Finite Group Databasebeta

← #36327411

Subgroups — #36327411

Order 21376.

528 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 151 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
1C2×#19128797C211yesmaximal, characteristic, complemented
2C2×#19128790C211yesmaximal, characteristic
3#24923204C211yesmaximal, characteristic
4#24923203C212yesmaximal, complemented
5#24923193C212yesmaximal
6#24923189C211yesmaximal, characteristic, complemented
7#24923182C211yesmaximal, characteristic
8#24923120C211yesmaximal, characteristic, complemented
9#24923120C212yesmaximal, complemented
10#24923116C212yesmaximal
11#400×C167C211yesmaximal, characteristic, complemented
12C22×Dic334C2211yescharacteristic
13C22×C4×D167C2211yescharacteristic
14#19128797—24no—
15#19128790—24no—
16C2×C4⋊Dic167C2211yescharacteristic
17C2×Dic167⋊C4C2212yes—
18C2×Dic167⋊C4C2212yes—
19C2×Dic167⋊C4C2212yes—
20C2×Dic167⋊C4C2212yes—
21C2×(C2×D167)⋊C4C2212yes—
22C2×(C2×D167)⋊C4C2211yescharacteristic
23C2×(C2×D167)⋊C4C2211yescharacteristic
24C2×(C2×D167)⋊C4C2212yes—
25C2×C4×Dic167C2211yescharacteristic
26C2×C4×Dic167C2212yes—
27C2×C4×Dic167C2212yes—
28C2×C4×Dic167C2211yescharacteristic
29C167⋊#95C2212yes—
30C167⋊#95C2212yes—
31C167⋊#95C2211yescharacteristic
32C2×C4⋊C4×C167C2211yescharacteristic
33C2×C4⋊C4×C167C2211yescharacteristic
34#95×C167C2211yescharacteristic
35#95×C167C2212yes—
36#95×C167C2212yes—
37C23×D167C2311yescharacteristic
38C2×Dic334—24no—
39C2×Dic334—22no—
40C22×Dic167C2311yescharacteristic
41C22×Dic167C2311yescharacteristic
42C22×Dic167C2312yes—
43C22×Dic167C2312yes—
44C22×Dic167C2311yescharacteristic
45C2×C4×D167—22no—
46C2×C4×D167—22no—
47C2×C4×D167Q814yescomplemented
48C4⋊Dic167—22no—
49Dic167⋊C4—24no—
50Dic167⋊C4—24no—
51Dic167⋊C4—24no—
52Dic167⋊C4—24no—
53(C2×D167)⋊C4—22no—
54(C2×D167)⋊C4—24no—
55(C2×D167)⋊C4—24no—
56(C2×D167)⋊C4—22no—
57C4×Dic167—22no—
58C4×Dic167—24no—
59C4×Dic167—24no—
60C4×Dic167D414yes—
61C22×C4×C167C2312yes—
62C22×C4×C167C2311yescharacteristic
63C22×C4×C167C2312yes—
64C22×C4×C167C2311yescharacteristic
65C22×C4×C167C2311yescharacteristic
66C4⋊C4×C167—22no—
67C4⋊C4×C167—22no—
68C22×D167—21no—
69C22×D167—22no—
70C22×D167—22no—
71C22×D167—21no—
72C22×D167C2×Q812yes—
73Dic334—28no—
74C2×Dic167—22no—
75C2×Dic167—22no—
76C2×Dic167—22no—
77C2×Dic167—22no—
78C2×Dic167—22no—
79C2×Dic167—21no—
80C2×Dic167—22no—
81C2×Dic167C2×D412yes—
82C2×Dic167C2×D412yes—
83C2×Dic167C2×Q812yes—
84C2×Dic167#4114yes—
85C2×Dic167#4114yes—
86C2×Dic167#4112yes—
87C4×D167—28no—
88C23×C167C2411yescharacteristic
89C2×C4×C167—21no—
90C2×C4×C167—21no—
91C2×C4×C167—22no—
92C2×C4×C167—21no—
93C2×C4×C167—22no—
94C2×C4×C167—22no—
95C2×C4×C167—22no—
96C2×C4×C167—22no—
97C2×C4×C167—22no—
98C2×C4×C167—22no—
99C2×C4×C167—22no—
100C2×C4×C167C2×D412yes—
101C2×C4×C167C2×Q812yes—
102C22×C167C22×D411yescharacteristic
103C22×C167C22×Q811yescharacteristic
104C22×C167C2×#4111yescharacteristic
105C22×C167C2×#4112yes—
106C22×C167C2×#4112yes—
107C2×D167—22no—
108C2×D167—24no—
109C2×D167—41no—
110C2×D167—42no—
111C2×D167—41no—
112C4×C167—22no—
113C4×C167—22no—
114C4×C167—41no—
115C4×C167—42no—
116C4×C167—42no—
117Dic167—24no—
118Dic167—24no—
119Dic167—22no—
120Dic167—41no—
121Dic167C22⋊Q814yes—
122C2×C167C2×C22⋊Q811yescharacteristic
