Finite Group Databasebeta

← #37509405

Subgroups — #37509405

Order 22912.

578 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 578 conjugacy classes of subgroups — too many to draw. Only its 83 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×#19153082C211yesmaximal, characteristic
2#26038912C211yesmaximal, characteristic, complemented
3C2×#19153027C211yesmaximal, characteristic, complemented
4#26038295C211yesmaximal, characteristic, complemented
5#26038206C211yesmaximal, characteristic, complemented
6#26038205C211yesmaximal, characteristic, complemented
7#379×C179C211yesmaximal, characteristic, complemented
8C22×C22⋊D179—21no—
9C22×C22⋊D179C2211yescharacteristic
10#19153082—22no—
11#19153082C414yescomplemented
12C4×C22⋊D179—22no—
13(C22⋊D179)⋊C4—22no—
14#19153069—21no—
15C2×C22⋊Dic179—21no—
16C2×C22⋊Dic179—22no—
17C2×C22⋊Dic179C2211yescharacteristic
18C2×C22⋊Dic179C2211yescharacteristic
19C2×C22⋊Dic179C2211yescharacteristic
20C22⋊C4×D179—21no—
21#19153027—22no—
22C25×C179C2211yescharacteristic
23C2×C22⋊C4×C179—22no—
24C2×C22⋊C4×C179C2211yescharacteristic
25C2×C22⋊C4×C179C2211yescharacteristic
26C23×D179D411yescharacteristic
27C2×C22⋊D179—24no—
28C2×C22⋊D179—21no—
29C2×C22⋊D179—21no—
30C2×C22⋊D179—21no—
31C2×C22⋊D179—21no—
32C2×C22⋊D179—44no—
33C2×C22⋊D179C2×C412yes—
34C2×C22⋊D179D412yes—
35C22×Dic179—21no—
36C22×Dic179—21no—
37C22×Dic179D411yescharacteristic
38C22×Dic179D412yescomplemented
39C2×C4×D179—22no—
40Dic179⋊C4—22no—
41C22⋊Dic179—24no—
42C22⋊Dic179—21no—
43C22⋊Dic179—21no—
44C22⋊Dic179—21no—
45C22⋊Dic179—21no—
46C22⋊Dic179—21no—
47C22⋊Dic179—44no—
48C22⋊Dic179C2×C412yescomplemented
49(C2×D179)⋊C4—22no—
50C4×Dic179—22no—
51C24×C179—21no—
52C24×C179—21no—
53C24×C179—21no—
54C24×C179—21no—
55C24×C179—21no—
56C24×C179—21no—
57C24×C179—44no—
58C24×C179C2×C412yescomplemented
59C24×C179C2311yescharacteristic
60C22×C4×C179—21no—
61C22×C4×C179D412yescomplemented
62C22⋊C4×C179—21no—
63C22⋊C4×C179—21no—
64C22⋊C4×C179—21no—
65C22⋊C4×C179—21no—
66C22⋊C4×C179—44no—
67C22×D179—21no—
68C22×D179—21no—
69C22×D179—42no—
70C22×D179C22⋊C412yescomplemented
71C22⋊D179—22no—
72C22⋊D179—22no—
73C22⋊D179—22no—
74C22⋊D179—22no—
75C22⋊D179—48no—
76C2×Dic179—21no—
77C2×Dic179—21no—
78C2×Dic179—22no—
79C2×Dic179—22no—
80C2×Dic179—42no—
81C2×Dic179—42no—
82C2×Dic179—42no—
83C2×Dic179—41no—
84C2×Dic179C22⋊C412yescomplemented
85C4×D179—82no—
86C23×C179—24no—
87C23×C179—24no—
88C23×C179—24no—
89C23×C179—22no—
90C23×C179—22no—
91C23×C179—22no—
92C23×C179—44no—
93C23×C179—42no—
94C23×C179—42no—
95C23×C179—42no—
96C23×C179—42no—
97C23×C179—44no—
98C23×C179—44no—
99C23×C179—44no—
100C23×C179—41no—
