Finite Group Databasebeta

← #36354664

Subgroups — #36354664

Order 22144.

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

normal subgroups only

This group has 370 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×Dic692C211yesmaximal, characteristic
2C22×#14583816C211yesmaximal, characteristic
3C2×#19149677C211yesmaximal, characteristic
4C2×#19149602C211yesmaximal, characteristic
5#26011396C211yesmaximal, characteristic
6#26011245C211yesmaximal, characteristic
7C2×#103×C173C211yesmaximal, characteristic
8C22×Dic346C2211yescharacteristic
9C22×Dic346C2211yescharacteristic
10C2×Dic692—22no—
11C2×Dic692—22no—
12C2×Dic692—22no—
13C2×#14583816—22no—
14C2×#14583816—22no—
15C2×#14583816—22no—
16C22×#13063436C2211yescharacteristic
17#19149677—22no—
18#19149677—22no—
19C2×#14583805—22no—
20C2×C4×Dic173C2211yescharacteristic
21#19149602—22no—
22#19149586—22no—
23C22×Q8×C173C2211yescharacteristic
24C22×C8×C173C2211yescharacteristic
25C2×C4⋊C4×C173C2211yescharacteristic
26#103×C173—22no—
27C2×Dic346—21no—
28C2×Dic346—21no—
29C2×Dic346—21no—
30C2×Dic346—21no—
31C2×Dic346—21no—
32C2×Dic346—21no—
33C2×Dic346—21no—
34C2×Dic346—21no—
35C2×Dic346—21no—
36C2×Dic346—21no—
37C2×Dic346D412yes—
38C2×Dic346D412yes—
39C22×Dic173—21no—
40C22×Dic173—21no—
41C22×Dic173D412yes—
42Dic692—28no—
43#14583816—28no—
44C2×#13063436—21no—
45C2×#13063436—21no—
46C2×#13063436D412yes—
47#14583805—28no—
48C4×Dic173—21no—
49C4×Dic173—21no—
50C2×Q8×C173—21no—
51C2×Q8×C173—21no—
52C2×Q8×C173—21no—
53C2×Q8×C173—21no—
54C2×Q8×C173—21no—
55C2×Q8×C173D412yes—
56C22×C4×C173—21no—
57C22×C4×C173—21no—
58C22×C4×C173C2311yescharacteristic
59C2×C8×C173—21no—
60C2×C8×C173—21no—
61C2×C8×C173D412yes—
62C4⋊C4×C173—21no—
63C4⋊C4×C173—21no—
64Dic346—22no—
65Dic346—22no—
66Dic346—22no—
67Dic346—22no—
68Dic346—41no—
69Dic346—41no—
70Dic346—41no—
71Dic346—41no—
72C2×Dic173—22no—
73C2×Dic173—22no—
74C2×Dic173—22no—
75C2×Dic173—22no—
76C2×Dic173—41no—
77C2×Dic173—41no—
78C2×Dic173—41no—
79C2×Dic173—41no—
80C2×Dic173Q1614yescomplemented
81#13063436—22no—
82C23×C173C2×D411yescharacteristic
83Q8×C173—22no—
84Q8×C173—22no—
85Q8×C173—41no—
86Q8×C173—41no—
87C2×C4×C173—22no—
88C2×C4×C173—41no—
89C2×C4×C173—41no—
90C2×C4×C173—41no—
91C2×C4×C173—42no—
92C2×C4×C173C2×D411yescharacteristic
93C2×C4×C173C2×D411yescharacteristic
94C2×C4×C173C2×D411yescharacteristic
95C2×C4×C173C2×D411yescharacteristic
96C2×C4×C173C2×D411yescharacteristic
97C2×C4×C173#4111yescharacteristic
98C8×C173—22no—
99C22×C173#12011yescharacteristic
100C22×C173#12111yescharacteristic
101C22×C173#12711yescharacteristic
102C22×C173C2×Q1612yes—
103C22×C173#13712yes—
104C4×C173—42no—
105C4×C173—81no—
106C4×C173#12011yescharacteristic
107C4×C173#12011yescharacteristic
108C4×C173#12111yescharacteristic
109C4×C173#12111yescharacteristic
110Dic173—24no—
111Dic173—42no—
112Dic173—42no—
113C2×C173#39211yescharacteristic
114C2×C173#45112yes—
115C2×C173#46212yes—
116C2×C173#49411yescharacteristic
