Finite Group Databasebeta

← #34098567

Subgroups — #34098567

Order 20096.

246 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 71 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
1C4×#14578953C211yesmaximal, characteristic
2#23757329C211yesmaximal, characteristic, complemented
3#23757324C211yesmaximal, characteristic, complemented
4#23757005C211yesmaximal, characteristic, complemented
5#23757003C211yesmaximal, characteristic, complemented
6D4⋊#11929434C211yesmaximal, characteristic, complemented
7D4⋊C8×C157C211yesmaximal, characteristic, complemented
8C2×#14578953C411yescharacteristic
9C4×Dic314C2211yescharacteristic
10#19097736C411yescharacteristic
11D4×Dic157C411yescharacteristic
12C4×C22⋊D157—21no—
13#19097731—21no—
14#19097729C411yescharacteristic
15#19097727—21no—
16C2×C4×Dic157—21no—
17C42×D157C2211yescharacteristic
18C4⋊#11929434C2211yescharacteristic
19#19097700—21no—
20#19097683—21no—
21#19097683—21no—
22#19097681—21no—
23C4×#11929434C2211yescharacteristic
24C4×D4×C157C2211yescharacteristic
25C4⋊C8×C157C2211yescharacteristic
26C4×C8×C157C2211yescharacteristic
27#14578953—22no—
28#14578953—22no—
29C2×Dic314C2×C411yescharacteristic
30C2×C22⋊D157—21no—
31C22×Dic157—21no—
32C22×Dic157—21no—
33C2×C4×D157C2×C411yescharacteristic
34C2×C4×D157D411yescharacteristic
35C2×C4×D157D411yescharacteristic
36C2×#11929434—21no—
37C2×#11929434—21no—
38C4⋊Dic157C2×C411yescharacteristic
39#14578933—21no—
40#14578933—21no—
41C22⋊Dic157—21no—
42#14578931—21no—
43C4×Dic157—21no—
44C4×Dic157—21no—
45C4×Dic157—21no—
46C4×Dic157C2×C411yescharacteristic
47C4×Dic157D411yescharacteristic
48C4×Dic157D411yescharacteristic
49C2×D4×C157C2×C411yescharacteristic
50C22×C4×C157—21no—
51C2×C8×C157—21no—
52C2×C8×C157—21no—
53C4⋊C4×C157C2×C411yescharacteristic
54C22⋊C4×C157—21no—
55C42×C157C2311yescharacteristic
56C22×D157C22⋊C411yescharacteristic
57Dic314—21no—
58Dic314M1612yes—
59C22⋊D157—24no—
60C2×Dic157—24no—
61C2×Dic157—21no—
62C2×Dic157—21no—
63C2×Dic157—42no—
64C2×Dic157C22⋊C411yescharacteristic
65C2×Dic157C22⋊C411yescharacteristic
66C2×Dic157C22⋊C411yescharacteristic
67C4×D157—21no—
68C4×D157—21no—
69C4×D157—21no—
70C4×D157—21no—
71C4×D157—41no—
72#11929434—41no—
73#11929434—41no—
74C23×C157—21no—
75D4×C157—21no—
76D4×C157M1612yes—
77C2×C4×C157—21no—
78C2×C4×C157—41no—
79C2×C4×C157—41no—
80C2×C4×C157C22×C411yescharacteristic
81C2×C4×C157C2×D411yescharacteristic
82C2×C4×C157C2×D411yescharacteristic
83C8×C157—41no—
84C8×C157—41no—
85C22×C157—22no—
86C22×C157—41no—
87C22×C157—41no—
88C22×C157C2×C22⋊C411yescharacteristic
89C2×D157—21no—
90C2×D157—21no—
91C2×D157—21no—
92C4×C157—21no—
93C4×C157—41no—
94C4×C157C2×M1611yescharacteristic
95C4×C157C2×M1611yescharacteristic
96C4×C157#13511yescharacteristic
97C4×C157#13511yescharacteristic
98Dic157—21no—
99Dic157—22no—
100Dic157—21no—
101Dic157—21no—
102Dic157—41no—
103C2×C157—42no—
104C2×C157#40711yescharacteristic
105C2×C157#41611yescharacteristic
