Finite Group Databasebeta

← #31796647

Subgroups — #31796647

Order 17536.

312 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 92 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×#17984155C211yesmaximal, characteristic, complemented
2C2×C4⋊#11764047C211yesmaximal, characteristic, complemented
3C2×C22⋊#11764047C212yesmaximal
4#22578269C212yesmaximal, complemented
5C2×#17984044C211yesmaximal, characteristic
6C22×#11764047C2212yes—
7C22×#11764047C2212yes—
8#17984155—22no—
9C4⋊#11764047—22no—
10C22⋊#11764047—24no—
11C23×Dic137C2211yescharacteristic
12#17984044—22no—
13#17984044—22no—
14C2×#14555489C411yescharacteristic
15C2×#14555489C411yescharacteristic
16C2×C22⋊Dic137C411yescharacteristic
17C2×C4×Dic137C2211yescharacteristic
18C2×C4×Dic137C2211yescharacteristic
19C2×C22⋊C4×C137C411yescharacteristic
20C2×#11764047—22no—
21C2×#11764047—22no—
22C2×#11764047—22no—
23C2×#11764047—22no—
24C2×#11764047—22no—
25C2×#11764047—22no—
26C22×Dic137—21no—
27C22×Dic137—21no—
28C22×Dic137—21no—
29C22×Dic137—21no—
30C22×Dic137C2×C411yescharacteristic
31C22×Dic137C2×C411yescharacteristic
32C22×Dic137C2×C411yescharacteristic
33C22×Dic137D412yes—
34C22×Dic137Q812yes—
35C22×Dic137C2311yescharacteristic
36#14555489—22no—
37#14555489—22no—
38C22⋊Dic137—22no—
39C4×Dic137—21no—
40C4×Dic137—21no—
41C4×Dic137D412yes—
42C4×Dic137D412yes—
43C24×C137C2×C411yescharacteristic
44C22×C4×C137C2×C411yescharacteristic
45C22×C4×C137C2×C411yescharacteristic
46C22⋊C4×C137—22no—
47#11764047—42no—
48#11764047—42no—
49C2×Dic137—21no—
50C2×Dic137—21no—
51C2×Dic137—22no—
52C2×Dic137—22no—
53C2×Dic137—22no—
54C2×Dic137—22no—
55C2×Dic137—22no—
56C2×Dic137—21no—
57C2×Dic137—22no—
58C2×Dic137C22⋊C412yes—
59C2×Dic137C4⋊C412yes—
60C2×Dic137C2×D411yescharacteristic
61C2×Dic137C2×D411yescharacteristic
62C2×Dic137C2×D411yescharacteristic
63C2×Dic137C2×Q811yescharacteristic
64C2×Dic137#4111yescharacteristic
65C2×Dic137#4111yescharacteristic
66C23×C137—21no—
67C23×C137—21no—
68C23×C137—22no—
69C23×C137—22no—
70C23×C137C4⋊C412yes—
71C23×C137C22×C411yescharacteristic
72C2×C4×C137—22no—
73C2×C4×C137—21no—
74C2×C4×C137—22no—
75C2×C4×C137C22⋊C412yes—
76C22×C137—22no—
77C22×C137—24no—
78C22×C137—41no—
79C22×C137—41no—
80C22×C137—42no—
81C22×C137C2×C22⋊C411yescharacteristic
82C22×C137C2×C4⋊C411yescharacteristic
83C22×C137#11711yescharacteristic
84C22×C137#13112yes—
85C22×C137#13112yes—
86C4×C137—22no—
87C4×C137—41no—
88Dic137—22no—
89Dic137—22no—
90Dic137—41no—
91Dic137#12111yescharacteristic
92Dic137#12111yescharacteristic
93Dic137C22⋊Q811yescharacteristic
94Dic137C22⋊Q811yescharacteristic
95C2×C137—42no—
96C2×C137#38011yescharacteristic
