Finite Group Databasebeta

← #34103424

Subgroups — #34103424

Order 20096.

568 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 147 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×#19097736C211yesmaximal, characteristic, complemented
2C2×D4×Dic157C211yesmaximal, characteristic
3#23757213C211yesmaximal, characteristic, complemented
4#23757212C211yesmaximal, characteristic, complemented
5#23757201C212yesmaximal, complemented
6#23757187C211yesmaximal, characteristic
7#23757185C212yesmaximal, complemented
8C22⋊C4×Dic157C212yesmaximal, complemented
9C22⋊C4×Dic157C212yesmaximal, complemented
10#23757118C211yesmaximal, characteristic, complemented
11#394×C157C211yesmaximal, characteristic
12C22×Dic314C2211yescharacteristic
13C23×Dic157C2212yes—
14#19097736—24no—
15D4×Dic157—24no—
16C2×C4⋊Dic157C2211yescharacteristic
17C2×#14578933C2211yescharacteristic
18C2×#14578933C2211yescharacteristic
19C2×C22⋊Dic157C2212yes—
20C2×C22⋊Dic157C2212yes—
21C2×C22⋊Dic157C2212yes—
22C2×C4×Dic157C2212yes—
23C2×C4×Dic157C2211yescharacteristic
24C2×C4×Dic157C2211yescharacteristic
25C2×C4×Dic157C2211yescharacteristic
26#19097678C2212yes—
27#19097678C2211yescharacteristic
28#19097678C2212yes—
29#19097678C2212yes—
30#19097678C2212yes—
31#19097678C2212yes—
32C22×D4×C157C2211yescharacteristic
33C2×C4⋊C4×C157C2211yescharacteristic
34C2×C22⋊C4×C157C2212yes—
35C2×C22⋊C4×C157C2212yes—
36#95×C157C2211yescharacteristic
37C2×Dic314—22no—
38C2×Dic314—22no—
39C2×Dic314—22no—
40C22×Dic157—24no—
41C22×Dic157—24no—
42C22×Dic157—24no—
43C22×Dic157C2312yes—
44C22×Dic157C2312yes—
45C22×Dic157C2311yescharacteristic
46C22×Dic157C2311yescharacteristic
47C22×Dic157C2311yescharacteristic
48C22×Dic157C2311yescharacteristic
49C4⋊Dic157—22no—
50#14578933—22no—
51#14578933—22no—
52C22⋊Dic157—24no—
53C22⋊Dic157—24no—
54C22⋊Dic157—24no—
55C4×Dic157—22no—
56C4×Dic157—22no—
57C4×Dic157—24no—
58C4×Dic157D414yes—
59C24×C157C2312yes—
60C2×D4×C157—24no—
61C2×D4×C157—22no—
62C22×C4×C157C2312yes—
63C22×C4×C157C2311yescharacteristic
64C22×C4×C157C2311yescharacteristic
65C22×C4×C157C2311yescharacteristic
66C4⋊C4×C157—22no—
67C22⋊C4×C157—24no—
68C22⋊C4×C157—24no—
69Dic314—28no—
70C2×Dic157—21no—
71C2×Dic157—21no—
72C2×Dic157—21no—
73C2×Dic157—21no—
74C2×Dic157—22no—
75C2×Dic157—21no—
76C2×Dic157—22no—
77C2×Dic157—22no—
78C2×Dic157—22no—
79C2×Dic157—22no—
80C2×Dic157—22no—
81C2×Dic157—22no—
82C2×Dic157—48no—
83C2×Dic157C2×D412yes—
84C2×Dic157C2×D412yes—
85C2×Dic157#4112yes—
86C2×Dic157#4114yes—
87C23×C157—22no—
88C23×C157—22no—
89C23×C157—22no—
90C23×C157—22no—
91C23×C157—22no—
92C23×C157—22no—
93C23×C157#4114yes—
94C23×C157C2411yescharacteristic
95D4×C157—28no—
96C2×C4×C157—21no—
97C2×C4×C157—21no—
98C2×C4×C157—22no—
99C2×C4×C157—21no—
100C2×C4×C157—22no—
101C2×C4×C157—22no—
102C2×C4×C157—22no—
103C2×C4×C157—22no—
104C2×C4×C157C2×D412yes—
105C2×C4×C157#4112yes—
106C22×C157—24no—
107C22×C157—24no—
108C22×C157—24no—
109C22×C157—42no—
110C22×C157—42no—
111C22×C157—42no—
112C22×C157—42no—
113C22×C157C22×D411yescharacteristic
114C22×C157C2×#4111yescharacteristic
115C22×C157C2×#4112yes—
116C22×C157C2×#4112yes—
117C22×C157C2×#4111yescharacteristic
118C4×C157—22no—
119C4×C157—22no—
120C4×C157—41no—
121C4×C157—42no—
122Dic157—22no—
123Dic157—22no—
124Dic157—41no—
125Dic157—42no—
126Dic157—42no—
127Dic157#12414yes—
128C2×C157—44no—
129C2×C157#52511yescharacteristic
