Finite Group Databasebeta

← #34005621

Subgroups — #34005621

Order 18688.

348 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 82 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#22626829C211yesmaximal, characteristic
2#22626104C211yesmaximal, characteristic
3#22625506C212yesmaximal
4#22624826C212yesmaximal
5#22608253C211yesmaximal, characteristic
6#17989637—22no—
7#17989637—22no—
8#17989608—21no—
9#17989608C2212yes—
10#17989603—21no—
11C2×C4⋊#10596833C2211yescharacteristic
12C2×#14556646C2211yescharacteristic
13#17989453—21no—
14#17989453—21no—
15#17989450—22no—
16#17989449—22no—
17C2×#14556556C2211yescharacteristic
18#17988637C2212yes—
19#17988313C411yescharacteristic
20#17988312C412yes—
21#324×C73C411yescharacteristic
22C22×#10596833—21no—
23C22×#10596833—21no—
24C4⋊#10596833—21no—
25C4⋊#10596833—21no—
26C4⋊#10596833—21no—
27C4⋊#10596833—21no—
28C4⋊#10596833—22no—
29C4⋊#10596833D412yes—
30C4⋊#10596833D412yes—
31C22⋊#10596833—22no—
32#14556646—22no—
33#14556646—22no—
34#14556646—22no—
35C4×#10596833—22no—
36C22×Dic146Q811yescharacteristic
37#14556556—22no—
38#14556556—22no—
39C2×#11924007C2×C411yescharacteristic
40C2×#11924007Q811yescharacteristic
41C2×C4×Dic73D411yescharacteristic
42C2×C4×Dic73D411yescharacteristic
43C2×C4×Dic73C2311yescharacteristic
44#14556520—22no—
45C22⋊#10596831C2×C412yes—
46C8×Dic73—22no—
47C2×C4⋊C4×C73C2×C411yescharacteristic
48C22⋊C8×C73C2×C412yes—
49C2×#10596833—22no—
50C2×#10596833—22no—
51C2×#10596833—22no—
52C2×#10596833—41no—
53C2×#10596833—41no—
54C2×#10596833—42no—
55C2×#10596833—41no—
56C2×Dic146—22no—
57C2×Dic146—22no—
58C2×Dic146—41no—
59C22×Dic73C4⋊C411yescharacteristic
60C22×Dic73C4⋊C411yescharacteristic
61C22×Dic73C2×D411yescharacteristic
62C22×Dic73C2×Q811yescharacteristic
63C2×#10596831—22no—
64#11924007—22no—
65#11924007—22no—
66C4×Dic73—21no—
67C4×Dic73—21no—
68C4×Dic73—21no—
69C4×Dic73—21no—
70C4×Dic73C2×D412yes—
71C4×Dic73#4112yes—
72C22×C4×C73C4⋊C411yescharacteristic
73C22×C4×C73C4⋊C411yescharacteristic
74C22×C4×C73C22×C411yescharacteristic
75C2×C8×C73—22no—
76C4⋊C4×C73—22no—
77C23×C73C2×C4⋊C411yescharacteristic
78C2×C4×C73—21no—
79C2×C4×C73—21no—
80C2×C4×C73—21no—
81C2×C4×C73—41no—
82C2×C4×C73—41no—
83C2×C4×C73#10812yes—
84C2×C4×C73C4×D412yes—
85C2×Dic73—21no—
86C2×Dic73—21no—
87C2×Dic73—21no—
88C2×Dic73—21no—
89C2×Dic73—21no—
90C2×Dic73—21no—
91C2×Dic73—21no—
92C2×Dic73—21no—
93C2×Dic73—41no—
94C2×Dic73#12011yescharacteristic
95C2×Dic73C22⋊Q812yes—
96C2×Dic73#12311yescharacteristic
97Dic146—44no—
98#10596833—42no—
99#10596833—42no—
100#10596833—42no—
101#10596833—81no—
102C8×C73—42no—
103#10596831—42no—
104C22×C73—21no—
105C22×C73—21no—
106C22×C73#38511yescharacteristic
107C22×C73C2×#10811yescharacteristic
108C22×C73#43011yescharacteristic
109C4×C73—22no—
110C4×C73—22no—
111C4×C73—41no—
112Dic73—22no—
113Dic73—21no—
114Dic73—41no—
115Dic73—41no—
