Finite Group Databasebeta

← #32990056

Subgroups — #32990056

Order 18688.

388 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 81 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#22610050C211yesmaximal, characteristic, complemented
2#22608626C212yesmaximal, complemented
3#22608439C212yesmaximal, complemented
4#22607801C211yesmaximal, characteristic, complemented
5C73×#3245C211yesmaximal, characteristic
6#17988644C2212yes—
7C2×C4⋊#10596831C2211yescharacteristic
8#17988433C2211yescharacteristic
9#17988341—22no—
10#17988336—22no—
11#17988331—22no—
12#17988321—24no—
13#532×C73C2212yes—
14#474×C73—22no—
15#466×C73—22no—
16#459×C73—22no—
17C2×C4⋊C8×C73C2211yescharacteristic
18C22×#10596831—21no—
19C2×C4×Dic73—21no—
20C4⋊#10596831—22no—
21C4⋊#10596831—22no—
22C22⋊#10596831—22no—
23#14556502—22no—
24#14556498—21no—
25#14556498D412yes—
26C2×#41×C73—22no—
27C2×D8×C73—22no—
28C22×C8×C73D412yes—
29C4⋊Q8×C73D412yes—
30#127×C73D412yes—
31#124×C73—22no—
32#121×C73—22no—
33#121×C73—22no—
34C4×D4×C73—22no—
35C4×D4×C73—22no—
36C2×C42×C73C2311yescharacteristic
37#107×C73—22no—
38C4⋊C8×C73—22no—
39#102×C73—24no—
40C22×Dic73—22no—
41C2×#10596831—42no—
42C2×#10596831—41no—
43C2×#10596831—42no—
44C4×Dic73—41no—
45C4×Dic73—41no—
46#41×C73—84no—
47C2×Q8×C73—22no—
48C2×D4×C73—22no—
49C2×D4×C73—22no—
50C2×D4×C73—22no—
51C2×D4×C73—42no—
52C22×C4×C73—42no—
53C22×C4×C73C2×D411yescharacteristic
54C22×C4×C73C2×D411yescharacteristic
55C22×C4×C73#4111yescharacteristic
56D8×C73—82no—
57C2×C8×C73—22no—
58C2×C8×C73—42no—
59C4⋊C4×C73—22no—
60C4⋊C4×C73—22no—
61C22⋊C4×C73—42no—
62C22⋊C4×C73—42no—
63C42×C73C2×D412yes—
64C42×C73#4112yes—
65C23×C73—42no—
66C23×C73#12111yescharacteristic
67Q8×C73—82no—
68D4×C73—42no—
69D4×C73—42no—
70D4×C73—82no—
71D4×C73—82no—
72D4×C73—84no—
73C2×C4×C73—22no—
74C2×C4×C73—21no—
75C2×C4×C73—21no—
76C2×C4×C73—41no—
77C2×C4×C73—42no—
78C2×C4×C73—84no—
79C2×C4×C73#12011yescharacteristic
80C2×C4×C73#12112yes—
81C2×C4×C73#12312yes—
82C2×C4×C73#12411yescharacteristic
83C2×Dic73—42no—
84C2×Dic73—42no—
85C2×Dic73—42no—
86C8×C73—42no—
87#10596831—82no—
88#10596831—81no—
89C22×C73—21no—
90C22×C73—21no—
91C22×C73—82no—
92C22×C73—82no—
93C22×C73—82no—
94C22×C73#39411yescharacteristic
95C22×C73#47212yes—
96C4×C73—22no—
97C4×C73—21no—
98C4×C73—22no—
99C4×C73—82no—
100C4×C73#45411yescharacteristic
101C4×C73#45511yescharacteristic
102Dic73—82no—
103#10513—731nomaximal, Sylow
104C2×C73—22no—
105C2×C73—162no—
106C2×C73#204011yescharacteristic
107C2×C73#204511yescharacteristic
108C2×C73#205311yescharacteristic
109#3245D7311yescharacteristic
110#1832—731no—
111#1551—732no—
112#1505—732no—
113#1290—731no—
114C73#1051311yescharacteristic, complemented, Sylow
115#532C2×D7312yes—
116#474—22no—
117#466—22no—
118#459—22no—
119#432—732no—
120C2×C4⋊C8—731no—
121C2×C4⋊C8C2×D7311yescharacteristic
122#376—731no—
123#336—1462no—
124#333—1462no—
125#331—1462no—
126#327—1464no—
127C2×#41—22no—
128C2×D8—22no—
129C22×C8—1461no—
130C22×C8C22⋊D7312yes—
131C4⋊Q8C22⋊D7312yes—
132#127C22⋊D7312yes—
133#124—22no—
134#121—22no—
135#121—22no—
136C4×D4—22no—
137C4×D4—22no—
138C2×C42—1461no—
139C2×C42C22×D7311yescharacteristic
140#107—22no—
141C4⋊C8—22no—
142C4⋊C8—1462no—
143C4⋊C8—1462no—
144#102—24no—
145C22⋊C8—1462no—
146#97—1462no—
147#95—732no—
148#95—1461no—
149#41—84no—
150C2×Q8—22no—
151C2×D4—22no—
152C2×D4—22no—
153C2×D4—22no—
154C2×D4—42no—
155C22×C4—42no—
156C22×C4—1462no—
157C22×C4D4×D7311yescharacteristic
158C22×C4C2×C22⋊D7311yescharacteristic
159C22×C4#1192402711yescharacteristic
160D8—82no—
161C2×C8—22no—
162C2×C8—42no—
163C2×C8—2922no—
164C2×C8—2921no—
165C2×C8—2922no—
166C4⋊C4—22no—
167C4⋊C4—22no—
168C22⋊C4—42no—
169C22⋊C4—42no—
170C42—2921no—
171C42—2921no—
172C42C2×C22⋊D7312yes—
173C42#1192402712yes—
174C23—42no—
175C23#1455656211yescharacteristic
176Q8—82no—
177D4—42no—
178D4—42no—
179D4—82no—
180D4—82no—
181D4—84no—
182C2×C4—22no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—41no—
186C2×C4—42no—
187C2×C4—84no—
188C2×C4—2922no—
189C2×C4—2922no—
190C2×C4—2922no—
191C2×C4#1455656011yescharacteristic
192C2×C4#1455656312yes—
193C2×C4#1455657212yes—
194C2×C4#1455657811yescharacteristic
195C8—42no—
196C8—5842no—
197C8—5841no—
198C22—21no—
199C22—21no—
200C22—82no—
201C22—82no—
202C22—82no—
203C22#1798850111yescharacteristic
204C22#1798885412yes—
205C4—22no—
206C4—21no—
207C4—22no—
208C4—82no—
209C4—5842no—
210C4#1798876011yescharacteristic
211C4#1798876511yescharacteristic
212C2—22no—
213C2—162no—
214C2#2261115711yescharacteristic
215C2#2261118411yescharacteristic
216C2#2261123411yescharacteristic
subgroupG/H
centerC22#17988501
commutator, cosocleC2×C42×C73C23
FrattiniC2×C42C22×D73
Fitting, prime radicalC73×#3245C2
socleC22×C73#394
hypercenter#3245D73
hypocenterC73#10513
LCSG ▷ C2×C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3245

Complements

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

NH ≅ G/Ntype
1. #22610050C2semidirect
2. #22608626C2semidirect
3. #22608439C2semidirect
4. #22607801C2semidirect
114. C73#10513semidirect

Maximal subgroups

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

subgroupindex
#226100502
#226086262
#226084392
#226078012
C73×#32452
#1051373