Finite Group Databasebeta

← #32995415

Subgroups — #32995415

Order 18688.

330 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 69 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#22618563C211yesmaximal, characteristic
2#22611740C212yesmaximal, complemented
3#22607882C211yesmaximal, characteristic, complemented
4#22607866C212yesmaximal
5C73×#1327C211yesmaximal, characteristic, complemented
6C2×C42×D73C411yescharacteristic
7#17988643—22no—
8#17988639—22no—
9#17988611C411yescharacteristic
10#17988550C411yescharacteristic
11#17988550C2211yescharacteristic
12M16×Dic73C411yescharacteristic
13M16×Dic73C2211yescharacteristic
14#17988407—22no—
15#17988369C2212yes—
16#17988364—22no—
17#423×C73C2211yescharacteristic
18#349×C73C2212yes—
19C22×C4×D73—21no—
20C22×C4×D73C2×C411yescharacteristic
21C2×#11924012C2×C411yescharacteristic
22C2×#11924012C2×C411yescharacteristic
23C2×C4×Dic73—21no—
24C2×C4×Dic73C2×C411yescharacteristic
25C42×D73—21no—
26C42×D73—21no—
27C42×D73—21no—
28C42×D73—21no—
29#14556531—22no—
30C4⋊#10596831—22no—
31#14556503—22no—
32#14556503—22no—
33#14556501—21no—
34#14556501D412yes—
35C8×Dic73—21no—
36C8×Dic73D412yes—
37C2×M16×C73C2×C411yescharacteristic
38C2×M16×C73C2311yescharacteristic
39C2×C42×C73C2×C411yescharacteristic
40M32×C73—22no—
41C4⋊C8×C73—22no—
42C23×D73M1611yescharacteristic
43C22×Dic73—21no—
44C22×Dic73M1611yescharacteristic
45C2×C4×D73—21no—
46C2×C4×D73—21no—
47C2×C4×D73—21no—
48C2×C4×D73—21no—
49C2×C4×D73—21no—
50C2×C4×D73—21no—
51C2×C4×D73—44no—
52C2×C4×D73—41no—
53C2×C4×D73—41no—
54#11924012—41no—
55#11924012—41no—
56C2×#10596831—21no—
57C2×#10596831C22⋊C412yes—
58C4×Dic73—21no—
59C4×Dic73—21no—
60C4×Dic73—21no—
61C4×Dic73—21no—
62C4×Dic73—21no—
63C4×Dic73C22⋊C411yescharacteristic
64C4×Dic73C22⋊C411yescharacteristic
65#11924003—42no—
66C22×C4×C73—21no—
67C22×C4×C73C22×C411yescharacteristic
68M16×C73—41no—
69M16×C73—41no—
70C2×C8×C73—21no—
71C2×C8×C73C2×D412yes—
72C42×C73—21no—
73C42×C73M1611yescharacteristic
74C42×C73M1611yescharacteristic
75C16×C73—42no—
76C23×C73C2×M1611yescharacteristic
77C22×D73—21no—
78C22×D73—42no—
79C22×D73—41no—
80C2×C4×C73—21no—
81C2×C4×C73—21no—
82C2×C4×C73—21no—
83C2×C4×C73—21no—
84C2×C4×C73—42no—
85C2×C4×C73C2×C22⋊C411yescharacteristic
86C2×C4×C73C2×M1611yescharacteristic
87C2×Dic73—21no—
88C2×Dic73—21no—
89C2×Dic73—21no—
90C2×Dic73—42no—
91C2×Dic73—41no—
92C4×D73—24no—
93C4×D73—42no—
94C4×D73—42no—
95C4×D73—42no—
96C4×D73—42no—
97C4×D73—41no—
98C4×D73—41no—
99C8×C73—22no—
100C8×C73—41no—
101#10596831—22no—
102#10596831—41no—
103C22×C73—21no—
104C22×C73—41no—
105C22×C73#40711yescharacteristic
106C2×D73—22no—
107C2×D73—41no—
108C2×D73—42no—
109C2×D73—41no—
110C2×D73—42no—
111C4×C73—22no—
112C4×C73—21no—
113C4×C73—41no—
114C4×C73#41011yescharacteristic
115C4×C73#41311yescharacteristic
116Dic73—22no—
117Dic73—41no—
118Dic73—41no—
119#11521—731nomaximal, Sylow
120C2×C73—21no—
121C2×C73—41no—
