Finite Group Databasebeta

← #32973751

Subgroups — #32973751

Order 18688.

270 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 59 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#22611596C211yesmaximal, characteristic
2#22610315C211yesmaximal, characteristic
3C73×#1536C211yesmaximal, characteristic
4#17989354C411yescharacteristic, complemented
5#17988604—21no—
6#17988561—21no—
7#17988561C411yescharacteristic, complemented
8#17988382—21no—
9#17988380C412yescomplemented
10#17988347—21no—
11#532×C73C2211yescharacteristic
12#432×C73—21no—
13#329×C73—21no—
14#329×C73C412yescomplemented
15#14556598—42no—
16C2×#11924012—21no—
17C2×#11924012—41no—
18#14556550—21no—
19#14556550—41no—
20#14556518—42no—
21#14556506—22no—
22#14556506—41no—
23#14556506—41no—
24C2×#41×C73C2×C411yescharacteristic
25C2×#41×C73D411yescharacteristic
26C4⋊Q8×C73Q811yescharacteristic
27#127×C73C2×C411yescharacteristic
28#124×C73C2×C411yescharacteristic
29#124×C73D411yescharacteristic
30#121×C73—41no—
31C4×D4×C73—41no—
32C2×C42×C73D411yescharacteristic
33C4⋊C8×C73—41no—
34C22⋊C8×C73—41no—
35#97×C73—22no—
36#11924012—22no—
37#11924012—41no—
38#11924012—42no—
39C2×#10596831—41no—
40C2×#10596831—42no—
41C4⋊Dic73—41no—
42C4⋊Dic73—81no—
43C22⋊Dic73—42no—
44C22⋊Dic73—41no—
45C4×Dic73—41no—
46C4×Dic73—81no—
47#41×C73—41no—
48#41×C73—41no—
49#41×C73—41no—
50#41×C73—41no—
51C2×Q8×C73C4⋊C411yescharacteristic
52C2×Q8×C73C4⋊C411yescharacteristic
53C2×D4×C73—21no—
54C2×D4×C73—21no—
55C2×D4×C73—41no—
56C2×D4×C73C4211yescharacteristic
57C2×D4×C73C22⋊C411yescharacteristic
58C22×C4×C73—21no—
59C22×C4×C73—21no—
60C22×C4×C73—21no—
61C22×C4×C73C22⋊C411yescharacteristic
62C2×C8×C73—42no—
63C4⋊C4×C73—41no—
64C22⋊C4×C73—41no—
65C22⋊C4×C73—41no—
66C42×C73—21no—
67C42×C73C22⋊C411yescharacteristic
68C42×C73C4⋊C411yescharacteristic
69C23×C73—21no—
70C23×C73—21no—
71C23×C73#9911yescharacteristic
72Q8×C73—41no—
73Q8×C73—41no—
74D4×C73—41no—
75D4×C73—41no—
76D4×C73—42no—
77D4×C73—81no—
78D4×C73—82no—
79C2×C4×C73—21no—
80C2×C4×C73—21no—
81C2×C4×C73—21no—
82C2×C4×C73—21no—
83C2×C4×C73—21no—
84C2×C4×C73—21no—
85C2×C4×C73—41no—
86C2×C4×C73—41no—
87C2×C4×C73—41no—
88C2×C4×C73—41no—
89C2×C4×C73—81no—
90C2×C4×C73#9511yescharacteristic
91C2×C4×C73#9911yescharacteristic
92C2×Dic73—81no—
93C2×Dic73—82no—
94C8×C73—82no—
95#10596831—42no—
96C22×C73—21no—
97C22×C73—41no—
98C22×C73—41no—
99C22×C73—41no—
100C22×C73—81no—
101C22×C73—81no—
102C22×C73#34211yescharacteristic
103C4×C73—21no—
104C4×C73—41no—
105C4×C73—41no—
106C4×C73—41no—
107C4×C73—41no—
108C4×C73AΣL(1,𝔽16)11yescharacteristic
109C4×C73#35211yescharacteristic
110Dic73—161no—
111Dic73—162no—
112#7494—731nomaximal, Sylow
113C2×C73—21no—
114C2×C73—41no—
115C2×C73—81no—
116C2×C73—81no—
117C2×C73#139011yescharacteristic
