Finite Group Databasebeta

← #31783877

Subgroups — #31783877

Order 17536.

496 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 209 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×#17984086C211yesmaximal, characteristic
2#22577310C211yesmaximal, characteristic
3#22577307C212yesmaximal
4#22577300C212yesmaximal
5#22577298C212yesmaximal
6C4×#14555489C212yesmaximal
7#22577254C212yesmaximal
8#22577254C212yesmaximal
9#391×C137C211yesmaximal, characteristic
10C22×Dic274C2211yescharacteristic
11#17984086C418yescomplemented
12C2×C4⋊Dic137C2211yescharacteristic
13C2×#14555489C2212yes—
14C2×#14555489C2212yes—
15C2×#14555489C2212yes—
16C2×#14555489C2212yes—
17C2×#14555489C2212yes—
18C2×C4×Dic137C2212yes—
19C2×C4×Dic137C2212yes—
20C2×C4×Dic137C2211yescharacteristic
21C2×C4×Dic137C2212yes—
22C2×C4×Dic137C2211yescharacteristic
23#17983995C2212yes—
24#17983995C2212yes—
25#17983995C2212yes—
26#17983995C2212yes—
27C22×Q8×C137C2211yescharacteristic
28C2×C4⋊C4×C137C2211yescharacteristic
29C2×C42×C137C2211yescharacteristic
30#95×C137C2212yes—
31#95×C137C2212yes—
32C2×Dic274—41no—
33C2×Dic274—41no—
34C2×Dic274C2×C414yes—
35C22×Dic137C2312yes—
36C22×Dic137C2312yes—
37C22×Dic137C2312yes—
38C22×Dic137C2312yes—
39C4⋊Dic137—41no—
40#14555489—24no—
41#14555489—42no—
42#14555489C2×C418yes—
43#14555489C2×C418yes—
44#14555489Q818yescomplemented
45C4×Dic137—24no—
46C4×Dic137—24no—
47C4×Dic137—42no—
48C4×Dic137—41no—
49C4×Dic137C2×C414yes—
50C2×Q8×C137—41no—
51C2×Q8×C137—41no—
52C2×Q8×C137C2×C414yes—
53C22×C4×C137C2312yes—
54C22×C4×C137C2311yescharacteristic
55C22×C4×C137C2311yescharacteristic
56C22×C4×C137C2311yescharacteristic
57C22×C4×C137C2312yes—
58C4⋊C4×C137—41no—
59C42×C137—41no—
60Dic274—44no—
61C2×Dic137—22no—
62C2×Dic137—22no—
63C2×Dic137—22no—
64C2×Dic137—22no—
65C2×Dic137—22no—
66C2×Dic137—22no—
67C2×Dic137—22no—
68C2×Dic137C22×C414yes—
69C2×Dic137C22×C414yes—
70C2×Dic137C2×Q814yes—
71C2×Dic137C2×Q814yes—
72C2×Dic137#4114yes—
73C23×C137C2411yescharacteristic
74Q8×C137—44no—
75C2×C4×C137—22no—
76C2×C4×C137—21no—
77C2×C4×C137—22no—
78C2×C4×C137—21no—
79C2×C4×C137—21no—
80C2×C4×C137—21no—
81C2×C4×C137—21no—
82C2×C4×C137—21no—
83C2×C4×C137—22no—
84C2×C4×C137—22no—
85C2×C4×C137—22no—
86C2×C4×C137C22×C412yes—
87C2×C4×C137C22×C412yes—
88C2×C4×C137C22×C412yes—
89C2×C4×C137C2×Q814yes—
90C22×C137C23×C411yescharacteristic
91C22×C137C22×Q812yes—
92C22×C137C2×#4112yes—
93C22×C137H(5,𝔽2)11yescharacteristic
94C22×C137H(5,𝔽2)11yescharacteristic
95C4×C137—22no—
96C4×C137—22no—
97C4×C137—22no—
98C4×C137—24no—
99C4×C137—42no—
100Dic137—24no—
101Dic137—24no—
102Dic137—42no—
103Dic137C4×Q818yes—
104C2×C137C2×C4×Q812yes—
105C2×C137#51811yescharacteristic
