Finite Group Databasebeta

← #31789269

Subgroups — #31789269

Order 17536.

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

normal subgroups only

This group has 704 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×#17984136C211yesmaximal, characteristic
2#22578169C212yesmaximal, complemented
3#22577386C214yesmaximal, complemented
4#22577385C212yesmaximal, complemented
5C2×#17984007C211yesmaximal, characteristic, complemented
6#101×D137C214yesmaximal, complemented
7C2×#101×C137C211yesmaximal, characteristic, complemented
8#17984136—24no—
9#17984136—24no—
10C2×#14555511C412yes—
11C2×#14555511C2212yes—
12C2×#14555510C414yes—
13C2×#14555510C2211yescharacteristic
14C2×#14555510C2211yescharacteristic
15C2×Q8×D137C412yes—
16C2×Q8×D137C2212yes—
17#17984096—24no—
18#17984094—24no—
19C2×#14555494C2212yes—
20M16×D137C2218yes—
21#17984007C2214yes—
22#17984006C2214yes—
23#17984004C2214yes—
24C22×Q8×C137C2211yescharacteristic
25C2×M16×C137C2212yes—
26#101×C137C2214yes—
27#14555511—28no—
28#14555511—28no—
29#14555510—28no—
30#14555510—28no—
31#14555510—22no—
32#14555510—22no—
33#14555510D414yes—
34#14555510D414yes—
35C2×Dic274C2×C414yes—
36C2×Dic274C2×C411yescharacteristic
37C2×Dic274C2311yescharacteristic
38Q8×D137—28no—
39Q8×D137—28no—
40C2×D548C2×C414yes—
41C2×D548C2×C411yescharacteristic
42C2×D548C2311yescharacteristic
43C2×C22⋊D137C2×C412yes—
44C2×C22⋊D137C2×C412yes—
45C2×C4×D137C2×C414yes—
46C2×C4×D137C2×C412yes—
47C2×C4×D137C2×C414yes—
48C2×C4×D137C2312yes—
49#14555494—22no—
50#14555494C2314yes—
51#14555493—28no—
52C2×#11764040—22no—
53C8×D137—28no—
54C2×Q8×C137—22no—
55C2×Q8×C137—22no—
56C2×Q8×C137C2×C412yes—
57C2×Q8×C137C2312yes—
58C22×C4×C137C2×C412yes—
59C22×C4×C137C2311yescharacteristic
60M16×C137—22no—
61M16×C137C2314yes—
62C2×C8×C137—22no—
63C22×D137C22×C412yes—
64C22×D137C22×C412yes—
65Dic274—22no—
66Dic274—28no—
67Dic274C22⋊C414yes—
68D548—28no—
69D548—22no—
70D548C22⋊C414yes—
71C22⋊D137—22no—
72C22⋊D137—22no—
73C22⋊D137C22⋊C414yes—
74C22⋊D137C22⋊C414yes—
75C2×Dic137C22×C412yes—
76C2×Dic137C22×C412yes—
77C4×D137—28no—
78C4×D137—24no—
79C4×D137—28no—
80C4×D137C2×D418yes—
81#11764040—24no—
82C23×C137C22×C411yescharacteristic
83Q8×C137—24no—
84Q8×C137—24no—
85C2×C4×C137—22no—
86C2×C4×C137—22no—
87C2×C4×C137C22×C414yes—
88C2×C4×C137C22×C411yescharacteristic
89C2×C4×C137C2×D412yes—
90C2×C4×C137C2×D412yes—
91C2×C4×C137C2411yescharacteristic
92C8×C137—24no—
93C22×C137—41no—
94C22×C137C2×C22⋊C411yescharacteristic
95C22×C137C2×C22⋊C411yescharacteristic
96C22×C137C23×C411yescharacteristic
97C2×D137—22no—
98C2×D137—42no—
99C2×D137—42no—
100C2×D137C2×C22⋊C414yes—
101C4×C137—24no—
102C4×C137C2×C22⋊C412yes—
103C4×C137C22×D412yes—
104Dic137—22no—
105Dic137C2×C22⋊C414yes—
106C2×C137—21no—
107C2×C137—21no—
108C2×C137—21no—
109C2×C137C22×C22⋊C411yescharacteristic
110D137—24no—
111D137—42no—
112C137#288911yescharacteristic, complemented, Sylow
