Finite Group Databasebeta

← #22588874

Subgroups — #22588874

Order 9088.

314 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 79 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#17986570C211yesmaximal, characteristic
2#17986476C211yesmaximal, characteristic, complemented
3#17986016C211yesmaximal, characteristic
4C2×#14556046C211yesmaximal, characteristic, complemented
5#17985756C211yesmaximal, characteristic, complemented
6#17985755C211yesmaximal, characteristic, complemented
7#421×C71C211yesmaximal, characteristic
8C22×Dic142C2211yescharacteristic
9C2×#11765849C2211yescharacteristic
10#14556126C412yescomplemented
11#14556115C412yescomplemented
12#14556109—22no—
13#14556107—22no—
14C4×Dic142—22no—
15#14556093—22no—
16#14556087C2211yescharacteristic
17#14556087C2211yescharacteristic
18#14556084—22no—
19#14556055—22no—
20#14556046—22no—
21C2×#41×C71C2211yescharacteristic
22C2×M16×C71C2211yescharacteristic
23#117×C71C2211yescharacteristic
24#104×C71—22no—
25C2×Dic142—21no—
26C2×Dic142—21no—
27C2×Dic142—21no—
28C2×Dic142—21no—
29C2×Dic142C2×C412yes—
30C2×Dic142D412yes—
31C22×Dic71D412yes—
32#11765849—21no—
33#11765849—21no—
34C2×C71⋊C8—21no—
35C4⋊Dic71—21no—
36C4⋊Dic71—21no—
37Dic71⋊C4—24no—
38Dic71⋊C4—24no—
39C22⋊Dic71—21no—
40C22⋊Dic71—21no—
41C22⋊Dic71—24no—
42C4×Dic71—24no—
43C4×Dic71C2×C412yes—
44C4×Dic71D412yes—
45#41×C71—21no—
46#41×C71—21no—
47#41×C71—21no—
48#41×C71—21no—
49C2×Q8×C71C2×C411yescharacteristic
50C2×D4×C71—21no—
51C2×D4×C71C2×C411yescharacteristic
52C22×C4×C71—21no—
53C22×C4×C71C2311yescharacteristic
54M16×C71—21no—
55M16×C71—21no—
56C2×C8×C71—21no—
57C4⋊C4×C71—21no—
58C22⋊C4×C71—21no—
59C42×C71D412yes—
60C23×C71—21no—
61C23×C71C2×D411yescharacteristic
62Q8×C71—21no—
63Q8×C71—21no—
64D4×C71—21no—
65D4×C71—21no—
66D4×C71—41no—
67D4×C71—41no—
68C2×C4×C71—21no—
69C2×C4×C71—22no—
70C2×C4×C71—41no—
71C2×C4×C71—41no—
72C2×C4×C71C22×C411yescharacteristic
73C2×C4×C71C2×D411yescharacteristic
74C2×C4×C71C2×D411yescharacteristic
75C2×C4×C71C2×D411yescharacteristic
76C2×C4×C71#4111yescharacteristic
77C2×C4×C71#4111yescharacteristic
78C2×Dic71—22no—
79C2×Dic71—22no—
80C2×Dic71—22no—
81C2×Dic71—22no—
82C2×Dic71C22⋊C414yes—
83Dic142—22no—
84Dic142—22no—
85Dic142—22no—
86Dic142—22no—
87C8×C71—41no—
88C71⋊C8—41no—
89C22×C71—41no—
90C22×C71—42no—
91C22×C71—41no—
92C22×C71C2×C22⋊C411yescharacteristic
93C22×C71#12011yescharacteristic
94C22×C71#12311yescharacteristic
95C4×C71—42no—
96C4×C71—41no—
97C4×C71C4×D411yescharacteristic
98C4×C71C4×D411yescharacteristic
99C4×C71#12111yescharacteristic
100C4×C71#12111yescharacteristic
101Dic71—24no—
102Dic71—42no—
103Dic71—42no—
104C2×C71—21no—
105C2×C71—21no—
106C2×C71—21no—
