Finite Group Databasebeta

← #34110666

Subgroups — #34110666

Order 20096.

340 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 149 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#23757904C211yesmaximal, characteristic
2#23757901C211yesmaximal, characteristic
3#23757681C212yesmaximal, complemented
4#23757680C212yesmaximal, complemented
5#23757674C212yesmaximal, complemented
6#23757670C212yesmaximal, complemented
7#23757315C211yesmaximal, characteristic
8#23757309C211yesmaximal, characteristic
9C2×#19097702C211yesmaximal, characteristic
10#23757231C211yesmaximal, characteristic
11#428×C157C211yesmaximal, characteristic
12C22×#11929434C2211yescharacteristic
13#19097771—21no—
14#19097770—21no—
15#19097770C2212yes—
16#19097770C2212yes—
17#19097763—21no—
18#19097762—21no—
19#19097762C2212yes—
20#19097762C2212yes—
21#19097750—22no—
22#19097748—22no—
23#19097730C2211yescharacteristic
24#19097730C2211yescharacteristic
25C2×C4⋊Dic157C2211yescharacteristic
26#19097705C2212yes—
27#19097704C2211yescharacteristic
28#19097704C2211yescharacteristic
29#19097703C2212yes—
30#19097702C2211yescharacteristic
31#19097702C2212yes—
32#19097702C2211yescharacteristic
33#19097702C2212yes—
34#19097681C2212yes—
35C8×Dic157C2212yes—
36C2×M16×C157C2211yescharacteristic
37#117×C157C2211yescharacteristic
38C2×C4⋊C4×C157C2211yescharacteristic
39#107×C157C2212yes—
40#106×C157C2212yes—
41C2×Dic314—22no—
42C2×Dic314—22no—
43C22×Dic157—21no—
44C2×#11929434Q814yescomplemented
45C2×#11929434C2311yescharacteristic
46C2×#11929434C2311yescharacteristic
47C4⋊Dic157—22no—
48C4⋊Dic157—22no—
49C4⋊Dic157—21no—
50C4⋊Dic157D411yescharacteristic
51C4⋊Dic157D411yescharacteristic
52C4⋊Dic157C2312yes—
53C4⋊Dic157C2311yescharacteristic
54C4⋊Dic157C2311yescharacteristic
55#14578933—22no—
56#14578933—22no—
57#14578933—22no—
58#14578933—22no—
59C22⋊Dic157—21no—
60C22⋊Dic157D412yes—
61C4×Dic157—21no—
62C4×Dic157C2312yes—
63C22×C4×C157—21no—
64C22×C4×C157C2311yescharacteristic
65M16×C157Q814yescomplemented
66C2×C8×C157C2312yes—
67C4⋊C4×C157—21no—
68C4⋊C4×C157C2311yescharacteristic
69C4⋊C4×C157C2312yes—
70C4⋊C4×C157C2311yescharacteristic
71C22⋊C4×C157—21no—
72C42×C157—21no—
73Dic314—82no—
74Dic314—82no—
75C2×Dic157—22no—
76C2×Dic157—21no—
77C2×Dic157—21no—
78C2×Dic157—81no—
79C2×Dic157C2×D412yes—
80C2×Dic157C2×D412yes—
81#11929434C2×Q812yes—
82#11929434C2×Q812yes—
83C23×C157C2×D411yescharacteristic
84C2×C4×C157—21no—
85C2×C4×C157—22no—
86C2×C4×C157—21no—
87C2×C4×C157—81no—
88C2×C4×C157C2×D411yescharacteristic
89C2×C4×C157C2×Q812yes—
90C2×C4×C157C2×Q812yes—
91C2×C4×C157C2411yescharacteristic
92C8×C157C2×Q814yes—
93C22×C157—21no—
94C22×C157—21no—
95C22×C157C4⋊Q812yes—
96C22×C157C22×D411yescharacteristic
97C4×C157—41no—
98C4×C157—42no—
99C4×C157—41no—
100C4×C157C4⋊Q812yes—
101C4×C157C22×Q811yescharacteristic
102C4×C157C22×Q811yescharacteristic
103Dic157—22no—
104Dic157—22no—
105Dic157—41no—
106Dic157—42no—
107Dic157—41no—
108C2×C157—22no—
