Finite Group Databasebeta

← #37510652

Subgroups — #37510652

Order 22912.

234 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 55 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#26039011C211yesmaximal, characteristic
2#26038420C211yesmaximal, characteristic
3#26038411C211yesmaximal, characteristic
4#26038230C211yesmaximal, characteristic, complemented
5#26038229C211yesmaximal, characteristic, complemented
6#26038228C211yesmaximal, characteristic, complemented
7#352×C179C211yesmaximal, characteristic, complemented
8C2×#15668258C2211yescharacteristic
9C22×Dic358C411yescharacteristic, complemented
10M16×D179—21no—
11#19153112—21no—
12#19153099—21no—
13#19153097C411yescharacteristic, complemented
14#19153096C411yescharacteristic, complemented
15#19153094C2211yescharacteristic
16#19153091—21no—
17#19153090—21no—
18#19153069—21no—
19#19153069C411yescharacteristic, complemented
20#19153031—21no—
21#19153030C2211yescharacteristic
22#19153028C2211yescharacteristic
23#19153027—21no—
24#123×C179C2211yescharacteristic
25#100×C179C2211yescharacteristic
26#99×C179C2211yescharacteristic
27#15668258—41no—
28#15668258—41no—
29C2×Dic358—21no—
30C2×Dic358—21no—
31C2×Dic358—41no—
32C2×Dic358—41no—
33C2×Dic358D411yescharacteristic
34C2×C22⋊D179C2×C411yescharacteristic
35C2×C22⋊D179D411yescharacteristic
36C22×Dic179C2×C411yescharacteristic
37C22×Dic179C2×C411yescharacteristic
38C22×Dic179D411yescharacteristic
39C2×C4×D179D411yescharacteristic
40#15668243—21no—
41#15668242—41no—
42C8×D179—41no—
43C4⋊Dic179—21no—
44Dic179⋊C4—21no—
45Dic179⋊C4—21no—
46Dic179⋊C4—21no—
47Dic179⋊C4—41no—
48C22⋊Dic179—21no—
49C22⋊Dic179—21no—
50C22⋊Dic179C2×C411yescharacteristic
51(C2×D179)⋊C4—21no—
52(C2×D179)⋊C4—21no—
53C4×Dic179—21no—
54C4×Dic179—41no—
55C2×D4×C179C2311yescharacteristic
56C22×C4×C179C2×C411yescharacteristic
57M16×C179—21no—
58C4⋊C4×C179—21no—
59C22⋊C4×C179—21no—
60C22⋊C4×C179—21no—
61C22⋊C4×C179C2×C411yescharacteristic
62C22×D179C22⋊C411yescharacteristic
63Dic358—41no—
64Dic358—42no—
65Dic358—41no—
66Dic358—41no—
67C22⋊D179—41no—
68C22⋊D179—41no—
69C2×Dic179—21no—
70C2×Dic179—21no—
71C2×Dic179—21no—
72C2×Dic179—21no—
73C2×Dic179—21no—
74C2×Dic179—21no—
75C2×Dic179—41no—
76C2×Dic179—41no—
77C2×Dic179C22⋊C411yescharacteristic
78C2×Dic179C22⋊C411yescharacteristic
79C2×Dic179C22⋊C411yescharacteristic
80C4×D179—21no—
81C4×D179—21no—
82C179⋊C8—41no—
83C23×C179C22×C411yescharacteristic
84C23×C179C2×D411yescharacteristic
85D4×C179—41no—
86C2×C4×C179—21no—
87C2×C4×C179—21no—
88C2×C4×C179—41no—
89C2×C4×C179—41no—
90C2×C4×C179C2×D411yescharacteristic
91C8×C179—41no—
92C22×C179—21no—
93C22×C179—21no—
94C22×C179—41no—
95C22×C179—41no—
96C22×C179C2×C22⋊C411yescharacteristic
97C2×D179—21no—
98C2×D179—41no—
99C4×C179—21no—
100C4×C179—41no—
101C4×C179—41no—
102C4×C179—81no—
103Dic179—21no—
104Dic179—21no—
105Dic179—41no—
106Dic179—41no—
107Dic179—81no—
108Dic179#9911yescharacteristic
109Dic179#9911yescharacteristic
110C2×C179—21no—
111C2×C179—41no—
