Finite Group Databasebeta

← #31783957

Subgroups — #31783957

Order 17536.

464 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 173 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×#17984067C211yesmaximal, characteristic
2#22577307C212yesmaximal, complemented
3#22577300C212yesmaximal, complemented
4#22577297C212yesmaximal
5#22577291C212yesmaximal
6#22577290C212yesmaximal, complemented
7#22577288C212yesmaximal, complemented
8#22577263C211yesmaximal, characteristic
9#381×C137C211yesmaximal, characteristic
10C22×Dic274C2211yescharacteristic
11#17984067C418yescomplemented
12C2×C4⋊Dic137C2212yes—
13C2×C4⋊Dic137C2211yescharacteristic
14C2×#14555489C2212yes—
15C2×#14555489C2212yes—
16C2×#14555489C2212yes—
17C2×C22⋊Dic137C2211yescharacteristic
18C2×C22⋊Dic137C2211yescharacteristic
19C2×C22⋊Dic137C2212yes—
20C2×C4×Dic137C2212yes—
21C2×C4×Dic137C2212yes—
22#17983995C2212yes—
23#17983995C2212yes—
24#17983995C2212yes—
25#17983995C2212yes—
26#17983995C2212yes—
27C23×C4×C137C2211yescharacteristic
28C2×C22⋊C4×C137C2212yes—
29#95×C137C2212yes—
30#95×C137C2211yescharacteristic
31#95×C137C2211yescharacteristic
32C2×Dic274—42no—
33C2×Dic274C2×C414yes—
34C22×Dic137C2312yes—
35C22×Dic137C2312yes—
36C22×Dic137C2312yes—
37C22×Dic137C2312yes—
38C4⋊Dic137—42no—
39C4⋊Dic137C2×C414yes—
40#14555489—42no—
41#14555489—42no—
42#14555489C2×C418yes—
43C22⋊Dic137—41no—
44C22⋊Dic137—41no—
45C22⋊Dic137C2×C418yes—
46C4×Dic137—42no—
47C4×Dic137—42no—
48C24×C137C2311yescharacteristic
49C22×C4×C137—42no—
50C22×C4×C137C2×C414yes—
51C22×C4×C137C2312yes—
52C22×C4×C137C2312yes—
53C22×C4×C137C2311yescharacteristic
54C22×C4×C137C2311yescharacteristic
55C22⋊C4×C137—42no—
56Dic274—44no—
57C2×Dic137—22no—
58C2×Dic137—22no—
59C2×Dic137—22no—
60C2×Dic137—22no—
61C2×Dic137—22no—
62C2×Dic137—22no—
63C2×Dic137—22no—
64C2×Dic137—22no—
65C2×Dic137—22no—
66C2×Dic137—22no—
67C2×Dic137C22×C414yes—
68C2×Dic137C22×C414yes—
69C23×C137—22no—
70C23×C137—21no—
71C23×C137—22no—
72C23×C137—21no—
73C23×C137C22×C412yes—
74C23×C137C2411yescharacteristic
75C2×C4×C137—22no—
76C2×C4×C137—22no—
77C2×C4×C137—22no—
78C2×C4×C137—22no—
79C2×C4×C137—22no—
80C2×C4×C137—22no—
81C2×C4×C137—22no—
82C2×C4×C137—22no—
83C2×C4×C137—44no—
84C2×C4×C137C22×C412yes—
85C2×C4×C137C22×C412yes—
86C22×C137—22no—
87C22×C137—22no—
88C22×C137—22no—
89C22×C137—42no—
90C22×C137—42no—
91C22×C137C23×C411yescharacteristic
92C22×C137H(5,𝔽2)12yes—
93C22×C137H(5,𝔽2)11yescharacteristic
94C22×C1372−1+412yes—
95C22×C1372−1+411yescharacteristic
96C4×C137—22no—
97C4×C137—22no—
98C4×C137—42no—
99C4×C137—42no—
100Dic137—24no—
101Dic137—24no—
102Dic137—42no—
103Dic137—42no—
104C2×C137—42no—
105C2×C137#51811yescharacteristic
106C2×C137#51911yescharacteristic
107C2×C137#52011yescharacteristic
108C2×C137#56212yes—
