Finite Group Databasebeta

← #32997507

Subgroups — #32997507

Order 18688.

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

normal subgroups only

This group has 268 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#22620590C211yesmaximal, characteristic
2#22611697C211yesmaximal, characteristic
3#22607991C212yesmaximal, complemented
4#22607969C212yesmaximal, complemented
5C73×#2166C211yesmaximal, characteristic
6#17989199C412yes—
7C2×#14556531—21no—
8#17988989C2211yescharacteristic
9#17988967C2211yescharacteristic
10#17988956C412yes—
11#17988845—21no—
12#17988828—21no—
13#17988695—21no—
14C4×Dic292—21no—
15#17988408—21no—
16#17988408C2212yes—
17#405×C73C2211yescharacteristic
18#364×C73C2212yes—
19#349×C73—21no—
20#347×C73—21no—
21C2×Dic292—21no—
22C2×Dic292—21no—
23#14556591C2×C412yes—
24#14556584—42no—
25#14556583C2×C412yes—
26#14556581—42no—
27#14556570—22no—
28C4×Dic146—22no—
29#14556531—22no—
30#14556531—41no—
31C2×#11924003—41no—
32#14556525—21no—
33#14556523—21no—
34#14556516—22no—
35#14556516—22no—
36C2×M16×C73D411yescharacteristic
37C22×C8×C73D411yescharacteristic
38#117×C73C2×C411yescharacteristic
39M32×C73—22no—
40#97×C73C2×C411yescharacteristic
41#97×C73D411yescharacteristic
42C4×C8×C73D411yescharacteristic
43C4×C8×C73C2311yescharacteristic
44C2×Dic146—22no—
45Dic292—42no—
46Dic292—41no—
47Dic292—41no—
48C4⋊Dic73—22no—
49C4⋊Dic73—42no—
50#11924007—42no—
51#11924007—42no—
52#11924007—42no—
53C22⋊Dic73—42no—
54C4×Dic73—42no—
55#11924003—42no—
56C22×C4×C73C22⋊C411yescharacteristic
57M16×C73—22no—
58C2×C8×C73—21no—
59C2×C8×C73—21no—
60C2×C8×C73C22⋊C411yescharacteristic
61C2×C8×C73C22⋊C411yescharacteristic
62C2×C8×C73C2×D411yescharacteristic
63C2×C8×C73C2×D411yescharacteristic
64C4⋊C4×C73—21no—
65C22⋊C4×C73—21no—
66C42×C73C22⋊C411yescharacteristic
67C42×C73C22×C411yescharacteristic
68C16×C73—42no—
69C23×C73#10011yescharacteristic
70C2×C4×C73—21no—
71C2×C4×C73—21no—
72C2×C4×C73—21no—
73C2×C4×C73—21no—
74C2×C4×C73#10011yescharacteristic
75C2×C4×C73C2×C22⋊C411yescharacteristic
76C2×Dic73—42no—
77C2×Dic73—42no—
78Dic146—42no—
79Dic146—42no—
80C8×C73—21no—
81C8×C73—21no—
82C8×C73#13512yes—
83C8×C73AGL(1,ℤ/8)11yescharacteristic
84C8×C73#13711yescharacteristic
85C22×C73—21no—
86C22×C73—41no—
87C22×C73C2×#10011yescharacteristic
88C4×C73—21no—
89C4×C73—41no—
90C4×C73—41no—
91C4×C73#41611yescharacteristic
92C4×C73#41711yescharacteristic
93Dic73—82no—
94Dic73—82no—
95#11990—731nomaximal, Sylow
96C2×C73—21no—
97C2×C73—41no—
98C2×C73#155111yescharacteristic
99#3177—731no—
100#2166D7311yescharacteristic
101#2153—731no—
102#1366—732no—
103#1358—732no—
104C73#1199011yescharacteristic, complemented, Sylow
105#541—732no—
106C2×M32—1461no—
107#499—731no—
108#494—731no—
109#491—732no—
110#470—1461no—
111#467—1461no—
112#441—1461no—
113C4×Q16—1461no—
114#405C2×D7311yescharacteristic
115#364—732no—
116#364—1461no—
117#364C2×D7312yes—
118#349—21no—
119#347—21no—
120C2×Q16—1461no—
121C2×Q16—1461no—
122C2×M16C22⋊D7311yescharacteristic
123C22×C8D29211yescharacteristic
124C4⋊Q8—732no—
125#126—2922no—
126#125—2922no—
127#125—732no—
128C22⋊Q8—1462no—
129C4×Q8—1462no—
130#117C4×D7311yescharacteristic
131M32—22no—
132M32—2921no—
133M32—1462no—
134C2×C16—2921no—
135#107—1461no—
136#106—1461no—
137#103—1462no—
138#103—1462no—
139#97C4×D7311yescharacteristic
140#97C22⋊D7311yescharacteristic
141C4×C8D29211yescharacteristic
142C4×C8C22×D7311yescharacteristic
143C2×Q8—1462no—
144C22×C4#1192400511yescharacteristic
145Q16—2922no—
146Q16—2921no—
147Q16—2921no—
148M16—22no—
149C2×C8—21no—
150C2×C8—21no—
151C2×C8#1192400511yescharacteristic
152C2×C8#1192400511yescharacteristic
153C2×C8C2×C22⋊D7311yescharacteristic
154C2×C8C2×D29211yescharacteristic
155C4⋊C4—21no—
156C4⋊C4—2922no—
157C4⋊C4—2922no—
158C4⋊C4—2922no—
159C4⋊C4—2922no—
160C4⋊C4—1462no—
161C22⋊C4—21no—
162C22⋊C4—2922no—
163C42—2922no—
164C42#1192400511yescharacteristic
165C42C2×C4×D7311yescharacteristic
166C16—42no—
167C16—2922no—
168C23#1455650711yescharacteristic
169Q8—2922no—
170Q8—2922no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—2922no—
176C2×C4—2922no—
177C2×C4#1455650711yescharacteristic
178C2×C4C2×#1192400511yescharacteristic
179C8—21no—
180C8—21no—
181C8#1455661112yes—
182C8#1455661911yescharacteristic
183C8#1455662411yescharacteristic
184C22—21no—
185C22—41no—
186C22C2×#1455650711yescharacteristic
187C4—21no—
188C4—41no—
189C4—41no—
190C4—5842no—
191C4—5842no—
192C4#1798858111yescharacteristic
193C4#1798858911yescharacteristic
194C2—21no—
195C2—41no—
196C2#2260862711yescharacteristic
subgroupG/H
centerC2#22608627
commutatorC42×C73C22×C4
FrattiniC4×C8C22×D73
FittingC73×#2166C2
socleC2×C73#1551
prime radicalC23×C73#100
cosocleC4×C8×C73C23
hypercenter#2166D73
hypocenterC73#11990
LCSG ▷ C42×C73 ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2 ◁ C2×C4 ◁ #405 ◁ #2166

Complements

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

NH ≅ G/Ntype
3. #22607991C2semidirect
4. #22607969C2semidirect
104. C73#11990semidirect

Maximal subgroups

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

subgroupindex
#226205902
#226116972
#226079912
#226079692
C73×#21662
#1199073