Finite Group Databasebeta

← #31796227

Subgroups — #31796227

Order 17536.

288 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 68 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
1C2×#17984145C211yesmaximal, characteristic
2#22578345C211yesmaximal, characteristic
3#22578257C212yesmaximal, complemented
4#22578256C212yesmaximal, complemented
5C22⋊Q8×D137C211yesmaximal, characteristic
6C22×#11764047—21no—
7C2×C4⋊#10587739—21no—
8C2×C22⋊#10587739—21no—
9#17984145—21no—
10#17984145C2212yes—
11(C2×#10587739)⋊C4C2212yes—
12C2×Q8×D137C411yescharacteristic, complemented
13C22×C4×D137C2211yescharacteristic
14#17984068C2211yescharacteristic
15#17984067C411yescharacteristic, complemented
16#17984066—21no—
17#17984065—21no—
18#17984064C2211yescharacteristic
19C4⋊C4×D137—21no—
20C4⋊C4×D137C411yescharacteristic, complemented
21C22⋊C4×D137—21no—
22C22⋊Q8×C137C411yescharacteristic, complemented
23C22×#10587739—22no—
24C2×#11764047—22no—
25C2×#11764047—42no—
26C4⋊#10587739—42no—
27C22⋊#10587739—42no—
28C23×D137D411yescharacteristic
29C2×Dic274—21no—
30C2×Dic274C2×C411yescharacteristic
31Q8×D137—42no—
32C22×Dic137D411yescharacteristic
33C2×C4×D137—21no—
34C2×C4×D137—21no—
35C2×C4×D137—42no—
36C2×C4×D137C2×C411yescharacteristic
37C2×C4×D137D411yescharacteristic
38C2×C4×D137D411yescharacteristic
39C2×C4×D137C2311yescharacteristic
40C4⋊Dic137—21no—
41C4⋊Dic137C2×C411yescharacteristic
42#14555489—21no—
43#14555489—21no—
44#14555489—21no—
45C22⋊Dic137—21no—
46#14555487—21no—
47#14555487—21no—
48C2×Q8×C137C2×C411yescharacteristic
49C22×C4×C137C2×C411yescharacteristic
50C4⋊C4×C137—21no—
51C4⋊C4×C137C2×C411yescharacteristic
52C22⋊C4×C137—21no—
53C2×#10587739—42no—
54C2×#10587739—42no—
55C2×#10587739—42no—
56#11764047—42no—
57C22×D137—41no—
58C22×D137—41no—
59C22×D137C22⋊C411yescharacteristic
60C22×D137SD1612yes—
61C22×D137Q1612yes—
62C22×D137C2×D411yescharacteristic
63Dic274—42no—
64Dic274—41no—
65C2×Dic137—22no—
66C2×Dic137—21no—
67C2×Dic137—21no—
68C2×Dic137C22⋊C411yescharacteristic
69C2×Dic137C2×D411yescharacteristic
70C4×D137—22no—
71C4×D137—22no—
72C4×D137—41no—
73C4×D137—41no—
74C4×D137—42no—
75C4×D137—42no—
76C23×C137C22⋊C411yescharacteristic
77Q8×C137—41no—
78C2×C4×C137—21no—
79C2×C4×C137—21no—
80C2×C4×C137—41no—
81C2×C4×C137C22⋊C411yescharacteristic
82C2×C4×C137C22×C411yescharacteristic
83C22×C137—21no—
84C22×C137—21no—
85C22×C137#10312yes—
86C22×C137C2×C22⋊C411yescharacteristic
87C2×D137—21no—
88C2×D137—21no—
89C2×D137—21no—
90C2×D137—21no—
91C2×D137—42no—
92C2×D137—42no—
93C2×D137#10312yes—
94C2×D137C2×SD1611yescharacteristic
95C2×D137C2×Q1611yescharacteristic
96#10587739—82no—
97C4×C137—21no—
98C4×C137—21no—
99C4×C137—41no—
100C4×C137—41no—
101Dic137—21no—
102Dic137—41no—
103Dic137—41no—
104Dic137#9912yes—
105C2×C137—22no—
106C2×C137C2×#9911yescharacteristic
107C2×C137C2×#10311yescharacteristic
108C2×C137#42111yescharacteristic
