Finite Group Databasebeta

← #37510036

Subgroups — #37510036

Order 22912.

362 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 75 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×#19153144C211yesmaximal, characteristic
2#26039013C211yesmaximal, characteristic
3C2×#19153041C211yesmaximal, characteristic
4#26038448C211yesmaximal, characteristic, complemented
5C2×#19153032C211yesmaximal, characteristic, complemented
6#26038199C211yesmaximal, characteristic, complemented
7C2×C4⋊Q8×C179C211yesmaximal, characteristic, complemented
8C2×#15668259C2211yescharacteristic
9#19153144—22no—
10#19153144—22no—
11C2×#15668249—21no—
12C2×#15668246—21no—
13C2×#15668243C2211yescharacteristic
14C2×#15668243C2211yescharacteristic
15#19153111—21no—
16#19153102—21no—
17#19153096—22no—
18#19153089—22no—
19#19153072C2211yescharacteristic
20#19153041—22no—
21#19153037—22no—
22#19153032—22no—
23C22×Q8×C179C2211yescharacteristic
24C22×Q8×C179C2211yescharacteristic
25C4⋊Q8×C179—22no—
26C4⋊Q8×C179—22no—
27C2×C4⋊C4×C179—22no—
28C2×C42×C179C2211yescharacteristic
29#15668259—21no—
30#15668259—21no—
31#15668259—21no—
32#15668259—21no—
33C2×Dic358D411yescharacteristic
34C2×D716D411yescharacteristic
35C2×C22⋊D179—21no—
36C2×C4×D179—21no—
37#15668249—24no—
38#15668246—24no—
39#15668243—21no—
40#15668243—21no—
41#15668243—21no—
42#15668243D412yes—
43C2×C179⋊C8—21no—
44C2×C179⋊C8—21no—
45C4⋊Dic179D412yes—
46Dic179⋊C4—24no—
47C22⋊Dic179—21no—
48(C2×D179)⋊C4—24no—
49C4×Dic179—21no—
50C2×Q8×C179—21no—
51C2×Q8×C179—21no—
52C2×Q8×C179—21no—
53C2×Q8×C179—21no—
54C2×Q8×C179—21no—
55C2×Q8×C179—21no—
56C2×Q8×C179—21no—
57C2×Q8×C179—21no—
58C2×Q8×C179—21no—
59C2×Q8×C179—21no—
60C2×Q8×C179—21no—
61C2×Q8×C179D412yes—
62C22×C4×C179—21no—
63C22×C4×C179—21no—
64C22×C4×C179D412yes—
65C22×C4×C179C2311yescharacteristic
66C4⋊C4×C179—28no—
67C42×C179—21no—
68C42×C179D412yes—
69C22×D179—21no—
70Dic358—21no—
71Dic358—21no—
72D716—21no—
73D716—21no—
74C22⋊D179—41no—
75C22⋊D179—41no—
76C2×Dic179—22no—
77C2×Dic179—21no—
78C4×D179—41no—
79C4×D179—41no—
80C179⋊C8—22no—
81C179⋊C8—41no—
82C23×C179C2×D411yescharacteristic
83Q8×C179—22no—
84Q8×C179—22no—
85Q8×C179—41no—
86Q8×C179—41no—
87Q8×C179—41no—
88Q8×C179—41no—
89Q8×C179—41no—
90Q8×C179—41no—
91C2×C4×C179—24no—
92C2×C4×C179—22no—
93C2×C4×C179—22no—
94C2×C4×C179—22no—
95C2×C4×C179—22no—
96C2×C4×C179—22no—
97C2×C4×C179—41no—
98C2×C4×C179—41no—
99C2×C4×C179C2×D411yescharacteristic
100C2×C4×C179C2×D411yescharacteristic
101C2×C4×C179C2×D411yescharacteristic
102C2×C4×C179C2×D411yescharacteristic
103C2×C4×C179C2×D411yescharacteristic
104C2×C4×C179#4111yescharacteristic
105C22×C179—22no—
106C22×C179#12011yescharacteristic
107C22×C179#12011yescharacteristic
108C22×C179#12111yescharacteristic
109C2×D179—41no—
110C2×D179—42no—
111C4×C179—24no—
112C4×C179—42no—
113C4×C179—42no—
114C4×C179#12011yescharacteristic
115C4×C179#12111yescharacteristic
116C4×C179#12111yescharacteristic
117C4×C179#12711yescharacteristic
118Dic179—41no—
119Dic179—42no—
120C2×C179—21no—
121C2×C179—21no—
122C2×C179#39211yescharacteristic
