Finite Group Databasebeta

← #36325222

Subgroups — #36325222

Order 21376.

406 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 93 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
1C22×#14581939C211yesmaximal, characteristic
2C22×#14581937C211yesmaximal, characteristic
3C2×#19128830C211yesmaximal, characteristic, complemented
4C2×#19128755C211yesmaximal, characteristic, complemented
5#24923170C211yesmaximal, characteristic
6#24923023C211yesmaximal, characteristic, complemented
7C2×#103×C167C211yesmaximal, characteristic, complemented
8C22×Dic334C2211yescharacteristic
9C22×D668C2211yescharacteristic
10C2×#14581939—22no—
11C2×#14581939—22no—
12C2×#14581939—22no—
13C2×#14581937—22no—
14C2×#14581937—22no—
15C2×#14581937—22no—
16C22×C167⋊C8C2211yescharacteristic
17#19128830—22no—
18#19128830—22no—
19C2×Dic167⋊C4—22no—
20C2×(C2×D167)⋊C4—22no—
21C2×C4×Dic167C2211yescharacteristic
22#19128755—22no—
23C22×Q8×C167C2211yescharacteristic
24C22×C8×C167C2211yescharacteristic
25C2×C4⋊C4×C167C2211yescharacteristic
26#103×C167—22no—
27C23×D167—21no—
28C2×Dic334—21no—
29C2×Dic334—21no—
30C2×Dic334—21no—
31C2×Dic334—21no—
32C2×Dic334—21no—
33C2×Dic334D412yes—
34C2×D668—21no—
35C2×D668—21no—
36C2×D668—21no—
37C2×D668—21no—
38C2×D668—21no—
39C2×D668D412yes—
40C22×Dic167—21no—
41C22×Dic167D412yes—
42#14581939—28no—
43#14581937—28no—
44C2×C167⋊C8—21no—
45C2×C167⋊C8—21no—
46C2×C167⋊C8D412yes—
47Dic167⋊C4—28no—
48(C2×D167)⋊C4—44no—
49C4×Dic167—21no—
50C4×Dic167—21no—
51C2×Q8×C167—21no—
52C2×Q8×C167—21no—
53C2×Q8×C167—21no—
54C2×Q8×C167—21no—
55C2×Q8×C167—21no—
56C2×Q8×C167D412yes—
57C22×C4×C167—21no—
58C22×C4×C167—21no—
59C22×C4×C167C2311yescharacteristic
60C2×C8×C167—21no—
61C2×C8×C167—21no—
62C2×C8×C167D412yes—
63C4⋊C4×C167—21no—
64C4⋊C4×C167—21no—
65C22×D167—22no—
66C22×D167—41no—
67C22×D167—41no—
68C22×D167—42no—
69C22×D167—42no—
70Dic334—22no—
71Dic334—22no—
72Dic334—41no—
73Dic334—41no—
74D668—22no—
75D668—22no—
76D668—41no—
77D668—41no—
78C2×Dic167—22no—
79C2×Dic167—22no—
80C2×Dic167—22no—
81C2×Dic167—41no—
82C2×Dic167—41no—
83C2×Dic167SD1614yescomplemented
84C167⋊C8—22no—
85C23×C167C2×D411yescharacteristic
86Q8×C167—22no—
87Q8×C167—22no—
88Q8×C167—41no—
89Q8×C167—41no—
90C2×C4×C167—22no—
91C2×C4×C167—41no—
92C2×C4×C167—41no—
93C2×C4×C167—41no—
94C2×C4×C167—42no—
95C2×C4×C167C2×D411yescharacteristic
96C2×C4×C167C2×D411yescharacteristic
97C2×C4×C167C2×D411yescharacteristic
98C2×C4×C167C2×D411yescharacteristic
99C2×C4×C167C2×D411yescharacteristic
100C2×C4×C167#4111yescharacteristic
101C8×C167—22no—
102C22×C167#12011yescharacteristic
103C22×C167#12111yescharacteristic
104C22×C167#12711yescharacteristic
105C22×C167C2×SD1612yes—
106C22×C167#13712yes—
107C2×D167—42no—
108C2×D167—44no—
109C2×D167—82no—
110C2×D167—81no—
111C2×D167—81no—
112C4×C167—42no—
113C4×C167—81no—
114C4×C167#12011yescharacteristic
115C4×C167#12011yescharacteristic
116C4×C167#12111yescharacteristic
117C4×C167#12111yescharacteristic
118Dic167—24no—
119Dic167—42no—
120C2×C167#39211yescharacteristic
121C2×C167#44812yes—
122C2×C167#46112yes—
123C2×C167#49211yescharacteristic
124C2×C167#49711yescharacteristic
125D167—82no—
126C167#201311yescharacteristic, complemented, Sylow
