Finite Group Databasebeta

← #31779419

Subgroups — #31779419

Order 17536.

200 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 57 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#22577938C211yesmaximal, characteristic
2#22577458C211yesmaximal, characteristic, complemented
3#22577458C211yesmaximal, characteristic, complemented
4#22577145C211yesmaximal, characteristic, complemented
5#22577142C211yesmaximal, characteristic, complemented
6#22577142C211yesmaximal, characteristic, complemented
7#330×C137C211yesmaximal, characteristic, complemented
8#17984083—21no—
9#17984076C2211yescharacteristic
10#17984073—21no—
11#17984068C411yescharacteristic
12#17984063C411yescharacteristic
13#17984053C411yescharacteristic
14#17984050—21no—
15#17984050C411yescharacteristic
16#17984046—21no—
17C42×D137C2211yescharacteristic
18#17984017—21no—
19#17984017—21no—
20#17984000—21no—
21#17984000—21no—
22#17983999C2211yescharacteristic
23#17983999C2211yescharacteristic
24#125×C137C2211yescharacteristic
25#97×C137C2211yescharacteristic
26#97×C137C2211yescharacteristic
27C2×Dic274C2×C411yescharacteristic
28C2×D548—21no—
29C2×D548C2×C411yescharacteristic
30C2×C4×D137—21no—
31C2×C4×D137C2×C411yescharacteristic
32C2×C4×D137D411yescharacteristic
33C2×C4×D137D411yescharacteristic
34C2×#11764040—21no—
35C2×#11764040—21no—
36C4⋊Dic137—21no—
37#14555489—21no—
38#14555487—21no—
39#14555487—21no—
40#14555487—21no—
41#14555487—21no—
42#14555487—21no—
43C4×Dic137—21no—
44C4×Dic137C2×C411yescharacteristic
45C4×Dic137D411yescharacteristic
46C4×Dic137D411yescharacteristic
47C2×C8×C137—21no—
48C2×C8×C137—21no—
49C4⋊C4×C137—21no—
50C4⋊C4×C137—21no—
51C4⋊C4×C137C2×C411yescharacteristic
52C4⋊C4×C137C2×C411yescharacteristic
53C42×C137C2311yescharacteristic
54C22×D137—21no—
55C22×D137C22⋊C411yescharacteristic
56Dic274—41no—
57D548—41no—
58D548—42no—
59C2×Dic137—21no—
60C2×Dic137C22⋊C411yescharacteristic
61C2×Dic137C22⋊C411yescharacteristic
62C2×Dic137C22⋊C411yescharacteristic
63C4×D137—41no—
64C4×D137—41no—
65C4×D137—41no—
66C4×D137—42no—
67#11764040—41no—
68#11764040—41no—
69C2×C4×C137—21no—
70C2×C4×C137—21no—
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—41no—
81C2×D137—41no—
82C2×D137—41no—
83C4×C137—21no—
84C4×C137—21no—
85C4×C137—21no—
86C4×C137—41no—
87C4×C137—41no—
88Dic137—21no—
89Dic137—21no—
90Dic137—21no—
91Dic137—41no—
92C2×C137#41311yescharacteristic
93C2×C137#42111yescharacteristic
94C2×C137#42111yescharacteristic
95D137—41no—
96D137—81no—
97C137#153511yescharacteristic, complemented, Sylow
98#1535—1371nomaximal, Sylow
99#529—1371no—
100#432—1371no—
101#432—1371no—
102#330—1371no—
103#330D13711yescharacteristic, complemented
104#329—1371no—
105#329—1371no—
106#126—2741no—
107#125C2×D13711yescharacteristic
108#124—1371no—
109#123—2741no—
110C22⋊Q8—1371no—
111#121—1371no—
112C4×Q8—1371no—
113C4×D4—2741no—
114C4×D4—1371no—
115#117—2741no—
116C2×C42—1371no—
117C4⋊C8—2741no—
118C4⋊C8—2741no—
119C22⋊C8—2741no—
120C22⋊C8—2741no—
121#97—1371no—
122#97—1371no—
123#97C2×D13711yescharacteristic
124#97C2×D13711yescharacteristic
125C2×Q8—1371no—
126C2×D4—2741no—
127C2×D4—1371no—
128C22×C4—2741no—
129C22×C4—1371no—
130C22×C4—1371no—
131C22×C4—1371no—
132C2×C8—21no—
133C2×C8—21no—
134C2×C8—2741no—
135C2×C8—2741no—
136C4⋊C4—21no—
137C4⋊C4—21no—
138C4⋊C4—2741no—
139C4⋊C4—2741no—
140C4⋊C4C4×D13711yescharacteristic
141C4⋊C4C4×D13711yescharacteristic
142C22⋊C4—2741no—
143C22⋊C4—2741no—
144C22⋊C4—2741no—
145C22⋊C4—2741no—
146C22⋊C4—2741no—
147C42—2741no—
148C42—1371no—
149C42—1371no—
150C42—1371no—
151C42C22×D13711yescharacteristic
152C23—2741no—
153C23—1371no—
154Q8—5481no—
155D4—5481no—
156D4—5482no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—2741no—
160C2×C4—5481no—
161C2×C4—5481no—
162C2×C4—5481no—
163C2×C4—5482no—
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—5481no—
178C22—5481no—
179C22—5481no—
180C22C22⋊C4×D13711yescharacteristic
181C4—21no—
182C4—21no—
183C4—21no—
184C4—41no—
185C4—41no—
186C4—2741no—
187C4—2741no—
188C4—2741no—
189C4—5481no—
190C2—5481no—
191C2—10961no—
192C2#2257738611yescharacteristic
193C2#2257741811yescharacteristic
194C2#2257741811yescharacteristic
subgroupG/H
centerC22C22⋊C4×D137
commutatorC2×C4×C137C22×C4
FrattiniC42C22×D137
Fitting#330×C137C2
socleC22×C137C2×C22⋊C4
prime radical#17984063C4
cosocleC42×C137C23
hypercenter#330D137
hypocenterC137#1535
LCSG ▷ C2×C4×C137 ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #330

Complements

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

NH ≅ G/Ntype
2. #22577458C2semidirect
3. #22577458C2semidirect
4. #22577145C2semidirect
5. #22577142C2semidirect
6. #22577142C2semidirect
7. #330×C137C2semidirect
97. C137#1535semidirect
103. #330D137semidirect

Maximal subgroups

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

subgroupindex
#225779382
#225774582
#225771452
#225771422
#330×C1372
#1535137