Finite Group Databasebeta

← #34100611

Subgroups — #34100611

Order 20096.

304 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 93 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#23758040C211yesmaximal, characteristic
2#23757290C211yesmaximal, characteristic
3#23757282C211yesmaximal, characteristic, complemented
4C2×#19097690C211yesmaximal, characteristic, complemented
5#23757044C211yesmaximal, characteristic, complemented
6#23757031C211yesmaximal, characteristic, complemented
7C2×#137×C157C211yesmaximal, characteristic
8#19097778—22no—
9#19097778—22no—
10C2×#14578938—22no—
11C2×#14578938C2211yescharacteristic
12C2×#14578938C2211yescharacteristic
13C2×#14578938C2211yescharacteristic
14#19097730C2211yescharacteristic
15#19097698—22no—
16#19097695—22no—
17#19097690—22no—
18C2×#41×C157C2211yescharacteristic
19C22×Q8×C157C2211yescharacteristic
20#137×C157—22no—
21#137×C157—22no—
22C2×Q16×C157C412yescomplemented
23C2×SD16×C157C412yescomplemented
24C2×M16×C157C2211yescharacteristic
25#14578938—24no—
26#14578938—21no—
27#14578938—21no—
28#14578938—21no—
29#14578938—21no—
30#14578938—21no—
31#14578938D412yes—
32C2×#11929434—24no—
33C2×#11929434—21no—
34C2×#11929434—21no—
35C2×#11929434D412yes—
36C4⋊Dic157—21no—
37C22⋊Dic157—21no—
38C4×Dic157—21no—
39#41×C157—21no—
40#41×C157—21no—
41#41×C157D412yes—
42#41×C157D412yes—
43C2×Q8×C157—21no—
44C2×Q8×C157—21no—
45C2×Q8×C157—21no—
46C2×Q8×C157—21no—
47C2×Q8×C157—21no—
48C2×Q8×C157C2×C412yes—
49C2×Q8×C157C2×C411yescharacteristic
50C2×D4×C157—21no—
51C2×D4×C157C2×C411yescharacteristic
52C22×C4×C157—21no—
53C22×C4×C157—21no—
54C22×C4×C157C2311yescharacteristic
55Q16×C157—24no—
56SD16×C157—24no—
57M16×C157—21no—
58M16×C157—21no—
59C2×C8×C157C2×C412yes—
60C2×Dic157—41no—
61#11929434—22no—
62#11929434—22no—
63#11929434—41no—
64C23×C157—21no—
65C23×C157#4111yescharacteristic
66Q8×C157—21no—
67Q8×C157—22no—
68Q8×C157—22no—
69Q8×C157—41no—
70Q8×C157—41no—
71Q8×C157C22⋊C412yes—
72D4×C157—22no—
73D4×C157—22no—
74D4×C157—21no—
75D4×C157C22⋊C412yes—
76C2×C4×C157—21no—
77C2×C4×C157—22no—
78C2×C4×C157—22no—
79C2×C4×C157—22no—
80C2×C4×C157—41no—
81C2×C4×C157—41no—
82C2×C4×C157C22×C411yescharacteristic
83C2×C4×C157C2×D411yescharacteristic
84C2×C4×C157C2×D411yescharacteristic
85C2×C4×C157C2×D411yescharacteristic
86C2×C4×C157C2×D411yescharacteristic
87C2×C4×C157#4111yescharacteristic
88C8×C157—22no—
89C22×C157—22no—
90C22×C157—41no—
91C22×C157—81no—
92C22×C157C4×D411yescharacteristic
93C22×C157#12111yescharacteristic
94C22×C157#12311yescharacteristic
95C4×C157—22no—
96C4×C157—42no—
97C4×C157C2×C22⋊C411yescharacteristic
98C4×C157C4×D411yescharacteristic
99C4×C157#12011yescharacteristic
100C4×C157#12111yescharacteristic
101Dic157—81no—
102C2×C157—21no—
103C2×C157—21no—
104C2×C157—21no—
105C2×C157—42no—
106C2×C157#38611yescharacteristic
