Finite Group Databasebeta

← #32996533

Subgroups — #32996533

Order 18688.

292 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 75 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#22609686C211yesmaximal, characteristic
3#22607926C212yesmaximal
4#22607920C212yesmaximal
5C73×#2166C211yesmaximal, characteristic
6#17989199—21no—
7#17988989C411yescharacteristic, complemented
8#17988967C411yescharacteristic, complemented
9#17988956—21no—
10#17988845C2212yes—
11#17988828—21no—
12#17988695C412yescomplemented
13C4×Dic292—21no—
14#17988542—21no—
15#17988345C2211yescharacteristic
16#17988344C2211yescharacteristic
17#17988343—21no—
18#405×C73C2211yescharacteristic
19#364×C73—21no—
20#349×C73C2212yes—
21#347×C73—21no—
22C2×Dic292—21no—
23C2×Dic292C2×C412yes—
24C2×#11924012—21no—
25#14556591—21no—
26#14556584—42no—
27#14556583—21no—
28#14556581—42no—
29#14556570—22no—
30C4×Dic146—22no—
31#14556550—21no—
32#14556525—21no—
33#14556523C2×C412yes—
34#14556516—22no—
35#14556516—22no—
36#14556501—22no—
37C8×Dic73—22no—
38C2×M16×C73C2311yescharacteristic
39C22×C8×C73D411yescharacteristic
40#117×C73D411yescharacteristic
41M32×C73—22no—
42#97×C73C2×C411yescharacteristic
43#97×C73D411yescharacteristic
44C4×C8×C73C2×C411yescharacteristic
45C4×C8×C73D411yescharacteristic
46C2×Dic146—22no—
47Dic292—42no—
48Dic292—41no—
49Dic292—41no—
50#11924012—41no—
51#11924012—41no—
52C2×#10596831—22no—
53C4⋊Dic73—22no—
54C4⋊Dic73—41no—
55C4⋊Dic73—42no—
56#11924007—42no—
57#11924007—42no—
58#11924007—42no—
59C22⋊Dic73—41no—
60C22⋊Dic73—42no—
61C4×Dic73—22no—
62C4×Dic73—42no—
63C22×C4×C73C2×D411yescharacteristic
64M16×C73—41no—
65C2×C8×C73—21no—
66C2×C8×C73—21no—
67C2×C8×C73C22⋊C411yescharacteristic
68C2×C8×C73C22⋊C411yescharacteristic
69C2×C8×C73C22×C411yescharacteristic
70C2×C8×C73C2×D411yescharacteristic
71C4⋊C4×C73—21no—
72C22⋊C4×C73—21no—
73C42×C73C22⋊C411yescharacteristic
74C42×C73C22⋊C411yescharacteristic
75C16×C73—42no—
76C23×C73AGL(1,ℤ/8)11yescharacteristic
77C2×C4×C73—21no—
78C2×C4×C73—21no—
79C2×C4×C73—21no—
80C2×C4×C73—21no—
81C2×C4×C73C2×C22⋊C411yescharacteristic
82C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
83C2×Dic73—42no—
84C2×Dic73—42no—
85C2×Dic73—42no—
86Dic146—42no—
87Dic146—42no—
88C8×C73—21no—
89C8×C73—21no—
90C8×C73—21no—
91C8×C73#10411yescharacteristic
92C8×C73#10411yescharacteristic
93#10596831—42no—
94C22×C73—21no—
95C22×C73—41no—
96C22×C73#41811yescharacteristic
97C4×C73—21no—
98C4×C73—41no—
99C4×C73—41no—
100C4×C73#41011yescharacteristic
101C4×C73C2×#10411yescharacteristic
102Dic73—82no—
103Dic73—82no—
104Dic73—82no—
105#11771—731nomaximal, Sylow
106C2×C73—21no—
107C2×C73—41no—
108C2×C73#150511yescharacteristic
109#3177—731no—
110#2166D7311yescharacteristic
111#1747—731no—
112#1342—732no—
113#1340—732no—
114C73#1177111yescharacteristic, complemented, Sylow
115#541—1461no—
116#499—731no—
117#494—731no—
118#491—1461no—
119#470—732no—
120#467—1461no—
121#441—732no—
122C4×Q16—1461no—
123#405—1461no—
124#405C2×D7311yescharacteristic
125#364—21no—
126#349C2×D7312yes—
127#347—21no—
128#338—731no—
129H(3,ℤ/4)—731no—
130H(3,ℤ/4)—1461no—
131C2×Q16—732no—
132C2×Q16—1461no—
133C2×M16—1461no—
134C2×M16C22×D7311yescharacteristic
135C22×C8D29211yescharacteristic
136C4⋊Q8—1461no—
137#126—2922no—
138#125—2922no—
139#125—1461no—
140C22⋊Q8—1462no—
141C4×Q8—1462no—
142#117—1461no—
143#117D29211yescharacteristic
144M32—22no—
145#107—1461no—
146#106—732no—
147#103—1462no—
148#103—1462no—
149#97—1462no—
150#97C4×D7311yescharacteristic
151#97C22⋊D7311yescharacteristic
152C4×C8—1462no—
153C4×C8C4×D7311yescharacteristic
154C4×C8C22⋊D7311yescharacteristic
155C2×Q8—1462no—
156C22×C4C2×D29211yescharacteristic
157Q16—2922no—
158Q16—2921no—
159Q16—2921no—
160M16—41no—
161M16—2921no—
162M16—2921no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—1462no—
166C2×C8#1192400511yescharacteristic
167C2×C8#1192400511yescharacteristic
168C2×C8C2×C4×D7311yescharacteristic
169C2×C8C2×C22⋊D7311yescharacteristic
170C4⋊C4—21no—
171C4⋊C4—2922no—
172C4⋊C4—2922no—
173C4⋊C4—2921no—
174C4⋊C4—2922no—
175C4⋊C4—2922no—
176C4⋊C4—1462no—
177C22⋊C4—21no—
178C22⋊C4—2921no—
179C22⋊C4—2922no—
180C42—2922no—
181C42—1462no—
182C42#1192400511yescharacteristic
183C42#1192400511yescharacteristic
184C16—42no—
185C23#1455661911yescharacteristic
186Q8—2922no—
187Q8—2922no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—2922no—
193C2×C4—2922no—
194C2×C4—2922no—
195C2×C4C2×#1192400511yescharacteristic
196C2×C4#1455661911yescharacteristic
197C8—21no—
198C8—21no—
199C8—21no—
200C8—2922no—
201C8#1455651711yescharacteristic
202C8#1455651711yescharacteristic
203C22—21no—
204C22—41no—
205C22#1798859311yescharacteristic
206C4—21no—
207C4—41no—
208C4—41no—
209C4—5842no—
210C4—5842no—
211C4—5842no—
212C4#1798856011yescharacteristic
213C4C2×#1455651711yescharacteristic
214C2—21no—
215C2—41no—
216C2#2260844011yescharacteristic
subgroupG/H
centerC2#22608440
commutatorC2×C8×C73C22×C4
FrattiniC2×M16C22×D73
FittingC73×#2166C2
socleC2×C73#1505
prime radicalC23×C73AGL(1,ℤ/8)
cosocleC2×M16×C73C23
hypercenter#2166D73
hypocenterC73#11771
LCSG ▷ C2×C8×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
7. #17988989C4semidirect
8. #17988967C4semidirect
12. #17988695C4semidirect
114. C73#11771semidirect

Maximal subgroups

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

subgroupindex
#226205902
#226096862
#226079262
#226079202
C73×#21662
#1177173