Finite Group Databasebeta

← #32987320

Subgroups — #32987320

Order 18688.

500 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 97 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×#17988997C212yesmaximal
2#22610750C211yesmaximal, characteristic
3C2×#17988329C212yesmaximal
4#22607812C211yesmaximal, characteristic
5C73×#2822C211yesmaximal, characteristic
6C2×#14556591C2211yescharacteristic
7#17988997—24no—
8#17988997—24no—
9C2×#14556524—22no—
10C2×#14556522—22no—
11C2×#14556514—22no—
12C2×#14556502C2211yescharacteristic
13C2×#14556502C2212yes—
14#17988329—24no—
15C2×#125×C73C2212yes—
16#400×C73—22no—
17#398×C73—22no—
18#398×C73—22no—
19#382×C73C2211yescharacteristic
20C22×Dic146—21no—
21C22×#10596831—22no—
22C22×#10596831—21no—
23#14556591—22no—
24#14556591—21no—
25#14556591D412yes—
26C2×C4⋊Dic73—22no—
27#14556524—28no—
28#14556522—28no—
29#14556514—44no—
30#14556502—22no—
31#14556502—22no—
32#14556502D414yes—
33#125×C73—22no—
34#125×C73—22no—
35#125×C73—22no—
36#125×C73—22no—
37C2×C4⋊C4×C73—22no—
38C2×C4⋊C4×C73—22no—
39C2×C4⋊C4×C73—22no—
40C2×C4⋊C4×C73D412yes—
41C2×C4⋊C4×C73D412yes—
42C2×C4⋊C4×C73D412yes—
43C2×C42×C73C2311yescharacteristic
44#95×C73—24no—
45#95×C73—22no—
46#95×C73—22no—
47C2×Dic146—22no—
48C2×Dic146—42no—
49C2×Dic146—42no—
50C2×Dic146—41no—
51C22×Dic73—41no—
52C2×#10596831—28no—
53C2×#10596831—24no—
54C2×#10596831—22no—
55C2×#10596831—41no—
56C2×#10596831—41no—
57C4⋊Dic73—24no—
58C4⋊Dic73—42no—
59C22×C4×C73—22no—
60C22×C4×C73—22no—
61C22×C4×C73—22no—
62C22×C4×C73C2×D412yes—
63C22×C4×C73C2×D411yescharacteristic
64C4⋊C4×C73—22no—
65C4⋊C4×C73—22no—
66C4⋊C4×C73—22no—
67C4⋊C4×C73—22no—
68C4⋊C4×C73—42no—
69C4⋊C4×C73—42no—
70C4⋊C4×C73—42no—
71C4⋊C4×C73—42no—
72C4⋊C4×C73—42no—
73C4⋊C4×C73—82no—
74C42×C73C2×D411yescharacteristic
75C42×C73C2×D412yes—
76C42×C73#4111yescharacteristic
77C23×C73#12011yescharacteristic
78C2×C4×C73—21no—
79C2×C4×C73—21no—
80C2×C4×C73—24no—
81C2×C4×C73—42no—
82C2×C4×C73—42no—
83C2×C4×C73—42no—
84C2×C4×C73—42no—
85C2×C4×C73—42no—
86C2×C4×C73—44no—
87C2×C4×C73—44no—
88C2×C4×C73#12012yes—
89C2×C4×C73#12111yescharacteristic
90C2×C4×C73#12112yes—
91C2×C4×C73#12711yescharacteristic
92C2×Dic73—42no—
93C2×Dic73—81no—
94C2×Dic73—81no—
95Dic146—44no—
96Dic146—82no—
97#10596831—28no—
98#10596831—42no—
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—
109C4×C73—82no—
110Dic73—82no—
111#10087—731nomaximal, Sylow
112C2×C73#166012yes—
113C2×C73#166312yes—
114C2×C73#200512yes—
115C2×C73#200611yescharacteristic
116C2×#501—732no—
117#2822D7311yescharacteristic
118#1960—731no—
119C2×#330—732no—
120#1295—731no—
121C73#1008711yescharacteristic, complemented, Sylow
122C2×C4⋊Q8—731no—
123C2×#125C2×D7312yes—
124#501—1464no—
125#501—1464no—
126C2×#107—1462no—
127C2×#106—1462no—
128C2×#103—1462no—
129C2×#97—732no—
130C2×#97—731no—
131#400—22no—
132#398—22no—
133#398—22no—
134#382C2×D7311yescharacteristic
135#330—1464no—
136C22×Q8—1461no—
137C22×C8—1461no—
138C22×C8—1462no—
139C4⋊Q8—732no—
140C4⋊Q8—1462no—
141C4⋊Q8—1461no—
142#125—22no—
143#125—22no—
144#125—22no—
145#125—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—1462no—
150C2×C4⋊C4C22⋊D7312yes—
151C2×C4⋊C4C22⋊D7312yes—
152C2×C4⋊C4C22⋊D7312yes—
153C2×C42C22×D7311yescharacteristic
154#107—1468no—
155#106—1468no—
156#103—2924no—
157#97—734no—
158#97—1462no—
159#97—1462no—
160#95—24no—
161#95—22no—
162#95—22no—
163C2×Q8—2922no—
164C2×Q8—2922no—
165C2×Q8—2921no—
166C2×Q8—1462no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—2921no—
171C22×C4C2×C22⋊D7312yes—
172C22×C4C2×C22⋊D7311yescharacteristic
173C2×C8—2921no—
174C2×C8—2921no—
175C2×C8—1462no—
176C2×C8—1468no—
177C2×C8—1464no—
178C4⋊C4—22no—
179C4⋊C4—22no—
180C4⋊C4—22no—
181C4⋊C4—22no—
182C4⋊C4—42no—
183C4⋊C4—42no—
184C4⋊C4—42no—
185C4⋊C4—42no—
186C4⋊C4—42no—
187C4⋊C4—82no—
188C4⋊C4—2922no—
189C4⋊C4—1464no—
190C42D4×D7311yescharacteristic
191C42D4×D7312yes—
192C42#1192402711yescharacteristic
193C23#1455656011yescharacteristic
194Q8—2924no—
195Q8—5842no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—24no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—44no—
205C2×C4—44no—
206C2×C4—2922no—
207C2×C4—5841no—
208C2×C4—5841no—
209C2×C4#1455655912yes—
210C2×C4#1455656211yescharacteristic
211C2×C4#1455656312yes—
212C2×C4#1455658711yescharacteristic
213C8—2922no—
214C8—1468no—
215C22#1798849511yescharacteristic
216C22#1798875511yescharacteristic
217C22#1798876011yescharacteristic
218C22#1798877014yes—
219C4—22no—
220C4—22no—
221C4—21no—
222C4—21no—
223C4—82no—
224C4—82no—
225C4—82no—
226C4—5842no—
227C2#2260924212yes—
228C2#2260925912yes—
229C2#2261096512yes—
230C2#2261097311yescharacteristic
subgroupG/H
centerC23#14556560
commutator, cosocleC2×C42×C73C23
FrattiniC2×C42C22×D73
FittingC73×#2822C2
socle, prime radicalC23×C73#120
hypercenter#2822D73
hypocenterC73#10087
LCSG ▷ C2×C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2822

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×#179889972
#226107502
C2×#179883292
#226078122
C73×#28222
#1008773