Finite Group Databasebeta

← #22610450

Subgroups — #22610450

Order 9344.

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

normal subgroups only

This group has 312 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#17989353C211yesmaximal, characteristic
2#17989161C211yesmaximal, characteristic
3#17988596C211yesmaximal, characteristic, complemented
4#17988560C211yesmaximal, characteristic
5#17988344C211yesmaximal, characteristic, complemented
6#17988343C211yesmaximal, characteristic, complemented
7#419×C73C211yesmaximal, characteristic, complemented
8C2×#11924028C2211yescharacteristic
9#14556599—22no—
10#14556597—22no—
11C2×#11924012C2211yescharacteristic
12M16×D73—22no—
13#14556589C412yescomplemented
14#14556580C412yescomplemented
15#14556574—22no—
16#14556567—22no—
17#14556550C2211yescharacteristic
18#14556550C2211yescharacteristic
19#14556515—22no—
20#14556505—22no—
21C22×Q8×C73C2211yescharacteristic
22C2×M16×C73C2211yescharacteristic
23#117×C73C2211yescharacteristic
24#103×C73—22no—
25#11924028—21no—
26#11924028—21no—
27#11924028—21no—
28#11924028—21no—
29C2×Dic146C2×C411yescharacteristic
30C2×D292C2×C411yescharacteristic
31C2×C22⋊D73D412yes—
32C2×C4×D73D412yes—
33#11924012—21no—
34#11924012—21no—
35#11924011—24no—
36C2×#10596831D412yes—
37C8×D73—24no—
38C4⋊Dic73—21no—
39C4⋊Dic73—21no—
40#11924007—24no—
41C22⋊Dic73—21no—
42C22⋊Dic73—21no—
43#11924005—24no—
44C4×Dic73—21no—
45C4×Dic73C2×C412yes—
46C2×Q8×C73—21no—
47C2×Q8×C73—21no—
48C2×Q8×C73—21no—
49C2×Q8×C73—21no—
50C2×Q8×C73—21no—
51C2×Q8×C73C2×C412yes—
52C22×C4×C73—21no—
53C22×C4×C73C2311yescharacteristic
54M16×C73—21no—
55M16×C73—21no—
56C2×C8×C73D412yes—
57C4⋊C4×C73—21no—
58C22⋊C4×C73—21no—
59C42×C73—21no—
60C23×C73C2×D411yescharacteristic
61C22×D73C22⋊C412yes—
62Q8×C73—22no—
63Q8×C73—22no—
64Q8×C73—41no—
65Q8×C73—41no—
66C2×C4×C73—22no—
67C2×C4×C73—41no—
68C2×C4×C73—41no—
69C2×C4×C73—41no—
70C2×C4×C73C22×C411yescharacteristic
71C2×C4×C73C2×D411yescharacteristic
72C2×C4×C73C2×D411yescharacteristic
73C2×C4×C73C2×D411yescharacteristic
74C2×C4×C73#4111yescharacteristic
75C2×C4×C73#4111yescharacteristic
76C22⋊D73—22no—
77C22⋊D73—22no—
78C2×Dic73—22no—
79C2×Dic73—41no—
80C2×Dic73C22⋊C412yes—
81D292—21no—
82D292—21no—
83C4×D73—22no—
84C4×D73—22no—
85Dic146—21no—
86Dic146—21no—
87C8×C73—22no—
88#10596831—22no—
89C22×C73—41no—
90C22×C73C2×C22⋊C411yescharacteristic
91C22×C73#12111yescharacteristic
92C22×C73#12111yescharacteristic
93C2×D73—22no—
94C2×D73—42no—
95C4×C73—42no—
96C4×C73—81no—
97C4×C73C4×D411yescharacteristic
98C4×C73C4×D411yescharacteristic
99C4×C73#12011yescharacteristic
100C4×C73#12311yescharacteristic
101Dic73—22no—