106C2×C137#55412yes—
107C2×C137#55511yescharacteristic
108C2×C137Q8211yescharacteristic
109C137#236911yescharacteristic, complemented, Sylow
110#2369—1371nomaximal, Sylow
111C2×C4⋊Q8—1371no—
112#391—1371no—
113#391—1372no—
114#391D13711yescharacteristic
115#389—1372no—
116#389—1372no—
117C4×C4⋊C4—1372no—
118C4×C4⋊C4—1372no—
119C4×C4⋊C4—1372no—
120C22×Q8—1371no—
121C22×Q8C2×D13711yescharacteristic
122C4⋊Q8—1378no—
123C2×C4⋊C4—1372no—
124C2×C4⋊C4—1372no—
125C2×C4⋊C4—1371no—
126C2×C4⋊C4—1372no—
127C2×C4⋊C4—1372no—
128C2×C4⋊C4—1372no—
129C2×C4⋊C4C2×D13711yescharacteristic
130C2×C42—1372no—
131C2×C42—1372no—
132C2×C42—1371no—
133C2×C42—1372no—
134C2×C42—1371no—
135C2×C42C2×D13711yescharacteristic
136#95—1372no—
137#95—1372no—
138#95—1372no—
139#95—1372no—
140#95C2×D13712yes—
141#95C2×D13712yes—
142C2×Q8—41no—
143C2×Q8—41no—
144C2×Q8—5481no—
145C2×Q8—5481no—
146C2×Q8—1374no—
147C2×Q8C4×D13714yes—
148C22×C4—1372no—
149C22×C4—1372no—
150C22×C4—1372no—
151C22×C4—1372no—
152C22×C4C22×D13712yes—
153C22×C4C22×D13711yescharacteristic
154C22×C4C22×D13711yescharacteristic
155C22×C4C22×D13711yescharacteristic
156C22×C4C22×D13712yes—
157C4⋊C4—41no—
158C4⋊C4—2744no—
159C4⋊C4—5482no—
160C4⋊C4—5481no—
161C4⋊C4—1378no—
162C4⋊C4—1378no—
163C4⋊C4—1378no—
164C42—41no—
165C42—2744no—
166C42—2744no—
167C42—5482no—
168C42—5481no—
169C42—1374no—
170C23C23×D13711yescharacteristic
171Q8—44no—
172Q8—5484no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—2742no—
185C2×C4—2742no—
186C2×C4—2742no—
187C2×C4—2742no—
188C2×C4—2742no—
189C2×C4—2742no—
190C2×C4—2742no—
191C2×C4—1374no—
192C2×C4—1374no—
193C2×C4—1374no—
194C2×C4—1374no—
195C2×C4—1374no—
196C2×C4C2×C4×D13712yes—
197C2×C4C2×C4×D13712yes—
198C2×C4C2×C4×D13712yes—
199C2×C4Q8×D13714yes—
200C22C22×C4×D13711yescharacteristic
201C22C2×Q8×D13712yes—
202C22#41×D13712yes—
203C22#1798413311yescharacteristic
204C22#1798413311yescharacteristic
205C4—24no—
206C4—22no—
207C4—22no—
208C4—22no—
209C4—42no—
210C4—2744no—
211C4—2744no—
212C4—5482no—
213C4—1378no—
214C2C4×Q8×D13712yes—
215C2#2257787611yescharacteristic
216C2#2257810712yes—
217C2#2257811511yescharacteristic
218C2#2257813311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D137
commutatorC22×C137C23×C4
Fitting#391×C137C2
socle, prime radical, cosocleC23×C137C24
hypercenter#391D137
hypocenterC137#2369
LCSG ▷ C22×C137 ▷ C137
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
11. #17984086C4semidirect
44. #14555489Q8semidirect
109. C137#2369semidirect

Maximal subgroups

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

subgroupindex
C2×#179840862
#225773102
#225773072
#225773002
#225772982
C4×#145554892
#225772542
#391×C1372
#2369137