109C2×C137#56312yes—
110C137#237811yescharacteristic, complemented, Sylow
111#2378—1371nomaximal, Sylow
112C2×C22⋊Q8—1371no—
113#391—1372no—
114#389—1372no—
115#388—1372no—
116#388—1372no—
117#387—1372no—
118#387—1372no—
119#381D13711yescharacteristic
120#380—1371no—
121C22×Q8—1371no—
122C23×C4C2×D13711yescharacteristic
123C22⋊Q8—1378no—
124C2×C4⋊C4—1372no—
125C2×C4⋊C4—1372no—
126C2×C4⋊C4—1372no—
127C2×C4⋊C4—1372no—
128C2×C4⋊C4—1371no—
129C2×C22⋊C4—1371no—
130C2×C22⋊C4—1371no—
131C2×C22⋊C4—1372no—
132C2×C22⋊C4C2×D13712yes—
133C2×C42—1372no—
134C2×C42—1372no—
135#95—1372no—
136#95—1372no—
137#95—1372no—
138#95—1372no—
139#95—1372no—
140#95C2×D13711yescharacteristic
141#95C2×D13712yes—
142#95C2×D13711yescharacteristic
143C24C22×D13711yescharacteristic
144C2×Q8—5482no—
145C2×Q8—1374no—
146C22×C4—42no—
147C22×C4—1372no—
148C22×C4—1372no—
149C22×C4—1372no—
150C22×C4—1372no—
151C22×C4C4×D13714yes—
152C22×C4C22×D13712yes—
153C22×C4C22×D13712yes—
154C22×C4C22×D13711yescharacteristic
155C22×C4C22×D13711yescharacteristic
156C4⋊C4—5482no—
157C4⋊C4—5482no—
158C4⋊C4—5482no—
159C4⋊C4—1378no—
160C4⋊C4—1374no—
161C22⋊C4—42no—
162C22⋊C4—5481no—
163C22⋊C4—5481no—
164C22⋊C4—1378no—
165C42—5482no—
166C42—5482no—
167C23—22no—
168C23—21no—
169C23—22no—
170C23—21no—
171C23C2×C4×D13712yes—
172C23C23×D13711yescharacteristic
173Q8—5484no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—44no—
183C2×C4—2742no—
184C2×C4—2742no—
185C2×C4—2742no—
186C2×C4—2742no—
187C2×C4—2742no—
188C2×C4—2742no—
189C2×C4—2742no—
190C2×C4—2742no—
191C2×C4—2742no—
192C2×C4—2742no—
193C2×C4—1374no—
194C2×C4—1374no—
195C2×C4C2×C4×D13712yes—
196C2×C4C2×C4×D13712yes—
197C22—22no—
198C22—22no—
199C22—22no—
200C22—42no—
201C22—42no—
202C22C22×C4×D13711yescharacteristic
203C22#1798413312yes—
204C22#1798413411yescharacteristic
205C22#1798413512yes—
206C22#1798413511yescharacteristic
207C4—22no—
208C4—22no—
209C4—42no—
210C4—42no—
211C4—2744no—
212C4—2744no—
213C4—5482no—
214C4—5482no—
215C2—42no—
216C2#2257787611yescharacteristic
217C2#2257788011yescharacteristic
218C2#2257788311yescharacteristic
219C2#2257815012yes—
220C2#2257815612yes—
subgroupG/H
center, FrattiniC23C23×D137
commutatorC22×C137C23×C4
Fitting#381×C137C2
socle, cosocleC23×C137C24
prime radicalC24×C137C23
hypercenter#381D137
hypocenterC137#2378
LCSG ▷ C22×C137 ▷ C137
UCSC1 ◁ C23 ◁ #381

Complements

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

NH ≅ G/Ntype
2. #22577307C2semidirect
3. #22577300C2semidirect
6. #22577290C2semidirect
7. #22577288C2semidirect
11. #17984067C4semidirect
110. C137#2378semidirect

Maximal subgroups

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

subgroupindex
C2×#179840672
#225773072
#225773002
#225772972
#225772912
#225772902
#225772882
#225772632
#381×C1372
#2378137