Finite Group Databasebeta

← #32999313

Subgroups — #32999313

Order 18688.

324 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 55 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#22620755C211yesmaximal, characteristic
2#22612150C212yesmaximal, complemented
3#22607924C212yesmaximal, complemented
4#22607867C211yesmaximal, characteristic
5C73×#2091C211yesmaximal, characteristic
6C2×#14556570—21no—
7#17988918—22no—
8#17988845—21no—
9#17988845C2212yes—
10#17988745—22no—
11#17988411—22no—
12#17988402—22no—
13#17988369C2212yes—
14#430×C73—21no—
15#430×C73C2211yescharacteristic
16#407×C73—21no—
17#407×C73C2211yescharacteristic
18#338×C73C2211yescharacteristic
19C22×Dic146—21no—
20C2×Dic292—21no—
21C2×Dic292D412yes—
22#14556570—22no—
23#14556570—22no—
24#14556570—41no—
25#14556570—41no—
26C2×C4⋊Dic73—21no—
27C2×#11924007—41no—
28C2×C22⋊Dic73—41no—
29#14556531—22no—
30#14556523—21no—
31#14556523D412yes—
32#14556516—22no—
33#14556516—22no—
34C23×C4×C73D411yescharacteristic
35C2×M16×C73—21no—
36C2×M16×C73—21no—
37C2×M16×C73D411yescharacteristic
38C2×M16×C73C2311yescharacteristic
39#108×C73—22no—
40#108×C73—42no—
41C22⋊C8×C73—22no—
42C22⋊C8×C73—42no—
43C2×Dic146—22no—
44C2×Dic146—41no—
45C2×Dic146—81no—
46C2×Dic146—81no—
47C22×Dic73—41no—
48C22×Dic73—41no—
49Dic292—42no—
50Dic292—81no—
51C4⋊Dic73—22no—
52C4⋊Dic73—41no—
53#11924007—42no—
54#11924007—42no—
55C22⋊Dic73—42no—
56C22⋊Dic73—42no—
57#11924003—42no—
58C24×C73SD1611yescharacteristic
59C22×C4×C73—21no—
60C22×C4×C73—21no—
61C22×C4×C73—41no—
62C22×C4×C73—41no—
63C22×C4×C73SD1611yescharacteristic
64C22×C4×C73C2×D411yescharacteristic
65M16×C73—41no—
66M16×C73—41no—
67M16×C73—41no—
68M16×C73—41no—
69M16×C73—41no—
70M16×C73—41no—
71C2×C8×C73—22no—
72C2×C8×C73—21no—
73C2×C8×C73—41no—
74C2×C8×C73C2×D412yes—
75C23×C73—21no—
76C23×C73—41no—
77C23×C73—42no—
78C23×C73C2×SD1611yescharacteristic
79C2×C4×C73—21no—
80C2×C4×C73—21no—
81C2×C4×C73—21no—
82C2×C4×C73—41no—
83C2×C4×C73—41no—
84C2×C4×C73—41no—
85C2×C4×C73—81no—
86C2×C4×C73#12011yescharacteristic
87C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
88C2×Dic73—42no—
89C2×Dic73—42no—
90C2×Dic73—82no—
91C2×Dic73—82no—
92Dic146—42no—
93Dic146—42no—
94Dic146—82no—
95C8×C73—22no—
96C8×C73—42no—
97C8×C73—41no—
98C8×C73—81no—
99C22×C73—22no—
100C22×C73—21no—
101C22×C73—41no—
102C22×C73—41no—
103C22×C73—42no—
104C22×C73—41no—
105C22×C73—81no—
106C22×C73#45011yescharacteristic
107C4×C73—21no—
108C4×C73—41no—
109C4×C73#45311yescharacteristic
110C4×C73#45811yescharacteristic
111Dic73—82no—
112Dic73—82no—
113#12299—731nomaximal, Sylow
114C2×C73—21no—
115C2×C73—42no—
116C2×C73—41no—
117C2×C73#161311yescharacteristic
118#3195—731no—