127#2013—1671nomaximal, Sylow
128C22×SD16—1671no—
129C22×SD16—1671no—
130C2×C4⋊Q8—1671no—
131C2×#103—1671no—
132C2×#103D16711yescharacteristic, complemented
133#390—1671no—
134#336—1671no—
135C22×Q8—1671no—
136C22×Q8C2×D16711yescharacteristic
137C22×D4—1671no—
138C2×SD16—3342no—
139C2×SD16—3342no—
140C2×SD16—3342no—
141C2×SD16—3342no—
142C2×SD16—3342no—
143C2×SD16—3342no—
144C22×C8—1671no—
145C22×C8C2×D16711yescharacteristic
146C4⋊Q8—3342no—
147C4⋊Q8—3342no—
148C2×C4⋊C4—3342no—
149C2×C4⋊C4C2×D16711yescharacteristic
150C2×C22⋊C4—3342no—
151C2×C42—1671no—
152#103—22no—
153#103—3342no—
154C24—3341no—
155C2×Q8—21no—
156C2×Q8—21no—
157C2×Q8—21no—
158C2×Q8—21no—
159C2×Q8—21no—
160C2×Q8—3341no—
161C2×Q8—3341no—
162C2×Q8—3341no—
163C2×Q8—3341no—
164C2×Q8—3341no—
165C2×Q8—1672no—
166C2×Q8D66812yes—
167C2×D4—3341no—
168C2×D4—3341no—
169C2×D4—3341no—
170C2×D4—3341no—
171C2×D4—3341no—
172C2×D4—1672no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—3341no—
176C22×C4—1672no—
177C22×C4C22×D16711yescharacteristic
178SD16—3348no—
179SD16—3348no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—3341no—
183C2×C8—3341no—
184C2×C8—1672no—
185C2×C8C22⋊D16712yes—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—3348no—
189C22⋊C4—6684no—
190C42—3341no—
191C42—3341no—
192C23—3342no—
193C23—6682no—
194C23—6682no—
195C23—6681no—
196C23—6681no—
197C23D4×D16711yescharacteristic
198Q8—22no—
199Q8—22no—
200Q8—41no—
201Q8—41no—
202Q8—3342no—
203Q8—3342no—
204Q8—6681no—
205Q8—6681no—
206D4—3342no—
207D4—3342no—
208D4—6681no—
209D4—6681no—
210C2×C4—22no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—42no—
214C2×C4—41no—
215C2×C4—3342no—
216C2×C4—3342no—
217C2×C4—3342no—
218C2×C4—6681no—
219C2×C4—6681no—
220C2×C4—1674no—
221C2×C4D4×D16711yescharacteristic
222C2×C4D4×D16711yescharacteristic
223C2×C4D4×D16711yescharacteristic
224C2×C4C2×C22⋊D16711yescharacteristic
225C2×C4C2×D66811yescharacteristic
226C2×C4#1458195011yescharacteristic
227C8—22no—
228C8—3342no—
229C22—13361no—
230C22—13362no—
231C22—13361no—
232C22—6682no—
233C22—6684no—
234C22#1912880211yescharacteristic
235C22#1912880511yescharacteristic
236C22#1912882811yescharacteristic
237C22SD16×D16712yes—
238C22#1912886412yes—
239C4—42no—
240C4—81no—
241C4—3344no—
242C4—6682no—
243C4#1912880011yescharacteristic
244C4#1912880211yescharacteristic
245C4#1912880511yescharacteristic
246C4#1912880611yescharacteristic
247C2—13362no—
248C2#2492317811yescharacteristic
249C2#2492340912yes—
250C2#2492347312yes—
251C2#2492364011yescharacteristic
252C2#2492366211yescharacteristic
subgroupG/H
centerC23D4×D167
commutator, cosocleC22×C4×C167C23
FrattiniC22×C4C22×D167
FittingC2×#103×C167C2
socleC23×C167C2×D4
prime radicalC22×D668C22
hypercenterC2×#103D167
hypocenterC167#2013
LCSG ▷ C22×C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
3. C2×#19128830C2semidirect
4. C2×#19128755C2semidirect
6. #24923023C2semidirect
7. C2×#103×C167C2semidirect
83. C2×Dic167SD16semidirect
126. C167#2013semidirect
132. C2×#103D167semidirect

Maximal subgroups

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

subgroupindex
C22×#145819392
C22×#145819372
C2×#191288302
C2×#191287552
#249231702
#249230232
C2×#103×C1672
#2013167