112C2×C179—41no—
113C2×C179C2×#9911yescharacteristic
114D179—41no—
115C179#212311yescharacteristic, complemented, Sylow
116#2123—1791nomaximal, Sylow
117#536—1791no—
118#413—1791no—
119#410—1791no—
120#352—1791no—
121#352—1791no—
122#352—1791no—
123#352D17911yescharacteristic, complemented
124C2×#41—1791no—
125C22×Q8—1791no—
126C2×M16—3581no—
127C4⋊Q8—3581no—
128#124—3581no—
129#123—1791no—
130#123—1791no—
131#123—1791no—
132#123C2×D17911yescharacteristic
133C22⋊Q8—3581no—
134C22⋊Q8—3581no—
135#117—3581no—
136#117—1791no—
137#101—3581no—
138#100—1791no—
139#100C2×D17911yescharacteristic
140#99—3581no—
141#99—1791no—
142#99C2×D17911yescharacteristic
143#41—7161no—
144#41—7161no—
145C2×Q8—3581no—
146C2×Q8—3581no—
147C2×Q8—1791no—
148C2×Q8—7161no—
149C2×Q8—7161no—
150C2×D4—1791no—
151C2×D4—1791no—
152C2×D4C22×D17911yescharacteristic
153C22×C4—1791no—
154C22×C4—1791no—
155C22×C4—1791no—
156C22×C4—1791no—
157C22×C4C4×D17911yescharacteristic
158M16—21no—
159M16—3581no—
160M16—7161no—
161C2×C8—7161no—
162C4⋊C4—21no—
163C4⋊C4—3581no—
164C4⋊C4—3581no—
165C4⋊C4—3581no—
166C4⋊C4—3581no—
167C4⋊C4—7161no—
168C22⋊C4—21no—
169C22⋊C4—21no—
170C22⋊C4—3581no—
171C22⋊C4—3581no—
172C22⋊C4—3581no—
173C22⋊C4—3581no—
174C22⋊C4—1791no—
175C22⋊C4C4×D17911yescharacteristic
176C42—3581no—
177C42—7161no—
178C23—1791no—
179C23C2×C4×D17911yescharacteristic
180C23D4×D17911yescharacteristic
181Q8—7161no—
182Q8—7162no—
183Q8—7161no—
184Q8—7161no—
185D4—41no—
186D4—7161no—
187D4—7161no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—3581no—
193C2×C4—3581no—
194C2×C4—3581no—
195C2×C4—3581no—
196C2×C4—3581no—
197C2×C4—3581no—
198C2×C4—3581no—
199C2×C4—3581no—
200C2×C4—1791no—
201C2×C4—1791no—
202C2×C4—1791no—
203C2×C4—7161no—
204C2×C4—7161no—
205C2×C4D4×D17911yescharacteristic
206C8—41no—
207C8—7161no—
208C22—21no—
209C22—21no—
210C22—41no—
211C22—41no—
212C22—3581no—
213C22—7161no—
214C22C22⋊C4×D17911yescharacteristic
215C4—21no—
216C4—41no—
217C4—41no—
218C4—81no—
219C4—3581no—
220C4—3581no—
221C4—14321no—
222C4—1791no—
223C4—1791no—
224C4—7161no—
225C4—7161no—
226C2—21no—
227C2—41no—
228C2—41no—
229C2—7161no—
230C2#99×D17911yescharacteristic
subgroupG/H
centerC2#99×D179
commutatorC23×C179C22×C4
FrattiniC2×D4C22×D179
Fitting#352×C179C2
socleC2×C179C2×#99
prime radicalC2×#15668258C22
cosocleC2×D4×C179C23
hypercenter#352D179
hypocenterC179#2123
LCSG ▷ C23×C179 ▷ C22×C179 ▷ C2×C179 ▷ C179
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #352

Complements

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

NH ≅ G/Ntype
4. #26038230C2semidirect
5. #26038229C2semidirect
6. #26038228C2semidirect
7. #352×C179C2semidirect
9. C22×Dic358C4semidirect
13. #19153097C4semidirect
14. #19153096C4semidirect
19. #19153069C4semidirect
115. C179#2123semidirect
123. #352D179semidirect

Maximal subgroups

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

subgroupindex
#260390112
#260384202
#260384112
#260382302
#260382292
#260382282
#352×C1792
#2123179