Finite Group Databasebeta

← #34011704

Subgroups — #34011704

Order 18688.

360 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 64 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#22626300C212yesmaximal, complemented
2#22624942C212yesmaximal, complemented
3#22624940C212yesmaximal, complemented
4#22620388C211yesmaximal, characteristic
5#17989510C2214yes—
6#17989483—22no—
7#17989479—22no—
8C2×#14556622—22no—
9#17989137C411yescharacteristic
10#17988979—22no—
11#17988966C411yescharacteristic
12#17988966C2211yescharacteristic
13#17988961C411yescharacteristic
14#17988961C2211yescharacteristic
15C4×#11924012C2211yescharacteristic
16C2×C4⋊Q8×C73C411yescharacteristic
17#14556665—24no—
18C2×#11924028—21no—
19#14556622—48no—
20C2×#11924018—22no—
21C2×#11924018—22no—
22C2×#11924015—22no—
23C2×#11924015—22no—
24C2×#11924012D412yes—
25#14556591C2×C411yescharacteristic
26#14556588C2×C411yescharacteristic
27#14556579—21no—
28#14556565—41no—
29C4×D292—41no—
30#14556502D412yes—
31C4×#10596831C2×C411yescharacteristic
32C4×#10596831C2311yescharacteristic
33C22×Q8×C73—21no—
34C4⋊Q8×C73—21no—
35C4⋊Q8×C73—41no—
36C4⋊Q8×C73C2×C411yescharacteristic
37C4⋊Q8×C73C2×C411yescharacteristic
38C2×C4⋊C4×C73—41no—
39C2×C42×C73C2×C411yescharacteristic
40#11924031—44no—
41#11924028—44no—
42C2×Dic146—21no—
43C2×D292—21no—
44C2×D292—41no—
45C2×C22⋊D73—41no—
46C2×C4×D73—41no—
47#11924018—44no—
48#11924018—44no—
49#11924015—44no—
50#11924015—44no—
51#11924012—24no—
52C2×#10596831C22⋊C412yes—
53C2×#10596831C2×D412yes—
54C4⋊Dic73—41no—
55#11924005—42no—
56C2×Q8×C73—21no—
57C2×Q8×C73—21no—
58C2×Q8×C73—44no—
59C2×Q8×C73—41no—
60C22×C4×C73—21no—
61C22×C4×C73—41no—
62C22×C4×C73C22⋊C411yescharacteristic
63C4⋊C4×C73—41no—
64C4⋊C4×C73—41no—
65C4⋊C4×C73—42no—
66C42×C73—21no—
67C42×C73C22⋊C411yescharacteristic
68C42×C73C22×C411yescharacteristic
69C23×C73#10011yescharacteristic
70C22×D73—41no—
71Q8×C73—42no—
72Q8×C73—42no—
73Q8×C73—82no—
74C2×C4×C73—21no—
75C2×C4×C73—22no—
76C2×C4×C73—21no—
77C2×C4×C73—42no—
78C2×C4×C73—41no—
79C2×C4×C73—41no—
80C2×C4×C73—82no—
81C2×C4×C73#10011yescharacteristic
82C2×C4×C73C2×C22⋊C411yescharacteristic
83C22⋊D73—82no—
84C2×Dic73—41no—
85D292—42no—
86D292—82no—
87C4×D73—82no—
88Dic146—42no—
89#10596831—22no—
90#10596831AGL(1,ℤ/8)14yes—
91C22×C73—21no—
92C22×C73—41no—
93C22×C73C2×#10011yescharacteristic
94C2×D73—82no—
95C2×D73—81no—
96C4×C73—22no—
97C4×C73—21no—
98C4×C73—41no—
99C4×C73—81no—
100C4×C73—81no—
101C4×C73#41812yes—
102Dic73—81no—
103#11993—731nomaximal, Sylow
104C2×C73—21no—
105C2×C73—41no—
106C2×C73#155511yescharacteristic
107D73—161no—
108#3152—731no—
109#2165—732no—
110#1359—732no—
111#1358—732no—
112C73#1199311yescharacteristic, complemented, Sylow
113C2×#137—1462no—
114#532—731no—
115C2×C4⋊Q8#6440811yescharacteristic
116#497—1462no—
117#494—731no—
118#494—731no—
119#492—731no—
120#492—731no—
121C4×M16—731no—
122#364—734no—
123#349—1462no—
124#347—1462no—
125C2×#41—1461no—
126C22×Q8—21no—
127#137—2928no—
128C2×Q16—1462no—
129C2×Q16—1462no—
130C2×SD16—1462no—
131C2×SD16—1462no—
132C2×M16—732no—
133C4⋊Q8—21no—
134C4⋊Q8—41no—
135C4⋊Q8—731no—
136C4⋊Q8C2×#6440811yescharacteristic
137C4⋊Q8C2×#6440811yescharacteristic
138#127—731no—
139#124—1461no—
140#121—2921no—
141C4×D4—2921no—
142C2×C4⋊C4—41no—
143C2×C42C2×#6440811yescharacteristic
144M32—1464no—
145#97—732no—
146C4×C8—731no—
147C4×C8—731no—
148#41—2924no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—44no—
152C2×Q8—41no—
153C2×Q8—1461no—
154C2×D4—2921no—
155C2×D4—2921no—
156C2×D4—1461no—
157C22×C4—21no—
158C22×C4—41no—
159C22×C4—2921no—
160C22×C4C22⋊#6440811yescharacteristic
161Q16—2924no—
162Q16—2924no—
163SD16—2924no—
164SD16—2924no—
165M16—1464no—
166C2×C8—732no—
167C2×C8—732no—
168C4⋊C4—41no—
169C4⋊C4—41no—
170C4⋊C4—42no—
171C4⋊C4—2921no—
172C22⋊C4—2922no—
173C42—21no—
174C42C22⋊#6440811yescharacteristic
175C42C22×#6440811yescharacteristic
176C16—2924no—
177C23—2921no—
178C23#1455665111yescharacteristic
179Q8—42no—
180Q8—42no—
181Q8—82no—
182Q8—2922no—
183D4—2922no—
184D4—5842no—
185D4—5842no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—21no—
189C2×C4—42no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—82no—
193C2×C4—2921no—
194C2×C4—5842no—
195C2×C4#1455665111yescharacteristic
196C2×C4C2×C22⋊#6440811yescharacteristic
197C8—734no—
198C8—1462no—
199C22—21no—
200C22—41no—
201C22—5842no—
202C22—5841no—
203C22C2×#1455665111yescharacteristic
204C4—22no—
205C4—21no—
206C4—41no—
207C4—81no—
208C4—81no—
209C4—5841no—
210C4#1798957912yes—
211C2—21no—
212C2—41no—
213C2—11681no—
214C2#2262545811yescharacteristic
subgroupG/H
centerC2#22625458
commutatorC42×C73C22×C4
FrattiniC42C22×#64408
FittingC2×C4⋊Q8×C73C4
socleC2×C73#1555
prime radical#17989137C4
cosocleC4×#10596831C23
hypercenterC2×C4⋊Q8#64408
hypocenterC73#11993
LCSG ▷ C42×C73 ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. #22626300C2semidirect
2. #22624942C2semidirect
3. #22624940C2semidirect
112. C73#11993semidirect

Maximal subgroups

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

subgroupindex
#226263002
#226249422
#226249402
#226203882
#1199373