Finite Group Databasebeta

← #32979571

Subgroups — #32979571

Order 18688.

378 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 107 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#22612896C211yesmaximal, characteristic
2#22609927C211yesmaximal, characteristic
3#22607833C212yesmaximal
4#22607790C212yesmaximal
5C73×#1461C211yesmaximal, characteristic
6#17988611C2212yes—
7C2×#14556510—21no—
8#17988494—21no—
9#17988473—22no—
10#17988455C412yes—
11#17988453C2211yescharacteristic
12C4⋊C4×Dic73—21no—
13C4×C4⋊Dic73C2211yescharacteristic
14#17988433C412yes—
15#17988356—21no—
16C2×C4×Q8×C73C2211yescharacteristic
17#431×C73C2212yes—
18#324×C73—22no—
19C2×#11924012—22no—
20C2×C4⋊Dic73—21no—
21C2×C4⋊Dic73—21no—
22C2×C4×Dic73—22no—
23C2×C4×Dic73—21no—
24C4⋊#10596831—22no—
25C4⋊#10596831—22no—
26#14556510—42no—
27#14556498—22no—
28#14556498—22no—
29#14556498—22no—
30#14556498—21no—
31#14556498C2×C411yescharacteristic
32#14556498C2×C412yes—
33#14556498C2×C411yescharacteristic
34C22×Q8×C73D411yescharacteristic
35C4×Q8×C73—22no—
36C4×Q8×C73—22no—
37C2×C4⋊C4×C73D411yescharacteristic
38C2×C4⋊C4×C73D412yes—
39C2×C42×C73C2×C412yes—
40C2×C42×C73C2311yescharacteristic
41C4⋊C8×C73—22no—
42C22⋊C8×C73—22no—
43C4×C8×C73—22no—
44C22×Dic73—21no—
45C22×Dic73—22no—
46C22×Dic73—21no—
47#11924012—42no—
48#11924012—42no—
49C2×#10596831—42no—
50C4⋊Dic73—42no—
51C4⋊Dic73—42no—
52C4×Dic73—44no—
53C4×Dic73—41no—
54C4×Dic73—41no—
55C2×Q8×C73—22no—
56C2×Q8×C73—22no—
57C2×Q8×C73—21no—
58C2×Q8×C73—21no—
59C22×C4×C73C22⋊C412yes—
60C22×C4×C73C22⋊C412yes—
61C22×C4×C73C22×C411yescharacteristic
62C22×C4×C73C2×D411yescharacteristic
63C22×C4×C73C2×D411yescharacteristic
64C2×C8×C73—42no—
65C4⋊C4×C73—24no—
66C4⋊C4×C73—22no—
67C42×C73—22no—
68C42×C73—22no—
69C42×C73#4112yes—
70C42×C73#4112yes—
71C23×C73C2×C22⋊C411yescharacteristic
72Q8×C73—42no—
73Q8×C73—42no—
74C2×C4×C73—22no—
75C2×C4×C73—22no—
76C2×C4×C73—22no—
77C2×C4×C73—22no—
78C2×C4×C73—41no—
79C2×C4×C73—41no—
80C2×C4×C73—42no—
81C2×C4×C73#11711yescharacteristic
82C2×C4×C73#11711yescharacteristic
83C2×C4×C73#12312yes—
84C2×C4×C73#12312yes—
85C2×C4×C73#13512yes—
86C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
87C2×C4×C73#13711yescharacteristic
88C2×Dic73—22no—
89C2×Dic73—41no—
90C2×Dic73—41no—
91C2×Dic73—42no—
92C2×Dic73—41no—
93C2×Dic73—41no—
94C2×Dic73—42no—
95C2×Dic73—42no—
96C2×Dic73—41no—
97C8×C73—82no—
98#10596831—82no—
99C22×C73—21no—
100C22×C73—21no—
101C22×C73#38111yescharacteristic
102C22×C73#41611yescharacteristic
103C22×C73#41711yescharacteristic
104C4×C73—22no—
105C4×C73—21no—
106C4×C73—21no—
107C4×C73—42no—
108C4×C73—42no—
109C4×C73#48212yes—
110C4×C73#48512yes—
111Dic73—42no—
112Dic73—81no—
113Dic73—82no—
114#8768—731nomaximal, Sylow
115C2×C73—21no—
116C2×C73—21no—
117C2×C73#182511yescharacteristic
118C2×C73#182711yescharacteristic
