Finite Group Databasebeta

← #32987319

Subgroups — #32987319

Order 18688.

500 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 95 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
1C2×#17988993C212yesmaximal
2#22617999C211yesmaximal, characteristic
3C2×#17988328C212yesmaximal
4#22607812C211yesmaximal, characteristic
5C73×#1960C211yesmaximal, characteristic
6C2×#14556583C2212yes—
7#17988993—24no—
8#17988993—24no—
9C2×#14556524—22no—
10C2×#14556522—22no—
11C2×#14556502C2212yes—
12#17988527—22no—
13#17988519—22no—
14#17988517—22no—
15#17988328—24no—
16C2×C4⋊Q8×C73C2211yescharacteristic
17C2×#103×C73—22no—
18C2×#97×C73C2211yescharacteristic
19#382×C73C2211yescharacteristic
20C22×#10596831—22no—
21#14556583—22no—
22#14556583—22no—
23#14556583—22no—
24#14556583—22no—
25C2×C4⋊Dic73—22no—
26C2×C4⋊Dic73D412yes—
27C2×C4⋊Dic73D412yes—
28C2×#11924007—22no—
29C2×#11924007—22no—
30#14556524—28no—
31#14556522—28no—
32#14556502—22no—
33#14556502D414yes—
34#14556498—24no—
35#14556498—22no—
36C22×Q8×C73—21no—
37C22×C8×C73—21no—
38C4⋊Q8×C73—22no—
39C4⋊Q8×C73—21no—
40C4⋊Q8×C73D412yes—
41C2×C4⋊C4×C73—22no—
42C2×C4⋊C4×C73D412yes—
43C2×C42×C73C2311yescharacteristic
44#103×C73—44no—
45#97×C73—22no—
46#95×C73—22no—
47C22×Dic73—22no—
48C22×Dic73—22no—
49C2×#10596831—28no—
50C2×#10596831—24no—
51C4⋊Dic73—22no—
52C4⋊Dic73—22no—
53C4⋊Dic73—22no—
54C4⋊Dic73—22no—
55C4⋊Dic73—82no—
56#11924007—42no—
57#11924007—42no—
58#11924007—42no—
59#11924007—42no—
60C2×Q8×C73—22no—
61C2×Q8×C73—42no—
62C2×Q8×C73—42no—
63C2×Q8×C73—41no—
64C22×C4×C73—22no—
65C22×C4×C73—41no—
66C22×C4×C73C2×D412yes—
67C22×C4×C73C2×D411yescharacteristic
68C2×C8×C73—22no—
69C2×C8×C73—41no—
70C2×C8×C73—41no—
71C4⋊C4×C73—24no—
72C4⋊C4×C73—42no—
73C4⋊C4×C73—42no—
74C42×C73C2×D412yes—
75C42×C73C2×D411yescharacteristic
76C42×C73#4111yescharacteristic
77C23×C73#12011yescharacteristic
78Q8×C73—44no—
79Q8×C73—82no—
80C2×C4×C73—21no—
81C2×C4×C73—21no—
82C2×C4×C73—24no—
83C2×C4×C73—42no—
84C2×C4×C73—42no—
85C2×C4×C73—44no—
86C2×C4×C73—81no—
87C2×C4×C73—81no—
88C2×C4×C73#12012yes—
89C2×C4×C73#12112yes—
90C2×C4×C73#12111yescharacteristic
91C2×C4×C73#12711yescharacteristic
92C2×Dic73—42no—
93C2×Dic73—42no—
94C2×Dic73—42no—
95C2×Dic73—42no—
96C2×Dic73—44no—
97C8×C73—42no—
98#10596831—28no—
99C22×C73#39211yescharacteristic
100C22×C73#45311yescharacteristic
101C22×C73#45411yescharacteristic
102C22×C73#45614yes—
103C4×C73—22no—
104C4×C73—22no—
105C4×C73—21no—
106C4×C73—21no—
107C4×C73—82no—
108C4×C73—82no—
109Dic73—82no—
110Dic73—82no—
111#10087—731nomaximal, Sylow
112C2×C73#166012yes—
113C2×C73#166312yes—
114C2×C73#200512yes—
115C2×C73#200611yescharacteristic
116C2×#501—732no—
117#2822—731no—
118#1960D7311yescharacteristic
