Finite Group Databasebeta

← #34010116

Subgroups — #34010116

Order 18688.

330 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 66 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#22626301C212yesmaximal, complemented
2#22624878C212yesmaximal, complemented
3#22624863C212yesmaximal
4#22618560C211yesmaximal, characteristic
5#17989509—22no—
6#17989483C2214yes—
7#17989480—22no—
8C22×C4×Dic73C411yescharacteristic
9#17988641—22no—
10#17988637—22no—
11#17988610C411yescharacteristic
12#17988610C2211yescharacteristic
13#17988551C2211yescharacteristic
14M16×Dic73C411yescharacteristic
15M16×Dic73C2211yescharacteristic
16#407×C73C411yescharacteristic
17#14556665—24no—
18C23×Dic73—21no—
19C2×#11924012C2×C411yescharacteristic
20C2×#11924012C2311yescharacteristic
21C2×C4×Dic73—21no—
22C2×C4×Dic73—21no—
23C2×C4×Dic73—21no—
24C2×C4×Dic73—21no—
25C2×C4×Dic73—21no—
26C2×C4×Dic73C2×C411yescharacteristic
27C2×C4×Dic73C2×C411yescharacteristic
28#14556520—22no—
29#14556520—22no—
30C22⋊#10596831—22no—
31#14556501—21no—
32#14556501D412yes—
33C8×Dic73—21no—
34C8×Dic73D412yes—
35C23×C4×C73C2×C411yescharacteristic
36C2×M16×C73C2×C411yescharacteristic
37C2×M16×C73C2×C411yescharacteristic
38C22⋊C8×C73—22no—
39#11924031—44no—
40C22×Dic73—21no—
41C22×Dic73—21no—
42C22×Dic73—21no—
43C22×Dic73—21no—
44C22×Dic73—21no—
45C22×Dic73—21no—
46C22×Dic73—44no—
47#11924012—41no—
48#11924012—41no—
49C2×#10596831—21no—
50C2×#10596831C2×D412yes—
51C4×Dic73—21no—
52C4×Dic73—21no—
53C4×Dic73—21no—
54C4×Dic73—21no—
55C4×Dic73—21no—
56C4×Dic73—21no—
57C4×Dic73C22⋊C411yescharacteristic
58C4×Dic73C22⋊C411yescharacteristic
59C4×Dic73M1611yescharacteristic
60C4×Dic73M1611yescharacteristic
61C24×C73M1611yescharacteristic
62C22×C4×C73—21no—
63C22×C4×C73—21no—
64C22×C4×C73—41no—
65C22×C4×C73—41no—
66C22×C4×C73M1611yescharacteristic
67C22×C4×C73C22×C411yescharacteristic
68M16×C73—41no—
69M16×C73—41no—
70C2×C8×C73—21no—
71C2×C8×C73C22⋊C412yes—
72C23×C73—21no—
73C23×C73—41no—
74C23×C73—42no—
75C23×C73C2×M1611yescharacteristic
76C2×C4×C73—21no—
77C2×C4×C73—21no—
78C2×C4×C73—21no—
79C2×C4×C73—42no—
80C2×C4×C73—41no—
81C2×C4×C73—41no—
82C2×C4×C73—41no—
83C2×C4×C73C2×C22⋊C411yescharacteristic
84C2×C4×C73C2×M1611yescharacteristic
85C2×Dic73—24no—
86C2×Dic73—21no—
87C2×Dic73—21no—
88C2×Dic73—21no—
89C2×Dic73—21no—
90C2×Dic73—42no—
91C2×Dic73—42no—
92C2×Dic73—42no—
93C2×Dic73—42no—
94C2×Dic73—42no—
95C8×C73—22no—
96C8×C73—41no—
97#10596831—22no—
98#10596831—41no—
99C22×C73—22no—
100C22×C73—21no—
101C22×C73—41no—
102C22×C73—41no—
103C22×C73—42no—
104C22×C73—42no—
105C22×C73—41no—
106C22×C73#40711yescharacteristic
107C4×C73—21no—
108C4×C73—41no—
109C4×C73#41011yescharacteristic
110C4×C73#41311yescharacteristic
111Dic73—22no—
112Dic73—22no—
113Dic73—41no—
114Dic73—41no—
115#11521—731nomaximal, Sylow
