q8—711no—
129C4⋊Q8—1421no—
130C4⋊Q8C2×D7111yescharacteristic
131#125—712no—
132#125—1422no—
133#125C2×D7112yes—
134C22⋊Q8—21no—
135C22⋊Q8—22no—
136C4×Q8—711no—
137C4×Q8—712no—
138C4×Q8—1421no—
139C4×Q8C2×D7111yescharacteristic
140C4×D4C2×D7111yescharacteristic
141C4×D4C2×D7112yes—
142C2×C4⋊C4—21no—
143#107—712no—
144#107—712no—
145#106—712no—
146#106—711no—
147#106—712no—
148C4⋊C8—711no—
149C4⋊C8—712no—
150#103—712no—
151#103—711no—
152#102—712no—
153#102—711no—
154C4×C8—711no—
155C4×C8—712no—
156C2×Q8—21no—
157C2×Q8—711no—
158C2×Q8—1421no—
159C2×Q8C22⋊D7111yescharacteristic
160C2×D4C22×D7111yescharacteristic
161C22×C4—21no—
162C22×C4—22no—
163SD16—714no—
164C2×C8—711no—
165C2×C8—711no—
166C2×C8—712no—
167C4⋊C4—22no—
168C4⋊C4—21no—
169C4⋊C4—22no—
170C4⋊C4—712no—
171C4⋊C4—711no—
172C4⋊C4—1421no—
173C4⋊C4—1422no—
174C4⋊C4—1422no—
175C4⋊C4—1422no—
176C4⋊C4—1421no—
177C4⋊C4C22⋊D7111yescharacteristic
178C4⋊C4C22⋊D7112yes—
179C4⋊C4C22×D7112yes—
180C4⋊C4C22×D7111yescharacteristic
181C22⋊C4—21no—
182C22⋊C4—22no—
183C42—1422no—
184C42—1421no—
185C42C22×D7112yes—
186C42C22×D7111yescharacteristic
187C23—21no—
188Q8—22no—
189Q8—81no—
190Q8—2842no—
191Q8—712no—
192Q8—1421no—
193D4—21no—
194D4Q8×D7112yes—
195C2×C4—21no—
196C2×C4—22no—
197C2×C4—42no—
198C2×C4—82no—
199C2×C4—1421no—
200C2×C4—1422no—
201C2×C4—1421no—
202C2×C4C2×C22⋊D7112yes—
203C2×C4C2×C22⋊D7112yes—
204C2×C4C2×C22⋊D7111yescharacteristic
205C2×C4C2×C22⋊D7111yescharacteristic
206C2×C4C23×D7111yescharacteristic
207C8—712no—
208C8—1421no—
209C8—1422no—
210C22—22no—
211C22—41no—
212C22—41no—
213C22C22×C22⋊D7111yescharacteristic
214C4—22no—
215C4—22no—
216C4—21no—
217C4—41no—
218C4—42no—
219C4—2842no—
220C4—2841no—
221C4—1422no—
222C4#1455614812yes—
223C4C2×Q8×D7111yescharacteristic
224C4#1455617611yescharacteristic
225C2—42no—
226C2Q8×C22⋊D7111yescharacteristic
227C2C2×#1455614811yescharacteristic
228C2#1798681011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#554×C71C2
socleC22×C71C22×D4
hypercenter#554D71
hypocenterC71#3381
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #554

Complements

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

NH ≅ G/Ntype
1. #17986727C2semidirect
2. #17986398C2semidirect
3. #17986395C2semidirect
4. #17986303C2semidirect
6. #17986291C2semidirect
8. Q8×C71⋊C8C2semidirect
43. #11765853Q8semidirect
115. C71#3381semidirect