116Dic73#45411yescharacteristic
117Dic73#45611yescharacteristic
118#8865—731nomaximal, Sylow
119C2×C73—21no—
120C2×C73—21no—
121C2×C73#185511yescharacteristic
122C2×C73#185711yescharacteristic
123C2×C73#186811yescharacteristic
124#2983—731no—
125#1985—731no—
126#1573—732no—
127#1461—731no—
128#1302—732no—
129C73#886511yescharacteristic, complemented, Sylow
130C2×C4×Q8—731no—
131#446—1462no—
132#446—1462no—
133#432—732no—
134#432—1461no—
135#431—732no—
136#431—1461no—
137C2×C4⋊C8—731no—
138C2×#97—731no—
139#336—1461no—
140#336—1461no—
141#335—1462no—
142#334—1462no—
143#324—731no—
144#324—732no—
145#324#6440811yescharacteristic
146C22×Q8—731no—
147C22×C8—1461no—
148C22×C8—1461no—
149C4×Q8—1462no—
150C4×Q8—1462no—
151C2×C4⋊C4—731no—
152C2×C4⋊C4—731no—
153C2×C4⋊C4C2×#6440811yescharacteristic
154C2×C42—731no—
155C2×C42—731no—
156C2×C42—731no—
157C4⋊C8—732no—
158C4⋊C8—732no—
159C4⋊C8—1462no—
160C4⋊C8—1462no—
161C4⋊C8—1461no—
162C4⋊C8—1461no—
163C4⋊C8—1461no—
164C4⋊C8—1461no—
165C22⋊C8—732no—
166C22⋊C8—1462no—
167C22⋊C8C2×#6440812yes—
168#97—1462no—
169#97—1462no—
170#97—1462no—
171C4×C8—1462no—
172C4×C8—1462no—
173C2×Q8—2921no—
174C2×Q8—1462no—
175C2×Q8—1462no—
176C22×C4—731no—
177C22×C4—731no—
178C22×C4—731no—
179C22×C4—731no—
180C22×C4C4⋊#6440811yescharacteristic
181C22×C4C4⋊#6440811yescharacteristic
182C22×C4C22×#6440811yescharacteristic
183C2×C8—22no—
184C2×C8—2921no—
185C2×C8—2921no—
186C2×C8—2921no—
187C2×C8—2922no—
188C2×C8—1462no—
189C2×C8—1462no—
190C2×C8—1462no—
191C2×C8—1462no—
192C4⋊C4—22no—
193C4⋊C4—1462no—
194C4⋊C4—1462no—
195C42—732no—
196C42—732no—
197C42—1461no—
198C42—1461no—
199C42—1461no—
200C42—1461no—
201C23C2×C4⋊#6440811yescharacteristic
202Q8—2924no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—2921no—
209C2×C4—731no—
210C2×C4—731no—
211C2×C4—732no—
212C2×C4—1461no—
213C2×C4—1461no—
214C2×C4—1461no—
215C2×C4—1461no—
216C2×C4—1461no—
217C2×C4—1461no—
218C2×C4—1461no—
219C2×C4—1461no—
220C2×C4#1455666212yes—
221C2×C4D4×#6440812yes—
222C8—42no—
223C8—2922no—
224C8—2922no—
225C8—2922no—
226C8—2922no—
227C8—5841no—
228C22—21no—
229C22—21no—
230C22#1798953411yescharacteristic
231C22C2×#1455666211yescharacteristic
232C22#1798960211yescharacteristic
233C4—22no—
234C4—22no—
235C4—41no—
236C4—2921no—
237C4—2921no—
238C4—731no—
239C4—731no—
240C4—1462no—
241C4—1461no—
242C2—21no—
243C2—21no—
244C2#2262588311yescharacteristic
245C2#2262588711yescharacteristic
246C2#2262590311yescharacteristic
subgroupG/H
centerC22#17989534
commutatorC22×C4×C73C22×C4
FrattiniC22×C4C22×#64408
Fitting#324×C73C4
socleC22×C73#385
prime radicalC23×C73C2×C4⋊C4
cosocleC2×C4×Dic73C23
hypercenter#324#64408
hypocenterC73#8865
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
129. C73#8865semidirect

Maximal subgroups

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

subgroupindex
#226268292
#226261042
#226255062
#226248262
#226082532
#886573