Finite Group Databasebeta

← #32999862

Subgroups — #32999862

Order 18688.

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

normal subgroups only

This group has 288 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#22612158C211yesmaximal, characteristic
2#22612073C211yesmaximal, characteristic
3#22609325C211yesmaximal, characteristic
4#22607980C211yesmaximal, characteristic
5Q8⋊#11924003C211yesmaximal, characteristic
6#22607895C211yesmaximal, characteristic
7C73×#3395C211yesmaximal, characteristic
8#17988992C2211yescharacteristic
9#17988797—21no—
10#17988414—21no—
11C4⋊#11924003C2211yescharacteristic
12#17988391—21no—
13#17988391—22no—
14C4×#11924003C2211yescharacteristic
15#17988340C2211yescharacteristic
16#17988332—21no—
17#557×C73—21no—
18#494×C73—21no—
19#494×C73C2211yescharacteristic
20#462×C73—22no—
21#462×C73—21no—
22C8×Q8×C73C2211yescharacteristic
23C4×Q16×C73—21no—
24C4×Q16×C73C2211yescharacteristic
25C2×#11924018—41no—
26#14556591—21no—
27C2×#11924003—21no—
28C2×#11924003—21no—
29#14556525—21no—
30#14556525—21no—
31C4⋊#10596831—21no—
32#14556514—41no—
33C2×Q16×C73—22no—
34C2×Q16×C73—21no—
35C2×Q16×C73D411yescharacteristic
36C2×Q16×C73D412yes—
37C4⋊Q8×C73—22no—
38C4⋊Q8×C73—21no—
39#125×C73—42no—
40#125×C73—41no—
41C4×Q8×C73—22no—
42C4×Q8×C73—21no—
43C4×Q8×C73—41no—
44C4×Q8×C73D411yescharacteristic
45#107×C73D411yescharacteristic
46C4⋊C8×C73—21no—
47C4⋊C8×C73D411yescharacteristic
48#103×C73—22no—
49#103×C73—21no—
50C4×C8×C73—21no—
51C4×C8×C73C2311yescharacteristic
52C2×Dic146—41no—
53#11924018—82no—
54C2×#10596831—41no—
55C4⋊Dic73—41no—
56C4⋊Dic73—41no—
57#11924003—41no—
58#11924003—41no—
59C2×Q8×C73—21no—
60C2×Q8×C73—22no—
61C2×Q8×C73D811yescharacteristic
62Q16×C73—21no—
63Q16×C73—21no—
64Q16×C73—42no—
65Q16×C73—41no—
66Q16×C73—41no—
67Q16×C73—82no—
68C2×C8×C73—21no—
69C2×C8×C73C2×D411yescharacteristic
70C2×C8×C73C2×D411yescharacteristic
71C4⋊C4×C73—21no—
72C4⋊C4×C73—21no—
73C4⋊C4×C73—42no—
74C4⋊C4×C73—42no—
75C4⋊C4×C73—42no—
76C4⋊C4×C73—41no—
77C4⋊C4×C73—41no—
78C4⋊C4×C73—41no—
79C4⋊C4×C73D811yescharacteristic
80C42×C73—21no—
81C42×C73—42no—
82C42×C73—41no—
83C42×C73C2×D411yescharacteristic
84Q8×C73—21no—
85Q8×C73—41no—
86Q8×C73—42no—
87Q8×C73—42no—
88Q8×C73—41no—
89Q8×C73SD3212yes—
90C2×C4×C73—21no—
91C2×C4×C73—21no—
92C2×C4×C73—41no—
93C2×C4×C73—42no—
94C2×C4×C73—41no—
95C2×C4×C73#12011yescharacteristic
96C2×C4×C73C2×D811yescharacteristic
97C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
98C2×Dic73—81no—
99Dic146—81no—
100Dic146—81no—
101C8×C73—21no—
102C8×C73—41no—
103C8×C73#13511yescharacteristic
104C8×C73#13711yescharacteristic
105#10596831—81no—
106C22×C73#44711yescharacteristic
107C4×C73—22no—
108C4×C73—21no—
109C4×C73—41no—
110C4×C73—81no—
111C4×C73—82no—
112C4×C73—81no—
113C4×C73#45211yescharacteristic
114C4×C73#45511yescharacteristic
115C4×C73C2×SD3211yescharacteristic
116C4×C73#51011yescharacteristic
117Dic73—161no—
118#12382—731nomaximal, Sylow
119C2×C73#164111yescharacteristic
120C2×C73#218511yescharacteristic
