Finite Group Databasebeta

← #37507518

Subgroups — #37507518

Order 22912.

294 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 65 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
1C22⋊Q8×D179C211yesmaximal, characteristic
2#26038401C211yesmaximal, characteristic
3#26038286C211yesmaximal, characteristic
4#26038177C211yesmaximal, characteristic, complemented
5#26038176C211yesmaximal, characteristic, complemented
6#26038175C211yesmaximal, characteristic, complemented
7#328×C179C211yesmaximal, characteristic, complemented
8C2×Q8×D179C411yescharacteristic, complemented
9C22×C4×D179C2211yescharacteristic
10C2×#15668242—21no—
11#19153093C411yescharacteristic, complemented
12#19153092C2211yescharacteristic
13#19153091—21no—
14#19153090—21no—
15#19153089C411yescharacteristic, complemented
16C4⋊C4×D179—21no—
17C4⋊C4×D179C411yescharacteristic, complemented
18C2×(C2×D179)⋊C4—21no—
19C22⋊C4×D179—21no—
20C2×C4×Dic179—21no—
21C22⋊(C179⋊C8)C2211yescharacteristic
22(C2×D179)⋊C8—21no—
23C179⋊#95C2211yescharacteristic
24C22⋊Q8×C179C2211yescharacteristic
25C22⋊C8×C179C2211yescharacteristic
26#95×C179C2211yescharacteristic
27C23×D179D411yescharacteristic
28C2×Dic358—21no—
29C2×Dic358C2×C411yescharacteristic
30Q8×D179—42no—
31C22×Dic179—21no—
32C22×Dic179D411yescharacteristic
33C2×C4×D179—21no—
34C2×C4×D179—21no—
35C2×C4×D179—42no—
36C2×C4×D179C2×C411yescharacteristic
37C2×C4×D179C2×C411yescharacteristic
38C2×C4×D179D411yescharacteristic
39C2×C4×D179D411yescharacteristic
40#15668242—42no—
41C2×C179⋊C8—21no—
42C4⋊Dic179—21no—
43C4⋊Dic179C2×C411yescharacteristic
44Dic179⋊C4—21no—
45Dic179⋊C4—21no—
46Dic179⋊C4—21no—
47C22⋊Dic179—21no—
48(C2×D179)⋊C4—24no—
49(C2×D179)⋊C4—21no—
50(C2×D179)⋊C4—21no—
51C4×Dic179—42no—
52C2×Q8×C179C2×C411yescharacteristic
53C22×C4×C179—21no—
54C22×C4×C179C2311yescharacteristic
55C2×C8×C179—21no—
56C4⋊C4×C179—21no—
57C4⋊C4×C179C2×C411yescharacteristic
58C22⋊C4×C179—21no—
59C22×D179—21no—
60C22×D179—21no—
61C22×D179—22no—
62C22×D179—41no—
63C22×D179C22⋊C411yescharacteristic
64C22×D179C22⋊C411yescharacteristic
65Dic358—42no—
66Dic358—41no—
67C2×Dic179—22no—
68C2×Dic179—22no—
69C2×Dic179—21no—
70C2×Dic179—21no—
71C2×Dic179—41no—
72C2×Dic179—41no—
73C2×Dic179C22⋊C411yescharacteristic
74C2×Dic179C22⋊C411yescharacteristic
75C4×D179—22no—
76C4×D179—22no—
77C4×D179—41no—
78C4×D179—41no—
79C4×D179—42no—
80C4×D179—42no—
81C179⋊C8—41no—
82C23×C179C2×D411yescharacteristic
83Q8×C179—41no—
84C2×C4×C179—22no—
85C2×C4×C179—21no—
86C2×C4×C179—21no—
87C2×C4×C179—41no—
88C2×C4×C179—41no—
89C2×C4×C179—41no—
90C2×C4×C179C22×C411yescharacteristic
91C2×C4×C179C2×D411yescharacteristic
92C8×C179—41no—
93C22×C179—21no—
94C22×C179—21no—
95C22×C179C2×C22⋊C411yescharacteristic
96C22×C179#13711yescharacteristic
97C22×C179#13711yescharacteristic
98C2×D179—21no—
99C2×D179—21no—
100C2×D179—21no—
101C2×D179—21no—
102C2×D179—21no—
103C2×D179—42no—
104C2×D179—42no—
105C2×D179#10412yes—
106C4×C179—21no—
107C4×C179—21no—
108C4×C179—42no—
109C4×C179—41no—
110C4×C179—41no—
111Dic179—21no—
112Dic179—42no—
113Dic179—41no—
114Dic179—41no—
115Dic179#9912yes—
116C2×C179—21no—
117C2×C179—21no—
118C2×C179C2×#9911yescharacteristic