113#2889—1371nomaximal, Sylow
114C2×2−1+4—1371no—
115#568—1372no—
116#413—1372no—
117#413—1374no—
118C2×#101—1371no—
119C2×#101—1374no—
120C2×#101D13711yescharacteristic, complemented
1212−1+4—2744no—
1222−1+4—2744no—
123C2×#41—1372no—
124C2×#41—1372no—
125C2×#41—1374no—
126C2×#41—1371no—
127C2×#41—1371no—
128C22×Q8—1372no—
129C22×Q8—1372no—
130C22×Q8C2×D13711yescharacteristic
131#131—2744no—
132#131—2744no—
133C2×M16—1372no—
134C2×M16—1378no—
135C2×M16C2×D13712yes—
136#101—1374no—
137#101—1374no—
138#101C2×D13714yes—
139#100—1374no—
140#41—2748no—
141#41—2748no—
142#41—2748no—
143#41—2748no—
144#41—2742no—
145#41—2742no—
146#41—1374no—
147#41—1374no—
148C2×Q8—22no—
149C2×Q8—22no—
150C2×Q8—2748no—
151C2×Q8—2748no—
152C2×Q8—1374no—
153C2×Q8—1371no—
154C2×Q8—1371no—
155C2×Q8C4×D13712yes—
156C2×Q8C22×D13712yes—
157C2×D4—1372no—
158C2×D4—1372no—
159C2×D4—1374no—
160C2×D4—1371no—
161C2×D4—1371no—
162C22×C4—1374no—
163C22×C4—1374no—
164C22×C4—1372no—
165C22×C4—1372no—
166C22×C4C4×D13712yes—
167C22×C4C22×D13711yescharacteristic
168M16—22no—
169M16—2748no—
170M16—2742no—
171M16—1374no—
172M16C22×D13714yes—
173C2×C8—22no—
174C2×C8—2748no—
175C2×C8—2742no—
176C23—1372no—
177C23—1372no—
178C23C2×C4×D13711yescharacteristic
179Q8—24no—
180Q8—24no—
181Q8—2748no—
182Q8—2742no—
183Q8—1374no—
184D4—2748no—
185D4—2742no—
186D4—2742no—
187D4—2742no—
188D4—1374no—
189D4—1374no—
190D4—1374no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—2748no—
194C2×C4—2748no—
195C2×C4—2744no—
196C2×C4—1372no—
197C2×C4—1372no—
198C2×C4—1378no—
199C2×C4C2×C4×D13714yes—
200C2×C4C2×C4×D13711yescharacteristic
201C2×C4D4×D13712yes—
202C2×C4D4×D13712yes—
203C2×C4C23×D13711yescharacteristic
204C8—24no—
205C8—2744no—
206C22—41no—
207C22—2742no—
208C22—5482no—
209C22—5482no—
210C22—1374no—
211C22C22⋊C4×D13711yescharacteristic
212C22C22⋊C4×D13711yescharacteristic
213C22C22×C4×D13711yescharacteristic
214C4—24no—
215C4—2742no—
216C4—1374no—
217C4C22⋊C4×D13712yes—
218C4C2×D4×D13712yes—
219C2—21no—
220C2—21no—
221C2—21no—
222C2—2744no—
223C2—5482no—
224C2C2×C22⋊C4×D13711yescharacteristic
subgroupG/H
centerC2C2×C22⋊C4×D137
commutatorC22×C137C23×C4
FrattiniC2×C4C23×D137
FittingC2×#101×C137C2
socleC2×C137C22×C22⋊C4
prime radicalC2×#17984136C2
cosocleC2×C4×C137C24
hypercenterC2×#101D137
hypocenterC137#2889
LCSG ▷ C22×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#101

Complements

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

NH ≅ G/Ntype
2. #22578169C2semidirect
3. #22577386C2semidirect
4. #22577385C2semidirect
5. C2×#17984007C2semidirect
6. #101×D137C2semidirect
7. C2×#101×C137C2semidirect
112. C137#2889semidirect
120. C2×#101D137semidirect

Maximal subgroups

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

subgroupindex
C2×#179841362
#225781692
#225773862
#225773852
C2×#179840072
#101×D1372
C2×#101×C1372
#2889137