Finite Group Databasebeta

← #31779751

Subgroups — #31779751

Order 17536.

244 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 49 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#22578002C211yesmaximal, characteristic
2#22577542C211yesmaximal, characteristic
3#22577322C211yesmaximal, characteristic, complemented
4#22577140C211yesmaximal, characteristic, complemented
5#22577138C211yesmaximal, characteristic
6#22577128C211yesmaximal, characteristic, complemented
7#451×C137C211yesmaximal, characteristic, complemented
8C2×#14555497—21no—
9#17984083—21no—
10#17984079—21no—
11#17984073—21no—
12#17984072—21no—
13#17984064—21no—
14#17984062C2211yescharacteristic
15#17984048—21no—
16C2×#14555489—21no—
17C2×#14555489C2211yescharacteristic
18#17984012—21no—
19C22⋊#11764040C2211yescharacteristic
20#17983995—21no—
21#17983995C2211yescharacteristic
22C22×Q8×C137C2211yescharacteristic
23C2×Q16×C137—21no—
24C22⋊Q8×C137C2211yescharacteristic
25#103×C137—21no—
26C22⋊C8×C137C2211yescharacteristic
27C2×D548D411yescharacteristic
28C2×C22⋊D137—21no—
29C22×Dic137—21no—
30C22×Dic137D411yescharacteristic
31C22×Dic137D411yescharacteristic
32C2×C4×D137—21no—
33#14555497—42no—
34C2×#11764040—21no—
35C4⋊Dic137—21no—
36C4⋊Dic137—21no—
37C4⋊Dic137D411yescharacteristic
38#14555489—21no—
39#14555489—21no—
40#14555489—21no—
41#14555489—42no—
42#14555487—21no—
43#14555487—21no—
44#14555487—21no—
45#14555487—21no—
46C4×Dic137—21no—
47C2×Q8×C137—21no—
48C2×Q8×C137—21no—
49C2×Q8×C137—42no—
50C2×Q8×C137D411yescharacteristic
51C22×C4×C137—21no—
52C22×C4×C137C2311yescharacteristic
53Q16×C137—42no—
54C2×C8×C137—21no—
55C4⋊C4×C137—21no—
56C4⋊C4×C137D411yescharacteristic
57C22⋊C4×C137—21no—
58C22×D137—21no—
59D548—41no—
60C22⋊D137—41no—
61C22⋊D137—41no—
62C2×Dic137—22no—
63C2×Dic137—21no—
64C2×Dic137—21no—
65C2×Dic137—21no—
66C2×Dic137—21no—
67C2×Dic137—21no—
68C2×Dic137—41no—
69C2×Dic137—41no—
70C2×Dic137—41no—
71C4×D137—81no—
72#11764040—41no—
73C23×C137C2×D411yescharacteristic
74Q8×C137—22no—
75Q8×C137—22no—
76Q8×C137—41no—
77Q8×C137—41no—
78Q8×C137—41no—
79C2×C4×C137—22no—
80C2×C4×C137—21no—
81C2×C4×C137—21no—
82C2×C4×C137—41no—
83C2×C4×C137—41no—
84C2×C4×C137—41no—
85C2×C4×C137C2×D411yescharacteristic
86C2×C4×C137C2×D411yescharacteristic
87C8×C137—41no—
88C22×C137—21no—
89C22×C137—21no—
90C22×C137#12011yescharacteristic
91C22×C137#13511yescharacteristic
92C22×C137#13711yescharacteristic
93C2×D137—41no—
94C2×D137—41no—
95C2×D137—41no—
96C4×C137—21no—
97C4×C137—21no—
98C4×C137—41no—
99C4×C137—42no—
100C4×C137—41no—
101Dic137—42no—
102Dic137—41no—
103Dic137—41no—
104Dic137—41no—
105C2×C137—21no—
106C2×C137—21no—
107C2×C137#45211yescharacteristic
108C2×C137#45611yescharacteristic