102Dic73—42no—
103Dic73—81no—
104C2×C73—21no—
105C2×C73—21no—
106C2×C73—21no—
107C2×C73#38611yescharacteristic
108D73—42no—
109#1909—731nomaximal, Sylow
110C73#190911yescharacteristic, complemented, Sylow
111#568—731no—
112#536—731no—
113#419—731no—
114#419D7311yescharacteristic, complemented
115#410—731no—
116H(3,ℤ/4)—731no—
117H(3,ℤ/4)—731no—
118C2×#41—731no—
119C22×Q8C2×D7311yescharacteristic
120#131—1462no—
121#131—1462no—
122C2×M16—731no—
123C2×M16—1462no—
124C2×M16C2×D7311yescharacteristic
125C4⋊Q8—732no—
126#124—732no—
127#123—1462no—
128C22⋊Q8—1462no—
129#117—731no—
130#117—731no—
131#117C2×D7311yescharacteristic
132#103—22no—
133#103—1462no—
134#99—1462no—
135#41—1461no—
136#41—1461no—
137#41—1461no—
138#41—1461no—
139C2×Q8—21no—
140C2×Q8—21no—
141C2×Q8—21no—
142C2×Q8—21no—
143C2×Q8—21no—
144C2×Q8—731no—
145C2×Q8C4×D7312yes—
146C2×D4—732no—
147C2×D4—731no—
148C22×C4—21no—
149C22×C4—732no—
150C22×C4C22×D7311yescharacteristic
151M16—21no—
152M16—21no—
153M16—1464no—
154M16—1461no—
155M16—1461no—
156C2×C8—732no—
157C2×C8—1464no—
158C2×C8C22⋊D7312yes—
159C4⋊C4—21no—
160C4⋊C4—1464no—
161C4⋊C4—1461no—
162C4⋊C4—1461no—
163C22⋊C4—21no—
164C22⋊C4—1464no—
165C22⋊C4—1461no—
166C22⋊C4—1461no—
167C42—21no—
168C42—732no—
169C42—1461no—
170C23—732no—
171C23D4×D7311yescharacteristic
172Q8—22no—
173Q8—22no—
174Q8—41no—
175Q8—41no—
176Q8—1461no—
177Q8—1461no—
178D4—1462no—
179D4—1462no—
180D4—1461no—
181D4—1461no—
182C2×C4—22no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—2921no—
187C2×C4—732no—
188C2×C4—1462no—
189C2×C4—1462no—
190C2×C4—1462no—
191C2×C4C2×C4×D7311yescharacteristic
192C2×C4D4×D7311yescharacteristic
193C2×C4D4×D7311yescharacteristic
194C2×C4C2×C22⋊D7311yescharacteristic
195C2×C4#1192402711yescharacteristic
196C2×C4#1192402811yescharacteristic
197C8—22no—
198C8—1462no—
199C22—41no—
200C22—2922no—
201C22—1462no—
202C22C22⋊C4×D7311yescharacteristic
203C22#1455656211yescharacteristic
204C22#1455656411yescharacteristic
205C4—42no—
206C4—81no—
207C4—2922no—
208C4—5841no—
209C4—1462no—
210C4#1455655111yescharacteristic
211C4C4×C22⋊D7311yescharacteristic
212C4#1455655911yescharacteristic
213C4#1455657311yescharacteristic
214C2—21no—
215C2—21no—
216C2—21no—
217C2—2922no—
218C2#1798846311yescharacteristic
subgroupG/H
centerC2#17988463
commutatorC2×C4×C73C22×C4
FrattiniC22×C4C22×D73
Fitting#419×C73C2
socleC2×C73#386
hypercenter#419D73
hypocenterC73#1909
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
3. #17988596C2semidirect
5. #17988344C2semidirect
6. #17988343C2semidirect
7. #419×C73C2semidirect
13. #14556589C4semidirect
14. #14556580C4semidirect
110. C73#1909semidirect
114. #419D73semidirect