Finite Group Databasebeta

← #34099201

Subgroups — #34099201

Order 20096.

278 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 61 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#23757649C211yesmaximal, characteristic, complemented
2#23757473C211yesmaximal, characteristic, complemented
3#23757468C211yesmaximal, characteristic, complemented
4#23757017C211yesmaximal, characteristic, complemented
5Q8⋊#11929434C211yesmaximal, characteristic, complemented
6D4⋊#11929434C211yesmaximal, characteristic, complemented
7#550×C157C211yesmaximal, characteristic, complemented
8C2×#14578941—21no—
9C2×#14578939—22no—
10C2×#14578939—21no—
11#19097768C2211yescharacteristic
12C4⋊#11929434C2211yescharacteristic
13C4⋊#11929434C2211yescharacteristic
14#19097691—21no—
15#19097691—21no—
16C4×#11929434C2211yescharacteristic
17C2×#41×C157—21no—
18#127×C157—21no—
19#127×C157C2211yescharacteristic
20#121×C157—22no—
21#121×C157—21no—
22C4×Q8×C157C2211yescharacteristic
23C4×D4×C157—21no—
24C4×D4×C157C2211yescharacteristic
25C2×D628—21no—
26C2×D628—21no—
27C2×D628—21no—
28#14578941—42no—
29#14578939—42no—
30#14578939—44no—
31C2×#11929434—21no—
32C2×#11929434—21no—
33C2×#11929434—21no—
34#41×C157—42no—
35#41×C157—42no—
36C2×Q8×C157D411yescharacteristic
37C2×D4×C157—22no—
38C2×D4×C157—22no—
39C2×D4×C157—21no—
40C2×D4×C157—21no—
41C2×D4×C157D411yescharacteristic
42C2×D4×C157D412yes—
43C22×C4×C157—22no—
44C22×C4×C157—21no—
45C4⋊C4×C157—21no—
46C4⋊C4×C157D411yescharacteristic
47C4⋊C4×C157D411yescharacteristic
48C22⋊C4×C157—22no—
49C22⋊C4×C157—21no—
50C42×C157—21no—
51C42×C157C2311yescharacteristic
52C22×D157—41no—
53D628—42no—
54D628—41no—
55D628—41no—
56D628—41no—
57D628—41no—
58#11929434—22no—
59#11929434—41no—
60#11929434—41no—
61C23×C157—21no—
62C23×C157—22no—
63Q8×C157—21no—
64Q8×C157D812yescomplemented
65D4×C157—21no—
66D4×C157—21no—
67D4×C157—42no—
68D4×C157—44no—
69D4×C157—41no—
70D4×C157—41no—
71D4×C157—41no—
72D4×C157—41no—
73D4×C157—82no—
74C2×C4×C157—21no—
75C2×C4×C157—21no—
76C2×C4×C157—21no—
77C2×C4×C157—44no—
78C2×C4×C157—41no—
79C2×C4×C157—41no—
80C2×C4×C157C2×D411yescharacteristic
81C2×C4×C157C2×D411yescharacteristic
82C2×C4×C157C2×D411yescharacteristic
83C22×C157—42no—
84C22×C157—41no—
85C22×C157—41no—
86C22×C157—42no—
87C22×C157—42no—
88C22×C157—41no—
89C22×C157#12011yescharacteristic
90C2×D157—81no—
91C2×D157—81no—
92C2×D157—81no—
93C4×C157—22no—
94C4×C157—21no—
95C4×C157—41no—
96C4×C157—41no—
97C4×C157C2×D811yescharacteristic
98C4×C157#13511yescharacteristic
99C4×C157AGL(1,ℤ/8)11yescharacteristic
100C4×C157AGL(1,ℤ/8)11yescharacteristic
101C2×C157—81no—
102C2×C157—82no—
103C2×C157#44711yescharacteristic
104C2×C157#44911yescharacteristic
105C2×C157#45311yescharacteristic
106D157—161no—
107C157#162111yescharacteristic, complemented, Sylow
108#1621—1571nomaximal, Sylow