123C2×C167#52511yescharacteristic
124C2×C167#52912yes—
125C2×C167#53211yescharacteristic
126C2×C167#53312yes—
127D167—42no—
128C167#239611yescharacteristic, complemented, Sylow
129#2396—1671nomaximal, Sylow
130C2×C4×Q8—1671no—
131C2×#117—1671no—
132#400D16711yescharacteristic, complemented
133#399—1671no—
134#398—1672no—
135#396—1672no—
136#395—1671no—
137#393—1671no—
138C4×C4⋊C4—1672no—
139C4×C4⋊C4—1671no—
140C4×C22⋊C4—1672no—
141C22×Q8—1671no—
142C23×C4—1671no—
143C4×Q8—3344no—
144#117—3344no—
145C2×C4⋊C4—1671no—
146C2×C4⋊C4—1672no—
147C2×C4⋊C4—1672no—
148C2×C4⋊C4—1672no—
149C2×C4⋊C4—1672no—
150C2×C4⋊C4C2×D16711yescharacteristic
151C2×C4⋊C4C2×D16711yescharacteristic
152C2×C22⋊C4—1671no—
153C2×C22⋊C4—1671no—
154C2×C22⋊C4—1672no—
155C2×C22⋊C4—1672no—
156C2×C42—1671no—
157C2×C42—1672no—
158C2×C42—1672no—
159C2×C42—1671no—
160#95—1672no—
161#95—1672no—
162#95—1671no—
163#95C2×D16712yes—
164#95C2×D16711yescharacteristic
165#95C2×D16712yes—
166C24—1671no—
167C2×Q8—3344no—
168C2×Q8—3342no—
169C22×C4—3342no—
170C22×C4—3342no—
171C22×C4—1671no—
172C22×C4—1672no—
173C22×C4—1672no—
174C22×C4—1671no—
175C22×C4—1674no—
176C22×C4—1671no—
177C22×C4C22×D16712yes—
178C22×C4C22×D16711yescharacteristic
179C22×C4C22×D16712yes—
180C22×C4C22×D16711yescharacteristic
181C22×C4C22×D16711yescharacteristic
182C4⋊C4—22no—
183C4⋊C4—22no—
184C4⋊C4—3342no—
185C4⋊C4—3344no—
186C4⋊C4—3344no—
187C4⋊C4—3344no—
188C4⋊C4—3344no—
189C22⋊C4—3342no—
190C22⋊C4—3344no—
191C22⋊C4—3344no—
192C22⋊C4—3342no—
193C42—3342no—
194C42—3344no—
195C42—3344no—
196C42—1674no—
197C23—3341no—
198C23—3342no—
199C23—3342no—
200C23—3341no—
201C23—1672no—
202C23C23×D16711yescharacteristic
203Q8—3348no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—22no—
207C2×C4—21no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—3348no—
216C2×C4—3342no—
217C2×C4—3342no—
218C2×C4—3342no—
219C2×C4—3342no—
220C2×C4—3342no—
221C2×C4—3341no—
222C2×C4—3342no—
223C2×C4—1672no—
224C2×C4—1674no—
225C2×C4—1674no—
226C2×C4—1672no—
227C2×C4—1672no—
228C2×C4—1672no—
229C2×C4D4×D16712yes—
230C2×C4Q8×D16712yes—
231C22—3342no—
232C22—3344no—
233C22—6681no—
234C22—6682no—
235C22—6681no—
236C22C2×D4×D16711yescharacteristic
237C22C2×Q8×D16711yescharacteristic
238C22#41×D16711yescharacteristic
239C22#41×D16712yes—
240C22#41×D16712yes—
241C4—22no—
242C4—22no—
243C4—41no—
244C4—42no—
245C4—42no—
246C4—3344no—
247C4—3344no—
248C4—3342no—
249C4—6681no—
250C4—1674no—
251C2—6682no—
252C2C22⋊Q8×D16711yescharacteristic
253C2#2492377711yescharacteristic
254C2#2492380212yes—
255C2#2492381811yescharacteristic
256C2#2492382512yes—
subgroupG/H
center, FrattiniC23C23×D167
commutator, socle, cosocleC23×C167C24
Fitting#400×C167C2
prime radicalC23×D167C23
hypercenter#400D167
hypocenterC167#2396
LCSG ▷ C23×C167 ▷ C167
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
1. C2×#19128797C2semidirect
4. #24923203C2semidirect
6. #24923189C2semidirect
8. #24923120C2semidirect
9. #24923120C2semidirect
11. #400×C167C2semidirect
47. C2×C4×D167Q8semidirect
128. C167#2396semidirect
132. #400D167semidirect

Maximal subgroups

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

subgroupindex
C2×#191287972
C2×#191287902
#249232042
#249232032
#249231932
#249231892
#249231822
#249231202
#249231162
#400×C1672
#2396167