Finite Group Databasebeta

← #31779510

Subgroups — #31779510

Order 17536.

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

normal subgroups only

This group has 246 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#22577425C211yesmaximal, characteristic, complemented
4#22577149C211yesmaximal, characteristic, complemented
5#22577146C211yesmaximal, characteristic, complemented
6#22577146C211yesmaximal, characteristic, complemented
7#332×C137C211yesmaximal, characteristic, complemented
8C2×#14555511—21no—
9C2×#14555493—21no—
10#17984085C2211yescharacteristic
11#17984077—21no—
12#17984063—21no—
13#17984063C412yes—
14#17984050—21no—
15#17984050C412yes—
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—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
35#14555493—42no—
36C2×#11764040—21no—
37C2×#11764040—21no—
38#14555487—22no—
39#14555487—22no—
40C4×Dic137C2×C411yescharacteristic
41C4×Dic137D411yescharacteristic
42C4×Dic137D411yescharacteristic
43C2×Q8×C137—21no—
44C2×C8×C137—21no—
45C2×C8×C137—21no—
46C4⋊C4×C137—21no—
47C4⋊C4×C137C2×C412yes—
48C42×C137C2311yescharacteristic
49C22×D137—22no—
50C22×D137C22⋊C411yescharacteristic
51D548—42no—
52D548—44no—
53D548—41no—
54D548—41no—
55D548—41no—
56D548—41no—
57C2×Dic137C22⋊C411yescharacteristic
58C2×Dic137C22⋊C411yescharacteristic
59C2×Dic137C22⋊C411yescharacteristic
60C4×D137—21no—
61C4×D137—21no—
62C4×D137—21no—
63C4×D137—21no—
64C4×D137—41no—
65C4×D137—44no—
66#11764040—41no—
67#11764040—41no—
68Q8×C137—41no—
69Q8×C137—41no—
70C2×C4×C137—22no—
71C2×C4×C137C22×C411yescharacteristic
72C2×C4×C137C2×D411yescharacteristic
73C2×C4×C137C2×D411yescharacteristic
74C8×C137—41no—
75C8×C137—41no—
76C22×C137C2×C22⋊C411yescharacteristic
77C2×D137—21no—
78C2×D137—21no—
79C2×D137—21no—
80C2×D137—42no—
81C2×D137—42no—
82C2×D137—42no—
83C4×C137—21no—
84C4×C137—42no—
85C4×C137AGL(1,ℤ/8)11yescharacteristic
86C4×C137AGL(1,ℤ/8)11yescharacteristic
87C4×C137AGL(1,ℤ/8)11yescharacteristic
88C4×C137#13711yescharacteristic
89Dic137—21no—
90Dic137—21no—
91Dic137—21no—
92C2×C137#41311yescharacteristic
93C2×C137#41711yescharacteristic
94C2×C137#41811yescharacteristic
95D137—41no—
96D137—82no—
97C137#155711yescharacteristic, complemented, Sylow
98#1557—1371nomaximal, Sylow
99#532—1371no—
100#432—1371no—
101#423—1371no—
102#332—1371no—
103#332D13711yescharacteristic, complemented
104#331—1371no—
105#331—1371no—
106C2×#41—2741no—
107C2×M16—2741no—
108C4⋊Q8C2×D13711yescharacteristic
109#127—1371no—
110#124—2741no—
111#121—2741no—
112#121—1372no—
113C4×D4—2741no—
114C4×D4—1372no—
115C2×C42—1371no—
116C4⋊C8—2741no—
117C4⋊C8—1371no—
118C4⋊C8—1371no—
119C4⋊C8C2×D13711yescharacteristic
120C4⋊C8C2×D13711yescharacteristic
121C22⋊C8—2741no—
122#97—2741no—
123#41—5482no—
124#41—5482no—
125C2×Q8—21no—
126C2×D4—2742no—
127C2×D4—2741no—
128C2×D4—2741no—
129C2×D4—1372no—
130C22×C4—2742no—
131C22×C4—1371no—
132C22×C4—1371no—
133C22×C4—1371no—
134M16—5482no—
135C2×C8—21no—
136C2×C8—21no—
137C2×C8—2741no—
138C2×C8—2741no—
139C4⋊C4—21no—
140C4⋊C4C4×D13712yes—
141C22⋊C4—2742no—
142C22⋊C4—2742no—
143C42—1371no—
144C42—1371no—
145C42—1371no—
146C42C22×D13711yescharacteristic
147C23—2742no—
148C23—1371no—
149Q8—41no—
150Q8—41no—
151D4—5482no—
152D4—5484no—
153D4—5481no—
154D4—5481no—
155D4—5481no—
156D4—5481no—
157C2×C4—22no—
158C2×C4—2741no—
159C2×C4—2741no—
160C2×C4—2741no—
161C2×C4—2741no—
162C2×C4—5481no—
163C2×C4—5484no—
164C2×C4—1371no—
165C2×C4—1371no—
166C2×C4—1371no—
167C2×C4C2×C4×D13711yescharacteristic
168C2×C4D4×D13711yescharacteristic
169C2×C4D4×D13711yescharacteristic
170C8—41no—
171C8—41no—
172C8—5481no—
173C8—5481no—
174C22—2741no—
175C22—2741no—
176C22—2741no—
177C22—5482no—
178C22—5482no—
179C22—5482no—
180C22C22⋊C4×D13711yescharacteristic
181C4—21no—
182C4—42no—
183C4—2741no—
184C4—2741no—
185C4—2741no—
186C4#1798411211yescharacteristic
187C4#1798411511yescharacteristic
188C4#1798411511yescharacteristic
189C4#1798412011yescharacteristic
190C2—5481no—
191C2—10962no—
192C2#2257738611yescharacteristic
193C2#2257740311yescharacteristic
194C2#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#1557
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. #22577425C2semidirect
4. #22577149C2semidirect
5. #22577146C2semidirect
6. #22577146C2semidirect
7. #332×C137C2semidirect
97. C137#1557semidirect
103. #332D137semidirect

Maximal subgroups

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

subgroupindex
#225779542
#225774592
#225774252
#225771492
#225771462
#332×C1372
#1557137