Finite Group Databasebeta

← #37509534

Subgroups — #37509534

Order 22912.

314 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 79 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#26039010C211yesmaximal, characteristic
2#26038916C211yesmaximal, characteristic, complemented
3#26038456C211yesmaximal, characteristic
4C2×#19153031C211yesmaximal, characteristic, complemented
5#26038196C211yesmaximal, characteristic, complemented
6#26038195C211yesmaximal, characteristic, complemented
7#421×C179C211yesmaximal, characteristic
8C22×Dic358C2211yescharacteristic
9C2×#15668243C2211yescharacteristic
10#19153111C412yescomplemented
11#19153100C412yescomplemented
12#19153094—22no—
13#19153092—22no—
14C4×Dic358—22no—
15#19153078—22no—
16#19153072C2211yescharacteristic
17#19153072C2211yescharacteristic
18#19153069—22no—
19#19153040—22no—
20#19153031—22no—
21C2×#41×C179C2211yescharacteristic
22C2×M16×C179C2211yescharacteristic
23#117×C179C2211yescharacteristic
24#104×C179—22no—
25C2×Dic358—21no—
26C2×Dic358—21no—
27C2×Dic358—21no—
28C2×Dic358—21no—
29C2×Dic358C2×C412yes—
30C2×Dic358D412yes—
31C22×Dic179D412yes—
32#15668243—21no—
33#15668243—21no—
34C2×C179⋊C8—21no—
35C4⋊Dic179—21no—
36C4⋊Dic179—21no—
37Dic179⋊C4—24no—
38Dic179⋊C4—24no—
39C22⋊Dic179—24no—
40C22⋊Dic179—21no—
41C22⋊Dic179—21no—
42C4×Dic179—24no—
43C4×Dic179C2×C412yes—
44C4×Dic179D412yes—
45#41×C179—21no—
46#41×C179—21no—
47#41×C179—21no—
48#41×C179—21no—
49C2×Q8×C179C2×C411yescharacteristic
50C2×D4×C179—21no—
51C2×D4×C179C2×C411yescharacteristic
52C22×C4×C179—21no—
53C22×C4×C179C2311yescharacteristic
54M16×C179—21no—
55M16×C179—21no—
56C2×C8×C179—21no—
57C4⋊C4×C179—21no—
58C22⋊C4×C179—21no—
59C42×C179D412yes—
60Dic358—22no—
61Dic358—22no—
62Dic358—22no—
63Dic358—22no—
64C2×Dic179—22no—
65C2×Dic179—22no—
66C2×Dic179—22no—
67C2×Dic179—22no—
68C2×Dic179C22⋊C414yes—
69C179⋊C8—41no—
70C23×C179—21no—
71C23×C179C2×D411yescharacteristic
72Q8×C179—21no—
73Q8×C179—21no—
74D4×C179—21no—
75D4×C179—21no—
76D4×C179—41no—
77D4×C179—41no—
78C2×C4×C179—21no—
79C2×C4×C179—22no—
80C2×C4×C179—41no—
81C2×C4×C179—41no—
82C2×C4×C179C22×C411yescharacteristic
83C2×C4×C179C2×D411yescharacteristic
84C2×C4×C179C2×D411yescharacteristic
85C2×C4×C179C2×D411yescharacteristic
86C2×C4×C179#4111yescharacteristic
87C2×C4×C179#4111yescharacteristic
88C8×C179—41no—
89C22×C179—41no—
90C22×C179—42no—
91C22×C179—41no—
92C22×C179C2×C22⋊C411yescharacteristic
93C22×C179#12011yescharacteristic
94C22×C179#12311yescharacteristic
95C4×C179—42no—
96C4×C179—41no—
97C4×C179C4×D411yescharacteristic
98C4×C179C4×D411yescharacteristic
99C4×C179#12111yescharacteristic
100C4×C179#12111yescharacteristic
101Dic179—24no—
102Dic179—42no—
103Dic179—42no—
104C2×C179—21no—
105C2×C179—21no—
106C2×C179—21no—
107C2×C179—81no—
108C2×C179#38611yescharacteristic
109C179#191211yescharacteristic, complemented, Sylow
