Finite Group Databasebeta

← #34116176

Subgroups — #34116176

Order 20096.

312 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 68 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
1D157⋊#131C211yesmaximal, characteristic
2#23758264C211yesmaximal, characteristic
3C2×#19097834C211yesmaximal, characteristic, complemented
4D157⋊#101C211yesmaximal, characteristic, complemented
5#23758152C211yesmaximal, characteristic, complemented
6#23758148C211yesmaximal, characteristic, complemented
7#137×D157C211yesmaximal, characteristic
8#19097859—21no—
9#19097858—22no—
10#19097858—21no—
11D157⋊M16—21no—
12D157⋊M16C2211yescharacteristic
13D157⋊M16C2211yescharacteristic
14C2×D157⋊C8—21no—
15C2×D157⋊C8C2211yescharacteristic
16C2×C4⋊#10597845C2211yescharacteristic
17#19097834—21no—
18#19097834—21no—
19#19097834—21no—
20#19097833—21no—
21#19097832—22no—
22#41×D157C2211yescharacteristic
23C2×Q8×D157C2211yescharacteristic
24#19097804C411yescharacteristic, complemented
25#19097803—21no—
26#19097802C411yescharacteristic, complemented
27#19097801C411yescharacteristic, complemented
28#19097800—21no—
29Q16×D157—21no—
30SD16×D157—21no—
31M16×D157C2211yescharacteristic
32#137×C157C411yescharacteristic, complemented
33C22×#10597845—21no—
34#14578964—22no—
35#14578964—21no—
36#14578964—21no—
37#14578963—22no—
38#14578963—21no—
39#14578963D412yes—
40D157⋊C8—22no—
41D157⋊C8—21no—
42D157⋊C8—21no—
43D157⋊C8—21no—
44D157⋊C8D411yescharacteristic
45D157⋊C8D411yescharacteristic
46C2×#11929441—22no—
47C2×#11929441—21no—
48C4⋊#10597845—21no—
49C4⋊#10597845—21no—
50#14578955D411yescharacteristic
51#14578954—21no—
52#14578954C2×C411yescharacteristic
53#14578953—21no—
54#14578953D411yescharacteristic
55C2×Dic314—21no—
56C2×Dic314C2×C411yescharacteristic
57Q8×D157—21no—
58Q8×D157—21no—
59Q8×D157—21no—
60Q8×D157—21no—
61Q8×D157D411yescharacteristic
62D4×D157—21no—
63D4×D157D411yescharacteristic
64C2×C4×D157—21no—
65C2×C4×D157—21no—
66C2×C4×D157C2311yescharacteristic
67Dic628—21no—
68#14578944—21no—
69#14578944—21no—
70#14578943—21no—
71#14578942—21no—
72#14578941—21no—
73#14578938C2×C411yescharacteristic
74#14578937—21no—
75C8×D157—21no—
76#41×C157C2×C411yescharacteristic
77C2×Q8×C157C2×C411yescharacteristic
78Q16×C157—21no—
79SD16×C157—21no—
80M16×C157C2×C411yescharacteristic
81C2×#10597845—41no—
82C2×#10597845—81no—
83#11929441—21no—
84#11929441—22no—
85#11929441—21no—
86#11929441—41no—
87C22×D157—21no—
88C22×D157#4111yescharacteristic
89Dic314—21no—
90Dic314—21no—
91Dic314—21no—
92Dic314—41no—
93Dic314—41no—
94Dic314C22⋊C411yescharacteristic
95D628—21no—
96D628C22⋊C411yescharacteristic
97C22⋊D157—21no—
98C22⋊D157—21no—
99C22⋊D157—21no—
100C2×Dic157—21no—
101C2×Dic157—21no—
102C2×Dic157C2×D411yescharacteristic
103C4×D157—21no—
104C4×D157—21no—
105C4×D157—21no—
106C4×D157—41no—
107C4×D157—41no—
108C4×D157C2×D411yescharacteristic
109C4×D157C2×D411yescharacteristic
110C4×D157C2×D411yescharacteristic
111C4×D157#4111yescharacteristic
112#11929434—21no—
113Q8×C157—21no—
114Q8×C157—21no—
115Q8×C157C22⋊C411yescharacteristic
116D4×C157—21no—
117D4×C157C22⋊C411yescharacteristic
118C2×C4×C157—21no—
119C2×C4×C157—21no—
120C2×C4×C157C22×C411yescharacteristic
121C8×C157—21no—
122C22×C157—21no—
123C22×C157C4×D411yescharacteristic
124C2×D157—21no—
125C2×D157—21no—
126C2×D157—21no—
127C2×D157—41no—
128C2×D157—41no—
129C2×D157#12111yescharacteristic
130C2×D157#12311yescharacteristic
131#10597845—81no—
132C4×C157—21no—
133C4×C157—41no—
134C4×C157C2×C22⋊C411yescharacteristic
135C4×C157C4×D411yescharacteristic
136Dic157—21no—
137Dic157—41no—
138Dic157#12011yescharacteristic
139Dic157#12111yescharacteristic
