Finite Group Databasebeta

← #31796791

Subgroups — #31796791

Order 17536.

332 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 66 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#22578483C211yesmaximal, characteristic, complemented
2#22578402C212yesmaximal, complemented
3#22578384C211yesmaximal, characteristic, complemented
4#22578274C212yesmaximal, complemented
5#22577954C211yesmaximal, characteristic
6#17984176—24no—
7D137⋊M16—22no—
8D137⋊M16C2212yes—
9#17984167C2212yes—
10#17984153—24no—
11#17984147—22no—
12C2×#14555511C2211yescharacteristic
13C2×#14555511C2211yescharacteristic
14#17984085C411yescharacteristic, complemented
15#17984077C411yescharacteristic, complemented
16#17984077C411yescharacteristic, complemented
17#17984063—22no—
18#17984050—22no—
19C42×D137C2211yescharacteristic
20C4⋊Q8×C137C411yescharacteristic, complemented
21#14555520—24no—
22#14555520D414yescomplemented
23#14555519—22no—
24#14555518—24no—
25#14555518—22no—
26C4⋊#10587739—22no—
27C22⋊#10587739—22no—
28C4×#10587739—22no—
29#14555511—22no—
30#14555511—22no—
31#14555511—22no—
32#14555511—22no—
33C2×D548—22no—
34C2×D548—21no—
35C2×D548C2×C411yescharacteristic
36C2×D548C2×C411yescharacteristic
37C2×D548D412yes—
38C2×C4×D137—21no—
39C2×C4×D137—21no—
40C2×C4×D137D412yes—
41C2×C4×D137C2311yescharacteristic
42#14555487—22no—
43#14555487—22no—
44C4×Dic137—21no—
45C4×Dic137C2×C411yescharacteristic
46C2×Q8×C137C2×C411yescharacteristic
47C2×Q8×C137C2×C411yescharacteristic
48C4⋊C4×C137—22no—
49C42×C137C2×C411yescharacteristic
50C2×#10587739—42no—
51#11764047—24no—
52C22×D137—21no—
53C22×D137C22⋊C412yes—
54C22×D137C2×D411yescharacteristic
55D548—24no—
56D548—22no—
57D548—22no—
58D548—44no—
59D548—42no—
60C2×Dic137—21no—
61C2×Dic137C2×D411yescharacteristic
62C4×D137—22no—
63C4×D137—22no—
64C4×D137—42no—
65C4×D137—41no—
66C4×D137—41no—
67C4×D137C2×D412yes—
68C4×D137#4112yes—
69Q8×C137—22no—
70Q8×C137—22no—
71C2×C4×C137—21no—
72C2×C4×C137—21no—
73C2×C4×C137C22⋊C412yes—
74C2×C4×C137C22×C411yescharacteristic
75C22×C137C2×C22⋊C411yescharacteristic
76C2×D137—22no—
77C2×D137—41no—
78C2×D137—42no—
79C2×D137—42no—
80C2×D137—41no—
81C2×D137#12112yes—
82#10587739—82no—
83C4×C137—22no—
84C4×C137—22no—
85C4×C137—41no—
86C4×C137C4×D412yes—
87Dic137—41no—
88Dic137#12011yescharacteristic
89Dic137#12311yescharacteristic
90C2×C137—21no—
91C2×C137#38611yescharacteristic
92D137—22no—
93D137—42no—
94D137—81no—
95C137#188411yescharacteristic, complemented, Sylow
96#1884—1371nomaximal, Sylow
97#568—1371no—
98#532—1371no—
99#421—1372no—
100#410—1371no—
101#339—1372no—
102C2×#41—1371no—
103C2×#41—1371no—
104#131—2744no—
105C2×M16—2742no—
106C2×M16—1372no—
107C4⋊Q8#1058773911yescharacteristic, complemented
108#127—1371no—
109#124—1371no—
110#124—1371no—
111#121—2742no—
112C4×D4—2742no—
113#117—1372no—
114C2×C42—1371no—
115#104—2744no—
116#99—2742no—
117#41—2742no—
118#41—2742no—
119#41—2742no—
120#41—2742no—
121C2×Q8C2×#1058773911yescharacteristic
122C2×Q8C2×#1058773911yescharacteristic
123C2×D4—2742no—
124C2×D4—2741no—
125C2×D4—1372no—
126C2×D4—1371no—
127C2×D4—1371no—
128C22×C4—2741no—
129C22×C4—2741no—
130C22×C4—1372no—
131C22×C4—1371no—
132M16—2744no—
133M16—2742no—
134M16—1374no—
135C2×C8—2744no—
136C2×C8—2742no—
137C4⋊C4—22no—
138C4⋊C4—2742no—
139C22⋊C4—2742no—
140C22⋊C4—2742no—
141C22⋊C4—2742no—
142C42—2741no—
143C42—2742no—
144C42—1371no—
145C42C2×#1058773911yescharacteristic
146C23—2741no—
147C23—1372no—
148C23—1371no—
149Q8—22no—
150Q8—22no—
151D4—2744no—
152D4—2742no—
153D4—2742no—
154D4—5484no—
155D4—5482no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—2742no—
159C2×C4—2742no—
160C2×C4—2741no—
161C2×C4—5482no—
162C2×C4—5481no—
163C2×C4—5481no—
164C2×C4—5482no—
165C2×C4—1371no—
166C2×C4—1372no—
167C2×C4—1372no—
168C2×C4C22⋊#1058773912yes—
169C2×C4C22×#1058773911yescharacteristic
170C8—2744no—
171C22—2742no—
172C22—5481no—
173C22—5482no—
174C22—5482no—
175C22—5481no—
176C22—1372no—
177C22C2×C22⋊#1058773911yescharacteristic
178C4—22no—
179C4—22no—
180C4—41no—
181C4—5481no—
182C4—10962no—
183C4—1371no—
184C4—1371no—
185C4D4×#1058773912yes—
186C2—21no—
187C2—2742no—
188C2—5482no—
189C2—10961no—
190C2#2257835111yescharacteristic
subgroupG/H
centerC2#22578351
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
FittingC4⋊Q8×C137C4
socleC2×C137#386
prime radical#22577954C2
cosocleC2×C4×D137C23
hypercenterC4⋊Q8#10587739
hypocenterC137#1884
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. #22578483C2semidirect
2. #22578402C2semidirect
3. #22578384C2semidirect
4. #22578274C2semidirect
14. #17984085C4semidirect
15. #17984077C4semidirect
16. #17984077C4semidirect
20. C4⋊Q8×C137C4semidirect
22. #14555520D4semidirect
95. C137#1884semidirect
107. C4⋊Q8#10587739semidirect

Maximal subgroups

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

subgroupindex
#225784832
#225784022
#225783842
#225782742
#225779542
#1884137