106C2×C157#42111yescharacteristic
107D157—41no—
108C157#148811yescharacteristic, complemented, Sylow
109#1488—1571nomaximal, Sylow
110C4×#41—1571no—
111#432—1571no—
112#431—1571no—
113Q8⋊C8—1571no—
114Q8⋊C8—1571no—
115D4⋊C8—1571no—
116D4⋊C8D15711yescharacteristic, complemented
117C2×#41—1571no—
118C4×Q8—1571no—
119C4×Q8—1571no—
120C4×D4—3141no—
121C4×D4—3141no—
122C4×D4—1571no—
123C4×D4C2×D15711yescharacteristic
124#117—3141no—
125#117—1571no—
126C2×C42—3141no—
127C2×C42—1571no—
128C4⋊C8—3141no—
129C4⋊C8—1571no—
130C4⋊C8C2×D15711yescharacteristic
131C22⋊C8—3141no—
132C22⋊C8—3141no—
133#97—3141no—
134C4×C8—1571no—
135C4×C8C2×D15711yescharacteristic
136#41—3142no—
137#41—3142no—
138C2×Q8—1571no—
139C2×D4—3141no—
140C2×D4C4×D15711yescharacteristic
141C22×C4—21no—
142C22×C4—3141no—
143C22×C4—3141no—
144C22×C4—1571no—
145C22×C4—1571no—
146C22×C4—1571no—
147C2×C8—21no—
148C2×C8—21no—
149C2×C8—3141no—
150C2×C8—3141no—
151C4⋊C4—3141no—
152C4⋊C4—3141no—
153C4⋊C4—1571no—
154C4⋊C4C4×D15711yescharacteristic
155C22⋊C4—21no—
156C22⋊C4—3141no—
157C22⋊C4—3141no—
158C42—3141no—
159C42—3141no—
160C42—3141no—
161C42—1571no—
162C42—1571no—
163C42—1571no—
164C42C22×D15711yescharacteristic
165C23—21no—
166C23—1571no—
167Q8—3141no—
168Q8—1572no—
169D4—21no—
170D4—3144no—
171D4#1457893712yes—
172C2×C4—21no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—3144no—
176C2×C4—3141no—
177C2×C4—3141no—
178C2×C4—3141no—
179C2×C4—3141no—
180C2×C4—3141no—
181C2×C4—3141no—
182C2×C4—6281no—
183C2×C4—6282no—
184C2×C4—1571no—
185C2×C4—1571no—
186C2×C4—1571no—
187C2×C4C2×C4×D15711yescharacteristic
188C2×C4D4×D15711yescharacteristic
189C2×C4D4×D15711yescharacteristic
190C8—41no—
191C8—41no—
192C8—6281no—
193C8—6281no—
194C22—22no—
195C22—41no—
196C22—41no—
197C22—3141no—
198C22—3141no—
199C22—3141no—
200C22C22⋊C4×D15711yescharacteristic
201C4—21no—
202C4—41no—
203C4—3141no—
204C4—3142no—
205C4—3141no—
206C4—3141no—
207C4—6281no—
208C4M16×D15711yescharacteristic
209C4C2×#1457893711yescharacteristic
210C4#1909779011yescharacteristic
211C4#1909779211yescharacteristic
212C2—42no—
213C2—6281no—
214C2#2375723511yescharacteristic
215C2#2375726611yescharacteristic
216C2#2375728811yescharacteristic
subgroupG/H
centerC2×C4D4×D157
commutatorC2×C4×C157C22×C4
FrattiniC42C22×D157
FittingD4⋊C8×C157C2
socleC22×C157C2×C22⋊C4
prime radicalC2×#14578953C4
cosocleC42×C157C23
hypercenterD4⋊C8D157
hypocenterC157#1488
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2×C4 ◁ C42 ◁ D4⋊C8

Complements

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

NH ≅ G/Ntype
2. #23757329C2semidirect
3. #23757324C2semidirect
4. #23757005C2semidirect
5. #23757003C2semidirect
6. D4⋊#11929434C2semidirect
7. D4⋊C8×C157C2semidirect
108. C157#1488semidirect
116. D4⋊C8D157semidirect

Maximal subgroups

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

subgroupindex
C4×#145789532
#237573292
#237573242
#237570052
#237570032
D4⋊#119294342
D4⋊C8×C1572
#1488157