109D137—21no—
110D137—22no—
111D137—21no—
112C137#152011yescharacteristic, complemented, Sylow
113#1520—1371nomaximal, Sylow
114C2×C22⋊Q8—1371no—
115C2×C22⋊C8—1371no—
116#380—1371no—
117#328—1372no—
118#328—1372no—
119C22×Q8—1371no—
120C23×C4—1371no—
121C22×C8—2741no—
122C22⋊Q8—2741no—
123C22⋊Q8—2741no—
124C22⋊Q8—1371no—
125C22⋊Q8—1371no—
126C22⋊Q8—1371no—
127C22⋊Q8#1058773911yescharacteristic, complemented
128C2×C4⋊C4—2741no—
129C2×C4⋊C4—2741no—
130C2×C4⋊C4—1371no—
131C2×C22⋊C4—2741no—
132C2×C22⋊C4—2741no—
133C22⋊C8—2741no—
134C22⋊C8—1372no—
135#95—1372no—
136C24—1371no—
137C2×Q8—2741no—
138C2×Q8—5482no—
139C2×Q8—1371no—
140C2×Q8C2×#1058773911yescharacteristic
141C22×C4—2741no—
142C22×C4—2741no—
143C22×C4—2742no—
144C22×C4—5482no—
145C22×C4—1371no—
146C22×C4—1371no—
147C22×C4—1371no—
148C22×C4—1371no—
149C22×C4—1371no—
150C22×C4C2×#1058773911yescharacteristic
151C2×C8—2742no—
152C2×C8—5482no—
153C4⋊C4—21no—
154C4⋊C4—2741no—
155C4⋊C4—2741no—
156C4⋊C4—2741no—
157C4⋊C4—2741no—
158C4⋊C4—5482no—
159C4⋊C4—1371no—
160C4⋊C4C2×#1058773911yescharacteristic
161C22⋊C4—21no—
162C22⋊C4—2741no—
163C22⋊C4—2741no—
164C22⋊C4—2741no—
165C22⋊C4—5482no—
166C23—5481no—
167C23—5481no—
168C23—1371no—
169C23—1371no—
170C23—1372no—
171C23—1372no—
172C23C22⋊#1058773911yescharacteristic
173Q8—41no—
174Q8—5481no—
175Q8—5482no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—41no—
179C2×C4—2742no—
180C2×C4—2742no—
181C2×C4—2742no—
182C2×C4—2741no—
183C2×C4—2741no—
184C2×C4—5481no—
185C2×C4—5481no—
186C2×C4—5482no—
187C2×C4—5482no—
188C2×C4—5482no—
189C2×C4—5482no—
190C2×C4—5482no—
191C2×C4—1371no—
192C2×C4—1371no—
193C2×C4C22⋊#1058773911yescharacteristic
194C2×C4C22×#1058773911yescharacteristic
195C8—5482no—
196C22—21no—
197C22—21no—
198C22—2741no—
199C22—2741no—
200C22—2741no—
201C22—2741no—
202C22—5482no—
203C22—5482no—
204C22—1371no—
205C22—1372no—
206C22—1371no—
207C22#1798415112yes—
208C22C2×C22⋊#1058773911yescharacteristic
209C4—21no—
210C4—21no—
211C4—41no—
212C4—41no—
213C4—2741no—
214C4—5481no—
215C4—5481no—
216C4—10962no—
217C4—1372no—
218C2—22no—
219C2—2741no—
220C2—2742no—
221C2—2741no—
222C2D137⋊#9911yescharacteristic
223C2C2×#1798415111yescharacteristic
224C2#2257840011yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10587739
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
FittingC22⋊Q8×C137C4
socleC22×C137C2×C22⋊C4
prime radicalC23×D137D4
cosocleC2×C4×D137C23
hypercenterC22⋊Q8#10587739
hypocenterC137#1520
LCSG ▷ C2×C4×C137 ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #22578257C2semidirect
4. #22578256C2semidirect
12. C2×Q8×D137C4semidirect
15. #17984067C4semidirect
20. C4⋊C4×D137C4semidirect
22. C22⋊Q8×C137C4semidirect
112. C137#1520semidirect
127. C22⋊Q8#10587739semidirect

Maximal subgroups

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

subgroupindex
C2×#179841452
#225783452
#225782572
#225782562
C22⋊Q8×D1372
#1520137