107C157#190211yescharacteristic, complemented, Sylow
108#1902—1571nomaximal, Sylow
109C2×#137D15711yescharacteristic
110#568—1571no—
111#421—1571no—
112#419—1571no—
113C2×#101—1571no—
114#344—1571no—
115#338—1571no—
116C2×#41C2×D15711yescharacteristic
117C22×Q8C2×D15711yescharacteristic
118#137—22no—
119#137—22no—
120C2×Q16Dic15712yescomplemented
121C2×SD16Dic15712yescomplemented
122#131—3142no—
123#131—3142no—
124C2×M16—3142no—
125C2×M16—1571no—
126C2×M16—1571no—
127C2×M16—1571no—
128C2×M16C2×D15711yescharacteristic
129#117—1571no—
130#104—3142no—
131#103—3142no—
132#101—3142no—
133#41—21no—
134#41—21no—
135#41C22⋊D15712yes—
136#41C22⋊D15712yes—
137C2×Q8—21no—
138C2×Q8—21no—
139C2×Q8—21no—
140C2×Q8—21no—
141C2×Q8—21no—
142C2×Q8C2×Dic15712yes—
143C2×Q8C2×Dic15711yescharacteristic
144C2×D4—21no—
145C2×D4C2×Dic15711yescharacteristic
146C22×C4—21no—
147C22×C4—21no—
148C22×C4C22×D15711yescharacteristic
149Q16—24no—
150SD16—24no—
151M16—21no—
152M16—21no—
153M16—3144no—
154M16—3141no—
155M16—3141no—
156M16—3141no—
157M16—3141no—
158M16—3141no—
159M16—1572no—
160C2×C8—3144no—
161C2×C8—3141no—
162C2×C8—3141no—
163C2×C8—1572no—
164C2×C8C2×Dic15712yes—
165C4⋊C4—3141no—
166C22⋊C4—3141no—
167C42—3141no—
168C23—21no—
169C23#1457895311yescharacteristic
170Q8—21no—
171Q8—22no—
172Q8—22no—
173Q8—41no—
174Q8—41no—
175Q8C22⋊Dic15712yes—
176D4—22no—
177D4—22no—
178D4—21no—
179D4C22⋊Dic15712yes—
180C2×C4—21no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—6281no—
187C2×C4C22×Dic15711yescharacteristic
188C2×C4D4×D15711yescharacteristic
189C2×C4D4×D15711yescharacteristic
190C2×C4C2×C22⋊D15711yescharacteristic
191C2×C4C2×C22⋊D15711yescharacteristic
192C2×C4#1457895311yescharacteristic
193C8—22no—
194C8—3142no—
195C8—3142no—
196C8—6281no—
197C22—22no—
198C22—41no—
199C22—81no—
200C22D4×Dic15711yescharacteristic
201C22#1909774311yescharacteristic
202C22#1909775211yescharacteristic
203C4—22no—
204C4—42no—
205C4—12561no—
206C4C2×C22⋊Dic15711yescharacteristic
207C4D4×Dic15711yescharacteristic
208C4#1909773911yescharacteristic
209C4#1909774311yescharacteristic
210C2—21no—
211C2—21no—
212C2—21no—
213C2—42no—
214C2#2375715311yescharacteristic
subgroupG/H
centerC2#23757153
commutatorC2×C4×C157C22×C4
FrattiniC22×C4C22×D157
FittingC2×#137×C157C2
socleC2×C157#386
prime radicalC2×#41×C157C22
cosocleC22×C4×C157C23
hypercenterC2×#137D157
hypocenterC157#1902
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
3. #23757282C2semidirect
4. C2×#19097690C2semidirect
5. #23757044C2semidirect
6. #23757031C2semidirect
22. C2×Q16×C157C4semidirect
23. C2×SD16×C157C4semidirect
107. C157#1902semidirect
120. C2×Q16Dic157semidirect
121. C2×SD16Dic157semidirect

Maximal subgroups

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

subgroupindex
#237580402
#237572902
#237572822
C2×#190976902
#237570442
#237570312
C2×#137×C1572
#1902157