Finite Group Databasebeta

← #34010124

Subgroups — #34010124

Order 18688.

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

normal subgroups only

This group has 280 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#22624880C212yesmaximal, complemented
3#22624865C212yesmaximal
4#22618578C211yesmaximal, characteristic
5#17989509—22no—
6#17989483C2214yes—
7#17989480—22no—
8#41×Dic73C411yescharacteristic
9#17988645—22no—
10#17988641—22no—
11#17988615C411yescharacteristic
12#17988613C2211yescharacteristic
13#17988554C2211yescharacteristic
14M16×Dic73C2211yescharacteristic
15C2×#14556501C411yescharacteristic
16#408×C73C411yescharacteristic
17#14556665—24no—
18C2×#11924012C2311yescharacteristic
19C22×#10596831C2×C411yescharacteristic
20Q8×Dic73—21no—
21D4×Dic73—22no—
22D4×Dic73—21no—
23#14556550—21no—
24#14556550C2×C411yescharacteristic
25C2×C4×Dic73—21no—
26C2×C4×Dic73C2×C411yescharacteristic
27#14556520—22no—
28#14556520—22no—
29C22⋊#10596831—22no—
30#14556501—21no—
31#14556501—21no—
32#14556501D412yes—
33C8×Dic73D412yes—
34C2×#41×C73C2×C411yescharacteristic
35C2×M16×C73C2×C411yescharacteristic
36C22×C8×C73C2×C411yescharacteristic
37C22⋊C8×C73—22no—
38#11924031—44no—
39C22×Dic73—22no—
40C22×Dic73—21no—
41#11924012—41no—
42C2×#10596831—21no—
43C2×#10596831—41no—
44C2×#10596831C2×D412yes—
45C4⋊Dic73—22no—
46C4⋊Dic73—21no—
47C22⋊Dic73—22no—
48C22⋊Dic73M1612yes—
49C4×Dic73—22no—
50C4×Dic73—21no—
51C4×Dic73—21no—
52C4×Dic73C22⋊C411yescharacteristic
53C4×Dic73C22⋊C411yescharacteristic
54#41×C73—41no—
55#41×C73—41no—
56C2×Q8×C73M1611yescharacteristic
57C2×D4×C73—21no—
58C2×D4×C73M1611yescharacteristic
59C22×C4×C73—21no—
60C22×C4×C73C22×C411yescharacteristic
61M16×C73—41no—
62C2×C8×C73—21no—
63C2×C8×C73—41no—
64C2×C8×C73C22⋊C412yes—
65C23×C73—21no—
66C23×C73C2×M1611yescharacteristic
67Q8×C73—41no—
68D4×C73—41no—
69D4×C73—42no—
70C2×C4×C73—21no—
71C2×C4×C73—21no—
72C2×C4×C73—21no—
73C2×C4×C73—41no—
74C2×C4×C73—41no—
75C2×C4×C73C2×C22⋊C411yescharacteristic
76C2×C4×C73C2×M1611yescharacteristic
77C2×Dic73—21no—
78C2×Dic73—22no—
79C2×Dic73—21no—
80C2×Dic73—81no—
81C2×Dic73—82no—
82C8×C73—22no—
83C8×C73—41no—
84#10596831—22no—
85#10596831—41no—
86C22×C73—21no—
87C22×C73—41no—
88C22×C73—42no—
89C22×C73—41no—
90C22×C73#40711yescharacteristic
91C4×C73—21no—
92C4×C73—41no—
93C4×C73#41011yescharacteristic
94C4×C73#41311yescharacteristic
95Dic73—42no—
96Dic73—41no—
97Dic73—41no—
98#11523—731nomaximal, Sylow
99C2×C73—21no—
100C2×C73—41no—
101C2×C73—81no—
102C2×C73#146411yescharacteristic
103#2923—731no—
104#2165—732no—
105#1328—732no—
106#1321—732no—
107C73#1152311yescharacteristic, complemented, Sylow
108C4×#41—731no—
109#433—1462no—
110#432—1462no—
111#424—731no—
112#424—731no—
113#408—731no—
114#408#6440811yescharacteristic
115C4×M16—731no—
116C2×#97—731no—
117#364—1462no—
118#349—734no—
119#347—1462no—
120C2×#41C2×#6440811yescharacteristic
121C2×M16—731no—
122C2×M16C2×#6440811yescharacteristic
123C22×C8—731no—
124C22×C8C2×#6440811yescharacteristic
125C4×Q8—1461no—
126C4×D4—1462no—
127C4×D4—1461no—
128#117—731no—
129#117—1461no—
130C2×C42—731no—
131C2×C42—1461no—
132M32—1464no—
133C4⋊C8—1462no—
134C4⋊C8—1462no—
135C22⋊C8—22no—
136C22⋊C8—1462no—
137#97—732no—
138#97—1461no—
139#97—1461no—
140C4×C8—732no—
141#41—41no—
142#41—41no—
143C2×Q8#1192403811yescharacteristic
144C2×D4—21no—
145C2×D4#1192403811yescharacteristic
146C22×C4—21no—
147C22×C4—1462no—
148C22×C4—1461no—
149C22×C4C22×#6440811yescharacteristic
150M16—41no—
151M16—2921no—
152C2×C8—21no—
153C2×C8—41no—
154C2×C8—2921no—
155C2×C8—732no—
156C2×C8—1461no—
157C2×C8C22⋊#6440812yes—
158C4⋊C4—1462no—
159C4⋊C4—1461no—
160C22⋊C4—732no—
161C22⋊C4—1462no—
162C42—731no—
163C42—731no—
164C42—1462no—
165C42—1461no—
166C42—1461no—
167C16—2924no—
168C23—21no—
169C23D73⋊M1611yescharacteristic
170Q8—41no—
171D4—41no—
172D4—42no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—5841no—
179C2×C4—5842no—
180C2×C4—1461no—
181C2×C4—1461no—
182C2×C4—1462no—
183C2×C4C2×C22⋊#6440811yescharacteristic
184C2×C4C2×#1192403811yescharacteristic
185C8—22no—
186C8—41no—
187C8—2921no—
188C8—1462no—
189C22—21no—
190C22—41no—
191C22—42no—
192C22—41no—
193C22#1798955811yescharacteristic
194C4—21no—
195C4—41no—
196C4—2921no—
197C4—2921no—
198C4—2922no—
199C4#1798956711yescharacteristic
200C4#1798957411yescharacteristic
201C2—21no—
202C2—41no—
203C2—81no—
204C2#2262518311yescharacteristic
subgroupG/H
centerC4#17989567
commutatorC22×C4×C73C22×C4
FrattiniC22×C4C22×#64408
Fitting#408×C73C4
socleC2×C73#1464
prime radicalC2×#41×C73C2×C4
cosocleC2×#11924012C23
hypercenter#408#64408
hypocenterC73#11523
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×M16 ◁ #408

Complements

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

NH ≅ G/Ntype
1. #22626301C2semidirect
2. #22624880C2semidirect
107. C73#11523semidirect

Maximal subgroups

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

subgroupindex
#226263012
#226248802
#226248652
#226185782
#1152373