119C2×C73#183011yescharacteristic
120#2356—731no—
121#1801—731no—
122#1461D7311yescharacteristic
123#1303—732no—
124#1286—732no—
125C73#876811yescharacteristic, complemented, Sylow
126C2×C4×Q8C2×D7311yescharacteristic
127#431C2×D7312yes—
128#423—732no—
129C2×#101—1461no—
130#391—1461no—
131#387—1462no—
132#383—732no—
133#382—731no—
134C4×C4⋊C4—731no—
135C4×C4⋊C4—1461no—
136#376—732no—
137#343—1461no—
138#324—22no—
139C22×Q8D29211yescharacteristic
140C2×M16—1462no—
141C4×Q8—22no—
142C4×Q8—22no—
143C2×C4⋊C4—1461no—
144C2×C4⋊C4—1461no—
145C2×C4⋊C4D29211yescharacteristic
146C2×C4⋊C4C22⋊D7312yes—
147C2×C42—1462no—
148C2×C42—1461no—
149C2×C42C4×D7312yes—
150C2×C42C22×D7311yescharacteristic
151C4⋊C8—22no—
152C4⋊C8—1462no—
153C4⋊C8—1462no—
154#101—2922no—
155C22⋊C8—22no—
156C4×C8—22no—
157#95—731no—
158#95—732no—
159#95—731no—
160#95—1462no—
161#95—1462no—
162#95—1462no—
163#95—1461no—
164C2×Q8—22no—
165C2×Q8—22no—
166C2×Q8—21no—
167C2×Q8—21no—
168C22×C4—1462no—
169C22×C4—1461no—
170C22×C4—1461no—
171C22×C4#1192400512yes—
172C22×C4#1192400512yes—
173C22×C4C2×C4×D7311yescharacteristic
174C22×C4C2×C22⋊D7311yescharacteristic
175C22×C4C2×D29211yescharacteristic
176M16—2922no—
177M16—2922no—
178C2×C8—42no—
179C2×C8—2922no—
180C4⋊C4—24no—
181C4⋊C4—22no—
182C4⋊C4—2922no—
183C4⋊C4—2922no—
184C42—22no—
185C42—22no—
186C42—2924no—
187C42—2921no—
188C42—2921no—
189C42#1192402712yes—
190C42#1192402712yes—
191C23C2×#1192400511yescharacteristic
192Q8—42no—
193Q8—42no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—42no—
201C2×C4—2921no—
202C2×C4—2921no—
203C2×C4—2922no—
204C2×C4—2921no—
205C2×C4—2921no—
206C2×C4—2922no—
207C2×C4—2922no—
208C2×C4—2921no—
209C2×C4—1462no—
210C2×C4#1455654711yescharacteristic
211C2×C4#1455654711yescharacteristic
212C2×C4#1455657212yes—
213C2×C4#1455657512yes—
214C2×C4#1455661312yes—
215C2×C4#1455661911yescharacteristic
216C2×C4#1455662411yescharacteristic
217C8—82no—
218C8—5842no—
219C22—21no—
220C22—21no—
221C22#1798844811yescharacteristic
222C22#1798858411yescharacteristic
223C22#1798858911yescharacteristic
224C4—22no—
225C4—21no—
226C4—21no—
227C4—42no—
228C4—42no—
229C4—2922no—
230C4—5841no—
231C4—5842no—
232C4#1798891212yes—
233C4#1798892912yes—
234C2—21no—
235C2—21no—
236C2#2261002711yescharacteristic
237C2#2261003211yescharacteristic
238C2#2261004311yescharacteristic
subgroupG/H
centerC22#17988448
commutatorC22×C4×C73C22×C4
FrattiniC2×C42C22×D73
FittingC73×#1461C2
socleC22×C73#381
prime radicalC23×C73C2×C22⋊C4
cosocleC2×C42×C73C23
hypercenter#1461D73
hypocenterC73#8768
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1461

Complements

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

NH ≅ G/Ntype
125. C73#8768semidirect

Maximal subgroups

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

subgroupindex
#226128962
#226099272
#226078332
#226077902
C73×#14612
#876873