130C2×C157#52511yescharacteristic
131C2×C157C2×#12411yescharacteristic
132C2×C157#52912yes—
133C2×C157#52912yes—
134C157#239811yescharacteristic, complemented, Sylow
135#2398—1571nomaximal, Sylow
136C2×C4×Q8—1571no—
137C2×C4×D4—1571no—
138#400—1571no—
139#400—1571no—
140#397—1572no—
141#394—1571no—
142#394D15711yescharacteristic
143#393—1572no—
144C4×C22⋊C4—1572no—
145C4×C22⋊C4—1572no—
146#376—1571no—
147C22×Q8—1571no—
148C22×D4C2×D15711yescharacteristic
149C23×C4—1572no—
150C4×Q8—3144no—
151C4×D4—3144no—
152C2×C4⋊C4—1571no—
153C2×C4⋊C4—1571no—
154C2×C4⋊C4—1571no—
155C2×C4⋊C4C2×D15711yescharacteristic
156C2×C22⋊C4—1572no—
157C2×C22⋊C4—1572no—
158C2×C22⋊C4—1572no—
159C2×C22⋊C4C2×D15712yes—
160C2×C22⋊C4C2×D15712yes—
161C2×C42—1571no—
162C2×C42—1571no—
163C2×C42—1572no—
164C2×C42—1571no—
165#95—1572no—
166#95—1571no—
167#95—1572no—
168#95—1572no—
169#95—1572no—
170#95—1572no—
171#95C2×D15711yescharacteristic
172C24C22×D15712yes—
173C2×Q8—3142no—
174C2×Q8—3142no—
175C2×Q8—3142no—
176C2×D4—24no—
177C2×D4—22no—
178C22×C4—3144no—
179C22×C4—3144no—
180C22×C4—3144no—
181C22×C4—1572no—
182C22×C4—1572no—
183C22×C4—1571no—
184C22×C4—1571no—
185C22×C4—1571no—
186C22×C4—1571no—
187C22×C4C22×D15712yes—
188C22×C4C22×D15711yescharacteristic
189C22×C4C22×D15711yescharacteristic
190C22×C4C22×D15711yescharacteristic
191C4⋊C4—22no—
192C4⋊C4—3142no—
193C4⋊C4—3142no—
194C4⋊C4—3142no—
195C22⋊C4—24no—
196C22⋊C4—24no—
197C22⋊C4—3144no—
198C22⋊C4—3144no—
199C22⋊C4—3144no—
200C42—3142no—
201C42—3142no—
202C42—3144no—
203C42—1574no—
204C23—22no—
205C23—22no—
206C23—22no—
207C23—22no—
208C23—22no—
209C23—22no—
210C23#1457895314yes—
211C23C23×D15711yescharacteristic
212Q8—3148no—
213D4—28no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—22no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—3141no—
223C2×C4—3141no—
224C2×C4—3141no—
225C2×C4—3141no—
226C2×C4—3141no—
227C2×C4—3142no—
228C2×C4—3142no—
229C2×C4—3142no—
230C2×C4—3142no—
231C2×C4—3142no—
232C2×C4—3142no—
233C2×C4—3142no—
234C2×C4—6288no—
235C2×C4—1572no—
236C2×C4—1572no—
237C2×C4—1572no—
238C2×C4—1574no—
239C2×C4D4×D15712yes—
240C2×C4#1457895312yes—
241C22—24no—
242C22—24no—
243C22—24no—
244C22—42no—
245C22—42no—
246C22—42no—
247C22—42no—
248C22C2×D4×D15711yescharacteristic
249C22#41×D15711yescharacteristic
250C22#41×D15712yes—
251C22C2×#1457895312yes—
252C22C2×#1457895311yescharacteristic
253C4—22no—
254C4—22no—
255C4—41no—
256C4—42no—
257C4—3142no—
258C4—3142no—
259C4—6281no—
260C4—6282no—
261C4—6282no—
262C4—1574no—
263C2—44no—
264C2#2375777911yescharacteristic
265C2#2375778311yescharacteristic
266C2#124×D15711yescharacteristic
267C2#2375780012yes—
268C2#2375780412yes—
subgroupG/H
center, FrattiniC23C23×D157
commutator, socle, cosocleC23×C157C24
Fitting#394×C157C2
prime radicalC22×D4×C157C22
hypercenter#394D157
hypocenterC157#2398
LCSG ▷ C23×C157 ▷ C157
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×#19097736C2semidirect
3. #23757213C2semidirect
4. #23757212C2semidirect
5. #23757201C2semidirect
7. #23757185C2semidirect
8. C22⋊C4×Dic157C2semidirect
9. C22⋊C4×Dic157C2semidirect
10. #23757118C2semidirect
134. C157#2398semidirect

Maximal subgroups

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

subgroupindex
C2×#190977362
C2×D4×Dic1572
#237572132
#237572122
#237572012
#237571872
#237571852
C22⋊C4×Dic1572
#237571182
#394×C1572
#2398157