Finite Group Databasebeta

← #36369434

Subgroups — #36369434

Order 22144.

292 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 84 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
1D173⋊(C4⋊C8)C211yesmaximal, characteristic
2C2×#19149742C212yesmaximal
3#26012435C211yesmaximal, characteristic
4#26012364C212yesmaximal
5C4×Q8×D173C211yesmaximal, characteristic
6C2×D173⋊C8C2212yes—
7C2×C4⋊#10621924C2212yes—
8#19149746—21no—
9C4⋊#13063443—21no—
10#19149742—24no—
11(C2×#10621924)⋊C4—21no—
12(C2×#10621924)⋊C4—21no—
13C2×Q8×D173C2211yescharacteristic
14Q8×Dic173C411yescharacteristic, complemented
15C4×Dic346—21no—
16C4×Dic346C411yescharacteristic, complemented
17#19149644—21no—
18#19149644C411yescharacteristic, complemented
19C4⋊C4×D173—21no—
20C4⋊C4×D173C2211yescharacteristic
21C42×D173—21no—
22C42×D173C2211yescharacteristic
23C4×Q8×C173C411yescharacteristic, complemented
24C22×#10621924—22no—
25D173⋊C8—22no—
26D173⋊C8—22no—
27C2×#13063443—22no—
28C4⋊#10621924—22no—
29C4⋊#10621924—22no—
30C2×Dic346—21no—
31C2×Dic346C2×C411yescharacteristic
32Q8×D173—21no—
33Q8×D173—21no—
34Q8×D173D412yes—
35Q8×D173Q812yes—
36C2×C4×D173—21no—
37C2×C4×D173—21no—
38C2×C4×D173D411yescharacteristic
39C2×C4×D173D411yescharacteristic
40C2×C4×D173C2311yescharacteristic
41C4⋊Dic173—21no—
42C4⋊Dic173C2×C411yescharacteristic
43#14583805—21no—
44#14583805—21no—
45#14583805—21no—
46C4×Dic173—21no—
47C4×Dic173—21no—
48C4×Dic173—21no—
49C4×Dic173C2×C411yescharacteristic
50C4×Dic173D411yescharacteristic
51C4×Dic173D411yescharacteristic
52C2×Q8×C173C2×C411yescharacteristic
53C4⋊C4×C173—21no—
54C4⋊C4×C173C2×C411yescharacteristic
55C42×C173—21no—
56C42×C173C2×C411yescharacteristic
57C2×#10621924—42no—
58C2×#10621924—42no—
59C2×#10621924—42no—
60#13063443—42no—
61C22×D173C2×D411yescharacteristic
62Dic346—21no—
63Dic346—22no—
64Dic346—22no—
65Dic346C4⋊C412yes—
66C2×Dic173—21no—
67C2×Dic173—21no—
68C2×Dic173C22⋊C411yescharacteristic
69C2×Dic173C22⋊C411yescharacteristic
70C2×Dic173C2×D411yescharacteristic
71C4×D173—22no—
72C4×D173—22no—
73C4×D173—22no—
74C4×D173—22no—
75C4×D173—42no—
76C4×D173C2×D411yescharacteristic
77C4×D173C2×Q811yescharacteristic
78C4×D173#4111yescharacteristic
79C4×D173#4111yescharacteristic
80Q8×C173—21no—
81Q8×C173C4⋊C412yes—
82C2×C4×C173—21no—
83C2×C4×C173—21no—
84C2×C4×C173C22⋊C411yescharacteristic
85C2×C4×C173C22⋊C411yescharacteristic
86C2×C4×C173C22×C411yescharacteristic
87C22×C173C2×C22⋊C411yescharacteristic
88C2×D173#12111yescharacteristic
89C2×D173C22⋊Q811yescharacteristic
90C2×D173#13512yes—
91C2×D173#13712yes—
92#10621924—82no—
93C4×C173—22no—
94C4×C173—21no—
95C4×C173—21no—
96C4×C173—41no—
97C4×C173C2×C4⋊C411yescharacteristic
98C4×C173#11711yescharacteristic
99Dic173—22no—
100Dic173—21no—
101Dic173—21no—
102Dic173—41no—
103Dic173#12111yescharacteristic
104Dic173C22⋊Q811yescharacteristic
105C2×C173#38011yescharacteristic
106C2×C173#41611yescharacteristic