109C2×C137B(4,𝔽2)11yescharacteristic
110D137—81no—
111C137#160311yescharacteristic, complemented, Sylow
112#1603—1371nomaximal, Sylow
113#539—1371no—
114#451D13711yescharacteristic, complemented
115#448—1371no—
116#395—1371no—
117#328—1371no—
118#327—1371no—
119#324—1371no—
120C22×Q8C2×D13711yescharacteristic
121C2×Q16—21no—
122C2×SD16—2741no—
123#126—2741no—
124#125—2741no—
125#123—2741no—
126#123—2741no—
127C22⋊Q8—2741no—
128C22⋊Q8C2×D13711yescharacteristic
129#121—1371no—
130C4×D4—2741no—
131C2×C4⋊C4—2741no—
132C2×C4⋊C4—1371no—
133#103—21no—
134#103—2741no—
135C22⋊C8—1371no—
136C22⋊C8C2×D13711yescharacteristic
137#95—2741no—
138#95—1371no—
139C2×Q8—21no—
140C2×Q8—21no—
141C2×Q8—42no—
142C2×Q8C22⋊D13711yescharacteristic
143C2×D4—2741no—
144C2×D4—1371no—
145C22×C4—21no—
146C22×C4—2741no—
147C22×C4—2741no—
148C22×C4—1371no—
149C22×C4—1371no—
150C22×C4C22×D13711yescharacteristic
151Q16—42no—
152SD16—5482no—
153C2×C8—21no—
154C2×C8—2741no—
155C4⋊C4—21no—
156C4⋊C4—2741no—
157C4⋊C4—2741no—
158C4⋊C4—2741no—
159C4⋊C4—2741no—
160C4⋊C4—2741no—
161C4⋊C4—5482no—
162C4⋊C4—1371no—
163C4⋊C4C22⋊D13711yescharacteristic
164C22⋊C4—21no—
165C22⋊C4—2741no—
166C22⋊C4—2741no—
167C22⋊C4—2741no—
168C22⋊C4—2741no—
169C42—2741no—
170C23—2741no—
171C23D4×D13711yescharacteristic
172Q8—22no—
173Q8—22no—
174Q8—41no—
175Q8—41no—
176Q8—41no—
177D4—5481no—
178D4—5481no—
179D4—5481no—
180C2×C4—22no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—2742no—
187C2×C4—2741no—
188C2×C4—2741no—
189C2×C4—2741no—
190C2×C4—2741no—
191C2×C4—2741no—
192C2×C4—5481no—
193C2×C4—5481no—
194C2×C4—5481no—
195C2×C4—10961no—
196C2×C4D4×D13711yescharacteristic
197C2×C4C2×C22⋊D13711yescharacteristic
198C8—41no—
199C8—5481no—
200C22—21no—
201C22—21no—
202C22—5481no—
203C22—5481no—
204C22—5481no—
205C22#1798405611yescharacteristic
206C22#1798411111yescharacteristic
207C22#1798412011yescharacteristic
208C4—21no—
209C4—21no—
210C4—41no—
211C4—42no—
212C4—41no—
213C4—5482no—
214C4—5481no—
215C4—5481no—
216C4—5481no—
217C2—21no—
218C2—21no—
219C2—10961no—
220C2#2257756411yescharacteristic
221C2#2257758411yescharacteristic
222C2#2257758911yescharacteristic
subgroupG/H
centerC22#17984056
commutator, cosocleC22×C4×C137C23
FrattiniC22×C4C22×D137
Fitting#451×C137C2
socleC22×C137#120
prime radical#17984062C22
hypercenter#451D137
hypocenterC137#1603
LCSG ▷ C22×C4×C137 ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
3. #22577322C2semidirect
4. #22577140C2semidirect
6. #22577128C2semidirect
7. #451×C137C2semidirect
111. C137#1603semidirect
114. #451D137semidirect

Maximal subgroups

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

subgroupindex
#225780022
#225775422
#225773222
#225771402
#225771382
#225771282
#451×C1372
#1603137