119C2×#330—732no—
120#1295—731no—
121C73#1008711yescharacteristic, complemented, Sylow
122C2×C4⋊Q8C2×D7311yescharacteristic
123C2×#125—732no—
124#501—1464no—
125#501—1464no—
126C2×#107—1462no—
127C2×#106—1462no—
128C2×#103—22no—
129C2×#97—732no—
130C2×#97C2×D7311yescharacteristic
131#400—1462no—
132#398—1462no—
133#398—1462no—
134#382C2×D7311yescharacteristic
135#330—1464no—
136C22×Q8—21no—
137C22×C8—21no—
138C22×C8—1462no—
139C4⋊Q8—22no—
140C4⋊Q8—21no—
141C4⋊Q8D29212yes—
142#125—1462no—
143#125—1462no—
144#125—1462no—
145#125—1462no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—732no—
148C2×C4⋊C4—732no—
149C2×C4⋊C4—1462no—
150C2×C4⋊C4—1462no—
151C2×C4⋊C4—1462no—
152C2×C4⋊C4C22⋊D7312yes—
153C2×C42C22×D7311yescharacteristic
154#107—1468no—
155#106—1468no—
156#103—44no—
157#97—22no—
158#97—734no—
159#97—1462no—
160#95—22no—
161#95—1464no—
162#95—1462no—
163C2×Q8—22no—
164C2×Q8—42no—
165C2×Q8—42no—
166C2×Q8—41no—
167C22×C4—22no—
168C22×C4—41no—
169C22×C4—1462no—
170C22×C4—1462no—
171C22×C4D4×D7312yes—
172C22×C4C2×C22⋊D7311yescharacteristic
173C2×C8—22no—
174C2×C8—41no—
175C2×C8—41no—
176C2×C8—1468no—
177C2×C8—1464no—
178C4⋊C4—24no—
179C4⋊C4—42no—
180C4⋊C4—42no—
181C4⋊C4—2922no—
182C4⋊C4—2922no—
183C4⋊C4—2922no—
184C4⋊C4—2922no—
185C4⋊C4—5842no—
186C4⋊C4—1462no—
187C4⋊C4—1462no—
188C4⋊C4—1462no—
189C4⋊C4—1462no—
190C42D4×D7312yes—
191C42C2×D29211yescharacteristic
192C42#1192402811yescharacteristic
193C23#1455655911yescharacteristic
194Q8—44no—
195Q8—82no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—24no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—44no—
202C2×C4—81no—
203C2×C4—81no—
204C2×C4—2922no—
205C2×C4—2922no—
206C2×C4—2922no—
207C2×C4—2922no—
208C2×C4—2924no—
209C2×C4#1455656112yes—
210C2×C4#1455656412yes—
211C2×C4#1455656511yescharacteristic
212C2×C4#1455658711yescharacteristic
213C8—42no—
214C8—1468no—
215C22#1798849611yescharacteristic
216C22#1798875411yescharacteristic
217C22#1798875911yescharacteristic
218C22#1798876814yes—
219C4—22no—
220C4—22no—
221C4—21no—
222C4—21no—
223C4—82no—
224C4—82no—
225C4—5842no—
226C4—5842no—
227C2#2260924012yes—
228C2#2260925712yes—
229C2#2261096812yes—
230C2#2261097211yescharacteristic
subgroupG/H
centerC23#14556559
commutator, cosocleC2×C42×C73C23
FrattiniC2×C42C22×D73
FittingC73×#1960C2
socle, prime radicalC23×C73#120
hypercenter#1960D73
hypocenterC73#10087
LCSG ▷ C2×C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1960

Complements

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

NH ≅ G/Ntype
121. C73#10087semidirect

Maximal subgroups

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

subgroupindex
C2×#179889932
#226179992
C2×#179883282
#226078122
C73×#19602
#1008773