107C2×C173#41911yescharacteristic
108D173#46412yes—
109D173#47912yes—
110C173#180511yescharacteristic, complemented, Sylow
111#1805—1731nomaximal, Sylow
112C2×C4×Q8—1731no—
113C2×C4⋊C8—1731no—
114C2×#103—1732no—
115#382—1731no—
116#340—1732no—
117C22×Q8—1731no—
118C22×C8—1732no—
119C4×Q8—3461no—
120C4×Q8—3461no—
121C4×Q8—1731no—
122C4×Q8—1731no—
123C4×Q8—1731no—
124C4×Q8#1062192411yescharacteristic, complemented
125C2×C4⋊C4—3461no—
126C2×C4⋊C4—1732no—
127C2×C4⋊C4—1731no—
128C2×C42—3461no—
129C2×C42—1731no—
130C4⋊C8—3461no—
131C4⋊C8—3461no—
132#103—3464no—
133#95—3461no—
134#95—3461no—
135C2×Q8—3461no—
136C2×Q8—3461no—
137C2×Q8—3461no—
138C2×Q8—1732no—
139C2×Q8—1732no—
140C2×Q8—1731no—
141C2×Q8C2×#1062192411yescharacteristic
142C22×C4—3461no—
143C22×C4—3461no—
144C22×C4—3462no—
145C22×C4—1731no—
146C22×C4—1731no—
147C22×C4—1731no—
148C2×C8—3462no—
149C2×C8—3462no—
150C2×C8—3462no—
151C4⋊C4—21no—
152C4⋊C4—3461no—
153C4⋊C4—3461no—
154C4⋊C4—3461no—
155C4⋊C4—3461no—
156C4⋊C4—3462no—
157C4⋊C4—3462no—
158C4⋊C4—1731no—
159C4⋊C4C2×#1062192411yescharacteristic
160C42—21no—
161C42—3461no—
162C42—3461no—
163C42—3461no—
164C42—1731no—
165C42—1731no—
166C42—1731no—
167C42C2×#1062192411yescharacteristic
168C23—1731no—
169Q8—21no—
170Q8—3462no—
171Q8—3462no—
172Q8—3461no—
173Q8—1732no—
174Q8C4⋊#1062192412yes—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—3462no—
178C2×C4—3462no—
179C2×C4—3462no—
180C2×C4—3462no—
181C2×C4—3461no—
182C2×C4—3461no—
183C2×C4—1731no—
184C2×C4—1731no—
185C2×C4—1731no—
186C2×C4—1731no—
187C2×C4—1731no—
188C2×C4—1731no—
189C2×C4—1731no—
190C2×C4—6922no—
191C2×C4—6922no—
192C2×C4—6922no—
193C2×C4—6922no—
194C2×C4C22⋊#1062192411yescharacteristic
195C2×C4C22⋊#1062192411yescharacteristic
196C2×C4C22×#1062192411yescharacteristic
197C8—6922no—
198C22—1731no—
199C22—1731no—
200C22—1732no—
201C22—1732no—
202C22C2×C22⋊#1062192411yescharacteristic
203C4—22no—
204C4—21no—
205C4—21no—
206C4—41no—
207C4—3462no—
208C4—3461no—
209C4—3461no—
210C4—13842no—
211C4—1731no—
212C4—1731no—
213C4—6921no—
214C4C2×C4⋊#1062192411yescharacteristic
215C4#1914975811yescharacteristic
216C2—1732no—
217C2—1732no—
218C2#2601243311yescharacteristic
219C2#2601248111yescharacteristic
220C2#2601248411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10621924
commutatorC2×C4×C173C22×C4
FrattiniC2×C4C22×#10621924
FittingC4×Q8×C173C4
socleC22×C173C2×C22⋊C4
prime radicalC22×D173C2×D4
cosocleC2×C4×D173C23
hypercenterC4×Q8#10621924
hypocenterC173#1805
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ C4×Q8

Complements

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

NH ≅ G/Ntype
14. Q8×Dic173C4semidirect
16. C4×Dic346C4semidirect
18. #19149644C4semidirect
23. C4×Q8×C173C4semidirect
110. C173#1805semidirect
124. C4×Q8#10621924semidirect

Maximal subgroups

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

subgroupindex
D173⋊(C4⋊C8)2
C2×#191497422
#260124352
#260123642
C4×Q8×D1732
#1805173