Finite Group Databasebeta

← #34100687

Subgroups — #34100687

Order 20096.

308 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 79 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#23758039C211yesmaximal, characteristic
2#23757849C211yesmaximal, characteristic
3#23757255C211yesmaximal, characteristic
4C2×#19097689C211yesmaximal, characteristic, complemented
5C2×#19097689C211yesmaximal, characteristic, complemented
6#23757043C211yesmaximal, characteristic, complemented
7#344×C157C211yesmaximal, characteristic, complemented
8C2×#14578954C2211yescharacteristic
9C22×Dic314C2211yescharacteristic
10#19097779—22no—
11#19097777—22no—
12C2×#14578938C2211yescharacteristic
13C2×#14578938C2211yescharacteristic
14M16×D157—22no—
15#19097771—21no—
16#19097759—21no—
17#19097755C414yes—
18#19097751—22no—
19#19097689—22no—
20#19097689—22no—
21#19097689—21no—
22#19097687—21no—
23C2×M16×C157C2211yescharacteristic
24C2×M16×C157C2211yescharacteristic
25#117×C157C2211yescharacteristic
26#14578954—21no—
27#14578954—21no—
28#14578954—21no—
29#14578954—21no—
30C2×Dic314—21no—
31C2×Dic314—21no—
32C2×Dic314—21no—
33C2×Dic314—21no—
34C2×Dic314—21no—
35C2×Dic314—21no—
36C2×Dic314D411yescharacteristic
37C2×D628D411yescharacteristic
38C2×C22⋊D157C2×C412yes—
39C22×Dic157C2×C412yes—
40C2×C4×D157D412yes—
41#14578938—21no—
42#14578938—21no—
43#14578938—21no—
44#14578938D412yes—
45#14578937—24no—
46C2×#11929434—21no—
47C2×#11929434—21no—
48C8×D157—24no—
49C4⋊Dic157—24no—
50#14578931—24no—
51C22×C4×C157C2311yescharacteristic
52M16×C157—21no—
53M16×C157—21no—
54M16×C157—21no—
55M16×C157D412yes—
56C2×C8×C157—21no—
57C2×C8×C157—21no—
58C4⋊C4×C157—21no—
59C22⋊C4×C157C2×C412yes—
60C42×C157—21no—
61C22×D157C22⋊C412yes—
62Dic314—21no—
63Dic314—21no—
64Dic314—41no—
65Dic314—41no—
66Dic314—41no—
67Dic314—41no—
68D628—21no—
69D628—21no—
70C22⋊D157—22no—
71C22⋊D157—22no—
72C2×Dic157—22no—
73C2×Dic157—22no—
74C2×Dic157—22no—
75C2×Dic157C22⋊C412yes—
76C4×D157—22no—
77C4×D157—22no—
78#11929434—22no—
79#11929434—41no—
80C23×C157C22×C411yescharacteristic
81C2×C4×C157—22no—
82C2×C4×C157C2×D411yescharacteristic
83C2×C4×C157C2×D411yescharacteristic
84C2×C4×C157C2×D411yescharacteristic
85C2×C4×C157C2×D411yescharacteristic
86C2×C4×C157#4111yescharacteristic
87C2×C4×C157#4111yescharacteristic
88C8×C157—22no—
89C8×C157—41no—
90C22×C157—41no—
91C22×C157C2×C22⋊C411yescharacteristic
92C22×C157C4×D411yescharacteristic
93C22×C157C4×D411yescharacteristic
94C2×D157—22no—
95C2×D157—42no—
96C4×C157—42no—
97C4×C157#12011yescharacteristic
98C4×C157#12111yescharacteristic
99C4×C157#12111yescharacteristic
100C4×C157#12311yescharacteristic
101Dic157—22no—
102Dic157—42no—
103C2×C157—21no—
104C2×C157—21no—
105C2×C157—21no—
106C2×C157#38611yescharacteristic
107D157—42no—
108C157#191511yescharacteristic, complemented, Sylow