121C2×C73#219511yescharacteristic
122#3395D7311yescharacteristic
123#2241—731no—
124#2225—731no—
125#1674—731no—
126#1362—731no—
127Q8⋊C16—731no—
128#1332—731no—
129C73#1238211yescharacteristic, complemented, Sylow
130#557—21no—
131#500—731no—
132#494—21no—
133#494C2×D7311yescharacteristic
134#462—22no—
135#462—21no—
136#462—1461no—
137C8×Q8C2×D7311yescharacteristic
138C4×Q16—21no—
139C4×Q16C2×D7311yescharacteristic
140#367—1461no—
141C4⋊C16—731no—
142#358—1461no—
143#358—1462no—
144C4×C16—731no—
145#335—731no—
146#332—1461no—
147C2×Q16—22no—
148C2×Q16—21no—
149C2×Q16—2921no—
150C2×Q16C22⋊D7311yescharacteristic
151C2×Q16C22⋊D7312yes—
152C4⋊Q8—22no—
153C4⋊Q8—21no—
154C4⋊Q8—1461no—
155#125—42no—
156#125—41no—
157C4×Q8—22no—
158C4×Q8—21no—
159C4×Q8—41no—
160C4×Q8C22⋊D7311yescharacteristic
161C2×C16—1461no—
162C2×C16—1461no—
163#107—1461no—
164#107—1461no—
165#107C22⋊D7311yescharacteristic
166C4⋊C8—21no—
167C4⋊C8—1461no—
168C4⋊C8C22⋊D7311yescharacteristic
169#103—22no—
170#103—21no—
171#103—2921no—
172C4×C8—21no—
173C4×C8C22×D7311yescharacteristic
174C2×Q8—21no—
175C2×Q8—22no—
176C2×Q8—2921no—
177C2×Q8#1192401311yescharacteristic
178Q16—21no—
179Q16—21no—
180Q16—42no—
181Q16—41no—
182Q16—41no—
183Q16—82no—
184Q16—5842no—
185C2×C8—21no—
186C2×C8—2921no—
187C2×C8C2×C22⋊D7311yescharacteristic
188C2×C8C2×C22⋊D7311yescharacteristic
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—42no—
192C4⋊C4—42no—
193C4⋊C4—42no—
194C4⋊C4—41no—
195C4⋊C4—41no—
196C4⋊C4—41no—
197C4⋊C4—2921no—
198C4⋊C4—2921no—
199C4⋊C4#1192401311yescharacteristic
200C42—21no—
201C42—42no—
202C42—41no—
203C42C2×C22⋊D7311yescharacteristic
204C16—2921no—
205C16—2921no—
206Q8—21no—
207Q8—41no—
208Q8—42no—
209Q8—42no—
210Q8—41no—
211Q8—5841no—
212Q8—5841no—
213Q8#1455653412yes—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—41no—
217C2×C4—42no—
218C2×C4—41no—
219C2×C4—5841no—
220C2×C4#1455656011yescharacteristic
221C2×C4C2×#1192401311yescharacteristic
222C2×C4#1455661711yescharacteristic
223C8—21no—
224C8—41no—
225C8—5841no—
226C8#1455661111yescharacteristic
227C8#1455662111yescharacteristic
228C22#1798872111yescharacteristic
229C4—22no—
230C4—21no—
231C4—41no—
232C4—81no—
233C4—81no—
234C4—82no—
235C4—11681no—
236C4#1798874911yescharacteristic
237C4#1798876511yescharacteristic
238C4C2×#1455653411yescharacteristic
239C4#1798902811yescharacteristic
240C2#2260912411yescharacteristic
241C2#2261184811yescharacteristic
242C2#2261190911yescharacteristic
subgroupG/H
centerC22#17988721
commutator, cosocleC4×C8×C73C23
FrattiniC4×C8C22×D73
FittingC73×#3395C2
socle, prime radicalC22×C73#447
hypercenter#3395D73
hypocenterC73#12382
LCSG ▷ C4×C8×C73 ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C42 ◁ C8×Q8 ◁ #3395

Complements

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

NH ≅ G/Ntype
129. C73#12382semidirect

Maximal subgroups

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

subgroupindex
#226121582
#226120732
#226093252
#226079802
Q8⋊#119240032
#226078952
C73×#33952
#1238273