Finite Group Databasebeta

← #36366216

Subgroups — #36366216

Order 22144.

350 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 117 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#26012215C211yesmaximal, characteristic, complemented
2#26012198C211yesmaximal, characteristic, complemented
3#26011847C212yesmaximal, complemented
4#26011841C211yesmaximal, characteristic, complemented
5#26011796C212yesmaximal, complemented
6#26011782C211yesmaximal, characteristic, complemented
7#26011708C212yesmaximal
8#26011615C211yesmaximal, characteristic, complemented
9SD16×Dic173C212yesmaximal, complemented
10C4×#14583811C211yesmaximal, characteristic
11#461×C173C211yesmaximal, characteristic, complemented
12C2×#14583811C2211yescharacteristic
13#19149679C2211yescharacteristic
14#19149674—22no—
15#19149670C2212yes—
16#19149668C2212yes—
17#19149659—21no—
18#19149658—21no—
19#19149656—22no—
20Q8×Dic173C2212yes—
21C4×Dic346C2211yescharacteristic
22C4×D692C2211yescharacteristic
23D4×Dic173C2212yes—
24#19149637—21no—
25C2×C4⋊Dic173—21no—
26#19149612C2211yescharacteristic
27#19149611C2212yes—
28#19149608C2212yes—
29#19149603C2212yes—
30#19149603C2212yes—
31#19149601C2212yes—
32#19149600C2211yescharacteristic
33#19149599C2211yescharacteristic
34C4×#13063436C2211yescharacteristic
35C8×Dic173C2212yes—
36C2×SD16×C173C2212yes—
37C4⋊Q8×C173C2211yescharacteristic
38C4×D4×C173C2211yescharacteristic
39C4⋊C8×C173C2211yescharacteristic
40#103×C173C2212yes—
41C2×Dic346—21no—
42C2×Dic346D411yescharacteristic
43C2×D692C2311yescharacteristic
44C22×Dic173—22no—
45C2×C4×D173—21no—
46#14583811—22no—
47C2×#13063436C2311yescharacteristic
48C2×#13063436C2311yescharacteristic
49C4⋊Dic173—22no—
50C4⋊Dic173—22no—
51C4⋊Dic173—21no—
52C4⋊Dic173D411yescharacteristic
53C4⋊Dic173C2311yescharacteristic
54C4⋊Dic173C2312yes—
55#14583805—22no—
56#14583805D412yes—
57C22⋊Dic173—22no—
58#14583803—22no—
59#14583803—22no—
60#14583803—21no—
61C4×Dic173—22no—
62C4×Dic173C2312yes—
63C2×Q8×C173C2312yes—
64C2×D4×C173C2311yescharacteristic
65C22×C4×C173—21no—
66SD16×C173—24no—
67C2×C8×C173C2312yes—
68C4⋊C4×C173—21no—
69C4⋊C4×C173—21no—
70C4⋊C4×C173C2311yescharacteristic
71C22⋊C4×C173—21no—
72C42×C173C2311yescharacteristic
73C22×D173—21no—
74Dic346—22no—
75Dic346—81no—
76D692—21no—
77D692—21no—
78C2×Dic173—21no—
79C2×Dic173—22no—
80C2×Dic173—82no—
81C2×Dic173C2×D412yes—
82C2×Dic173C2×D412yes—
83C4×D173—42no—
84#13063436—21no—
85#13063436—21no—
86C23×C173—21no—
87Q8×C173—22no—
88Q8×C173—22no—
89D4×C173—21no—
90D4×C173#4112yes—
91C2×C4×C173—21no—
92C2×C4×C173—22no—
93C2×C4×C173—42no—
94C2×C4×C173C2×D411yescharacteristic
95C2×C4×C173C2×D411yescharacteristic
96C2×C4×C173C2411yescharacteristic
97C8×C173—22no—
98C22×C173—22no—
99C22×C173—41no—
100C22×C173—41no—
101C22×C173C22×D411yescharacteristic
102C2×D173—41no—
103C2×D173—41no—
104C2×D173—41no—
105C4×C173—21no—
106C4×C173—42no—
107C4×C173—41no—
108C4×C173#13512yes—
109C4×C173C2×#4111yescharacteristic
110C4×C1732−1+411yescharacteristic
111Dic173—22no—
112Dic173—22no—
113Dic173—41no—
114Dic173—42no—
115C2×C173—42no—
116C2×C173#54811yescharacteristic
