Finite Group Databasebeta

← #32995490

Subgroups — #32995490

Order 18688.

270 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 79 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#22618863C211yesmaximal, characteristic
2#22611747C211yesmaximal, characteristic
3#22607883C212yesmaximal
4#22607881C212yesmaximal
5C73×#2166C211yesmaximal, characteristic
6#17989140C411yescharacteristic
7#17988715—21no—
8#17988715—21no—
9C8×Dic146—21no—
10C8×Dic146—21no—
11#17988615C2211yescharacteristic
12#17988614C411yescharacteristic
13#17988610C2211yescharacteristic
14#17988542C411yescharacteristic
15C2×C8×Dic73C411yescharacteristic
16#17988408—21no—
17#17988369C2212yes—
18#17988365—21no—
19#405×C73C2211yescharacteristic
20#364×C73—21no—
21#349×C73C2212yes—
22#347×C73—21no—
23C2×#11924012C2×C411yescharacteristic
24C22×#10596831C2×C411yescharacteristic
25#14556590—21no—
26#14556582—21no—
27#14556568—21no—
28#14556568—21no—
29C4×Dic146—21no—
30C4×Dic146—21no—
31#14556550C2×C411yescharacteristic
32C2×C4×Dic73C2×C411yescharacteristic
33#14556531—22no—
34C4⋊#10596831—21no—
35C4⋊#10596831—21no—
36#14556520—21no—
37#14556520—21no—
38#14556520—21no—
39#14556520—21no—
40#14556501—21no—
41C8×Dic73—21no—
42C8×Dic73—21no—
43C8×Dic73—21no—
44C2×M16×C73C2311yescharacteristic
45C22×C8×C73C2×C411yescharacteristic
46#117×C73C2×C411yescharacteristic
47M32×C73—22no—
48#97×C73D411yescharacteristic
49#97×C73D411yescharacteristic
50C4×C8×C73D411yescharacteristic
51C4×C8×C73D411yescharacteristic
52C2×Dic146—21no—
53C2×Dic146—21no—
54C22×Dic73—21no—
55#11924012—41no—
56C2×#10596831—21no—
57C2×#10596831—21no—
58C2×#10596831—41no—
59C4⋊Dic73—21no—
60C4⋊Dic73—21no—
61C4⋊Dic73—21no—
62#11924007—21no—
63#11924007—21no—
64#11924007—21no—
65#11924007—21no—
66C22⋊Dic73M1611yescharacteristic
67C22⋊Dic73M1611yescharacteristic
68C4×Dic73—21no—
69C4×Dic73—21no—
70C4×Dic73M1611yescharacteristic
71C4×Dic73M1611yescharacteristic
72#11924003—42no—
73C22×C4×C73C22×C411yescharacteristic
74M16×C73—41no—
75C2×C8×C73—22no—
76C2×C8×C73C22⋊C411yescharacteristic
77C2×C8×C73C22⋊C411yescharacteristic
78C2×C8×C73C2×D411yescharacteristic
79C2×C8×C73C2×D411yescharacteristic
80C4⋊C4×C73—21no—
81C22⋊C4×C73—21no—
82C42×C73C22⋊C411yescharacteristic
83C42×C73C22⋊C411yescharacteristic
84C16×C73—42no—
85C23×C73C2×M1611yescharacteristic
86C2×C4×C73—21no—
87C2×C4×C73—21no—
88C2×C4×C73—21no—
89C2×C4×C73—21no—
90C2×C4×C73C2×C22⋊C411yescharacteristic
91C2×C4×C73C2×M1611yescharacteristic
92C2×Dic73—21no—
93C2×Dic73—21no—
94C2×Dic73—21no—
95C2×Dic73—21no—
96C2×Dic73—42no—
97Dic146—42no—
98Dic146—42no—
99C8×C73—21no—
100C8×C73—21no—
101C8×C73—21no—
102C8×C73#10112yes—
103#10596831—41no—
104#10596831—41no—
105C22×C73—21no—
106C22×C73—41no—
107C22×C73#40711yescharacteristic
108C4×C73—21no—
109C4×C73—41no—
110C4×C73—41no—
111C4×C73#41011yescharacteristic
112C4×C73C2×#10111yescharacteristic
113Dic73—22no—
114Dic73—41no—
115Dic73—41no—
116Dic73—41no—
117#11539—731nomaximal, Sylow