119C2×C179#41911yescharacteristic
120C2×C179C2×#10411yescharacteristic
121D179—22no—
122D179—41no—
123C179#150811yescharacteristic, complemented, Sylow
124#1508—1791nomaximal, Sylow
125C2×C22⋊Q8—1791no—
126#407—1791no—
127C4×C22⋊C4—1791no—
128#328—1791no—
129#328—1791no—
130#328—1791no—
131#328D17911yescharacteristic, complemented
132C22×Q8—1791no—
133C23×C4—1791no—
134C2×M16—3581no—
135C22⋊Q8—3581no—
136C22⋊Q8—3581no—
137C22⋊Q8—1791no—
138C22⋊Q8—1791no—
139C22⋊Q8—1791no—
140C22⋊Q8C2×D17911yescharacteristic
141C2×C4⋊C4—3581no—
142C2×C4⋊C4—1791no—
143C2×C22⋊C4—3581no—
144C2×C22⋊C4—3581no—
145C2×C42—3581no—
146C22⋊C8—3581no—
147C22⋊C8—1791no—
148C22⋊C8C2×D17911yescharacteristic
149#95—1791no—
150#95C2×D17911yescharacteristic
151C24—1791no—
152C2×Q8—3581no—
153C2×Q8—1791no—
154C2×Q8—7162no—
155C2×Q8C4×D17911yescharacteristic
156C22×C4—21no—
157C22×C4—3581no—
158C22×C4—3581no—
159C22×C4—3581no—
160C22×C4—1791no—
161C22×C4—1791no—
162C22×C4—1791no—
163C22×C4—1791no—
164C22×C4—1791no—
165C22×C4—7162no—
166C22×C4C22×D17911yescharacteristic
167M16—7162no—
168C2×C8—21no—
169C2×C8—3581no—
170C4⋊C4—21no—
171C4⋊C4—3581no—
172C4⋊C4—3581no—
173C4⋊C4—3581no—
174C4⋊C4—3581no—
175C4⋊C4—1791no—
176C4⋊C4C4×D17911yescharacteristic
177C22⋊C4—21no—
178C22⋊C4—3584no—
179C22⋊C4—3581no—
180C22⋊C4—3581no—
181C22⋊C4—3581no—
182C42—7162no—
183C23—3582no—
184C23—3581no—
185C23—3581no—
186C23—1791no—
187C23—1791no—
188C23—7161no—
189C23D4×D17911yescharacteristic
190Q8—41no—
191Q8—7161no—
192Q8—7162no—
193C2×C4—22no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—3582no—
200C2×C4—3582no—
201C2×C4—3582no—
202C2×C4—3582no—
203C2×C4—3581no—
204C2×C4—3581no—
205C2×C4—1791no—
206C2×C4—1791no—
207C2×C4—7161no—
208C2×C4—7161no—
209C2×C4—7162no—
210C2×C4—7162no—
211C2×C4—7161no—
212C2×C4—7161no—
213C2×C4C2×C4×D17911yescharacteristic
214C2×C4D4×D17911yescharacteristic
215C8—41no—
216C8—7161no—
217C22—21no—
218C22—21no—
219C22—3581no—
220C22—3581no—
221C22—3581no—
222C22—3581no—
223C22—3581no—
224C22—1792no—
225C22—7162no—
226C22—7162no—
227C22C22⋊C4×D17911yescharacteristic
228C22#1915314211yescharacteristic
229C22#1915314511yescharacteristic
230C4—21no—
231C4—21no—
232C4—42no—
233C4—41no—
234C4—41no—
235C4—3581no—
236C4—1792no—
237C4—7162no—
238C4—7161no—
239C4—7161no—
240C2—21no—
241C2—21no—
242C2—3582no—
243C2—7161no—
244C2#99×D17911yescharacteristic
245C2#2603844611yescharacteristic
246C2#104×D17911yescharacteristic
subgroupG/H
centerC22C22⋊C4×D179
commutatorC2×C4×C179C22×C4
FrattiniC22×C4C22×D179
Fitting#328×C179C2
socleC22×C179C2×C22⋊C4
prime radicalC23×D179D4
cosocleC22×C4×C179C23
hypercenter#328D179
hypocenterC179#1508
LCSG ▷ C2×C4×C179 ▷ C22×C179 ▷ C179
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
4. #26038177C2semidirect
5. #26038176C2semidirect
6. #26038175C2semidirect
7. #328×C179C2semidirect
8. C2×Q8×D179C4semidirect
11. #19153093C4semidirect
15. #19153089C4semidirect
17. C4⋊C4×D179C4semidirect
123. C179#1508semidirect
131. #328D179semidirect

Maximal subgroups

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

subgroupindex
C22⋊Q8×D1792
#260384012
#260382862
#260381772
#260381762
#260381752
#328×C1792
#1508179