Finite Group Databasebeta

← #34111938

Subgroups — #34111938

Order 20096.

444 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 125 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#23758088C212yesmaximal, complemented
2#23758087C211yesmaximal, characteristic, complemented
3#23757651C211yesmaximal, characteristic, complemented
4#23757548C211yesmaximal, characteristic, complemented
5#23757546C212yesmaximal, complemented
6#23757545C211yesmaximal, characteristic, complemented
7#23757456C212yesmaximal, complemented
8#23757454C212yesmaximal, complemented
9#23757390C211yesmaximal, characteristic, complemented
10#23757388C211yesmaximal, characteristic, complemented
11#578×C157C211yesmaximal, characteristic, complemented
12#19097818C2212yes—
13#19097802—22no—
14#19097802C2212yes—
15#19097801—22no—
16#19097801C2212yes—
17#19097801C2212yes—
18#19097800—22no—
19#19097791—22no—
20#19097791C2211yescharacteristic
21#19097790—21no—
22C2×Dic628C2211yescharacteristic
23C2×#14578944—22no—
24C2×#14578944C2212yes—
25Q16×D157—21no—
26#19097778C2212yes—
27#19097777C2211yescharacteristic
28#19097769C2212yes—
29#19097707C2211yescharacteristic
30#19097706C2211yescharacteristic
31#19097698C2211yescharacteristic
32#19097698C2212yes—
33#19097697C2211yescharacteristic
34#19097690C2212yes—
35#19097689C2212yes—
36C8×Dic157C2211yescharacteristic
372−1+4×C157C2212yes—
38#137×C157—22no—
39#137×C157C2212yes—
40#135×C157—21no—
41#135×C157C2211yescharacteristic
42C2×Q16×C157—21no—
43C2×Q16×C157C2211yescharacteristic
44#131×C157C2211yescharacteristic
45#14578954—22no—
46#14578954C2311yescharacteristic
47#14578953—22no—
48#14578953—42no—
49C2×Dic314—22no—
50C2×Dic314C2312yes—
51Q8×D157—22no—
52Dic628—21no—
53Dic628—21no—
54#14578944—22no—
55#14578944—22no—
56#14578944—22no—
57#14578944—22no—
58#14578944—22no—
59#14578944D411yescharacteristic
60#14578942—22no—
61#14578942—22no—
62#14578942D411yescharacteristic
63#14578941—22no—
64#14578941D411yescharacteristic
65#14578939D411yescharacteristic
66#14578938—22no—
67#14578938C2312yes—
68#14578938C2311yescharacteristic
69#14578937—21no—
70C2×#11929434—22no—
71C2×#11929434C2311yescharacteristic
72C8×D157—21no—
73#14578933—42no—
74C4×Dic157C2311yescharacteristic
75#41×C157—22no—
76#41×C157—22no—
77#41×C157—42no—
78#41×C157C2311yescharacteristic
79#41×C157C2312yes—
80C2×Q8×C157—22no—
81C2×Q8×C157—22no—
82C2×Q8×C157C2312yes—
83Q16×C157—22no—
84Q16×C157—21no—
85Q16×C157—21no—
86Q16×C157D411yescharacteristic
87SD16×C157—22no—
88SD16×C157D412yes—
89D8×C157D411yescharacteristic
90M16×C157—21no—
91M16×C157C2311yescharacteristic
92C2×C8×C157—21no—
93C2×C8×C157C2311yescharacteristic
94Dic314—22no—
95Dic314—22no—
96Dic314—42no—
97Dic314C2×D411yescharacteristic
98D628C2×D411yescharacteristic
99C22⋊D157—21no—
100C22⋊D157—42no—
101C2×Dic157—21no—
102C2×Dic157—22no—
103C2×Dic157—42no—
104C4×D157—21no—
105C4×D157—42no—
106#11929434—21no—
107#11929434—22no—
108#11929434C2×D411yescharacteristic
109#11929434C2×D411yescharacteristic
110Q8×C157—22no—
111Q8×C157—22no—
112Q8×C157—42no—
113Q8×C157C2×D411yescharacteristic
114Q8×C157C2×D412yes—
115D4×C157—22no—
116D4×C157—21no—
117D4×C157—42no—
118D4×C157C2×D411yescharacteristic
119D4×C157C2×D412yes—
120C2×C4×C157—22no—
121C2×C4×C157—21no—
122C2×C4×C157—22no—
123C2×C4×C157—42no—
124C2×C4×C157—42no—
125C2×C4×C157C2411yescharacteristic
126C8×C157—21no—
127C8×C157C2×D411yescharacteristic
128C8×C157C2×D411yescharacteristic
129C22×C157—21no—
130C22×C157—22no—
131C22×C157H(5,𝔽2)11yescharacteristic
132C2×D157—21no—
133C4×C157—21no—
134C4×C157—22no—
135C4×C157—42no—
136C4×C157C22×D411yescharacteristic
137C4×C157C22×D411yescharacteristic