118C2×C73—21no—
119C2×C73—41no—
120C2×C73#146711yescharacteristic
121#2960—731no—
122#2166—731no—
123#2166D7311yescharacteristic
124#1328—732no—
125#1327—732no—
126C73#1153911yescharacteristic, complemented, Sylow
127#533—731no—
128#446—1461no—
129#446—1461no—
130C8×Q8—1461no—
131C8×Q8—1461no—
132#424—731no—
133#424—731no—
134#423—731no—
135#405—731no—
136#405C2×D7311yescharacteristic
137C2×C4×C8—731no—
138#364—21no—
139#364—1461no—
140#349—732no—
141#349C2×D7312yes—
142#347—21no—
143#347—1461no—
144C2×M16—731no—
145C2×M16C22×D7311yescharacteristic
146C22×C8—731no—
147C22×C8C4×D7311yescharacteristic
148C4⋊Q8—1461no—
149#125—1461no—
150C22⋊Q8—1461no—
151C22⋊Q8—1461no—
152C4×Q8—1461no—
153C4×Q8—1461no—
154#117—731no—
155#117C4×D7311yescharacteristic
156C2×C42—731no—
157M32—22no—
158M32—1462no—
159C4⋊C8—1461no—
160C4⋊C8—1461no—
161C4⋊C8—1461no—
162C4⋊C8—1461no—
163C4⋊C8—1461no—
164C4⋊C8—1461no—
165#97—1461no—
166#97D29211yescharacteristic
167#97C22⋊D7311yescharacteristic
168C4×C8—1461no—
169C4×C8—1461no—
170C4×C8—1461no—
171C4×C8D29211yescharacteristic
172C4×C8C22⋊D7311yescharacteristic
173C2×Q8—1461no—
174C2×Q8—1461no—
175C22×C4—1461no—
176C22×C4C2×C4×D7311yescharacteristic
177M16—41no—
178M16—2921no—
179C2×C8—22no—
180C2×C8—2921no—
181C2×C8—1461no—
182C2×C8—1461no—
183C2×C8#1192400511yescharacteristic
184C2×C8#1192400511yescharacteristic
185C2×C8C2×C22⋊D7311yescharacteristic
186C2×C8C2×D29211yescharacteristic
187C4⋊C4—21no—
188C4⋊C4—1461no—
189C4⋊C4—1461no—
190C4⋊C4—1461no—
191C4⋊C4—1461no—
192C4⋊C4—1461no—
193C4⋊C4—1461no—
194C4⋊C4—1461no—
195C22⋊C4—21no—
196C22⋊C4—731no—
197C22⋊C4—731no—
198C42—731no—
199C42—731no—
200C42—1461no—
201C42—1461no—
202C42#1192400511yescharacteristic
203C42#1192400511yescharacteristic
204C16—42no—
205C16—2922no—
206C23M16×D7311yescharacteristic
207Q8—2922no—
208Q8—2922no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—2922no—
214C2×C4—1461no—
215C2×C4—1461no—
216C2×C4—1461no—
217C2×C4—1461no—
218C2×C4C2×#1192400511yescharacteristic
219C2×C4M16×D7311yescharacteristic
220C8—21no—
221C8—21no—
222C8—21no—
223C8—2921no—
224C8—2921no—
225C8#1455650912yes—
226C22—21no—
227C22—41no—
228C22#1798855011yescharacteristic
229C4—21no—
230C4—41no—
231C4—41no—
232C4—2921no—
233C4—2921no—
234C4—2921no—
235C4—1462no—
236C4#1798856011yescharacteristic
237C4C2×#1455650911yescharacteristic
238C2—21no—
239C2—41no—
240C2#2260827411yescharacteristic
subgroupG/H
centerC4#17988560
commutatorC22×C4×C73C22×C4
FrattiniC2×M16C22×D73
FittingC73×#2166C2
socleC2×C73#1467
prime radicalC23×C73C2×M16
cosocleC2×M16×C73C23
hypercenter#2166D73
hypocenterC73#11539
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C4 ◁ C2×C8 ◁ #405 ◁ #2166

Complements

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

NH ≅ G/Ntype
126. C73#11539semidirect

Maximal subgroups

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

subgroupindex
#226188632
#226117472
#226078832
#226078812
C73×#21662
#1153973