110#1912—1791nomaximal, Sylow
111#536—1791no—
112#519—1791no—
113#421—1791no—
114#421D17911yescharacteristic
115C2×#101—1791no—
116H(3,ℤ/4)—1791no—
117H(3,ℤ/4)—1791no—
118C2×#41C2×D17911yescharacteristic
119C22×Q8—1791no—
120C2×M16—1791no—
121C2×M16C2×D17911yescharacteristic
122C4⋊Q8—1792no—
123#124—1792no—
124#123—3582no—
125C22⋊Q8—3582no—
126C4×Q8—3582no—
127C4×Q8—3582no—
128#117—3582no—
129#117—1791no—
130#117—1791no—
131#117C2×D17911yescharacteristic
132#104—22no—
133#104—3582no—
134#101—3582no—
135#41—21no—
136#41—21no—
137#41—21no—
138#41—21no—
139C2×Q8—3581no—
140C2×Q8—3581no—
141C2×Q8—3581no—
142C2×Q8—3581no—
143C2×Q8—1792no—
144C2×Q8—1792no—
145C2×Q8C4×D17911yescharacteristic
146C2×D4—21no—
147C2×D4C4×D17911yescharacteristic
148C22×C4—21no—
149C22×C4—1792no—
150C22×C4C22×D17911yescharacteristic
151M16—21no—
152M16—21no—
153M16—3581no—
154M16—3581no—
155C2×C8—21no—
156C2×C8—3581no—
157C4⋊C4—21no—
158C4⋊C4—3584no—
159C4⋊C4—3584no—
160C4⋊C4—3581no—
161C4⋊C4—3581no—
162C22⋊C4—21no—
163C22⋊C4—3581no—
164C22⋊C4—3581no—
165C22⋊C4—3584no—
166C42—3584no—
167C42—1792no—
168C42—1792no—
169C42C22⋊D17912yes—
170C23—21no—
171C23D4×D17911yescharacteristic
172Q8—21no—
173Q8—21no—
174Q8—3582no—
175Q8—3582no—
176Q8—3582no—
177Q8—3582no—
178D4—21no—
179D4—21no—
180D4—41no—
181D4—41no—
182C2×C4—21no—
183C2×C4—22no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—3582no—
187C2×C4—3582no—
188C2×C4—3582no—
189C2×C4—3582no—
190C2×C4—1794no—
191C2×C4C2×C4×D17911yescharacteristic
192C2×C4D4×D17911yescharacteristic
193C2×C4D4×D17911yescharacteristic
194C2×C4C2×C22⋊D17911yescharacteristic
195C2×C4#1566825811yescharacteristic
196C2×C4#1566825911yescharacteristic
197C8—41no—
198C8—7161no—
199C22—41no—
200C22—42no—
201C22—41no—
202C22C22⋊C4×D17911yescharacteristic
203C22#1915308111yescharacteristic
204C22#1915309511yescharacteristic
205C4—42no—
206C4—41no—
207C4—3584no—
208C4—7162no—
209C4—7162no—
210C4(C22⋊D179)⋊C411yescharacteristic
211C4C4×C22⋊D17911yescharacteristic
212C4#1915308411yescharacteristic
213C4#1915308611yescharacteristic
214C2—21no—
215C2—21no—
216C2—21no—
217C2—81no—
218C2#2603831511yescharacteristic
subgroupG/H
centerC2#26038315
commutatorC2×C4×C179C22×C4
FrattiniC22×C4C22×D179
Fitting#421×C179C2
socleC2×C179#386
prime radicalC2×#41×C179C22
cosocleC22×C4×C179C23
hypercenter#421D179
hypocenterC179#1912
LCSG ▷ C2×C4×C179 ▷ C2×C179 ▷ C179
UCSC1 ◁ C2 ◁ C22×C4 ◁ #421

Complements

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

NH ≅ G/Ntype
2. #26038916C2semidirect
4. C2×#19153031C2semidirect
5. #26038196C2semidirect
6. #26038195C2semidirect
10. #19153111C4semidirect
11. #19153100C4semidirect
109. C179#1912semidirect

Maximal subgroups

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

subgroupindex
#260390102
#260389162
#260384562
C2×#191530312
#260381962
#260381952
#421×C1792
#1912179