97C2×C137#40811yescharacteristic
98C2×C137#40811yescharacteristic
99C2×C137#42412yes—
100C2×C137#43312yes—
101C137#179811yescharacteristic, complemented, Sylow
102#1798—1371nomaximal, Sylow
103C2×#117—1371no—
104C2×C4⋊C8—1371no—
105C2×C4⋊C8—1371no—
106C2×C22⋊C8—1372no—
107#336—1372no—
108C23×C4—1371no—
109C22×C8—1372no—
110C22×C8—1372no—
111#117—2742no—
112#117—2742no—
113C2×C4⋊C4—1371no—
114C2×C4⋊C4—1371no—
115C2×C22⋊C4—1371no—
116C2×C22⋊C4#1058773911yescharacteristic
117C2×C42—1371no—
118C2×C42—1371no—
119C4⋊C8—2742no—
120C4⋊C8—2742no—
121C22⋊C8—2744no—
122C24C2×#1058773911yescharacteristic
123C22×C4—2741no—
124C22×C4—2741no—
125C22×C4—2741no—
126C22×C4—2741no—
127C22×C4—1371no—
128C22×C4—1372no—
129C22×C4—1372no—
130C22×C4—1371no—
131C22×C4—1371no—
132C22×C4—1371no—
133C22×C4C2×#1058773911yescharacteristic
134C22×C4C2×#1058773911yescharacteristic
135C2×C8—2742no—
136C2×C8—2742no—
137C2×C8—2742no—
138C2×C8—2742no—
139C2×C8—2742no—
140C2×C8—2742no—
141C4⋊C4—2742no—
142C4⋊C4—2742no—
143C22⋊C4—22no—
144C22⋊C4—2742no—
145C42—2741no—
146C42—2741no—
147C42—1372no—
148C42—1372no—
149C23—21no—
150C23—21no—
151C23—22no—
152C23—22no—
153C23C4⋊#1058773912yes—
154C23C22×#1058773911yescharacteristic
155C2×C4—22no—
156C2×C4—21no—
157C2×C4—22no—
158C2×C4—2741no—
159C2×C4—2741no—
160C2×C4—2742no—
161C2×C4—2742no—
162C2×C4—2742no—
163C2×C4—2742no—
164C2×C4—2742no—
165C2×C4—2741no—
166C2×C4—2742no—
167C2×C4—1372no—
168C2×C4—1372no—
169C2×C4—1371no—
170C2×C4—1371no—
171C2×C4—1371no—
172C2×C4—1371no—
173C2×C4—1371no—
174C2×C4—1371no—
175C2×C4C22⋊#1058773912yes—
176C8—5482no—
177C8—5482no—
178C22—22no—
179C22—24no—
180C22—41no—
181C22—41no—
182C22—42no—
183C22C2×C22⋊#1058773911yescharacteristic
184C22C2×C4⋊#1058773911yescharacteristic
185C22#1798416711yescharacteristic
186C22#1798417512yes—
187C22#1798417512yes—
188C4—22no—
189C4—41no—
190C4—2742no—
191C4—2742no—
192C4—5481no—
193C4—1371no—
194C4—1371no—
195C4—1371no—
196C4—1371no—
197C2—42no—
198C2#2257834511yescharacteristic
199C2#2257837611yescharacteristic
200C2#2257837711yescharacteristic
201C2#2257840912yes—
202C2#2257842712yes—
subgroupG/H
center, FrattiniC23C22×#10587739
commutator, socleC23×C137C22×C4
FittingC2×C22⋊C4×C137C4
prime radicalC24×C137C2×C4
cosocleC22×Dic137C23
hypercenterC2×C22⋊C4#10587739
hypocenterC137#1798
LCSG ▷ C23×C137 ▷ C137
UCSC1 ◁ C23 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#17984155C2semidirect
2. C2×C4⋊#11764047C2semidirect
4. #22578269C2semidirect
101. C137#1798semidirect

Maximal subgroups

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

subgroupindex
C2×#179841552
C2×C4⋊#117640472
C2×C22⋊#117640472
#225782692
C2×#179840442
#1798137