Finite Group Databasebeta

← #34099299

Subgroups — #34099299

Order 20096.

228 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 59 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
1Q8×D628C211yesmaximal, characteristic
2#23757623C211yesmaximal, characteristic, complemented
3#23757560C211yesmaximal, characteristic
4#23757020C211yesmaximal, characteristic, complemented
5Q8⋊#11929434C211yesmaximal, characteristic, complemented
6#23757000C211yesmaximal, characteristic
7#477×C157C211yesmaximal, characteristic, complemented
8C2×Q8×D157—21no—
9#19097771—21no—
10#19097771C2211yescharacteristic
11#19097751—22no—
12#19097751—21no—
13C4×D628—21no—
14C4×D628C2211yescharacteristic
15#19097702—21no—
16C4⋊#11929434C2211yescharacteristic
17#19097695—22no—
18#19097691—21no—
19C4×#11929434C2211yescharacteristic
20C4⋊Q8×C157C2211yescharacteristic
21C4×Q8×C157C2211yescharacteristic
22#107×C157—21no—
23C4⋊C8×C157C2211yescharacteristic
24#103×C157—21no—
25C2×Dic314—22no—
26C2×Dic314—21no—
27Q8×D157—42no—
28Q8×D157—42no—
29C2×D628D411yescharacteristic
30C2×C4×D157—22no—
31C2×C4×D157—21no—
32C2×#11929434—21no—
33C2×#11929434—21no—
34C4⋊Dic157—22no—
35C4⋊Dic157—21no—
36C4⋊Dic157D412yes—
37C4⋊Dic157D411yescharacteristic
38#14578931—22no—
39#14578931—21no—
40C2×Q8×C157—21no—
41C2×Q8×C157D411yescharacteristic
42C2×C8×C157—21no—
43C4⋊C4×C157—21no—
44C4⋊C4×C157—21no—
45C4⋊C4×C157—21no—
46C4⋊C4×C157D411yescharacteristic
47C42×C157—21no—
48C42×C157C2311yescharacteristic
49C22×D157—21no—
50Dic314—44no—
51Dic314—41no—
52Dic314—41no—
53D628—21no—
54D628Q1612yes—
55C2×Dic157—22no—
56C2×Dic157—21no—
57C4×D157—44no—
58C4×D157—41no—
59C4×D157—41no—
60#11929434—41no—
61#11929434—41no—
62Q8×C157—21no—
63Q8×C157—42no—
64Q8×C157SD1612yes—
65C2×C4×C157—21no—
66C2×C4×C157—21no—
67C2×C4×C157—41no—
68C2×C4×C157C2×D411yescharacteristic
69C2×C4×C157C2×D411yescharacteristic
70C2×C4×C157C2×D411yescharacteristic
71C8×C157—41no—
72C22×C157#12011yescharacteristic
73C2×D157—22no—
74C2×D157—41no—
75C2×D157—41no—
76C4×C157—22no—
77C4×C157—21no—
78C4×C157—41no—
79C4×C157—81no—
80C4×C157C2×SD1611yescharacteristic
81C4×C157C2×Q1611yescharacteristic
82C4×C157AGL(1,ℤ/8)11yescharacteristic
83C4×C157#13711yescharacteristic
84Dic157—42no—
85Dic157—41no—
86C2×C157#45011yescharacteristic
87C2×C157#45111yescharacteristic
88C2×C157#45411yescharacteristic
89D157—42no—
90C157#163711yescharacteristic, complemented, Sylow
91#1637—1571nomaximal, Sylow
92D4×Q8—1571no—
93#487—1571no—
94#477D15711yescharacteristic, complemented
95#476—1571no—
96#332—1571no—
97Q8⋊C8—1571no—
98D4⋊C8—1571no—
99C22×Q8—3141no—
100C4⋊Q8—3141no—
101C4⋊Q8—1571no—
102C4⋊Q8C2×D15711yescharacteristic
103C22⋊Q8—3142no—
104C22⋊Q8—3141no—
105C4×Q8C2×D15711yescharacteristic
106C4×D4—3141no—
107C4×D4—1571no—
108#107—21no—
109#106—3141no—
110C4⋊C8—1571no—
111C4⋊C8C2×D15711yescharacteristic
112#103—21no—
113#103—3142no—
114#102—3141no—
115C4×C8—1571no—
116C2×Q8—21no—
117C2×Q8—3142no—
118C2×Q8—3141no—
119C2×Q8—6282no—
120C2×Q8—6282no—
121C2×Q8C22⋊D15711yescharacteristic
122C2×D4—1571no—
123C22×C4—3142no—
124C22×C4—3141no—
125C2×C8—21no—
126C2×C8—3141no—
127C2×C8—3141no—
128C4⋊C4—21no—
129C4⋊C4—21no—
130C4⋊C4—21no—
131C4⋊C4—3142no—
132C4⋊C4—3141no—
133C4⋊C4—1572no—
134C4⋊C4—1571no—
135C4⋊C4C22⋊D15711yescharacteristic
136C22⋊C4—3142no—
137C22⋊C4—3141no—
138C42—21no—
139C42C22×D15711yescharacteristic
140C23—3141no—
141Q8—21no—
142Q8—42no—
143Q8—6284no—
144Q8—6281no—
145Q8—6281no—
146Q8#1457894112yes—
147D4—3141no—
148D4—1572no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—41no—
152C2×C4—3142no—
153C2×C4—3141no—
154C2×C4—6284no—
155C2×C4—6281no—
156C2×C4—6281no—
157C2×C4D4×D15711yescharacteristic
158C2×C4D4×D15711yescharacteristic
159C2×C4C2×C22⋊D15711yescharacteristic
160C8—41no—
161C8—6281no—
162C8—6281no—
163C22—3142no—
164C22—6281no—
165C22—6281no—
166C22#1909773911yescharacteristic
167C4—22no—
168C4—21no—
169C4—41no—
170C4—81no—
171C4—6282no—
172C4—6281no—
173C4C2×#1457894111yescharacteristic
174C4Q16×D15711yescharacteristic
175C4#1909779611yescharacteristic
176C4#1909780311yescharacteristic
177C2—6282no—
178C2#2375742311yescharacteristic
179C2#2375742711yescharacteristic
180C2#2375744411yescharacteristic
subgroupG/H
centerC22#19097739
commutator, cosocleC42×C157C23
FrattiniC42C22×D157
Fitting#477×C157C2
socleC22×C157#120
prime radicalC2×D628D4
hypercenter#477D157
hypocenterC157#1637
LCSG ▷ C42×C157 ▷ C22×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #477

Complements

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

NH ≅ G/Ntype
2. #23757623C2semidirect
4. #23757020C2semidirect
5. Q8⋊#11929434C2semidirect
7. #477×C157C2semidirect
90. C157#1637semidirect
94. #477D157semidirect

Maximal subgroups

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

subgroupindex
Q8×D6282
#237576232
#237575602
#237570202
Q8⋊#119294342
#237570002
#477×C1572
#1637157