140C2×C157—21no—
141C2×C157—41no—
142C2×C157#38611yescharacteristic
143D157—21no—
144D157—41no—
145D157#47111yescharacteristic
146D157#47311yescharacteristic
147C157#190811yescharacteristic, complemented, Sylow
148#1908—1571nomaximal, Sylow
149C2×#137—1571no—
150C2×#131—1571no—
151#417—1571no—
152C2×#103—1571no—
153C2×#101—1571no—
154#343—1571no—
155#341—1571no—
156C2×#41—1571no—
157C22×Q8—1571no—
158#137—3141no—
159#137—3141no—
160#137—1571no—
161#137—1571no—
162#137—1571no—
163#137#1059784511yescharacteristic, complemented
164C2×Q16—3141no—
165C2×SD16—3141no—
166#131—3141no—
167#131—3141no—
168#131—3142no—
169C2×M16—3141no—
170C2×M16—1571no—
171C2×M16—1571no—
172C2×M16—1571no—
173C22×C8—3141no—
174C22×C8—1571no—
175C2×C4⋊C4—1571no—
176#103—3141no—
177#103—3141no—
178#103—3141no—
179#102—3141no—
180#101—3142no—
181#41—3141no—
182#41—3141no—
183#41—1571no—
184#41—1571no—
185#41—1571no—
186#41C2×#1059784511yescharacteristic
187C2×Q8—3141no—
188C2×Q8—3141no—
189C2×Q8—3141no—
190C2×Q8—3141no—
191C2×Q8—3141no—
192C2×Q8—1571no—
193C2×Q8—1571no—
194C2×Q8C2×#1059784511yescharacteristic
195C2×D4—3141no—
196C2×D4—1571no—
197C22×C4—3141no—
198C22×C4—3141no—
199C22×C4—3141no—
200C22×C4—1571no—
201Q16—21no—
202Q16—3141no—
203Q16—3141no—
204Q16—3141no—
205SD16—21no—
206SD16—3141no—
207SD16—3141no—
208SD16—3141no—
209M16—3141no—
210M16—3142no—
211M16—3142no—
212M16—3141no—
213M16—3141no—
214M16—3141no—
215M16—1572no—
216M16—1571no—
217M16C2×#1059784511yescharacteristic
218C2×C8—3141no—
219C2×C8—3142no—
220C2×C8—3142no—
221C2×C8—3141no—
222C2×C8—3141no—
223C2×C8—3141no—
224C2×C8—3141no—
225C2×C8—1571no—
226C2×C8—1571no—
227C4⋊C4—3141no—
228C4⋊C4—3141no—
229C23—3141no—
230C23—1571no—
231Q8—21no—
232Q8—21no—
233Q8—3141no—
234Q8—3141no—
235Q8—3141no—
236Q8—6281no—
237Q8—6281no—
238Q8—1571no—
239Q8C22⋊#1059784511yescharacteristic
240D4—21no—
241D4—3141no—
242D4—3141no—
243D4—3141no—
244D4—3141no—
245D4—1571no—
246D4C22⋊#1059784511yescharacteristic
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—3141no—
250C2×C4—3141no—
251C2×C4—3141no—
252C2×C4—3141no—
253C2×C4—3141no—
254C2×C4—6281no—
255C2×C4—6281no—
256C2×C4—6281no—
257C2×C4—1571no—
258C2×C4—1571no—
259C2×C4—1571no—
260C2×C4—1571no—
261C2×C4—1571no—
262C2×C4—12561no—
263C2×C4C22×#1059784511yescharacteristic
264C8—21no—
265C8—3141no—
266C8—3142no—
267C8—3141no—
268C8—3141no—
269C8—6281no—
270C22—21no—
271C22—3141no—
272C22—3141no—
273C22—3141no—
274C22—6281no—
275C22—6281no—
276C22—1571no—
277C22—1571no—
278C22D4×#1059784511yescharacteristic
279C4—21no—
280C4—41no—
281C4—3141no—
282C4—6281no—
283C4—1571no—
284C4—1571no—
285C4—12561no—
286C4C2×C22⋊#1059784511yescharacteristic
287C4D4×#1059784511yescharacteristic
288C2—21no—
289C2—41no—
290C2—3141no—
291C2—6281no—
292C2—1571no—
293C2—1571no—
294C2#2375822111yescharacteristic
subgroupG/H
centerC2#23758221
commutatorC2×C4×C157C22×C4
FrattiniC2×C4C22×#10597845
Fitting#137×C157C4
socleC2×C157#386
prime radical#41×D157C22
cosocleC2×C4×D157C23
hypercenter#137#10597845
hypocenterC157#1908
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C2×C4 ◁ #137

Complements

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

NH ≅ G/Ntype
3. C2×#19097834C2semidirect
4. D157⋊#101C2semidirect
5. #23758152C2semidirect
6. #23758148C2semidirect
24. #19097804C4semidirect
26. #19097802C4semidirect
27. #19097801C4semidirect
32. #137×C157C4semidirect
147. C157#1908semidirect
163. #137#10597845semidirect

Maximal subgroups

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

subgroupindex
D157⋊#1312
#237582642
C2×#190978342
D157⋊#1012
#237581522
#237581482
#137×D1572
#1908157