Finite Group Databasebeta

← #31779483

Subgroups — #31779483

Order 17536.

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

normal subgroups only

This group has 252 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#22577954C211yesmaximal, characteristic
2#22577459C211yesmaximal, characteristic, complemented
3C4⋊C8×D137C211yesmaximal, characteristic, complemented
4#22577152C211yesmaximal, characteristic, complemented
5#22577146C211yesmaximal, characteristic, complemented
6#22577146C211yesmaximal, characteristic, complemented
7#333×C137C211yesmaximal, characteristic, complemented
8C2×#14555511C412yes—
9C2×C8×D137—21no—
10#17984085C2211yescharacteristic
11#17984077C412yes—
12#17984063—21no—
13#17984063—21no—
14#17984050—21no—
15#17984050—21no—
16C42×D137C2211yescharacteristic
17C4⋊#11764040C2211yescharacteristic
18C4⋊#11764040C2211yescharacteristic
19#17984017—21no—
20#17984000—21no—
21#17983998—21no—
22C4⋊Q8×C137C2211yescharacteristic
23C4⋊C8×C137C2211yescharacteristic
24C4⋊C8×C137C2211yescharacteristic
25#14555511—44no—
26C2×D548—22no—
27C2×D548—21no—
28C2×D548—21no—
29C2×D548C2×C412yes—
30C2×C4×D137—22no—
31C2×C4×D137C2×C411yescharacteristic
32C2×C4×D137D411yescharacteristic
33C2×C4×D137D411yescharacteristic
34C2×#11764040—21no—
35C2×#11764040—21no—
36C8×D137—42no—
37#14555487—22no—
38#14555487—22no—
39C4×Dic137C2×C411yescharacteristic
40C4×Dic137D411yescharacteristic
41C4×Dic137D411yescharacteristic
42C2×Q8×C137C2×C412yes—
43C2×C8×C137—21no—
44C2×C8×C137—21no—
45C4⋊C4×C137—21no—
46C4⋊C4×C137—21no—
47C42×C137C2311yescharacteristic
48C22×D137—22no—
49C22×D137C22⋊C411yescharacteristic
50D548—42no—
51D548—44no—
52D548—41no—
53D548—41no—
54D548—41no—
55D548—41no—
56C2×Dic137C22⋊C411yescharacteristic
57C2×Dic137C22⋊C411yescharacteristic
58C2×Dic137C22⋊C411yescharacteristic
59C4×D137—22no—
60C4×D137—21no—
61C4×D137—21no—
62C4×D137—21no—
63C4×D137—21no—
64C4×D137—44no—
65#11764040—41no—
66#11764040—41no—
67Q8×C137—42no—
68C2×C4×C137—22no—
69C2×C4×C137C22×C411yescharacteristic
70C2×C4×C137C2×D411yescharacteristic
71C2×C4×C137C2×D411yescharacteristic
72C8×C137—41no—
73C8×C137—41no—
74C22×C137C2×C22⋊C411yescharacteristic
75C2×D137—21no—
76C2×D137—21no—
77C2×D137—42no—
78C2×D137—42no—
79C2×D137—42no—
80C2×D137#10012yes—
81C4×C137—21no—
82C4×C137—42no—
83C4×C137#13511yescharacteristic
84C4×C137#13511yescharacteristic
85C4×C137AGL(1,ℤ/8)11yescharacteristic
86C4×C137#13711yescharacteristic
87Dic137—21no—
88Dic137—21no—
89Dic137—21no—
90C2×C137C2×#10011yescharacteristic
91C2×C137#41611yescharacteristic
92C2×C137#41711yescharacteristic
93D137—22no—
94D137—82no—
95C137#155111yescharacteristic, complemented, Sylow
96#1551—1371nomaximal, Sylow
97#532—1371no—
98#432—1371no—
99C2×C4⋊C8—1371no—
100#333—1371no—
101#333D13711yescharacteristic, complemented
102#331—1371no—
103#331—1371no—
104C2×#41—1372no—
105C22×C8—2741no—
106C4⋊Q8C2×D13711yescharacteristic
107#127—1371no—
108#124—1372no—
109#121—2741no—
110#121—2741no—
111C4×D4—2741no—
112C4×D4—2741no—
113C2×C42—1371no—
114C4⋊C8—2741no—
115C4⋊C8—1371no—
116C4⋊C8—1371no—
117C4⋊C8C2×D13711yescharacteristic
118C4⋊C8C2×D13711yescharacteristic
119C22⋊C8—2741no—
120#97—2741no—
121#41—5484no—
122C2×Q8C4×D13712yes—
123C2×D4—2742no—
124C2×D4—2741no—
125C2×D4—2741no—
126C2×D4—1372no—
127C22×C4—2742no—
128C22×C4—1371no—
129C22×C4—1371no—
130C22×C4—1371no—
131C2×C8—21no—
132C2×C8—21no—
133C2×C8—2741no—
134C2×C8—2741no—
135C2×C8—5482no—
136C4⋊C4—21no—
137C4⋊C4—21no—
138C22⋊C4—2742no—
139C22⋊C4—2742no—
140C42—1371no—
141C42—1371no—
142C42—1371no—
143C42C22×D13711yescharacteristic
144C23—2742no—
145C23—1371no—
146Q8—42no—
147D4—5482no—
148D4—5484no—
149D4—5481no—
150D4—5481no—
151D4—5481no—
152D4—5481no—
153C2×C4—22no—
154C2×C4—2742no—
155C2×C4—2741no—
156C2×C4—2741no—
157C2×C4—2741no—
158C2×C4—2741no—
159C2×C4—5484no—
160C2×C4—1371no—
161C2×C4—1371no—
162C2×C4—1371no—
163C2×C4C2×C4×D13711yescharacteristic
164C2×C4D4×D13711yescharacteristic
165C2×C4D4×D13711yescharacteristic
166C8—41no—
167C8—41no—
168C8—5481no—
169C8—5481no—
170C22—2741no—
171C22—2741no—
172C22—5482no—
173C22—5482no—
174C22—5482no—
175C22—1372no—
176C22C22⋊C4×D13711yescharacteristic
177C4—21no—
178C4—42no—
179C4—2741no—
180C4—2741no—
181C4—2741no—
182C4#1798410911yescharacteristic
183C4#1798411111yescharacteristic
184C4#1798411511yescharacteristic
185C4#1798412011yescharacteristic
186C2—2742no—
187C2—10962no—
188C2#100×D13711yescharacteristic
189C2#2257739811yescharacteristic
190C2#2257740311yescharacteristic
subgroupG/H
centerC22C22⋊C4×D137
commutatorC2×C4×C137C22×C4
FrattiniC42C22×D137
Fitting#333×C137C2
socleC22×C137C2×C22⋊C4
prime radical#22577954C2
cosocleC42×C137C23
hypercenter#333D137
hypocenterC137#1551
LCSG ▷ C2×C4×C137 ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #333

Complements

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

NH ≅ G/Ntype
2. #22577459C2semidirect
3. C4⋊C8×D137C2semidirect
4. #22577152C2semidirect
5. #22577146C2semidirect
6. #22577146C2semidirect
7. #333×C137C2semidirect
95. C137#1551semidirect
101. #333D137semidirect

Maximal subgroups

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

subgroupindex
#225779542
#225774592
C4⋊C8×D1372
#225771522
#225771462
#333×C1372
#1551137