Finite Group Databasebeta

← #31779201

Subgroups — #31779201

Order 17536.

234 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 67 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#22577940C211yesmaximal, characteristic
2C4⋊C8×D137C211yesmaximal, characteristic, complemented
3C4×#14555493C211yesmaximal, characteristic, complemented
4Q8⋊#11764040C211yesmaximal, characteristic, complemented
5#22577132C211yesmaximal, characteristic, complemented
6#22577130C211yesmaximal, characteristic, complemented
7Q8⋊C8×C137C211yesmaximal, characteristic, complemented
8C2×#14555493—21no—
9C2×C8×D137—21no—
10#17984083—21no—
11#17984079C411yescharacteristic
12#17984077C411yescharacteristic
13#17984073—21no—
14#17984064C411yescharacteristic
15#17984063C411yescharacteristic
16C4×D548—21no—
17C4×D548C2211yescharacteristic
18#17984045—21no—
19C42×D137C2211yescharacteristic
20C4⋊#11764040C2211yescharacteristic
21#17984017—21no—
22#17983998—21no—
23C4×#11764040C2211yescharacteristic
24C4×Q8×C137C2211yescharacteristic
25C4⋊C8×C137C2211yescharacteristic
26C4×C8×C137C2211yescharacteristic
27C2×D548—21no—
28C2×D548C2×C411yescharacteristic
29C2×C4×D137—21no—
30C2×C4×D137C2×C411yescharacteristic
31C2×C4×D137D411yescharacteristic
32C2×C4×D137D411yescharacteristic
33#14555493—24no—
34C2×#11764040—21no—
35C2×#11764040—21no—
36C8×D137—42no—
37C4⋊Dic137—21no—
38C4⋊Dic137C2×C411yescharacteristic
39#14555489—21no—
40#14555489—21no—
41#14555487—21no—
42#14555487—21no—
43#14555487—21no—
44#14555487—21no—
45#14555487—21no—
46C4×Dic137C2×C411yescharacteristic
47C4×Dic137D411yescharacteristic
48C4×Dic137D411yescharacteristic
49C2×Q8×C137C2×C411yescharacteristic
50C2×C8×C137—21no—
51C2×C8×C137—21no—
52C4⋊C4×C137—21no—
53C4⋊C4×C137C2×C411yescharacteristic
54C42×C137—21no—
55C42×C137C2311yescharacteristic
56C22×D137—21no—
57C22×D137C22⋊C411yescharacteristic
58D548—21no—
59D548—21no—
60D548—42no—
61C2×Dic137—21no—
62C2×Dic137C22⋊C411yescharacteristic
63C2×Dic137C22⋊C411yescharacteristic
64C2×Dic137C22⋊C411yescharacteristic
65C4×D137—22no—
66C4×D137—21no—
67C4×D137—21no—
68C4×D137—21no—
69C4×D137—21no—
70C4×D137—41no—
71C4×D137—41no—
72#11764040—22no—
73#11764040—41no—
74Q8×C137—21no—
75Q8×C137—21no—
76C2×C4×C137—21no—
77C2×C4×C137—21no—
78C2×C4×C137C22×C411yescharacteristic
79C2×C4×C137C2×D411yescharacteristic
80C2×C4×C137C2×D411yescharacteristic
81C8×C137—22no—
82C8×C137—41no—
83C22×C137C2×C22⋊C411yescharacteristic
84C2×D137—21no—
85C2×D137—21no—
86C2×D137—41no—
87C2×D137—41no—
88C2×D137—41no—
89C2×D137#10412yes—
90C4×C137—21no—
91C4×C137—41no—
92C4×C137—41no—
93C4×C137#13111yescharacteristic
94C4×C137#13111yescharacteristic
95C4×C137AGL(1,ℤ/8)11yescharacteristic
96C4×C137#13711yescharacteristic
97Dic137—21no—
98Dic137—21no—
99Dic137—21no—
100Dic137—41no—
101C2×C137#40811yescharacteristic
102C2×C137#41711yescharacteristic
103C2×C137C2×#10411yescharacteristic
104D137—22no—