118#2126—731no—
119#1884—731no—
120#1536D7311yescharacteristic
121C73#749411yescharacteristic, complemented, Sylow
122#568—731no—
123#532C2×D7311yescharacteristic
124#432—21no—
125#421—1461no—
126#410—731no—
127#410—1461no—
128#354—1461no—
129#353—732no—
130#339—1461no—
131#329—21no—
132#329Dic7312yescomplemented
133C2×#41C4×D7311yescharacteristic
134C2×#41D29211yescharacteristic
135#131—2922no—
136C2×M16—2921no—
137C2×M16—1461no—
138C4⋊Q8Dic14611yescharacteristic
139#127C4×D7311yescharacteristic
140#124C2×Dic7311yescharacteristic
141#124C22⋊D7311yescharacteristic
142#121—41no—
143C4×D4—41no—
144#117—2921no—
145#117—1461no—
146C2×C42C22⋊D7311yescharacteristic
147C4⋊C8—41no—
148#104—2922no—
149#99—2921no—
150#99—2921no—
151#99—1462no—
152C22⋊C8—41no—
153#97—22no—
154#41—41no—
155#41—41no—
156#41—41no—
157#41—41no—
158C2×Q8#1192400711yescharacteristic
159C2×Q8C4⋊Dic7311yescharacteristic
160C2×D4—21no—
161C2×D4—21no—
162C2×D4—41no—
163C2×D4C4×Dic7311yescharacteristic
164C2×D4#1192400511yescharacteristic
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4#1192400511yescharacteristic
169M16—2921no—
170M16—2922no—
171M16—1462no—
172C2×C8—42no—
173C2×C8—2921no—
174C2×C8—2922no—
175C4⋊C4—41no—
176C4⋊C4—2921no—
177C4⋊C4—5841no—
178C22⋊C4—41no—
179C22⋊C4—41no—
180C22⋊C4—2922no—
181C22⋊C4—2921no—
182C42—21no—
183C42—2921no—
184C42—5841no—
185C42C22⋊Dic7311yescharacteristic
186C42#1192400711yescharacteristic
187C23—21no—
188C23—21no—
189C23#1455650511yescharacteristic
190Q8—41no—
191Q8—41no—
192D4—41no—
193D4—41no—
194D4—42no—
195D4—81no—
196D4—82no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—81no—
208C2×C4—5841no—
209C2×C4—5842no—
210C2×C4#1455649811yescharacteristic
211C2×C4#1455650511yescharacteristic
212C8—82no—
213C8—2922no—
214C22—21no—
215C22—41no—
216C22—41no—
217C22—41no—
218C22—81no—
219C22—81no—
220C22#1798835311yescharacteristic
221C4—21no—
222C4—41no—
223C4—41no—
224C4—41no—
225C4—41no—
226C4—11681no—
227C4—11682no—
228C4#1798837511yescharacteristic
229C4#1798837811yescharacteristic
230C2—21no—
231C2—41no—
232C2—81no—
233C2—81no—
234C2#2260805011yescharacteristic
subgroupG/H
centerC2#22608050
commutatorC2×D4×C73C42
Frattini#532C2×D73
FittingC73×#1536C2
socleC2×C73#1390
prime radical, cosocle#532×C73C22
hypercenter#1536D73
hypocenterC73#7494
LCSG ▷ C2×D4×C73 ▷ C2×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ #1536

Complements

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

NH ≅ G/Ntype
4. #17989354C4semidirect
7. #17988561C4semidirect
9. #17988380C4semidirect
14. #329×C73C4semidirect
121. C73#7494semidirect
132. #329Dic73semidirect

Maximal subgroups

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

subgroupindex
#226115962
#226103152
C73×#15362
#749473