123C2×C179#45612yes—
124D179—81no—
125C179#200511yescharacteristic, complemented, Sylow
126#2005—1791nomaximal, Sylow
127C2×#137—1791no—
128#536—1791no—
129C2×C4⋊Q8D17911yescharacteristic, complemented
130C2×#104—1791no—
131#419—1791no—
132C2×#101—1791no—
133#339—1791no—
134C2×#41—1791no—
135C22×Q8C2×D17911yescharacteristic
136C22×Q8C2×D17911yescharacteristic
137#137—3582no—
138#137—3582no—
139C2×Q16—3581no—
140C2×SD16—3581no—
141C2×M16—1791no—
142C2×M16—1791no—
143C4⋊Q8—22no—
144C4⋊Q8—22no—
145C4⋊Q8—3581no—
146#124—3581no—
147#123—3582no—
148C22⋊Q8—3582no—
149#117—1791no—
150C2×C4⋊C4—22no—
151C2×C42C2×D17911yescharacteristic
152#104—3582no—
153#103—3582no—
154#101—3582no—
155#41—3581no—
156#41—3581no—
157#41—3581no—
158#41—3581no—
159C2×Q8—21no—
160C2×Q8—21no—
161C2×Q8—21no—
162C2×Q8—21no—
163C2×Q8—21no—
164C2×Q8—21no—
165C2×Q8—21no—
166C2×Q8—21no—
167C2×Q8—21no—
168C2×Q8—21no—
169C2×Q8—21no—
170C2×Q8—1791no—
171C2×Q8C22⋊D17912yes—
172C2×D4—3581no—
173C2×D4—1791no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—3581no—
177C22×C4C22⋊D17912yes—
178C22×C4C22×D17911yescharacteristic
179Q16—3584no—
180SD16—3584no—
181M16—3581no—
182M16—3581no—
183M16—3581no—
184M16—1792no—
185C2×C8—3581no—
186C2×C8—3581no—
187C4⋊C4—28no—
188C4⋊C4—3584no—
189C4⋊C4—1792no—
190C22⋊C4—3584no—
191C22⋊C4—3581no—
192C42—21no—
193C42—3581no—
194C42C22⋊D17912yes—
195C23—3581no—
196C23C2×C22⋊D17911yescharacteristic
197Q8—22no—
198Q8—22no—
199Q8—41no—
200Q8—41no—
201Q8—41no—
202Q8—41no—
203Q8—41no—
204Q8—41no—
205Q8—3581no—
206Q8—3581no—
207D4—3581no—
208D4—3581no—
209D4—7161no—
210D4—7161no—
211C2×C4—24no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—3582no—
220C2×C4—3581no—
221C2×C4—7161no—
222C2×C4—7161no—
223C2×C4D4×D17911yescharacteristic
224C2×C4D4×D17911yescharacteristic
225C2×C4D4×D17911yescharacteristic
226C2×C4C2×C22⋊D17911yescharacteristic
227C2×C4C2×C22⋊D17911yescharacteristic
228C2×C4#1566825811yescharacteristic
229C8—3582no—
230C8—7161no—
231C22—22no—
232C22—7161no—
233C22—7162no—
234C22#1915308111yescharacteristic
235C22#1915308211yescharacteristic
236C22#1915308511yescharacteristic
237C4—24no—
238C4—42no—
239C4—42no—
240C4—7161no—
241C4—7162no—
242C4#1915308111yescharacteristic
243C4#1915308411yescharacteristic
244C4#1915308511yescharacteristic
245C4#1915310911yescharacteristic
246C2—21no—
247C2—21no—
248C2—14321no—
249C2#2603834711yescharacteristic
250C2#2603862112yes—
subgroupG/H
centerC22#19153081
commutator, cosocleC22×C4×C179C23
FrattiniC22×C4C22×D179
FittingC2×C4⋊Q8×C179C2
socleC22×C179#120
prime radicalC2×#15668259C22
hypercenterC2×C4⋊Q8D179
hypocenterC179#2005
LCSG ▷ C22×C4×C179 ▷ C2×C179 ▷ C179
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
4. #26038448C2semidirect
5. C2×#19153032C2semidirect
6. #26038199C2semidirect
7. C2×C4⋊Q8×C179C2semidirect
125. C179#2005semidirect
129. C2×C4⋊Q8D179semidirect

Maximal subgroups

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

subgroupindex
C2×#191531442
#260390132
C2×#191530412
#260384482
C2×#191530322
#260381992
C2×C4⋊Q8×C1792
#2005179