Finite Group Databasebeta

← #34010120

Subgroups — #34010120

Order 18688.

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

normal subgroups only

This group has 284 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#22618566C211yesmaximal, characteristic
5#17989509—22no—
6#17989483C2214yes—
7#17989480—22no—
8C2×#14556550C411yescharacteristic
9#17988645—22no—
10#17988645—22no—
11#17988610C411yescharacteristic
12#17988610C2211yescharacteristic
13#17988551C2211yescharacteristic
14M16×Dic73C411yescharacteristic
15M16×Dic73C2211yescharacteristic
16#407×C73C411yescharacteristic
17#14556665—24no—
18C2×#11924012C2×C411yescharacteristic
19C2×#11924012C2311yescharacteristic
20#14556550—21no—
21#14556550—21no—
22#14556550—22no—
23C2×C4⋊Dic73—21no—
24C2×C22⋊Dic73—21no—
25C2×C4×Dic73C2×C411yescharacteristic
26C2×C4×Dic73C2×C411yescharacteristic
27#14556520—22no—
28#14556520—22no—
29C22⋊#10596831—22no—
30#14556501—21no—
31#14556501D412yes—
32C8×Dic73—21no—
33C8×Dic73D412yes—
34C23×C4×C73C2×C411yescharacteristic
35C2×M16×C73C2×C411yescharacteristic
36C2×M16×C73C2×C411yescharacteristic
37C22⋊C8×C73—22no—
38#11924031—44no—
39C22×Dic73—21no—
40C22×Dic73—21no—
41#11924012—41no—
42#11924012—41no—
43C2×#10596831—21no—
44C2×#10596831C2×D412yes—
45C4⋊Dic73—22no—
46C4⋊Dic73—22no—
47C22⋊Dic73—22no—
48C22⋊Dic73—22no—
49C4×Dic73—21no—
50C4×Dic73—21no—
51C4×Dic73C22⋊C411yescharacteristic
52C4×Dic73C22⋊C411yescharacteristic
53C4×Dic73M1612yes—
54C24×C73M1611yescharacteristic
55C22×C4×C73—21no—
56C22×C4×C73—21no—
57C22×C4×C73—41no—
58C22×C4×C73—41no—
59C22×C4×C73M1611yescharacteristic
60C22×C4×C73C22×C411yescharacteristic
61M16×C73—41no—
62M16×C73—41no—
63C2×C8×C73—21no—
64C2×C8×C73C22⋊C412yes—
65C23×C73—21no—
66C23×C73—41no—
67C23×C73—81no—
68C23×C73C2×M1611yescharacteristic
69C2×C4×C73—21no—
70C2×C4×C73—21no—
71C2×C4×C73—21no—
72C2×C4×C73—42no—
73C2×C4×C73—41no—
74C2×C4×C73—41no—
75C2×C4×C73—41no—
76C2×C4×C73C2×C22⋊C411yescharacteristic
77C2×C4×C73C2×M1611yescharacteristic
78C2×Dic73—21no—
79C2×Dic73—21no—
80C2×Dic73—22no—
81C2×Dic73—81no—
82C2×Dic73—81no—
83C8×C73—22no—
84C8×C73—41no—
85#10596831—22no—
86#10596831—41no—
87C22×C73—21no—
88C22×C73—41no—
89C22×C73—42no—
90C22×C73—41no—
91C22×C73—82no—
92C22×C73#40711yescharacteristic
93C4×C73—21no—
94C4×C73—41no—
95C4×C73#41011yescharacteristic
96C4×C73#41311yescharacteristic
97Dic73—41no—
98Dic73—41no—
99Dic73—42no—
100#11522—731nomaximal, Sylow
101C2×C73—21no—
102C2×C73—41no—
103C2×C73—81no—
104C2×C73#146411yescharacteristic
105#2921—731no—
106#2165—732no—
107#1327—732no—
108#1320—732no—
109C73#1152211yescharacteristic, complemented, Sylow
110C2×#117—731no—
111#433—1462no—
112#433—1462no—
113#423—731no—
114#423—731no—
115#407—731no—
116#407#6440811yescharacteristic
117C4×M16—731no—
118C4×M16—731no—
119#364—1462no—
120#349—734no—
121#347—1462no—
122C23×C4C2×#6440811yescharacteristic
123C2×M16—731no—
124C2×M16—731no—
125C2×M16C2×#6440811yescharacteristic
126C2×M16C2×#6440811yescharacteristic
127#117—1462no—
128#117—1461no—
129#117—1461no—
130C2×C4⋊C4—1461no—
131C2×C22⋊C4—1461no—
132C2×C42—731no—
133C2×C42—731no—
134M32—1464no—
135C4⋊C8—1462no—
136C4⋊C8—1462no—
137C22⋊C8—22no—
138C22⋊C8—1462no—
139#97—732no—
140#97—1461no—
141C4×C8—732no—
142C4×C8—1461no—
143C24#1192403811yescharacteristic
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—41no—
147C22×C4—41no—
148C22×C4—1461no—
149C22×C4—1461no—
150C22×C4#1192403811yescharacteristic
151C22×C4C22×#6440811yescharacteristic
152M16—41no—
153M16—41no—
154M16—2921no—
155M16—2921no—
156C2×C8—21no—
157C2×C8—732no—
158C2×C8—1461no—
159C2×C8C22⋊#6440812yes—
160C4⋊C4—1462no—
161C4⋊C4—1462no—
162C22⋊C4—1462no—
163C22⋊C4—1462no—
164C42—732no—
165C42—731no—
166C42—731no—
167C42—1461no—
168C42—1461no—
169C16—2924no—
170C23—21no—
171C23—41no—
172C23—81no—
173C23C2×#1192403811yescharacteristic
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—42no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—5841no—
182C2×C4—5841no—
183C2×C4—1462no—
184C2×C4—1461no—
185C2×C4—1461no—
186C2×C4C2×C22⋊#6440811yescharacteristic
187C2×C4D73⋊M1611yescharacteristic
188C8—22no—
189C8—41no—
190C8—2921no—
191C8—1462no—
192C22—21no—
193C22—41no—
194C22—42no—
195C22—41no—
196C22—82no—
197C22#1798955811yescharacteristic
198C4—21no—
199C4—41no—
200C4—2922no—
201C4—2921no—
202C4—2921no—
203C4#1798956711yescharacteristic
204C4#1798957411yescharacteristic
205C2—21no—
206C2—41no—
207C2—81no—
208C2#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#11522
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
109. C73#11522semidirect

Maximal subgroups

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

subgroupindex
#226263012
#226248782
#226248632
#226185662
#1152273