101C23×C179—41no—
102C23×C179—41no—
103C23×C179—41no—
104C23×C179C22×C411yescharacteristic
105C23×C179C2×D411yescharacteristic
106C23×C179C2×D411yescharacteristic
107C23×C179C2×D411yescharacteristic
108C23×C179C2×D411yescharacteristic
109C23×C179#4111yescharacteristic
110C23×C179#4111yescharacteristic
111C2×C4×C179—22no—
112C2×C4×C179—42no—
113C2×C4×C179—42no—
114C2×C4×C179—41no—
115C22×C179—22no—
116C22×C179—22no—
117C22×C179—22no—
118C22×C179—24no—
119C22×C179—24no—
120C22×C179—24no—
121C22×C179—41no—
122C22×C179—41no—
123C22×C179—41no—
124C22×C179—41no—
125C22×C179—42no—
126C22×C179—44no—
127C22×C179—44no—
128C22×C179—44no—
129C22×C179—42no—
130C22×C179—42no—
131C22×C179—42no—
132C22×C179—44no—
133C22×C179C2×C22⋊C411yescharacteristic
134C22×C179C4×D411yescharacteristic
135C22×C179C4×D411yescharacteristic
136C22×C179#12011yescharacteristic
137C22×C179#12111yescharacteristic
138C22×C179#12111yescharacteristic
139C22×C179#12311yescharacteristic
140C2×D179—22no—
141C2×D179—42no—
142C2×D179—42no—
143C2×D179—81no—
144C4×C179—42no—
145C4×C179—81no—
146Dic179—22no—
147Dic179—42no—
148Dic179—42no—
149Dic179—81no—
150C2×C179—21no—
151C2×C179—21no—
152C2×C179—21no—
153C2×C179—21no—
154C2×C179—21no—
155C2×C179—21no—
156C2×C179—44no—
157C2×C179#38611yescharacteristic
158C2×C179B(4,𝔽2)12yes—
159D179—42no—
160C179#188911yescharacteristic, complemented, Sylow
161#1889—1791nomaximal, Sylow
162C2×#120—1791no—
163#518—1791no—
164C2×#99—1791no—
165#379—1791no—
166#379D17911yescharacteristic, complemented
167#342—1791no—
168#342—1791no—
169C25C2×D17911yescharacteristic
170C22×D4—3581no—
171C22×D4—1791no—
172#120—3582no—
173#120—1794no—
174C4×D4—3582no—
175C4×D4—3582no—
176#117—3581no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—3581no—
179C2×C22⋊C4—3581no—
180C2×C22⋊C4—3582no—
181C2×C22⋊C4—1791no—
182C2×C22⋊C4—1791no—
183C2×C22⋊C4—1791no—
184C2×C22⋊C4C2×D17911yescharacteristic
185C2×C22⋊C4C2×D17911yescharacteristic
186#99—3582no—
187C24—21no—
188C24—21no—
189C24—21no—
190C24—21no—
191C24—21no—
192C24—21no—
193C24—44no—
194C24—1791no—
195C24C4×D17912yescomplemented
196C24C22×D17911yescharacteristic
197C2×D4—3581no—
198C2×D4—3581no—
199C2×D4—3581no—
200C2×D4—3581no—
201C2×D4—3584no—
202C2×D4—1792no—
203C2×D4—1792no—
204C2×D4—7164no—
205C22×C4—21no—
206C22×C4—3582no—
207C22×C4—3581no—
208C22×C4—3581no—
209C22×C4—1791no—
210C22×C4—1792no—
211C22×C4C22⋊D17912yescomplemented
212C4⋊C4—3582no—
213C22⋊C4—21no—
214C22⋊C4—21no—
215C22⋊C4—21no—
216C22⋊C4—21no—
217C22⋊C4—44no—
218C22⋊C4—3582no—
219C22⋊C4—3581no—
220C22⋊C4—3581no—
221C22⋊C4—3581no—
222C22⋊C4—3584no—
223C22⋊C4—3581no—
224C22⋊C4—3581no—
225C22⋊C4—1792no—
