Finite Group Databasebeta

← #31779493

Subgroups — #31779493

Order 17536.

242 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 65 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
3#22577455C211yesmaximal, characteristic, complemented
4#22577149C211yesmaximal, characteristic, complemented
5#22577146C211yesmaximal, characteristic, complemented
6#22577146C211yesmaximal, characteristic, complemented
7#332×C137C211yesmaximal, characteristic, complemented
8C2×#14555511—21no—
9#17984085C2211yescharacteristic
10#17984077—21no—
11#17984063—21no—
12#17984063C412yes—
13#17984050—21no—
14#17984050C412yes—
15C42×D137C2211yescharacteristic
16C4⋊#11764040C2211yescharacteristic
17C4⋊#11764040C2211yescharacteristic
18#17984000—21no—
19#17984000—21no—
20#17983998—21no—
21C8×Dic137—21no—
22C4⋊Q8×C137C2211yescharacteristic
23C4⋊C8×C137C2211yescharacteristic
24C4⋊C8×C137C2211yescharacteristic
25#14555511—42no—
26#14555511—42no—
27C2×D548—22no—
28C2×D548—21no—
29C2×D548—21no—
30C2×D548C2×C412yes—
31C2×C4×D137—22no—
32C2×C4×D137C2×C411yescharacteristic
33C2×C4×D137D411yescharacteristic
34C2×C4×D137D411yescharacteristic
35C2×#11764040—21no—
36C2×#11764040—21no—
37#14555487—22no—
38#14555487—22no—
39C4×Dic137C2×C411yescharacteristic
40C4×Dic137D411yescharacteristic
41C4×Dic137D411yescharacteristic
42C2×Q8×C137—21no—
43C2×C8×C137—21no—
44C2×C8×C137—21no—
45C4⋊C4×C137—21no—
46C4⋊C4×C137C2×C412yes—
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—21no—
60C4×D137—21no—
61C4×D137—21no—
62C4×D137—21no—
63C4×D137—41no—
64C4×D137—44no—
65#11764040—41no—
66#11764040—41no—
67Q8×C137—41no—
68Q8×C137—41no—
69C2×C4×C137—22no—
70C2×C4×C137C22×C411yescharacteristic
71C2×C4×C137C2×D411yescharacteristic
72C2×C4×C137C2×D411yescharacteristic
73C8×C137—41no—
74C8×C137—41no—
75C22×C137C2×C22⋊C411yescharacteristic
76C2×D137—21no—
77C2×D137—21no—
78C2×D137—21no—
79C2×D137—42no—
80C2×D137—42no—
81C2×D137—42no—
82C4×C137—21no—
83C4×C137—42no—
84C4×C137#13511yescharacteristic
85C4×C137#13511yescharacteristic
86C4×C137AGL(1,ℤ/8)11yescharacteristic
87C4×C137AGL(1,ℤ/8)11yescharacteristic
88Dic137—21no—
89Dic137—21no—
90Dic137—21no—
91C2×C137#41311yescharacteristic
92C2×C137#41611yescharacteristic
93C2×C137#41811yescharacteristic
94D137—41no—
95D137—82no—
96C137#155311yescharacteristic, complemented, Sylow
97#1553—1371nomaximal, Sylow
98#532—1371no—
99#432—1371no—
100#431—1371no—
101#332—1371no—
102#332D13711yescharacteristic, complemented
103#331—1371no—
104#331—1371no—
105C2×#41—2741no—
106C4⋊Q8C2×D13711yescharacteristic
107#127—1371no—
108#124—2741no—
109#121—2741no—
110#121—1372no—
111C4×D4—2741no—
112C4×D4—1372no—
113C2×C42—1371no—
114C4⋊C8—1371no—
115C4⋊C8—1371no—
116C4⋊C8C2×D13711yescharacteristic
117C4⋊C8C2×D13711yescharacteristic
118C22⋊C8—2741no—
119C22⋊C8—2741no—
120#97—2741no—
121C4×C8—2741no—
122#41—5482no—
123#41—5482no—
124C2×Q8—21no—
125C2×D4—2742no—
126C2×D4—2741no—
127C2×D4—2741no—
128C2×D4—1372no—
129C22×C4—2742no—
130C22×C4—1371no—
131C22×C4—1371no—
132C22×C4—1371no—
133C2×C8—21no—
134C2×C8—21no—
135C2×C8—2741no—
136C2×C8—2741no—
137C4⋊C4—21no—
138C4⋊C4C4×D13712yes—
139C22⋊C4—2742no—
140C22⋊C4—2742no—
141C42—1371no—
142C42—1371no—
143C42—1371no—
144C42C22×D13711yescharacteristic
145C23—2742no—
146C23—1371no—
147Q8—41no—
148Q8—41no—
149D4—5482no—
150D4—5484no—
151D4—5481no—
152D4—5481no—
153D4—5481no—
154D4—5481no—
155C2×C4—22no—
156C2×C4—2741no—
157C2×C4—2741no—
158C2×C4—2741no—
159C2×C4—2741no—
160C2×C4—5481no—
161C2×C4—5484no—
162C2×C4—1371no—
163C2×C4—1371no—
164C2×C4—1371no—
165C2×C4C2×C4×D13711yescharacteristic
166C2×C4D4×D13711yescharacteristic
167C2×C4D4×D13711yescharacteristic
168C8—41no—
169C8—41no—
170C8—5481no—
171C8—5481no—
172C22—2741no—
173C22—2741no—
174C22—2741no—
175C22—5482no—
176C22—5482no—
177C22—5482no—
178C22C22⋊C4×D13711yescharacteristic
179C4—21no—
180C4—42no—
181C4—2741no—
182C4—2741no—
183C4—2741no—
184C4#1798410911yescharacteristic
185C4#1798411111yescharacteristic
186C4#1798411211yescharacteristic
187C4#1798411511yescharacteristic
188C2—5481no—
189C2—10962no—
190C2#2257738611yescharacteristic
191C2#2257739811yescharacteristic
192C2#2257740611yescharacteristic
subgroupG/H
centerC22C22⋊C4×D137
commutatorC2×C4×C137C22×C4
FrattiniC42C22×D137
Fitting#332×C137C2
socleC22×C137C2×C22⋊C4
prime radical#22577954C2
cosocleC42×C137C23
hypercenter#332D137
hypocenterC137#1553
LCSG ▷ C2×C4×C137 ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
2. #22577459C2semidirect
3. #22577455C2semidirect
4. #22577149C2semidirect
5. #22577146C2semidirect
6. #22577146C2semidirect
7. #332×C137C2semidirect
96. C137#1553semidirect
102. #332D137semidirect

Maximal subgroups

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

subgroupindex
#225779542
#225774592
#225774552
#225771492
#225771462
#332×C1372
#1553137