Finite Group Databasebeta

← #34014122

Subgroups — #34014122

Order 18688.

316 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 96 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#22626274C211yesmaximal, characteristic
2#22626152C211yesmaximal, characteristic
3#22624981C212yesmaximal
4#22624975C212yesmaximal
5#22618866C211yesmaximal, characteristic
6C2×#14556665C2212yes—
7C2×#14556665C2212yes—
8#17989510—22no—
9#17989479—22no—
10#17988718—22no—
11Q8×#10596831—22no—
12#17988614C411yescharacteristic
13#17988614C411yescharacteristic
14#17988611C411yescharacteristic
15#17988543C2211yescharacteristic
16#17988543C2211yescharacteristic
17C2×C4×#10596831C2211yescharacteristic
18#533×C73C411yescharacteristic
19#14556665—22no—
20#14556665—22no—
21#14556665—22no—
22#14556665Q814yescomplemented
23C2×#11924031—22no—
24C2×#11924031—22no—
25C2×#11924012C2×C411yescharacteristic
26C2×#11924012C2×C411yescharacteristic
27C22×#10596831C2×C411yescharacteristic
28C22×#10596831C2311yescharacteristic
29C4⋊#10596831—22no—
30C4⋊#10596831—22no—
31C4⋊#10596831—22no—
32#14556502—21no—
33#14556502D412yes—
34C4×#10596831—21no—
35C4×#10596831—21no—
36C4×#10596831—21no—
37C4×#10596831D412yes—
38C4⋊Q8×C73—21no—
39#125×C73—21no—
40C22⋊Q8×C73—22no—
41C4×Q8×C73—22no—
42#117×C73C2×C411yescharacteristic
43#117×C73C2×C411yescharacteristic
44C2×C42×C73C2×C411yescharacteristic
45#11924031—24no—
46#11924031—42no—
47#11924012—22no—
48#11924012—22no—
49C2×#10596831—21no—
50C2×#10596831—21no—
51C2×#10596831—22no—
52C2×#10596831C22⋊C412yes—
53C2×#10596831C2×D411yescharacteristic
54C2×#10596831C2×D411yescharacteristic
55C2×#10596831C2×Q812yes—
56C2×#10596831#4112yes—
57C2×Q8×C73—22no—
58C22×C4×C73—21no—
59C22×C4×C73C22×C411yescharacteristic
60C4⋊C4×C73—24no—
61C4⋊C4×C73—22no—
62C4⋊C4×C73—21no—
63C4⋊C4×C73—21no—
64C22⋊C4×C73—21no—
65C22⋊C4×C73M1612yes—
66C42×C73—21no—
67C42×C73—21no—
68C42×C73C22⋊C412yes—
69C42×C73M1611yescharacteristic
70C42×C73M1611yescharacteristic
71C23×C73C2×M1611yescharacteristic
72Q8×C73—44no—
73C2×C4×C73—21no—
74C2×C4×C73—21no—
75C2×C4×C73—22no—
76C2×C4×C73—22no—
77C2×C4×C73—42no—
78C2×C4×C73C2×C22⋊C411yescharacteristic
79C2×C4×C73C4×Q812yes—
80C2×C4×C73C2×M1611yescharacteristic
81C2×C4×C73#13112yes—
82#10596831—22no—
83#10596831—41no—
84#10596831—41no—
85#10596831C22⋊Q812yes—
86#10596831#12411yescharacteristic
87#10596831C4⋊Q811yescharacteristic
88C22×C73—41no—
89C22×C73#40711yescharacteristic
90C22×C73#44612yes—
91C4×C73—22no—
92C4×C73—41no—
93C4×C73—42no—
94C4×C73—42no—
95C4×C73#39111yescharacteristic
96C4×C73#40811yescharacteristic
97C4×C73#44612yes—
98#13270—731nomaximal, Sylow
99C2×C73—21no—
100C2×C73—22no—
101C2×C73#198511yescharacteristic
102#2960—731no—
103#2153—731no—
104#2110—731no—
105#1378—732no—
106#1376—732no—
107C73#1327011yescharacteristic, complemented, Sylow
108#533#6440811yescharacteristic
109C2×M32—732no—
110C2×M32—732no—
111#446—1462no—
112C8×Q8—1462no—
113#424—731no—
114#424—731no—
115#423—731no—
116#405—731no—
117#405—731no—
118C2×C4×C8—731no—
119#364—1462no—
120#347—1462no—
121C2×M16—731no—
122C2×M16—731no—
123C22×C8—731no—
124C22×C8—731no—
125C4⋊Q8—21no—
126#125—21no—
127C22⋊Q8—22no—
128C4×Q8—22no—
129#117C2×#6440811yescharacteristic
130#117C2×#6440811yescharacteristic
131C2×C42C2×#6440811yescharacteristic
132M32—734no—
133M32—1462no—
134M32—1462no—
135M32—1462no—
136C2×C16—1462no—
137C2×C16—1462no—
138C4⋊C8—1462no—
139C4⋊C8—1462no—
140C4⋊C8—1462no—
141#97—732no—
142#97—1461no—
143C4×C8—732no—
144C4×C8—1461no—
145C4×C8—1461no—
146C4×C8—1461no—
147C2×Q8—22no—
148C22×C4—21no—
149C22×C4C22×#6440811yescharacteristic
150M16—1462no—
151M16—1462no—
152C2×C8—732no—
153C2×C8—732no—
154C2×C8—731no—
155C2×C8—731no—
156C2×C8—732no—
157C2×C8—1462no—
158C2×C8—1461no—
159C2×C8—1461no—
160C4⋊C4—24no—
161C4⋊C4—22no—
162C4⋊C4—21no—
163C4⋊C4—21no—
164C22⋊C4—21no—
165C22⋊C4#1192403712yes—
166C42—21no—
167C42—21no—
168C42C22⋊#6440812yes—
169C42#1192403711yescharacteristic
170C42#1192403711yescharacteristic
171C16—2922no—
172C16—1464no—
173C23C2×#1192403711yescharacteristic
174Q8—44no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—42no—
180C2×C4C2×C22⋊#6440811yescharacteristic
181C2×C4Q8×#6440812yes—
182C2×C4C2×#1192403711yescharacteristic
183C2×C4#1455667912yes—
184C8—2921no—
185C8—2921no—
186C8—731no—
187C8—732no—
188C8—731no—
189C8—1462no—
190C22—41no—
191C22#1798955711yescharacteristic
192C22#1798963812yes—
193C4—22no—
194C4—41no—
195C4—42no—
196C4—42no—
197C4(C2×#64408)⋊Q811yescharacteristic
198C4#1798956211yescharacteristic
199C4#1798963812yes—
200C2—21no—
201C2—22no—
202C2#2262610511yescharacteristic
subgroupG/H
centerC4(C2×#64408)⋊Q8
commutatorC22×C4×C73C22×C4
FrattiniC22×C4C22×#64408
Fitting#533×C73C4
socleC2×C73#1985
prime radicalC23×C73C2×M16
cosocleC22×#10596831C23
hypercenter#533#64408
hypocenterC73#13270
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C4 ◁ C22×C4 ◁ #533

Complements

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

NH ≅ G/Ntype
22. #14556665Q8semidirect
107. C73#13270semidirect

Maximal subgroups

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

subgroupindex
#226262742
#226261522
#226249812
#226249752
#226188662
#1327073