116C2×C73—21no—
117C2×C73—42no—
118C2×C73—41no—
119C2×C73#146411yescharacteristic
120#2920—731no—
121#2165—732no—
122#1327—732no—
123#1320—732no—
124C73#1152111yescharacteristic, complemented, Sylow
125C22×C42—731no—
126#432—1462no—
127#431—1462no—
128#423—731no—
129#423—731no—
130#407—731no—
131#407#6440811yescharacteristic
132C4×M16—731no—
133C4×M16—731no—
134#364—1462no—
135#349—734no—
136#347—1462no—
137C23×C4—1461no—
138C23×C4C2×#6440811yescharacteristic
139C2×M16—731no—
140C2×M16—731no—
141C2×M16C2×#6440811yescharacteristic
142C2×M16C2×#6440811yescharacteristic
143C2×C42—731no—
144C2×C42—731no—
145C2×C42—1461no—
146C2×C42—1461no—
147C2×C42—1461no—
148C2×C42—1461no—
149C2×C42—1461no—
150M32—1464no—
151C4⋊C8—1462no—
152C4⋊C8—1462no—
153C22⋊C8—22no—
154C22⋊C8—1462no—
155#97—732no—
156#97—1461no—
157C4×C8—732no—
158C4×C8—1461no—
159C24#1192403811yescharacteristic
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—2924no—
165C22×C4—1461no—
166C22×C4—1461no—
167C22×C4—1461no—
168C22×C4—1461no—
169C22×C4—1461no—
170C22×C4—1461no—
171C22×C4#1192403811yescharacteristic
172C22×C4C22×#6440811yescharacteristic
173M16—41no—
174M16—41no—
175M16—2921no—
176M16—2921no—
177C2×C8—21no—
178C2×C8—732no—
179C2×C8—1461no—
180C2×C8C22⋊#6440812yes—
181C42—731no—
182C42—731no—
183C42—731no—
184C42—731no—
185C42—1461no—
186C42—1461no—
187C42—1461no—
188C42—1461no—
189C42—1461no—
190C42—1461no—
191C16—2924no—
192C23—21no—
193C23—41no—
194C23—42no—
195C23C2×#1192403811yescharacteristic
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—42no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—2922no—
204C2×C4—2922no—
205C2×C4—2922no—
206C2×C4—2922no—
207C2×C4—2922no—
208C2×C4—1461no—
209C2×C4—1461no—
210C2×C4—1464no—
211C2×C4—1461no—
212C2×C4—1461no—
213C2×C4C2×C22⋊#6440811yescharacteristic
214C2×C4D73⋊M1611yescharacteristic
215C8—22no—
216C8—41no—
217C8—2921no—
218C8—1462no—
219C22—22no—
220C22—21no—
221C22—41no—
222C22—41no—
223C22—42no—
224C22—42no—
225C22—41no—
226C22#1798955811yescharacteristic
227C4—21no—
228C4—41no—
229C4—2921no—
230C4—2921no—
231C4—1462no—
232C4—1462no—
233C4#1798956711yescharacteristic
234C4#1798957411yescharacteristic
235C2—21no—
236C2—42no—
237C2—41no—
238C2#2262518311yescharacteristic
subgroupG/H
centerC4#17989567
commutatorC22×C4×C73C22×C4
FrattiniC22×C4C22×#64408
Fitting#407×C73C4
socleC2×C73#1464
prime radicalC24×C73M16
cosocleC2×#11924012C23
hypercenter#407#64408
hypocenterC73#11521
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×M16 ◁ #407

Complements

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

NH ≅ G/Ntype
1. #22626301C2semidirect
2. #22624878C2semidirect
124. C73#11521semidirect

Maximal subgroups

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

subgroupindex
#226263012
#226248782
#226248632
#226185602
#1152173