117C2×C173C2×#13511yescharacteristic
118C2×C173#57711yescharacteristic
119D173—81no—
120C173#336011yescharacteristic, complemented, Sylow
121#3360—1731nomaximal, Sylow
122#556—1731no—
123#554—1731no—
124#488—1732no—
125#487—1731no—
126#479—1732no—
127#476—1731no—
128#463—1732no—
129#461D17311yescharacteristic, complemented
130C8×Q8—1731no—
131C4×SD16—1732no—
132C4×D8—1731no—
133C2×SD16C2×D17312yes—
134C2×D8—1731no—
135C4⋊Q8—1731no—
136C4⋊Q8C2×D17311yescharacteristic
137#126—3462no—
138#125—1732no—
139#124—1732no—
140C22⋊Q8—3462no—
141C22⋊Q8—3461no—
142C22⋊Q8—3461no—
143C4×Q8—1732no—
144C4×Q8—1731no—
145C4×D4—1732no—
146C4×D4—1731no—
147C4×D4C2×D17311yescharacteristic
148#117—3461no—
149C2×C4⋊C4—3461no—
150#107—1731no—
151#106—1732no—
152C4⋊C8—1732no—
153C4⋊C8C2×D17311yescharacteristic
154#103—1732no—
155#103—1732no—
156#103C2×D17312yes—
157#102—1731no—
158#102—1732no—
159#102—1731no—
160C4×C8—1732no—
161C4×C8—1731no—
162C2×Q8—3461no—
163C2×Q8—1731no—
164C2×Q8C22×D17312yes—
165C2×D4—1731no—
166C2×D4C22×D17311yescharacteristic
167C22×C4—21no—
168C22×C4—3462no—
169C22×C4—3461no—
170SD16—24no—
171D8—3462no—
172C2×C8—1731no—
173C2×C8—1731no—
174C2×C8C22×D17312yes—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—3462no—
178C4⋊C4—3462no—
179C4⋊C4—3462no—
180C4⋊C4—3461no—
181C4⋊C4—1732no—
182C4⋊C4—1731no—
183C4⋊C4—1731no—
184C4⋊C4—1732no—
185C4⋊C4C22×D17311yescharacteristic
186C22⋊C4—21no—
187C22⋊C4—3462no—
188C22⋊C4—3462no—
189C22⋊C4—3462no—
190C22⋊C4—3461no—
191C42—3462no—
192C42—1732no—
193C42C22×D17311yescharacteristic
194C23—21no—
195C23—3461no—
196Q8—22no—
197Q8—22no—
198Q8—3462no—
199Q8—13841no—
200D4—21no—
201D4—3461no—
202D4—3461no—
203D4#1458382512yes—
204C2×C4—22no—
205C2×C4—21no—
206C2×C4—42no—
207C2×C4—3461no—
208C2×C4—3462no—
209C2×C4—13842no—
210C2×C4—1732no—
211C2×C4—1732no—
212C2×C4—6922no—
213C2×C4D4×D17311yescharacteristic
214C2×C4D4×D17311yescharacteristic
215C2×C4C23×D17311yescharacteristic
216C8—22no—
217C8—3461no—
218C8—3461no—
219C22—22no—
220C22—41no—
221C22—41no—
222C22—6921no—
223C22—6921no—
224C22—6921no—
225C22C2×D4×D17311yescharacteristic
226C4—21no—
227C4—41no—
228C4—42no—
229C4—3462no—
230C4—3462no—
231C4—6921no—
232C4—6922no—
233C4#1914970012yes—
234C4C2×#1458382511yescharacteristic
235C4#1914972611yescharacteristic
236C2—42no—
237C2—13841no—
238C2#2601214711yescharacteristic
239C2C2×#1914970011yescharacteristic
240C2#2601230011yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#461×C173C2
socleC22×C173C22×D4
prime radicalC2×#14583811C22
hypercenter#461D173
hypocenterC173#3360
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. #26012215C2semidirect
2. #26012198C2semidirect
3. #26011847C2semidirect
4. #26011841C2semidirect
5. #26011796C2semidirect
6. #26011782C2semidirect
8. #26011615C2semidirect
9. SD16×Dic173C2semidirect
11. #461×C173C2semidirect
120. C173#3360semidirect
129. #461D173semidirect

Maximal subgroups

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

subgroupindex
#260122152
#260121982
#260118472
#260118412
#260117962
#260117822
#260117082
#260116152
SD16×Dic1732
C4×#145838112
#461×C1732
#3360173