Finite Group Databasebeta

← #34014332

Subgroups — #34014332

Order 18688.

406 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 60 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#22626204C212yesmaximal
2#22625064C214yesmaximal, complemented
3#22620388C211yesmaximal, characteristic
4#17989566—22no—
5#17989494—22no—
6#17989492C2214yes—
7C2×#14556622—22no—
8#17989137C2211yescharacteristic
9#17988979—22no—
10#17988966C412yescomplemented
11#17988961C2212yes—
12C4×#11924012C411yescharacteristic, complemented
13C2×C4⋊Q8×C73C411yescharacteristic, complemented
14#14556670—42no—
15#14556650—24no—
16#14556649—42no—
17C2×#11924028D412yes—
18#14556622—44no—
19#14556622—44no—
20C2×#11924018—24no—
21C2×#11924015—24no—
22C2×#11924012—21no—
23#14556591C2×C411yescharacteristic
24#14556588C2311yescharacteristic
25#14556579D412yes—
26#14556565—41no—
27C4×D292—41no—
28#14556502—21no—
29C4×#10596831C2×C412yes—
30C22×Q8×C73—21no—
31C4⋊Q8×C73—21no—
32C4⋊Q8×C73—41no—
33C4⋊Q8×C73C2×C412yes—
34C2×C4⋊C4×C73—41no—
35C2×C42×C73C2×C411yescharacteristic
36C4⋊#64408—82no—
37C22⋊#64408—44no—
38C4×#64408—82no—
39#11924028—42no—
40#11924028—42no—
41C2×Dic146C22⋊C412yes—
42C2×D292—41no—
43C2×D292C2×D412yes—
44C2×C22⋊D73—22no—
45C2×C4×D73—22no—
46#11924018—48no—
47#11924015—48no—
48#11924012—24no—
49C2×#10596831—22no—
50C4⋊Dic73—41no—
51#11924005—42no—
52C2×Q8×C73—22no—
53C2×Q8×C73—42no—
54C2×Q8×C73—42no—
55C2×Q8×C73—41no—
56C22×C4×C73—21no—
57C22×C4×C73—41no—
58C22×C4×C73C22⋊C411yescharacteristic
59C4⋊C4×C73—42no—
60C4⋊C4×C73—42no—
61C42×C73—21no—
62C42×C73C22⋊C411yescharacteristic
63C42×C73C22×C411yescharacteristic
64C23×C73#9911yescharacteristic
65C22×D73—22no—
66C2×#64408—84no—
67Q8×C73—44no—
68Q8×C73—82no—
69C2×C4×C73—21no—
70C2×C4×C73—22no—
71C2×C4×C73—21no—
72C2×C4×C73—42no—
73C2×C4×C73—42no—
74C2×C4×C73—81no—
75C2×C4×C73—81no—
76C2×C4×C73#9911yescharacteristic
77C2×C4×C73C2×C22⋊C411yescharacteristic
78C22⋊D73—82no—
79C2×Dic73—22no—
80D292—42no—
81D292—82no—
82C4×D73—42no—
83C4×D73—42no—
84Dic146—42no—
85#10596831—24no—
86C22×C73—21no—
87C22×C73—41no—
88C22×C73C2×#9911yescharacteristic
89C2×D73—42no—
90C2×D73—82no—
91#64408—164no—
92C4×C73—22no—
93C4×C73—21no—
94C4×C73—41no—
95C4×C73—82no—
96C4×C73#35312yes—
97Dic73—42no—
98#13586—731nomaximal, Sylow
99C2×C73—21no—
100C2×C73—41no—
101C2×C73C2×#35311yescharacteristic
102D73—82no—
103#3152—731no—
104#2126—732no—
105#1409—734no—
106C73#1358611yescharacteristic, complemented, Sylow
107C2×#137—1462no—
108#532—731no—
109C2×C4⋊Q8#6440811yescharacteristic, complemented
110#497—1462no—
111#494—732no—
112#492—732no—
113#410—1462no—
114C4×M16—731no—
115#354—1462no—
116#353—734no—
117C2×#41—732no—
118C22×Q8—21no—
119#137—2924no—
120#137—2924no—
121C2×Q16—1464no—
122C2×SD16—1464no—
123C2×M16—1461no—
124C4⋊Q8—21no—
125C4⋊Q8—41no—
126C4⋊Q8—731no—
127C4⋊Q8C2×#6440812yes—
128#127—731no—
129#124—732no—
130#121—2921no—
131C4×D4—2921no—
132#117—2922no—
133C2×C4⋊C4—41no—
134C2×C42C2×#6440811yescharacteristic
135#99—2922no—
136#99—1464no—
137#97—1461no—
138C4×C8—732no—
139#41—2922no—
140#41—2922no—
141C2×Q8—22no—
142C2×Q8—42no—
143C2×Q8—42no—
144C2×Q8—41no—
145C2×Q8—732no—
146C2×D4—2921no—
147C2×D4—732no—
148C2×D4—1462no—
149C22×C4—21no—
150C22×C4—41no—
151C22×C4—1462no—
152C22×C4C22⋊#6440811yescharacteristic
153Q16—2928no—
154SD16—2928no—
155M16—1464no—
156C2×C8—1462no—
157C4⋊C4—42no—
158C4⋊C4—42no—
159C4⋊C4—2921no—
160C4⋊C4—5842no—
161C22⋊C4—2922no—
162C22⋊C4—2924no—
163C42—21no—
164C42—5842no—
165C42C22⋊#6440811yescharacteristic
166C42C22×#6440811yescharacteristic
167C23—1462no—
168C23#1455664911yescharacteristic
169Q8—44no—
170Q8—82no—
171Q8—2922no—
172D4—2922no—
173D4—5842no—
174D4—5842no—
175C2×C4—21no—
176C2×C4—22no—
177C2×C4—21no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—81no—
181C2×C4—81no—
182C2×C4—2922no—
183C2×C4—2922no—
184C2×C4—5844no—
185C2×C4—1462no—
186C2×C4#1455664911yescharacteristic
187C2×C4C2×C22⋊#6440811yescharacteristic
188C8—1464no—
189C22—21no—
190C22—41no—
191C22—2922no—
192C22—5842no—
193C22C2×#1455664911yescharacteristic
194C4—22no—
195C4—21no—
196C4—41no—
197C4—82no—
198C4—2922no—
199C4—11684no—
200C4#1798949112yes—
201C2—21no—
202C2—41no—
203C2—5842no—
204C2C2×#1798949111yescharacteristic
subgroupG/H
centerC2C2×#17989491
commutatorC42×C73C22×C4
FrattiniC42C22×#64408
FittingC2×C4⋊Q8×C73C4
socleC2×C73C2×#353
prime radical#17989137C22
cosocle#14556588C23
hypercenterC2×C4⋊Q8#64408
hypocenterC73#13586
LCSG ▷ C42×C73 ▷ C2×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ C2×C42 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
2. #22625064C2semidirect
10. #17988966C4semidirect
12. C4×#11924012C4semidirect
13. C2×C4⋊Q8×C73C4semidirect
106. C73#13586semidirect
109. C2×C4⋊Q8#64408semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#226262042
#226250642
#226203882
#1358673