Finite Group Databasebeta

← #34100618

Subgroups — #34100618

Order 20096.

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

normal subgroups only

This group has 324 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, complemented
2#23757290C211yesmaximal, characteristic, complemented
3#23757278C211yesmaximal, characteristic, complemented
4C2×#19097688C211yesmaximal, characteristic, complemented
5#23757044C211yesmaximal, characteristic, complemented
6#23757031C211yesmaximal, characteristic, complemented
7C2×AGL(1,ℤ/8)×C157C211yesmaximal, characteristic
8#19097778—22no—
9#19097778—22no—
10C2×#14578938—22no—
11C2×#14578938C2211yescharacteristic
12C2×#14578938C2211yescharacteristic
13C2×#14578938C2211yescharacteristic
14#19097730C2211yescharacteristic
15#19097698—22no—
16#19097692—22no—
17#19097688—22no—
18C2×#41×C157C2211yescharacteristic
19C22×D4×C157C2211yescharacteristic
20AGL(1,ℤ/8)×C157—22no—
21AGL(1,ℤ/8)×C157—22no—
22C2×SD16×C157C412yescomplemented
23C2×D8×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—
39C24×C157—21no—
40#41×C157—21no—
41#41×C157—21no—
42#41×C157D412yes—
43#41×C157D412yes—
44C2×Q8×C157C2×C411yescharacteristic
45C2×D4×C157—21no—
46C2×D4×C157—21no—
47C2×D4×C157—21no—
48C2×D4×C157—21no—
49C2×D4×C157—21no—
50C2×D4×C157—21no—
51C2×D4×C157C2×C412yes—
52C2×D4×C157C2×C411yescharacteristic
53C22×C4×C157—21no—
54C22×C4×C157C2311yescharacteristic
55SD16×C157—24no—
56D8×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—22no—
66C23×C157—41no—
67C23×C157—41no—
68C23×C157—82no—
69C23×C157#4111yescharacteristic
70Q8×C157—21no—
71Q8×C157C22⋊C412yes—
72D4×C157—22no—
73D4×C157—22no—
74D4×C157—22no—
75D4×C157—21no—
76D4×C157—22no—
77D4×C157—41no—
78D4×C157—41no—
79D4×C157C22⋊C412yes—
80C2×C4×C157—21no—
81C2×C4×C157—22no—
82C2×C4×C157—22no—
83C2×C4×C157C22×C411yescharacteristic
84C2×C4×C157C2×D411yescharacteristic
85C2×C4×C157C2×D411yescharacteristic
86C2×C4×C157C2×D411yescharacteristic
87C2×C4×C157C2×D411yescharacteristic
88C2×C4×C157#4111yescharacteristic
89C8×C157—22no—
90C22×C157—22no—
91C22×C157—41no—
92C22×C157—42no—
93C22×C157—81no—
94C22×C157—82no—
95C22×C157—82no—
96C22×C157—82no—
97C22×C157C4×D411yescharacteristic
98C22×C157#12111yescharacteristic
99C22×C157#12311yescharacteristic
100C4×C157—22no—
101C4×C157C2×C22⋊C411yescharacteristic
102C4×C157C4×D411yescharacteristic
103C4×C157#12011yescharacteristic
104C4×C157#12111yescharacteristic
105Dic157—81no—
106C2×C157—21no—
107C2×C157—21no—
108C2×C157—21no—
109C2×C157—42no—
110C2×C157—82no—
111C2×C157#38611yescharacteristic
112C157#190311yescharacteristic, complemented, Sylow
113#1903—1571nomaximal, Sylow
114C2×AGL(1,ℤ/8)D15711yescharacteristic
115#568—1571no—
116#421—1571no—
117#418—1571no—
118C2×#100—1571no—
119#344—1571no—
120#338—1571no—