122C2×C73#146411yescharacteristic
123D73—42no—
124#2920—731no—
125#2165—732no—
126#1327—731no—
127#1327D7311yescharacteristic, complemented
128#1320—732no—
129C73#1152111yescharacteristic, complemented, Sylow
130C22×C42—731no—
131#432—1462no—
132#431—1462no—
133#423—731no—
134#423C2×D7311yescharacteristic
135#407—731no—
136#407—731no—
137C4×M16—731no—
138C4×M16—731no—
139#364—1462no—
140#349—732no—
141#349C2×D7312yes—
142#347—1462no—
143C23×C4—731no—
144C23×C4—1461no—
145C2×M16—731no—
146C2×M16—731no—
147C2×M16C4×D7311yescharacteristic
148C2×M16C22×D7311yescharacteristic
149C2×C42—731no—
150C2×C42—1461no—
151C2×C42—1461no—
152C2×C42—1461no—
153C2×C42—1461no—
154C2×C42—1461no—
155C2×C42C4×D7311yescharacteristic
156M32—22no—
157M32—1462no—
158C4⋊C8—22no—
159C4⋊C8—1462no—
160C22⋊C8—1462no—
161C22⋊C8—1462no—
162#97—732no—
163#97—1461no—
164C4×C8—732no—
165C4×C8—1461no—
166C24—731no—
167C22×C4—21no—
168C22×C4—2921no—
169C22×C4—2921no—
170C22×C4—2924no—
171C22×C4—731no—
172C22×C4—1461no—
173C22×C4—1461no—
174C22×C4—1461no—
175C22×C4—1461no—
176C22×C4—1461no—
177C22×C4—1461no—
178C22×C4—1461no—
179C22×C4C2×C4×D7311yescharacteristic
180M16—41no—
181M16—41no—
182M16—2921no—
183M16—2921no—
184C2×C8—21no—
185C2×C8—732no—
186C2×C8—1461no—
187C2×C8D4×D7312yes—
188C42—21no—
189C42—731no—
190C42—731no—
191C42—1461no—
192C42—1461no—
193C42—1461no—
194C42—1461no—
195C42—1461no—
196C42#1192401111yescharacteristic
197C42#1192401111yescharacteristic
198C16—42no—
199C16—2922no—
200C23—2921no—
201C23—2922no—
202C23—1461no—
203C23M16×D7311yescharacteristic
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—42no—
209C2×C4—2922no—
210C2×C4—2922no—
211C2×C4—2922no—
212C2×C4—2922no—
213C2×C4—2922no—
214C2×C4—2921no—
215C2×C4—2921no—
216C2×C4—2921no—
217C2×C4—1464no—
218C2×C4—1461no—
219C2×C4—1461no—
220C2×C4—1461no—
221C2×C4C22⋊C4×D7311yescharacteristic
222C2×C4C2×#1192401111yescharacteristic
223C8—22no—
224C8—41no—
225C8—2921no—
226C8—1462no—
227C22—21no—
228C22—41no—
229C22—2921no—
230C22—2922no—
231C22—2922no—
232C22—2921no—
233C22—1462no—
234C22#1798854911yescharacteristic
235C4—22no—
236C4—21no—
237C4—41no—
238C4—2921no—
239C4—2921no—
240C4—1462no—
241C4#1798855911yescharacteristic
242C4#1798857011yescharacteristic
243C2—21no—
244C2—41no—
245C2—2922no—
246C2#2260826411yescharacteristic
subgroupG/H
centerC4#17988559
commutatorC22×C4×C73C22×C4
FrattiniC2×M16C22×D73
FittingC73×#1327C2
socleC2×C73#1464
prime radicalC23×D73M16
cosocleC2×M16×C73C23
hypercenter#1327D73
hypocenterC73#11521
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×M16 ◁ #1327

Complements

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

NH ≅ G/Ntype
2. #22611740C2semidirect
3. #22607882C2semidirect
5. C73×#1327C2semidirect
127. #1327D73semidirect
129. C73#11521semidirect

Maximal subgroups

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

subgroupindex
#226185632
#226117402
#226078822
#226078662
C73×#13272
#1152173