138Dic157—22no—
139Dic157—21no—
140Dic157—42no—
141C2×C157—21no—
142C2×C157—41no—
143C2×C157—42no—
144C2×C157D4211yescharacteristic
145D157—41no—
146C157#329311yescharacteristic, complemented, Sylow
147#3293—1571nomaximal, Sylow
148#578—1572no—
149#578—1571no—
150#578D15711yescharacteristic, complemented
151#494—1571no—
152#473—1572no—
153#473—1571no—
154#473—1571no—
155#456—1572no—
156#456—1572no—
157#443—1571no—
158#443—1571no—
1592−1+4—1572no—
1602−1+4C2×D15712yes—
161#137—22no—
162#137—3142no—
163#137—3142no—
164#137—3142no—
165#137—1572no—
166#137—1572no—
167#137—1572no—
168#137C2×D15712yes—
169#135—21no—
170#135—3142no—
171#135—3141no—
172#135—1571no—
173#135C2×D15711yescharacteristic
174C2×Q16—21no—
175C2×Q16—3142no—
176C2×Q16—3141no—
177C2×Q16—1572no—
178C2×Q16—1571no—
179C2×Q16C2×D15711yescharacteristic
180#131—1572no—
181#131—1571no—
182#131C2×D15711yescharacteristic
183C4⋊Q8—1572no—
184#108—1571no—
185#108—1571no—
186#104—1571no—
187#104—1572no—
188#104—1571no—
189#101—1572no—
190#101—1572no—
191C4×C8—1571no—
192#41—22no—
193#41—22no—
194#41—42no—
195#41—3142no—
196#41—3142no—
197#41—6282no—
198#41—1571no—
199#41C22×D15711yescharacteristic
200#41C22×D15712yes—
201C2×Q8—22no—
202C2×Q8—22no—
203C2×Q8—3142no—
204C2×Q8—3142no—
205C2×Q8—1572no—
206C2×Q8C22×D15712yes—
207Q16—22no—
208Q16—21no—
209Q16—21no—
210Q16—3142no—
211Q16—3142no—
212Q16—3142no—
213Q16—3142no—
214Q16—3142no—
215Q16—3141no—
216Q16—3141no—
217Q16—1571no—
218Q16C22⋊D15711yescharacteristic
219SD16—22no—
220SD16—3142no—
221SD16—3142no—
222SD16—3142no—
223SD16—1571no—
224SD16—1571no—
225SD16C22⋊D15712yes—
226D8—1571no—
227D8C22⋊D15711yescharacteristic
228M16—21no—
229M16—3142no—
230M16—3141no—
231M16—1572no—
232M16—1571no—
233M16C22×D15711yescharacteristic
234C2×C8—21no—
235C2×C8—3142no—
236C2×C8—3141no—
237C2×C8—1571no—
238C2×C8C22×D15711yescharacteristic
239C4⋊C4—6282no—
240C42—1571no—
241Q8—22no—
242Q8—22no—
243Q8—42no—
244Q8—3142no—
245Q8—3142no—
246Q8—6282no—
247Q8—1571no—
248Q8D4×D15711yescharacteristic
249Q8C2×C22⋊D15712yes—
250D4—21no—
251D4—22no—
252D4—42no—
253D4—3141no—
254D4—6282no—
255D4—1571no—
256D4D4×D15711yescharacteristic
257D4C2×C22⋊D15712yes—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—3141no—
264C2×C4—3141no—
265C2×C4—3142no—
266C2×C4—6282no—
267C2×C4—6282no—
268C2×C4C23×D15711yescharacteristic
269C8—21no—
270C8—3142no—
271C8—3141no—
272C8—1571no—
273C8—1571no—
274C8C2×C22⋊D15711yescharacteristic
275C8C2×C22⋊D15711yescharacteristic
276C22—21no—
277C22—22no—
278C22—3141no—
279C22#1909781611yescharacteristic
280C4—21no—
281C4—22no—
282C4—42no—
283C4—3142no—
284C4—3141no—
285C4—6282no—
286C4C2×D4×D15711yescharacteristic
287C4C22×C22⋊D15711yescharacteristic
288C2—21no—
289C2—41no—
290C2—42no—
291C2—6281no—
292C2D4×C22⋊D15711yescharacteristic
subgroupG/H
centerC2D4×C22⋊D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#578×C157C2
socleC2×C157D42
hypercenter#578D157
hypocenterC157#3293
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C2×C4 ◁ #578

Complements

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

NH ≅ G/Ntype
1. #23758088C2semidirect
2. #23758087C2semidirect
3. #23757651C2semidirect
4. #23757548C2semidirect
5. #23757546C2semidirect
6. #23757545C2semidirect
7. #23757456C2semidirect
8. #23757454C2semidirect
9. #23757390C2semidirect
10. #23757388C2semidirect
11. #578×C157C2semidirect
146. C157#3293semidirect
150. #578D157semidirect

Maximal subgroups

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

subgroupindex
#237580882
#237580872
#237576512
#237575482
#237575462
#237575452
#237574562
#237574542
#237573902
#237573882
#578×C1572
#3293157