105D137—81no—
106C137#148311yescharacteristic, complemented, Sylow
107#1483—1371nomaximal, Sylow
108#529—1371no—
109C2×C4⋊C8—1371no—
110C4×M16—1371no—
111Q8⋊C8—1371no—
112Q8⋊C8D13711yescharacteristic, complemented
113D4⋊C8—1371no—
114D4⋊C8—1371no—
115C2×M16—2741no—
116C22×C8—2741no—
117#126—2741no—
118#125—1371no—
119#124—1371no—
120#123—2741no—
121C22⋊Q8—1371no—
122#121—1371no—
123C4×Q8C2×D13711yescharacteristic
124C4×D4—2741no—
125C4×D4—1371no—
126#117—2741no—
127C2×C42—1371no—
128C4⋊C8—2741no—
129C4⋊C8—1371no—
130C4⋊C8C2×D13711yescharacteristic
131#97—2741no—
132C4×C8—1371no—
133C4×C8C2×D13711yescharacteristic
134C2×Q8C4×D13711yescharacteristic
135C2×D4—2741no—
136C2×D4—1371no—
137C22×C4—2741no—
138C22×C4—1371no—
139C22×C4—1371no—
140C22×C4—1371no—
141M16—2744no—
142C2×C8—21no—
143C2×C8—21no—
144C2×C8—2741no—
145C2×C8—2741no—
146C2×C8—5482no—
147C4⋊C4—21no—
148C4⋊C4—2741no—
149C4⋊C4—2741no—
150C4⋊C4—2741no—
151C4⋊C4—1371no—
152C4⋊C4C4×D13711yescharacteristic
153C22⋊C4—2741no—
154C22⋊C4—2741no—
155C22⋊C4—2741no—
156C22⋊C4—2741no—
157C22⋊C4—2741no—
158C42—21no—
159C42—1371no—
160C42—1371no—
161C42—1371no—
162C42C22×D13711yescharacteristic
163C23—2741no—
164C23—1371no—
165Q8—21no—
166Q8—21no—
167D4—2741no—
168D4—2741no—
169D4—5482no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—2742no—
173C2×C4—2741no—
174C2×C4—2741no—
175C2×C4—2741no—
176C2×C4—2741no—
177C2×C4—2741no—
178C2×C4—5481no—
179C2×C4—5481no—
180C2×C4—1371no—
181C2×C4—1371no—
182C2×C4—1371no—
183C2×C4C2×C4×D13711yescharacteristic
184C2×C4D4×D13711yescharacteristic
185C2×C4D4×D13711yescharacteristic
186C8—22no—
187C8—41no—
188C8—2742no—
189C8—5481no—
190C22—2741no—
191C22—2741no—
192C22—5481no—
193C22—5481no—
194C22—5481no—
195C22—1372no—
196C22C22⋊C4×D13711yescharacteristic
197C4—21no—
198C4—41no—
199C4—41no—
200C4—2741no—
201C4—2741no—
202C4—2741no—
203C4—5481no—
204C4#1798409411yescharacteristic
205C4#1798409611yescharacteristic
206C4#1798411511yescharacteristic
207C4#1798412011yescharacteristic
208C2—2742no—
209C2—10961no—
210C2#2257736811yescharacteristic
211C2#2257740311yescharacteristic
212C2#104×D13711yescharacteristic
subgroupG/H
centerC2×C4D4×D137
commutatorC2×C4×C137C22×C4
FrattiniC42C22×D137
FittingQ8⋊C8×C137C2
socleC22×C137C2×C22⋊C4
prime radical#17984063C4
cosocleC42×C137C23
hypercenterQ8⋊C8D137
hypocenterC137#1483
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2×C4 ◁ C42 ◁ Q8⋊C8

Complements

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

NH ≅ G/Ntype
2. C4⋊C8×D137C2semidirect
3. C4×#14555493C2semidirect
4. Q8⋊#11764040C2semidirect
5. #22577132C2semidirect
6. #22577130C2semidirect
7. Q8⋊C8×C137C2semidirect
106. C137#1483semidirect
112. Q8⋊C8D137semidirect

Maximal subgroups

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

subgroupindex
#225779402
C4⋊C8×D1372
C4×#145554932
Q8⋊#117640402
#225771322
#225771302
Q8⋊C8×C1372
#1483137