Finite Group Databasebeta

← #46810131

Subgroups — #46810131

Order 34432.

234 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 57 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#31771270C211yesmaximal, characteristic
2#31770882C211yesmaximal, characteristic
3#31770812C211yesmaximal, characteristic, complemented
4#31770351C211yesmaximal, characteristic, complemented
5#31770337C211yesmaximal, characteristic, complemented
6#31770334C211yesmaximal, characteristic
7#500×C269C211yesmaximal, characteristic, complemented
8C2×Q8×D269—21no—
9C2×#16864392—21no—
10#22575452—21no—
11#22575431—22no—
12#22575429—21no—
13C4×Dic538C2211yescharacteristic
14C4×D1076C2211yescharacteristic
15#22575415—21no—
16#22575386—21no—
17C4⋊#14524130C2211yescharacteristic
18C4⋊#14524130C2211yescharacteristic
19#22575377—21no—
20#22575373—21no—
21C4⋊Q8×C269C2211yescharacteristic
22C4⋊Q8×C269C2211yescharacteristic
23#107×C269—22no—
24C4×C8×C269C2211yescharacteristic
25C2×Dic538—22no—
26C2×Dic538D411yescharacteristic
27Q8×D269—42no—
28Q8×D269—42no—
29C2×D1076D411yescharacteristic
30C2×C4×D269—22no—
31C2×C4×D269—21no—
32#16864392—42no—
33C2×#14524130—21no—
34C2×#14524130—21no—
35C4⋊Dic269D411yescharacteristic
36C4⋊Dic269D411yescharacteristic
37#16864381—22no—
38#16864381—21no—
39#16864379—22no—
40#16864379—21no—
41C4×Dic269—21no—
42C2×Q8×C269—21no—
43C2×Q8×C269—21no—
44C2×C8×C269—21no—
45C4⋊C4×C269—21no—
46C4⋊C4×C269—21no—
47C4⋊C4×C269—21no—
48C4⋊C4×C269D412yes—
49C42×C269C2311yescharacteristic
50C22×D269—21no—
51Dic538—21no—
52Dic538—44no—
53Dic538D812yes—
54D1076—21no—
55D1076Q1612yescomplemented
56C2×Dic269—21no—
57C2×Dic269—21no—
58C2×Dic269—21no—
59C4×D269—44no—
60C4×D269—41no—
61C4×D269—41no—
62#14524130—41no—
63#14524130—41no—
64Q8×C269—41no—
65Q8×C269—41no—
66Q8×C269—41no—
67Q8×C269—41no—
68C2×C4×C269—22no—
69C2×C4×C269—41no—
70C2×C4×C269C2×D411yescharacteristic
71C2×C4×C269C2×D411yescharacteristic
72C2×C4×C269C2×D411yescharacteristic
73C8×C269—22no—
74C22×C269#12011yescharacteristic
75C2×D269—22no—
76C2×D269—41no—
77C2×D269—41no—
78C4×C269—21no—
79C4×C269—42no—
80C4×C269—81no—
81C4×C269C2×D811yescharacteristic
82C4×C269C2×Q1611yescharacteristic
83C4×C269AGL(1,ℤ/8)11yescharacteristic
84C4×C269#13711yescharacteristic
85Dic269—22no—
86Dic269—41no—
87Dic269—41no—
88C2×C269#44711yescharacteristic
89C2×C269#45111yescharacteristic
90C2×C269#45611yescharacteristic
91D269—42no—
92C269#164911yescharacteristic, complemented, Sylow
93#1649—2691nomaximal, Sylow
94D4×Q8—2691no—
95#500D26911yescharacteristic, complemented
96#474—2691no—
97#462—2691no—
98#332—2691no—
99Q8⋊C8—2691no—
100D4⋊C8—2691no—
101C22×Q8—5381no—
102C2×Q16—5381no—
103C4⋊Q8—5381no—
104C4⋊Q8C2×D26911yescharacteristic
105C4⋊Q8C2×D26911yescharacteristic
106C22⋊Q8—5381no—
107C22⋊Q8—5382no—
108C4×Q8—2691no—
109C4×D4—2691no—
110C4×D4—5381no—
111#107—22no—
112#107—5381no—
113C4⋊C8—2691no—
114C4⋊C8—2691no—
115#103—5381no—
116#102—5381no—
117C4×C8C2×D26911yescharacteristic
118C2×Q8—21no—
119C2×Q8—21no—
120C2×Q8—2691no—
121C2×Q8—5382no—
122C2×Q8—10762no—
123C2×Q8—10762no—
124C2×D4—2691no—
125C22×C4—5382no—
126C22×C4—5381no—
127Q16—10762no—
128C2×C8—21no—
129C2×C8—5381no—
130C2×C8—5381no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—21no—
134C4⋊C4—2691no—
135C4⋊C4—2691no—
136C4⋊C4—5382no—
137C4⋊C4—5381no—
138C4⋊C4C22⋊D26912yes—
139C22⋊C4—5382no—
140C22⋊C4—5381no—
141C42—5381no—
142C42C22×D26911yescharacteristic
143C23—5381no—
144Q8—41no—
145Q8—41no—
146Q8—41no—
147Q8—41no—
148Q8—2692no—
149Q8—5381no—
150Q8—10764no—
151D4—2692no—
152D4—5381no—
153C2×C4—22no—
154C2×C4—41no—
155C2×C4—5381no—
156C2×C4—5381no—
157C2×C4—5381no—
158C2×C4—10764no—
159C2×C4—10761no—
160C2×C4—10761no—
161C2×C4D4×D26911yescharacteristic
162C2×C4D4×D26911yescharacteristic
163C2×C4C2×C22⋊D26911yescharacteristic
164C8—22no—
165C8—10761no—
166C8—10761no—
167C22—5382no—
168C22—10761no—
169C22—10761no—
170C22#2257542111yescharacteristic
171C4—21no—
172C4—42no—
173C4—81no—
174C4—5382no—
175C4—10761no—
176C4—10761no—
177C4D8×D26911yescharacteristic
178C4Q16×D26911yescharacteristic
179C4#2257547711yescharacteristic
180C4#2257548511yescharacteristic
181C2—10762no—
182C2#3177073711yescharacteristic
183C2#3177075911yescharacteristic
184C2#3177078711yescharacteristic
subgroupG/H
centerC22#22575421
commutator, cosocleC42×C269C23
FrattiniC42C22×D269
Fitting#500×C269C2
socleC22×C269#120
prime radicalC2×D1076D4
hypercenter#500D269
hypocenterC269#1649
LCSG ▷ C42×C269 ▷ C22×C269 ▷ C269
UCSC1 ◁ C22 ◁ C42 ◁ #500

Complements

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

NH ≅ G/Ntype
3. #31770812C2semidirect
4. #31770351C2semidirect
5. #31770337C2semidirect
7. #500×C269C2semidirect
55. D1076Q16semidirect
92. C269#1649semidirect
95. #500D269semidirect

Maximal subgroups

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

subgroupindex
#317712702
#317708822
#317708122
#317703512
#317703372
#317703342
#500×C2692
#1649269