Finite Group Databasebeta

← #31796792

Subgroups — #31796792

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#22578481C211yesmaximal, characteristic, complemented
2#22578401C212yesmaximal, complemented
3#22578383C211yesmaximal, characteristic, complemented
4#22578274C212yesmaximal, complemented
5#22577951C211yesmaximal, characteristic
6#17984175—24no—
7D137⋊M16C2212yes—
8C2×#14555519—22no—
9#17984167C2212yes—
10#17984152—24no—
11#17984146—22no—
12C2×#14555509C2211yescharacteristic
13C2×#14555509C2211yescharacteristic
14#17984088C411yescharacteristic, complemented
15#17984075C411yescharacteristic, complemented
16#17984075C411yescharacteristic, complemented
17#17984059—22no—
18D4×Dic137—22no—
19C42×D137C2211yescharacteristic
20#127×C137C411yescharacteristic, complemented
21#14555520D414yescomplemented
22#14555519—24no—
23#14555519—22no—
24#14555518—22no—
25C2×#11764047—24no—
26C4⋊#10587739—22no—
27C22⋊#10587739—22no—
28C4×#10587739—22no—
29#14555509—22no—
30#14555509—22no—
31#14555509—22no—
32#14555509—22no—
33C2×Dic274C2×C411yescharacteristic
34C2×Dic274C2×C411yescharacteristic
35C2×C22⋊D137—21no—
36C2×C22⋊D137D412yes—
37C22×Dic137—21no—
38C22×Dic137D412yes—
39C2×C4×D137—21no—
40C2×C4×D137C2311yescharacteristic
41C4⋊Dic137—22no—
42C22⋊Dic137—22no—
43C22⋊Dic137—22no—
44C4×Dic137—21no—
45C4×Dic137C2×C411yescharacteristic
46C2×D4×C137—22no—
47C2×D4×C137C2×C411yescharacteristic
48C2×D4×C137C2×C411yescharacteristic
49C42×C137C2×C411yescharacteristic
50C2×#10587739—42no—
51#11764047—24no—
52C22×D137C2×D411yescharacteristic
53Dic274—22no—
54Dic274—22no—
55C22⋊D137—24no—
56C22⋊D137—42no—
57C2×Dic137—22no—
58C2×Dic137—22no—
59C2×Dic137—21no—
60C2×Dic137—21no—
61C2×Dic137—41no—
62C2×Dic137—41no—
63C2×Dic137C22⋊C412yes—
64C2×Dic137C2×D411yescharacteristic
65C4×D137—42no—
66C4×D137C2×D412yes—
67C4×D137#4112yes—
68C23×C137—21no—
69C23×C137C22⋊C412yes—
70D4×C137—22no—
71D4×C137—22no—
72D4×C137—44no—
73C2×C4×C137—21no—
74C2×C4×C137C22×C411yescharacteristic
75C22×C137—22no—
76C22×C137—42no—
77C22×C137—42no—
78C22×C137—41no—
79C22×C137C2×C22⋊C411yescharacteristic
80C2×D137—41no—
81C2×D137#12112yes—
82#10587739—82no—
83C4×C137—22no—
84C4×C137C4×D412yes—
85Dic137—22no—
86Dic137—41no—
87Dic137—41no—
88Dic137#12011yescharacteristic
89Dic137#12311yescharacteristic
90C2×C137—21no—
91C2×C137—42no—
92C2×C137—81no—
93C2×C137#38611yescharacteristic
94D137—22no—
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—1371no—
108#127#1058773911yescharacteristic, complemented
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×Q8—1371no—
122C2×Q8—1371no—
123C2×D4—22no—
124C2×D4—2741no—
125C2×D4—1372no—
126C2×D4C2×#1058773911yescharacteristic
127C2×D4C2×#1058773911yescharacteristic
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—2742no—
138C4⋊C4—2742no—
139C22⋊C4—2742no—
140C22⋊C4—2742no—
141C22⋊C4—2742no—
142C42—2741no—
143C42—2742no—
144C42—1371no—
145C42C2×#1058773911yescharacteristic
146C23—21no—
147C23—1371no—
148C23C22⋊#1058773912yes—
149Q8—2742no—
150Q8—2742no—
151D4—22no—
152D4—22no—
153D4—44no—
154D4—2744no—
155D4—5482no—
156C2×C4—21no—
157C2×C4—2742no—
158C2×C4—2742no—
159C2×C4—2741no—
160C2×C4—2741no—
161C2×C4—5482no—
162C2×C4—5481no—
163C2×C4—5481no—
164C2×C4—5482no—
165C2×C4—1372no—
166C2×C4—1371no—
167C2×C4—1372no—
168C2×C4—1372no—
169C2×C4C22×#1058773911yescharacteristic
170C8—2744no—
171C22—22no—
172C22—42no—
173C22—42no—
174C22—41no—
175C22—5481no—
176C22—1372no—
177C22C2×C22⋊#1058773911yescharacteristic
178C4—22no—
179C4—2742no—
180C4—5481no—
181C4—5481no—
182C4—10962no—
183C4—1371no—
184C4—1371no—
185C4D4×#1058773912yes—
186C2—21no—
187C2—42no—
188C2—81no—
189C2—2742no—
190C2#2257835111yescharacteristic
subgroupG/H
centerC2#22578351
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
Fitting#127×C137C4
socleC2×C137#386
prime radical#22577951C2
cosocleC2×C4×D137C23
hypercenter#127#10587739
hypocenterC137#1884
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2 ◁ C2×C4 ◁ #127

Complements

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

NH ≅ G/Ntype
1. #22578481C2semidirect
2. #22578401C2semidirect
3. #22578383C2semidirect
4. #22578274C2semidirect
14. #17984088C4semidirect
15. #17984075C4semidirect
16. #17984075C4semidirect
20. #127×C137C4semidirect
21. #14555520D4semidirect
95. C137#1884semidirect
108. #127#10587739semidirect

Maximal subgroups

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

subgroupindex
#225784812
#225784012
#225783832
#225782742
#225779512
#1884137