226C22⋊C4—7164no—
227C42—3582no—
228C23—24no—
229C23—24no—
230C23—24no—
231C23—22no—
232C23—22no—
233C23—22no—
234C23—44no—
235C23—42no—
236C23—42no—
237C23—42no—
238C23—42no—
239C23—44no—
240C23—44no—
241C23—44no—
242C23—41no—
243C23—41no—
244C23—41no—
245C23—41no—
246C23—3581no—
247C23—3581no—
248C23—1792no—
249C23—7162no—
250C23C2×C4×D17911yescharacteristic
251C23D4×D17911yescharacteristic
252C23D4×D17911yescharacteristic
253C23D4×D17911yescharacteristic
254C23C2×C22⋊D17911yescharacteristic
255C23#1566825811yescharacteristic
256C23#1566825911yescharacteristic
257D4—3582no—
258D4—3582no—
259D4—3582no—
260D4—3582no—
261D4—7168no—
262C2×C4—22no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—41no—
266C2×C4—3581no—
267C2×C4—3581no—
268C2×C4—3582no—
269C2×C4—3582no—
270C2×C4—14322no—
271C2×C4—1792no—
272C2×C4—7162no—
273C2×C4—7161no—
274C2×C4—7162no—
275C2×C4—7162no—
276C22—22no—
277C22—22no—
278C22—22no—
279C22—24no—
280C22—24no—
281C22—24no—
282C22—41no—
283C22—41no—
284C22—41no—
285C22—41no—
286C22—44no—
287C22—44no—
288C22—42no—
289C22—42no—
290C22—42no—
291C22—44no—
292C22—42no—
293C22—44no—
294C22—3582no—
295C22—14321no—
296C22—7162no—
297C22—7162no—
298C22C22⋊C4×D17911yescharacteristic
299C22(C22⋊D179)⋊C411yescharacteristic
300C22C4×C22⋊D17911yescharacteristic
301C22#1915308111yescharacteristic
302C22#1915308411yescharacteristic
303C22#1915308611yescharacteristic
304C22#1915309511yescharacteristic
305C4—42no—
306C4—81no—
307C4—3582no—
308C4—14321no—
309C4—7162no—
310C4—7162no—
311C2—21no—
312C2—21no—
313C2—21no—
314C2—21no—
315C2—21no—
316C2—21no—
317C2—44no—
318C2—7162no—
319C2#2603831511yescharacteristic
320C2#2603862512yes—
subgroupG/H
centerC22#19153081
commutatorC23×C179C22×C4
FrattiniC24C22×D179
Fitting#379×C179C2
socleC22×C179#120
prime radicalC2×#19153082C2
cosocleC24×C179C23
hypercenter#379D179
hypocenterC179#1889
LCSG ▷ C23×C179 ▷ C2×C179 ▷ C179
UCSC1 ◁ C22 ◁ C24 ◁ #379

Complements

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

NH ≅ G/Ntype
2. #26038912C2semidirect
3. C2×#19153027C2semidirect
4. #26038295C2semidirect
5. #26038206C2semidirect
6. #26038205C2semidirect
7. #379×C179C2semidirect
11. #19153082C4semidirect
38. C22×Dic179D4semidirect
48. C22⋊Dic179C2×C4semidirect
58. C24×C179C2×C4semidirect
61. C22×C4×C179D4semidirect
70. C22×D179C22⋊C4semidirect
84. C2×Dic179C22⋊C4semidirect
160. C179#1889semidirect
166. #379D179semidirect
195. C24C4×D179semidirect
211. C22×C4C22⋊D179semidirect

Maximal subgroups

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

subgroupindex
C2×#191530822
#260389122
C2×#191530272
#260382952
#260382062
#260382052
#379×C1792
#1889179