Finite Group Databasebeta

← #32975285

Subgroups — #32975285

Order 18688.

394 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 87 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#22616963C211yesmaximal, characteristic
2#22611495C212yesmaximal
3#22611469C211yesmaximal, characteristic
4#22607753C212yesmaximal
5C73×#1272C211yesmaximal, characteristic
6C2×#14556589—21no—
7C2×#14556524—21no—
8C2×#14556515—21no—
9#17988517—21no—
10#17988508—22no—
11#17988485C2212yes—
12#17988472—22no—
13#17988457C2211yescharacteristic
14#17988350—22no—
15#17988350—22no—
16#17988342C2211yescharacteristic
17#391×C73C2211yescharacteristic
18#336×C73C2212yes—
19C22×Dic146—21no—
20C22×#10596831—21no—
21#14556589—42no—
22#14556589—42no—
23C2×C4⋊Dic73—21no—
24C2×C4⋊Dic73—21no—
25C2×C4⋊Dic73—22no—
26C2×C4⋊Dic73D412yes—
27C2×C4⋊Dic73D412yes—
28C2×#11924007—22no—
29C2×#11924007—22no—
30C2×C4×Dic73—21no—
31#14556524—42no—
32#14556515—42no—
33#14556498—21no—
34#14556498—22no—
35#14556498—22no—
36C22×Q8×C73D411yescharacteristic
37C22×C8×C73—22no—
38C2×C4⋊C4×C73D411yescharacteristic
39C2×C42×C73C2311yescharacteristic
40#95×C73—22no—
41C2×Dic146—42no—
42C2×Dic146—81no—
43C2×Dic146—81no—
44C22×Dic73—22no—
45C22×Dic73—22no—
46C22×Dic73—41no—
47C2×#10596831—22no—
48C2×#10596831—41no—
49C2×#10596831—41no—
50C4⋊Dic73—22no—
51C4⋊Dic73—42no—
52C4⋊Dic73—41no—
53C4⋊Dic73—41no—
54C4⋊Dic73—44no—
55C4⋊Dic73—81no—
56C4⋊Dic73Q1614yes—
57#11924007—44no—
58#11924007—42no—
59#11924007—42no—
60C4×Dic73—42no—
61C2×Q8×C73—21no—
62C2×Q8×C73—41no—
63C2×Q8×C73—41no—
64C2×Q8×C73SD1612yes—
65C22×C4×C73—21no—
66C22×C4×C73—21no—
67C22×C4×C73C2×D412yes—
68C22×C4×C73C2×D411yescharacteristic
69C2×C8×C73—42no—
70C2×C8×C73—42no—
71C2×C8×C73—42no—
72C4⋊C4×C73—41no—
73C42×C73—41no—
74C23×C73#12011yescharacteristic
75Q8×C73—42no—
76Q8×C73—81no—
77C2×C4×C73—21no—
78C2×C4×C73—22no—
79C2×C4×C73—22no—
80C2×C4×C73—21no—
81C2×C4×C73—22no—
82C2×C4×C73—41no—
83C2×C4×C73—41no—
84C2×C4×C73—41no—
85C2×C4×C73—42no—
86C2×C4×C73C2×SD1611yescharacteristic
87C2×C4×C73C2×Q1612yes—
88C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
89C2×C4×C73#13712yes—
90C2×Dic73—24no—
91C2×Dic73—42no—
92C2×Dic73—42no—
93C2×Dic73—42no—
94C2×Dic73—42no—
95C2×Dic73—42no—
96C2×Dic73—81no—
97C2×Dic73—82no—
98Dic146—84no—
99C8×C73—82no—
100#10596831—42no—
101C22×C73#45011yescharacteristic
102C22×C73#45112yes—
103C22×C73#45311yescharacteristic
104C22×C73#45412yes—
105C22×C73#45811yescharacteristic
106C4×C73—21no—
107C4×C73—21no—
108C4×C73—22no—
109C4×C73—22no—
110C4×C73—42no—
111C4×C73—81no—
112Dic73—44no—
113Dic73—82no—
114Dic73—161no—
115#7829—731nomaximal, Sylow
116C2×C73#161311yescharacteristic
117C2×C73#161712yes—
118C2×C73#163311yescharacteristic
119C2×C73#163712yes—
120C2×C73#165711yescharacteristic
121#2734—731no—
122#2100—732no—