109#1915—1571nomaximal, Sylow
110#568—1571no—
111#536—1571no—
112#413—1571no—
113C2×#101—1571no—
114C2×#101—1571no—
115#344—1571no—
116#344D15711yescharacteristic, complemented
117C2×#41—1571no—
118C22×Q8—1571no—
119#131—3142no—
120#131—3142no—
121C2×M16—3142no—
122C2×M16—1571no—
123C2×M16—1571no—
124C2×M16C2×D15711yescharacteristic
125C2×M16C2×D15711yescharacteristic
126C4⋊Q8—3141no—
127#124—3141no—
128#123—1574no—
129C22⋊Q8—3142no—
130#117C2×D15711yescharacteristic
131#101—3141no—
132#101—3142no—
133#101—3142no—
134#100—3141no—
135#41—3141no—
136#41—3141no—
137#41—3141no—
138#41—3141no—
139C2×Q8—3141no—
140C2×Q8—3141no—
141C2×Q8—3141no—
142C2×Q8—3141no—
143C2×Q8—3141no—
144C2×Q8—3141no—
145C2×Q8—1571no—
146C2×D4—1572no—
147C2×D4—1571no—
148C22×C4—1572no—
149C22×C4—1572no—
150C22×C4C22×D15711yescharacteristic
151M16—21no—
152M16—21no—
153M16—21no—
154M16—3144no—
155M16—3141no—
156M16—3141no—
157M16—3141no—
158M16—1572no—
159M16D62812yes—
160C2×C8—21no—
161C2×C8—21no—
162C2×C8—3144no—
163C2×C8—3141no—
164C2×C8—3141no—
165C4⋊C4—21no—
166C4⋊C4—3144no—
167C22⋊C4—3144no—
168C22⋊C4C4×D15712yes—
169C42—21no—
170C23—1572no—
171C23C2×C4×D15711yescharacteristic
172Q8—3141no—
173Q8—3141no—
174Q8—6281no—
175Q8—6281no—
176Q8—6281no—
177Q8—6281no—
178D4—3142no—
179D4—3142no—
180D4—3141no—
181D4—3141no—
182C2×C4—22no—
183C2×C4—3142no—
184C2×C4—3142no—
185C2×C4—3142no—
186C2×C4—3142no—
187C2×C4—3142no—
188C2×C4—1572no—
189C2×C4D4×D15711yescharacteristic
190C2×C4D4×D15711yescharacteristic
191C2×C4D4×D15711yescharacteristic
192C2×C4C2×D62811yescharacteristic
193C2×C4#1457895411yescharacteristic
194C2×C4#1457895511yescharacteristic
195C8—22no—
196C8—41no—
197C8—3142no—
198C8—6281no—
199C22—41no—
200C22—3142no—
201C22—6282no—
202C22C22⋊C4×D15711yescharacteristic
203C22#1909773311yescharacteristic
204C22C4×D62811yescharacteristic
205C4—42no—
206C4—3142no—
207C4—6282no—
208C4#1909774111yescharacteristic
209C4#1909774411yescharacteristic
210C4#1909774611yescharacteristic
211C4#1909775611yescharacteristic
212C2—21no—
213C2—21no—
214C2—21no—
215C2—6282no—
216C2#2375715111yescharacteristic
subgroupG/H
centerC2#23757151
commutatorC23×C157C22×C4
FrattiniC22×C4C22×D157
Fitting#344×C157C2
socleC2×C157#386
prime radicalC2×#14578954C22
cosocleC22×C4×C157C23
hypercenter#344D157
hypocenterC157#1915
LCSG ▷ C23×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C22×C4 ◁ #344

Complements

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

NH ≅ G/Ntype
4. C2×#19097689C2semidirect
5. C2×#19097689C2semidirect
6. #23757043C2semidirect
7. #344×C157C2semidirect
108. C157#1915semidirect
116. #344D157semidirect

Maximal subgroups

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

subgroupindex
#237580392
#237578492
#237572552
C2×#190976892
#237570432
#344×C1572
#1915157