109C2×C157C2×C4⋊Q811yescharacteristic
110C2×C157#57711yescharacteristic
111C2×C157#57711yescharacteristic
112C157#316311yescharacteristic, complemented, Sylow
113#3163—1571nomaximal, Sylow
114#544—1571no—
115#544—1571no—
116#501—1572no—
117#501—1572no—
118#499—1572no—
119#498—1572no—
120#428—1571no—
121#428D15711yescharacteristic
122#427—1571no—
123C2×#106—1571no—
124#405—1571no—
125C2×M16C2×D15711yescharacteristic
126C22×C8—1571no—
127C4⋊Q8—3141no—
128C4⋊Q8—3141no—
129C4⋊Q8—1572no—
130C4⋊Q8—1572no—
131#125—3141no—
132#125—3141no—
133#125—1572no—
134#125—1572no—
135C22⋊Q8—3142no—
136C22⋊Q8—3142no—
137#117—1571no—
138#117—1571no—
139#117C2×D15711yescharacteristic
140C2×C4⋊C4—1571no—
141C2×C4⋊C4C2×D15711yescharacteristic
142#107—1572no—
143#107—1571no—
144#107—1571no—
145#107C2×D15712yes—
146#106—1572no—
147#106—1572no—
148#106—1572no—
149#106—1571no—
150#106—1571no—
151#106C2×D15712yes—
152#97—1572no—
153C4×C8—1572no—
154C2×Q8—3142no—
155C2×Q8—3142no—
156C22×C4—21no—
157C22×C4—3141no—
158C22×C4C22×D15711yescharacteristic
159M16Dic31414yescomplemented
160C2×C8—1574no—
161C2×C8—1571no—
162C2×C8—1571no—
163C2×C8C22×D15712yes—
164C4⋊C4—21no—
165C4⋊C4—3142no—
166C4⋊C4—3142no—
167C4⋊C4—3142no—
168C4⋊C4—3142no—
169C4⋊C4—3142no—
170C4⋊C4—3142no—
171C4⋊C4—3141no—
172C4⋊C4—1572no—
173C4⋊C4—1571no—
174C4⋊C4—1571no—
175C4⋊C4—1571no—
176C4⋊C4—1571no—
177C4⋊C4C22×D15712yes—
178C4⋊C4C22×D15711yescharacteristic
179C4⋊C4C22×D15711yescharacteristic
180C22⋊C4—21no—
181C22⋊C4—3141no—
182C22⋊C4—1572no—
183C42—21no—
184C42—3141no—
185C42—1572no—
186C23D4×D15711yescharacteristic
187Q8—12562no—
188Q8—12562no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—22no—
192C2×C4—81no—
193C2×C4—3141no—
194C2×C4—3141no—
195C2×C4—3142no—
196C2×C4—1572no—
197C2×C4—1572no—
198C2×C4—12561no—
199C2×C4D4×D15711yescharacteristic
200C2×C4Q8×D15712yes—
201C2×C4C2×Dic31412yes—
202C2×C4C23×D15711yescharacteristic
203C8—1572no—
204C8—1572no—
205C8C2×Dic31414yes—
206C22—21no—
207C22—21no—
208C22#1909777012yes—
209C22C2×D4×D15711yescharacteristic
210C4—41no—
211C4—42no—
212C4—41no—
213C4—3142no—
214C4—3142no—
215C4—6282no—
216C4—6281no—
217C4—6281no—
218C4#1909777012yes—
219C4C2×Q8×D15711yescharacteristic
220C4C22×Dic31411yescharacteristic
221C2—22no—
222C2C2×#1909777011yescharacteristic
223C2#2375808211yescharacteristic
224C2#2375808311yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#428×C157C2
socleC22×C157C22×D4
prime radicalC23×C157C2×D4
hypercenter#428D157
hypocenterC157#3163
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
3. #23757681C2semidirect
4. #23757680C2semidirect
5. #23757674C2semidirect
6. #23757670C2semidirect
44. C2×#11929434Q8semidirect
65. M16×C157Q8semidirect
112. C157#3163semidirect
159. M16Dic314semidirect

Maximal subgroups

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

subgroupindex
#237579042
#237579012
#237576812
#237576802
#237576742
#237576702
#237573152
#237573092
C2×#190977022
#237572312
#428×C1572
#3163157