123#2095—731no—
124#1272D7311yescharacteristic
125#1271—732no—
126C73#782911yescharacteristic, complemented, Sylow
127C2×C4⋊Q8—1461no—
128C2×#107—1461no—
129C2×#103—1461no—
130#398—1461no—
131#395—1462no—
132#391C2×D7311yescharacteristic
133#389—732no—
134#387—1462no—
135#384—731no—
136#340—1462no—
137#340—1462no—
138#336—731no—
139#336C2×D7312yes—
140C22×Q8—1461no—
141C22×Q8D29211yescharacteristic
142C22×C8—22no—
143C22×C8—1461no—
144C4⋊Q8—2922no—
145C4⋊Q8—2922no—
146C2×C4⋊C4—732no—
147C2×C4⋊C4—732no—
148C2×C4⋊C4—1462no—
149C2×C4⋊C4—1462no—
150C2×C4⋊C4—1461no—
151C2×C4⋊C4—1462no—
152C2×C4⋊C4—1461no—
153C2×C4⋊C4D29211yescharacteristic
154C2×C42—1461no—
155C2×C42C22×D7311yescharacteristic
156#107—2922no—
157#103—2922no—
158#95—22no—
159#95—1461no—
160#95—1462no—
161#95—1462no—
162C2×Q8—21no—
163C2×Q8—41no—
164C2×Q8—41no—
165C2×Q8—2922no—
166C2×Q8—5841no—
167C2×Q8—5841no—
168C2×Q8#1192401712yes—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—2921no—
172C22×C4—1462no—
173C22×C4—1462no—
174C22×C4D4×D7312yes—
175C22×C4C2×D29211yescharacteristic
176C2×C8—42no—
177C2×C8—42no—
178C2×C8—42no—
179C2×C8—2921no—
180C2×C8—2921no—
181C2×C8—1462no—
182C4⋊C4—41no—
183C4⋊C4—2922no—
184C4⋊C4—2924no—
185C4⋊C4—2921no—
186C4⋊C4—2921no—
187C4⋊C4—2924no—
188C4⋊C4—2922no—
189C4⋊C4—2922no—
190C4⋊C4—5841no—
191C4⋊C4—734no—
192C4⋊C4—1462no—
193C42—41no—
194C42—2922no—
195C23#1455656111yescharacteristic
196Q8—42no—
197Q8—81no—
198Q8—5844no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—21no—
203C2×C4—22no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—42no—
208C2×C4—2922no—
209C2×C4—2922no—
210C2×C4—2922no—
211C2×C4—2922no—
212C2×C4—2922no—
213C2×C4—5841no—
214C2×C4—5842no—
215C2×C4—1464no—
216C2×C4C2×#1192401711yescharacteristic
217C2×C4Q16×D7312yes—
218C2×C4#1455661911yescharacteristic
219C2×C4#1455662012yes—
220C8—82no—
221C8—2922no—
222C22#1798874011yescharacteristic
223C22#1798874412yes—
224C22#1798875711yescharacteristic
225C22#1798876112yes—
226C22#1798877811yescharacteristic
227C4—21no—
228C4—21no—
229C4—22no—
230C4—22no—
231C4—42no—
232C4—81no—
233C4—2924no—
234C4—5842no—
235C4—11681no—
236C2#2260895311yescharacteristic
237C2#2260897412yes—
238C2#2260907211yescharacteristic
239C2#2260909812yes—
240C2#2260923011yescharacteristic
subgroupG/H
centerC23#14556561
commutator, cosocleC2×C42×C73C23
FrattiniC2×C42C22×D73
FittingC73×#1272C2
socle, prime radicalC23×C73#120
hypercenter#1272D73
hypocenterC73#7829
LCSG ▷ C2×C42×C73 ▷ C23×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1272

Complements

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

NH ≅ G/Ntype
126. C73#7829semidirect

Maximal subgroups

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

subgroupindex
#226169632
#226114952
#226114692
#226077532
C73×#12722
#782973