Finite Group Databasebeta

← #31796122

Subgroups — #31796122

Order 17536.

246 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 64 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#22578424C211yesmaximal, characteristic, complemented
2#22578420C211yesmaximal, characteristic, complemented
3#22578253C212yesmaximal, complemented
4#22578251C212yesmaximal, complemented
5#22577871C211yesmaximal, characteristic
6#17984155—21no—
7C4⋊#11764047C2212yes—
8C22⋊#11764047—21no—
9C22⋊#11764047—21no—
10#17984143—21no—
11C4×#11764047C2212yes—
12C2×#14555510C411yescharacteristic
13#17984053C411yescharacteristic
14#17984053C2211yescharacteristic
15#17984050C411yescharacteristic
16#17984050C2211yescharacteristic
17#17984048—21no—
18#17984048—21no—
19#17984045—21no—
20C2×C4×Dic137C2211yescharacteristic
21C42×D137—21no—
22#117×C137C411yescharacteristic
23C2×#11764047—22no—
24C2×#11764047—22no—
25#14555510—22no—
26#14555510—22no—
27C2×Dic274C2×C411yescharacteristic
28C2×D548C2×C411yescharacteristic
29C2×C22⋊D137—21no—
30C22×Dic137D411yescharacteristic
31C22×Dic137D411yescharacteristic
32C2×C4×D137—21no—
33C2×C4×D137—21no—
34C2×C4×D137—21no—
35#14555489—21no—
36#14555489—21no—
37#14555487—21no—
38#14555487—21no—
39C4×Dic137—21no—
40C4×Dic137—21no—
41C4×Dic137C2×C411yescharacteristic
42C4×Dic137D411yescharacteristic
43C4×Dic137D411yescharacteristic
44C4×Dic137C2311yescharacteristic
45C22×C4×C137C2×C411yescharacteristic
46C4⋊C4×C137C2×C411yescharacteristic
47C4⋊C4×C137C2×C411yescharacteristic
48C22⋊C4×C137—21no—
49C42×C137—21no—
50#11764047—42no—
51#11764047—42no—
52C22×D137—21no—
53Dic274—21no—
54Dic274M1612yes—
55D548—21no—
56D548M1612yes—
57C22⋊D137—24no—
58C2×Dic137—21no—
59C2×Dic137—22no—
60C2×Dic137—41no—
61C2×Dic137C22⋊C411yescharacteristic
62C2×Dic137C22⋊C411yescharacteristic
63C2×Dic137C2×D411yescharacteristic
64C2×Dic137C2×D411yescharacteristic
65C4×D137—24no—
66C4×D137—42no—
67C4×D137—42no—
68C23×C137C22⋊C411yescharacteristic
69C2×C4×C137—21no—
70C2×C4×C137—21no—
71C2×C4×C137—21no—
72C2×C4×C137—21no—
73C2×C4×C137C22⋊C411yescharacteristic
74C2×C4×C137C22×C411yescharacteristic
75C22×C137—21no—
76C22×C137—21no—
77C22×C137—21no—
78C22×C137C2×C22⋊C411yescharacteristic
79C2×D137—22no—
80C2×D137—41no—
81C2×D137—41no—
82C4×C137—21no—
83C4×C137—41no—
84C4×C137—41no—
85C4×C137C2×M1611yescharacteristic
86C4×C137C2×M1611yescharacteristic
87Dic137—22no—
88Dic137—21no—
89Dic137—21no—
90Dic137—21no—
91Dic137#13512yes—
92C2×C137—41no—
93C2×C137#40711yescharacteristic
94C2×C137#41611yescharacteristic
95C2×C137#42111yescharacteristic
96D137—42no—
97C137#148811yescharacteristic, complemented, Sylow
98#1488—1371nomaximal, Sylow
99C4×#41—1371no—
100#432—1371no—
101#431—1371no—
102Q8⋊C8—1372no—
103D4⋊C8—1372no—
104C2×#41—1371no—
105C4×Q8—1371no—
106C4×Q8—1371no—
107C4×D4—2741no—
108C4×D4—2741no—
109C4×D4—1371no—
110C4×D4—1371no—
111#117—2741no—
112#117#1058773911yescharacteristic
113C2×C42—2741no—
114C2×C42—1371no—
115C4⋊C8—2741no—
116C4⋊C8—1372no—
117C22⋊C8—2741no—
118C22⋊C8—2741no—
119#97—2741no—
120C4×C8—1372no—
121#41—2742no—
122#41—2742no—
123C2×Q8—1371no—
124C2×D4—2741no—
125C2×D4—1371no—
126C22×C4—2741no—
127C22×C4—2741no—
128C22×C4—2741no—
129C22×C4—1371no—
130C22×C4—1371no—
131C22×C4C2×#1058773911yescharacteristic
132C2×C8—2742no—
133C2×C8—2742no—
134C4⋊C4—2741no—
135C4⋊C4—2741no—
136C4⋊C4C2×#1058773911yescharacteristic
137C4⋊C4C2×#1058773911yescharacteristic
138C22⋊C4—21no—
139C22⋊C4—2741no—
140C22⋊C4—2741no—
141C42—21no—
142C42—2741no—
143C42—2741no—
144C42—1371no—
145C42—1371no—
146C42—1371no—
147C42—1371no—
148C23—2741no—
149C23C22⋊#1058773911yescharacteristic
150Q8—2741no—
151Q8—1372no—
152D4—2744no—
153D4—2741no—
154D4—1372no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—2744no—
160C2×C4—2741no—
161C2×C4—2742no—
162C2×C4—5482no—
163C2×C4—5482no—
164C2×C4—5481no—
165C2×C4—1371no—
166C2×C4—1371no—
167C2×C4—1371no—
168C2×C4—1371no—
169C2×C4C22⋊#1058773911yescharacteristic
170C2×C4C22×#1058773911yescharacteristic
171C8—5482no—
172C8—5482no—
173C22—21no—
174C22—21no—
175C22—21no—
176C22—2742no—
177C22—5481no—
178C22—5481no—
179C22C2×C22⋊#1058773911yescharacteristic
180C4—21no—
181C4—41no—
182C4—41no—
183C4—2742no—
184C4—2741no—
185C4—2741no—
186C4—2741no—
187C4—1372no—
188C4D137⋊M1611yescharacteristic
189C4D137⋊M1611yescharacteristic
190C2—41no—
191C2—5482no—
192C2#2257837511yescharacteristic
193C2#2257839311yescharacteristic
194C2#2257840011yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10587739
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
Fitting#117×C137C4
socleC22×C137C2×C22⋊C4
prime radicalC2×#14555510C4
cosocleC4×Dic137C23
hypercenter#117#10587739
hypocenterC137#1488
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
1. #22578424C2semidirect
2. #22578420C2semidirect
3. #22578253C2semidirect
4. #22578251C2semidirect
97. C137#1488semidirect

Maximal subgroups

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

subgroupindex
#225784242
#225784202
#225782532
#225782512
#225778712
#1488137