109#550D15711yescharacteristic, complemented
110#493—1571no—
111#460—1571no—
112#459—1571no—
113#331—1571no—
114Q8⋊C8—1571no—
115D4⋊C8—1571no—
116C2×#41—21no—
117C2×SD16—3141no—
118C2×D8—3141no—
119C2×D8—3142no—
120#127—21no—
121#127—1571no—
122#127C2×D15711yescharacteristic
123#121—22no—
124#121—21no—
125C4×Q8C2×D15711yescharacteristic
126C4×D4—21no—
127C4×D4C2×D15711yescharacteristic
128C4⋊C8—1571no—
129C4⋊C8—1571no—
130#102—3141no—
131#102—3141no—
132C4×C8—1571no—
133#41—42no—
134#41—42no—
135C2×Q8C22⋊D15711yescharacteristic
136C2×D4—22no—
137C2×D4—22no—
138C2×D4—21no—
139C2×D4—21no—
140C2×D4—3141no—
141C2×D4—3141no—
142C2×D4—3141no—
143C2×D4C22⋊D15711yescharacteristic
144C2×D4C22⋊D15712yes—
145C22×C4—22no—
146C22×C4—21no—
147SD16—6282no—
148D8—6284no—
149D8—6282no—
150C2×C8—3141no—
151C2×C8—3141no—
152C2×C8—3141no—
153C4⋊C4—21no—
154C4⋊C4C22⋊D15711yescharacteristic
155C4⋊C4C22⋊D15711yescharacteristic
156C22⋊C4—22no—
157C22⋊C4—21no—
158C42—21no—
159C42C22×D15711yescharacteristic
160C23—21no—
161C23—22no—
162C23—6281no—
163Q8—21no—
164Q8#1457893912yescomplemented
165D4—21no—
166D4—21no—
167D4—42no—
168D4—44no—
169D4—41no—
170D4—41no—
171D4—41no—
172D4—41no—
173D4—82no—
174D4—6282no—
175D4—6281no—
176D4—6281no—
177D4—6281no—
178D4—6281no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—44no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4C2×C22⋊D15711yescharacteristic
186C2×C4C2×C22⋊D15711yescharacteristic
187C2×C4C2×C22⋊D15711yescharacteristic
188C8—3142no—
189C8—6281no—
190C8—6281no—
191C22—42no—
192C22—41no—
193C22—41no—
194C22—42no—
195C22—42no—
196C22—41no—
197C22—12561no—
198C22—12561no—
199C22—12561no—
200C22#1909774011yescharacteristic
201C4—22no—
202C4—21no—
203C4—41no—
204C4—41no—
205C4C2×#1457893911yescharacteristic
206C4#1909779111yescharacteristic
207C4#1909779611yescharacteristic
208C4#1909779711yescharacteristic
209C2—81no—
210C2—82no—
211C2—25121no—
212C2#2375740711yescharacteristic
213C2#2375741811yescharacteristic
214C2#2375744111yescharacteristic
subgroupG/H
centerC22#19097740
commutator, cosocleC42×C157C23
FrattiniC42C22×D157
Fitting#550×C157C2
socleC22×C157#120
hypercenter#550D157
hypocenterC157#1621
LCSG ▷ C42×C157 ▷ C22×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #550

Complements

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

NH ≅ G/Ntype
1. #23757649C2semidirect
2. #23757473C2semidirect
3. #23757468C2semidirect
4. #23757017C2semidirect
5. Q8⋊#11929434C2semidirect
6. D4⋊#11929434C2semidirect
7. #550×C157C2semidirect
64. Q8×C157D8semidirect
107. C157#1621semidirect
109. #550D157semidirect
164. Q8#14578939semidirect

Maximal subgroups

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

subgroupindex
#237576492
#237574732
#237574682
#237570172
Q8⋊#119294342
D4⋊#119294342
#550×C1572
#1621157