107C2×C71—81no—
108C2×C71#38611yescharacteristic
109#1912—711nomaximal, Sylow
110C71#191211yescharacteristic, complemented, Sylow
111#536—711no—
112#519—711no—
113#421—711no—
114#421D7111yescharacteristic
115C2×#101—711no—
116H(3,ℤ/4)—711no—
117H(3,ℤ/4)—711no—
118C2×#41C2×D7111yescharacteristic
119C22×Q8—711no—
120C2×M16—711no—
121C2×M16C2×D7111yescharacteristic
122C4⋊Q8—712no—
123#124—712no—
124#123—1422no—
125C22⋊Q8—1422no—
126C4×Q8—1422no—
127C4×Q8—1422no—
128#117—711no—
129#117—711no—
130#117—1422no—
131#117C2×D7111yescharacteristic
132#104—22no—
133#104—1422no—
134#101—1422no—
135#41—21no—
136#41—21no—
137#41—21no—
138#41—21no—
139C2×Q8—712no—
140C2×Q8—712no—
141C2×Q8—1421no—
142C2×Q8—1421no—
143C2×Q8—1421no—
144C2×Q8—1421no—
145C2×Q8C4×D7111yescharacteristic
146C2×D4—21no—
147C2×D4C4×D7111yescharacteristic
148C22×C4—21no—
149C22×C4—712no—
150C22×C4C22×D7111yescharacteristic
151M16—21no—
152M16—21no—
153M16—1421no—
154M16—1421no—
155C2×C8—21no—
156C2×C8—1421no—
157C4⋊C4—21no—
158C4⋊C4—1424no—
159C4⋊C4—1424no—
160C4⋊C4—1421no—
161C4⋊C4—1421no—
162C22⋊C4—21no—
163C22⋊C4—1424no—
164C22⋊C4—1421no—
165C22⋊C4—1421no—
166C42—712no—
167C42—712no—
168C42—1424no—
169C42C22⋊D7112yes—
170C23—21no—
171C23D4×D7111yescharacteristic
172Q8—21no—
173Q8—21no—
174Q8—1422no—
175Q8—1422no—
176Q8—1422no—
177Q8—1422no—
178D4—21no—
179D4—21no—
180D4—41no—
181D4—41no—
182C2×C4—21no—
183C2×C4—22no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—714no—
187C2×C4—1422no—
188C2×C4—1422no—
189C2×C4—1422no—
190C2×C4—1422no—
191C2×C4C2×C4×D7111yescharacteristic
192C2×C4D4×D7111yescharacteristic
193C2×C4D4×D7111yescharacteristic
194C2×C4C2×C22⋊D7111yescharacteristic
195C2×C4#1176586411yescharacteristic
196C2×C4#1176586511yescharacteristic
197C8—41no—
198C8—2841no—
199C22—41no—
200C22—42no—
201C22—41no—
202C22C22⋊C4×D7111yescharacteristic
203C22#1455609611yescharacteristic
204C22#1455611011yescharacteristic
205C4—42no—
206C4—41no—
207C4—2842no—
208C4—2842no—
209C4—1424no—
210C4(C22⋊D71)⋊C411yescharacteristic
211C4C4×C22⋊D7111yescharacteristic
212C4#1455609911yescharacteristic
213C4#1455610111yescharacteristic
214C2—21no—
215C2—21no—
216C2—21no—
217C2—81no—
218C2#1798587511yescharacteristic
subgroupG/H
centerC2#17985875
commutatorC2×C4×C71C22×C4
FrattiniC22×C4C22×D71
Fitting#421×C71C2
socleC2×C71#386
hypercenter#421D71
hypocenterC71#1912
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2 ◁ C22×C4 ◁ #421

Complements

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

NH ≅ G/Ntype
2. #17986476C2semidirect
4. C2×#14556046C2semidirect
5. #17985756C2semidirect
6. #17985755C2semidirect
10. #14556126C4semidirect
11. #14556115C4semidirect
110. C71#1912semidirect