121C2×#41C2×D15711yescharacteristic
122C22×D4C2×D15711yescharacteristic
123AGL(1,ℤ/8)—22no—
124AGL(1,ℤ/8)—22no—
125C2×SD16Dic15712yescomplemented
126C2×D8Dic15712yescomplemented
127#131—3142no—
128#131—3142no—
129C2×M16—3142no—
130C2×M16—1571no—
131C2×M16—1571no—
132C2×M16—1571no—
133C2×M16C2×D15711yescharacteristic
134#117—1571no—
135#104—3142no—
136#102—3142no—
137#100—3142no—
138C24—21no—
139#41—21no—
140#41—21no—
141#41C22⋊D15712yes—
142#41C22⋊D15712yes—
143C2×Q8C2×Dic15711yescharacteristic
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4—21no—
148C2×D4—21no—
149C2×D4—21no—
150C2×D4C2×Dic15712yes—
151C2×D4C2×Dic15711yescharacteristic
152C22×C4—21no—
153C22×C4C22×D15711yescharacteristic
154SD16—24no—
155D8—24no—
156M16—21no—
157M16—21no—
158M16—3144no—
159M16—3141no—
160M16—3141no—
161M16—3141no—
162M16—3141no—
163M16—3141no—
164M16—1572no—
165C2×C8—3144no—
166C2×C8—3141no—
167C2×C8—3141no—
168C2×C8—1572no—
169C2×C8C2×Dic15712yes—
170C4⋊C4—3141no—
171C22⋊C4—3141no—
172C42—3141no—
173C23—21no—
174C23—22no—
175C23—41no—
176C23—41no—
177C23—82no—
178C23#1457895311yescharacteristic
179Q8—21no—
180Q8C22⋊Dic15712yes—
181D4—22no—
182D4—22no—
183D4—22no—
184D4—21no—
185D4—22no—
186D4—41no—
187D4—41no—
188D4C22⋊Dic15712yes—
189C2×C4—21no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—6281no—
193C2×C4C22×Dic15711yescharacteristic
194C2×C4D4×D15711yescharacteristic
195C2×C4D4×D15711yescharacteristic
196C2×C4C2×C22⋊D15711yescharacteristic
197C2×C4C2×C22⋊D15711yescharacteristic
198C2×C4#1457895311yescharacteristic
199C8—22no—
200C8—3142no—
201C8—3142no—
202C8—6281no—
203C22—22no—
204C22—41no—
205C22—42no—
206C22—81no—
207C22—82no—
208C22—82no—
209C22—82no—
210C22D4×Dic15711yescharacteristic
211C22#1909774311yescharacteristic
212C22#1909775211yescharacteristic
213C4—22no—
214C4—12561no—
215C4C2×C22⋊Dic15711yescharacteristic
216C4D4×Dic15711yescharacteristic
217C4#1909773911yescharacteristic
218C4#1909774311yescharacteristic
219C2—21no—
220C2—21no—
221C2—21no—
222C2—42no—
223C2—82no—
224C2#2375715311yescharacteristic
subgroupG/H
centerC2#23757153
commutatorC2×C4×C157C22×C4
FrattiniC22×C4C22×D157
Fitting, prime radicalC2×AGL(1,ℤ/8)×C157C2
socleC2×C157#386
cosocleC22×C4×C157C23
hypercenterC2×AGL(1,ℤ/8)D157
hypocenterC157#1903
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #23758040C2semidirect
2. #23757290C2semidirect
3. #23757278C2semidirect
4. C2×#19097688C2semidirect
5. #23757044C2semidirect
6. #23757031C2semidirect
22. C2×SD16×C157C4semidirect
23. C2×D8×C157C4semidirect
112. C157#1903semidirect
125. C2×SD16Dic157semidirect
126. C2×D8Dic157semidirect

Maximal subgroups

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

subgroupindex
#237580402
#237572902
#237572782
C2×#190976882
#237570442
#237570312
C2×AGL(1,ℤ/8)×C1572
#1903157