Finite Group Databasebeta

← #31796118

Subgroups — #31796118

Order 17536.

248 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 76 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#22578422C211yesmaximal, characteristic
2#22578407C211yesmaximal, characteristic
3#22578253C212yesmaximal, complemented
4Q8⋊#11764047C212yesmaximal, complemented
5C4×Q8×D137C211yesmaximal, characteristic
6C2×#14555520—21no—
7C4⋊#11764047—21no—
8C4⋊#11764047—21no—
9C4⋊#11764047C2212yes—
10#17984145—21no—
11C4×#11764047C2212yes—
12C2×Q8×D137C411yescharacteristic
13Q8×Dic137C2211yescharacteristic
14C4×Dic274—21no—
15C4×Dic274C411yescharacteristic
16#17984053—21no—
17#17984053C2211yescharacteristic
18C4⋊C4×D137—21no—
19C4⋊C4×D137C411yescharacteristic
20C42×D137—21no—
21C42×D137C2211yescharacteristic
22C4×Q8×C137C411yescharacteristic
23#14555520—42no—
24C2×#11764047—22no—
25C2×#11764047—22no—
26C2×Dic274—21no—
27C2×Dic274C2×C411yescharacteristic
28Q8×D137—22no—
29Q8×D137—22no—
30C2×C4×D137—21no—
31C2×C4×D137—21no—
32C2×C4×D137C2×C411yescharacteristic
33C2×C4×D137D411yescharacteristic
34C2×C4×D137D411yescharacteristic
35C4⋊Dic137—21no—
36C4⋊Dic137C2×C411yescharacteristic
37#14555489—21no—
38#14555489—21no—
39#14555489—21no—
40C4×Dic137—21no—
41C4×Dic137—21no—
42C4×Dic137—21no—
43C4×Dic137D411yescharacteristic
44C4×Dic137D411yescharacteristic
45C4×Dic137C2311yescharacteristic
46C2×Q8×C137C2×C411yescharacteristic
47C4⋊C4×C137—21no—
48C4⋊C4×C137C2×C411yescharacteristic
49C42×C137—21no—
50C42×C137C2×C411yescharacteristic
51#11764047—42no—
52#11764047—42no—
53C22×D137C22⋊C411yescharacteristic
54Dic274—24no—
55Dic274—21no—
56Dic274M1612yescomplemented
57C2×Dic137—21no—
58C2×Dic137—21no—
59C2×Dic137C22⋊C411yescharacteristic
60C2×Dic137C2×D411yescharacteristic
61C2×Dic137C2×D411yescharacteristic
62C4×D137—24no—
63C4×D137—21no—
64C4×D137—21no—
65C4×D137—41no—
66C4×D137—42no—
67C4×D137SD1612yes—
68C4×D137Q1612yes—
69Q8×C137—21no—
70Q8×C137M1612yescomplemented
71C2×C4×C137—21no—
72C2×C4×C137—21no—
73C2×C4×C137C22⋊C411yescharacteristic
74C2×C4×C137C22⋊C411yescharacteristic
75C2×C4×C137C22×C411yescharacteristic
76C22×C137C2×C22⋊C411yescharacteristic
77C2×D137—21no—
78C2×D137—21no—
79C2×D137#10312yes—
80C4×C137—22no—
81C4×C137—21no—
82C4×C137—41no—
83C4×C137#10312yes—
84C4×C137C2×M1611yescharacteristic
85C4×C137C2×M1611yescharacteristic
86Dic137—22no—
87Dic137—21no—
88Dic137—21no—
89Dic137—41no—
90Dic137C2×SD1611yescharacteristic
91Dic137C2×Q1611yescharacteristic
92C2×C137#40711yescharacteristic
93C2×C137C2×#10311yescharacteristic
94C2×C137#42111yescharacteristic
95D137—22no—
96C137#148711yescharacteristic, complemented, Sylow
97#1487—1371nomaximal, Sylow
98C2×C4×Q8—1371no—
99#431—1371no—
100#423—1371no—
101Q8⋊C8—1372no—
102Q8⋊C8—1372no—
103C22×Q8—1371no—
104C2×M16—2741no—
105C4×Q8—2741no—
106C4×Q8—2741no—
107C4×Q8—1371no—
108C4×Q8—1371no—
109C4×Q8—1371no—
110C4×Q8#1058773911yescharacteristic
111C2×C4⋊C4—2741no—
112C2×C4⋊C4—1371no—
113C2×C42—2741no—
114C2×C42—1371no—
115C4⋊C8—2741no—
116C4⋊C8—2741no—
117C4⋊C8—1372no—
118C22⋊C8—2741no—
119C4×C8—1372no—
120C2×Q8—2741no—
121C2×Q8—2742no—
122C2×Q8—2742no—
123C2×Q8—1371no—
124C2×Q8C2×#1058773911yescharacteristic
125C22×C4—2741no—
126C22×C4—2741no—
127C22×C4—1371no—
128C22×C4—1371no—
129C22×C4—1371no—
130M16—5482no—
131C2×C8—2742no—
132C2×C8—2742no—
133C4⋊C4—21no—
134C4⋊C4—2741no—
135C4⋊C4—2741no—
136C4⋊C4—2741no—
137C4⋊C4—2741no—
138C4⋊C4—1371no—
139C4⋊C4C2×#1058773911yescharacteristic
140C42—21no—
141C42—2741no—
142C42—2741no—
143C42—2741no—
144C42—1371no—
145C42—1371no—
146C42—1371no—
147C42C2×#1058773911yescharacteristic
148C23—1371no—
149Q8—21no—
150Q8—2744no—
151Q8—2741no—
152Q8—1372no—
153Q8#1455552012yescomplemented
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—2744no—
157C2×C4—2741no—
158C2×C4—2741no—
159C2×C4—2741no—
160C2×C4—2741no—
161C2×C4—5481no—
162C2×C4—5482no—
163C2×C4—1371no—
164C2×C4—1371no—
165C2×C4—1371no—
166C2×C4—1372no—
167C2×C4—1372no—
168C2×C4C22⋊#1058773911yescharacteristic
169C2×C4C22⋊#1058773911yescharacteristic
170C2×C4C22×#1058773911yescharacteristic
171C8—5482no—
172C8—5482no—
173C22—2741no—
174C22—2741no—
175C22—1372no—
176C22C2×C22⋊#1058773911yescharacteristic
177C4—22no—
178C4—21no—
179C4—41no—
180C4—2742no—
181C4—2741no—
182C4—2741no—
183C4—5481no—
184C4—1371no—
185C4—1371no—
186C4#1798415112yes—
187C4D137⋊M1611yescharacteristic
188C4C2×#1455552011yescharacteristic
189C2—2742no—
190C2#2257837411yescharacteristic
191C2C2×#1798415111yescharacteristic
192C2#2257840011yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10587739
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
FittingC4×Q8×C137C4
socleC22×C137C2×C22⋊C4
prime radicalC22×D137C22⋊C4
cosocleC4×Dic137C23
hypercenterC4×Q8#10587739
hypocenterC137#1487
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ C4×Q8

Complements

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

NH ≅ G/Ntype
3. #22578253C2semidirect
4. Q8⋊#11764047C2semidirect
56. Dic274M16semidirect
70. Q8×C137M16semidirect
96. C137#1487semidirect
153. Q8#14555520semidirect

Maximal subgroups

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

subgroupindex
#225784222
#225784072
#225782532
Q8⋊#117640472
C4×Q8×D1372
#1487137