119#2239—732no—
120#2091D7311yescharacteristic
121#1342—732no—
122#1320—731no—
123C73#1229911yescharacteristic, complemented, Sylow
124C2×C22⋊Q8—1461no—
125#483—1462no—
126#470—732no—
127#470—1461no—
128#451—1462no—
129#430—21no—
130#430C2×D7311yescharacteristic
131#407—21no—
132#407C2×D7311yescharacteristic
133#365—1462no—
134#362—1462no—
135#349—732no—
136#338C2×D7311yescharacteristic
137C22×Q8—1461no—
138C23×C4C22⋊D7311yescharacteristic
139C2×Q16—732no—
140C2×Q16—1461no—
141C2×M16—21no—
142C2×M16—21no—
143C2×M16C22⋊D7311yescharacteristic
144C2×M16C22×D7311yescharacteristic
145C22⋊Q8—2921no—
146C22⋊Q8—2921no—
147C22⋊Q8—1462no—
148C22⋊Q8—1462no—
149C2×C4⋊C4—2921no—
150C2×C4⋊C4—1461no—
151C2×C22⋊C4—2921no—
152M32—1462no—
153#108—22no—
154#108—42no—
155#106—732no—
156#106—1461no—
157#103—1462no—
158#103—1462no—
159C22⋊C8—22no—
160C22⋊C8—42no—
161C24#1192401511yescharacteristic
162C2×Q8—2921no—
163C2×Q8—5841no—
164C2×Q8—5841no—
165C2×Q8—1462no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—2921no—
171C22×C4—2921no—
172C22×C4#1192401511yescharacteristic
173C22×C4C2×C22⋊D7311yescharacteristic
174Q16—2922no—
175Q16—5841no—
176M16—41no—
177M16—41no—
178M16—41no—
179M16—41no—
180M16—41no—
181M16—41no—
182C2×C8—22no—
183C2×C8—21no—
184C2×C8—41no—
185C2×C8D4×D7312yes—
186C4⋊C4—2922no—
187C4⋊C4—2921no—
188C4⋊C4—2922no—
189C4⋊C4—1462no—
190C22⋊C4—2922no—
191C22⋊C4—2922no—
192C16—2922no—
193C23—21no—
194C23—41no—
195C23—42no—
196C23C2×#1192401511yescharacteristic
197Q8—2922no—
198Q8—2922no—
199Q8—5842no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—81no—
207C2×C4—2922no—
208C2×C4—2922no—
209C2×C4—5842no—
210C2×C4—5842no—
211C2×C4#1455655911yescharacteristic
212C2×C4#1455661611yescharacteristic
213C8—22no—
214C8—42no—
215C8—41no—
216C8—81no—
217C22—22no—
218C22—21no—
219C22—41no—
220C22—41no—
221C22—42no—
222C22—41no—
223C22—81no—
224C22#1798873711yescharacteristic
225C4—21no—
226C4—41no—
227C4—5842no—
228C4—5842no—
229C4#1798875411yescharacteristic
230C4#1798877611yescharacteristic
231C2—21no—
232C2—42no—
233C2—41no—
234C2#2260895011yescharacteristic
subgroupG/H
centerC2#22608950
commutator, cosocleC2×M16×C73C23
FrattiniC2×M16C22×D73
FittingC73×#2091C2
socleC2×C73#1613
prime radicalC24×C73SD16
hypercenter#2091D73
hypocenterC73#12299
LCSG ▷ C2×M16×C73 ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #2091

Complements

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

NH ≅ G/Ntype
2. #22612150C2semidirect
3. #22607924C2semidirect
123. C73#12299semidirect

Maximal subgroups

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

subgroupindex
#226207552
#226121502
#226079242
#226078672
C73×#20912
#1229973