117C2×C173#49711yescharacteristic
118C173#201611yescharacteristic, complemented, Sylow
119#2016—1731nomaximal, Sylow
120C22×Q16—1731no—
121C22×Q16—1731no—
122C2×C4⋊Q8—1731no—
123C2×#103—1731no—
124C2×#103D17311yescharacteristic
125#391—1731no—
126#336—1731no—
127C22×Q8—1731no—
128C22×Q8—1731no—
129C22×Q8C2×D17311yescharacteristic
130C2×Q16—3462no—
131C2×Q16—3462no—
132C2×Q16—3462no—
133C2×Q16—3462no—
134C2×Q16—3462no—
135C2×Q16—3462no—
136C22×C8—1731no—
137C22×C8C2×D17311yescharacteristic
138C4⋊Q8—3462no—
139C4⋊Q8—3462no—
140C2×C4⋊C4—3462no—
141C2×C4⋊C4C2×D17311yescharacteristic
142C2×C42—1731no—
143#103—22no—
144#103—3462no—
145#95—3462no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—3461no—
152C2×Q8—3461no—
153C2×Q8—3461no—
154C2×Q8—3461no—
155C2×Q8—3461no—
156C2×Q8—3461no—
157C2×Q8—3461no—
158C2×Q8—3461no—
159C2×Q8—3461no—
160C2×Q8—3461no—
161C2×Q8—1732no—
162C2×Q8—1732no—
163C2×Q8D69212yes—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—3461no—
167C22×C4—3461no—
168C22×C4—1732no—
169C22×C4C22×D17311yescharacteristic
170Q16—3468no—
171Q16—3468no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—3461no—
175C2×C8—3461no—
176C2×C8—1732no—
177C2×C8C22⋊D17312yes—
178C4⋊C4—21no—
179C4⋊C4—21no—
180C4⋊C4—3468no—
181C42—3461no—
182C42—3461no—
183C23D4×D17311yescharacteristic
184Q8—22no—
185Q8—22no—
186Q8—41no—
187Q8—41no—
188Q8—3462no—
189Q8—3462no—
190Q8—3462no—
191Q8—3462no—
192Q8—6921no—
193Q8—6921no—
194Q8—6921no—
195Q8—6921no—
196C2×C4—22no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—42no—
201C2×C4—3462no—
202C2×C4—3462no—
203C2×C4—3462no—
204C2×C4—3462no—
205C2×C4—1734no—
206C2×C4—6921no—
207C2×C4—6921no—
208C2×C4—6921no—
209C2×C4—6921no—
210C2×C4D4×D17311yescharacteristic
211C2×C4D4×D17311yescharacteristic
212C2×C4D4×D17311yescharacteristic
213C2×C4C2×C22⋊D17311yescharacteristic
214C2×C4C2×D69211yescharacteristic
215C2×C4#1458382611yescharacteristic
216C8—22no—
217C8—3462no—
218C22#1914964911yescharacteristic
219C22#1914965211yescharacteristic
220C22#1914967511yescharacteristic
221C22Q16×D17312yes—
222C22#1914970812yes—
223C4—42no—
224C4—81no—
225C4—3464no—
226C4—6922no—
227C4—6922no—
228C4#1914964711yescharacteristic
229C4#1914964911yescharacteristic
230C4#1914965211yescharacteristic
231C4#1914965311yescharacteristic
232C2#2601140011yescharacteristic
233C2#2601164812yes—
234C2#2601170012yes—
235C2#2601186911yescharacteristic
236C2#2601188211yescharacteristic
subgroupG/H
centerC23D4×D173
commutator, cosocleC22×C4×C173C23
FrattiniC22×C4C22×D173
FittingC2×#103×C173C2
socle, prime radicalC23×C173C2×D4
hypercenterC2×#103D173
hypocenterC173#2016
LCSG ▷ C22×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
80. C2×Dic173Q16semidirect
118. C173#2016semidirect

Maximal subgroups

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

subgroupindex
C22×Dic6922
C22×#145838162
C2×#191496772
C2×#191496022
#260113962
#260112452
C2×#103×C1732
#2016173