Finite Group Databasebeta

← #31779251

Subgroups — #31779251

Order 17536.

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

normal subgroups only

This group has 216 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#22577935C211yesmaximal, characteristic
2#22577454C211yesmaximal, characteristic, complemented
3#22577425C211yesmaximal, characteristic, complemented
4#22577135C211yesmaximal, characteristic, complemented
5#22577133C211yesmaximal, characteristic, complemented
6D4⋊#11764040C211yesmaximal, characteristic, complemented
7D4⋊C8×C137C211yesmaximal, characteristic, complemented
8C2×#14555493—21no—
9#17984081—21no—
10#17984079C411yescharacteristic
11#17984075C411yescharacteristic
12#17984070—21no—
13#17984068C411yescharacteristic
14#17984059C411yescharacteristic
15C4×Dic274C2211yescharacteristic
16C4×C22⋊D137—21no—
17#17984047—21no—
18C42×D137C2211yescharacteristic
19C4⋊#11764040C2211yescharacteristic
20#17984017—21no—
21#17984017—21no—
22#17984000—21no—
23C4×#11764040C2211yescharacteristic
24C4×D4×C137C2211yescharacteristic
25C4⋊C8×C137C2211yescharacteristic
26C4×C8×C137C2211yescharacteristic
27C2×Dic274C2×C411yescharacteristic
28C2×C22⋊D137—21no—
29C2×C4×D137C2×C411yescharacteristic
30C2×C4×D137D411yescharacteristic
31C2×C4×D137D411yescharacteristic
32#14555493—42no—
33C2×#11764040—21no—
34C2×#11764040—21no—
35C4⋊Dic137—21no—
36C4⋊Dic137—21no—
37C4⋊Dic137C2×C411yescharacteristic
38#14555489—21no—
39#14555489—21no—
40C22⋊Dic137—21no—
41C22⋊Dic137—21no—
42C22⋊Dic137—21no—
43#14555487—21no—
44C4×Dic137—21no—
45C4×Dic137C2×C411yescharacteristic
46C4×Dic137D411yescharacteristic
47C4×Dic137D411yescharacteristic
48C2×D4×C137C2×C411yescharacteristic
49C22×C4×C137—21no—
50C2×C8×C137—21no—
51C2×C8×C137—21no—
52C4⋊C4×C137C2×C411yescharacteristic
53C22⋊C4×C137—21no—
54C42×C137C2311yescharacteristic
55C22×D137C22⋊C411yescharacteristic
56Dic274—21no—
57Dic274—21no—
58C22⋊D137—42no—
59C2×Dic137—21no—
60C2×Dic137—21no—
61C2×Dic137C22⋊C411yescharacteristic
62C2×Dic137C22⋊C411yescharacteristic
63C2×Dic137C22⋊C411yescharacteristic
64C4×D137—21no—
65C4×D137—21no—
66C4×D137—21no—
67C4×D137—21no—
68C4×D137—41no—
69#11764040—41no—
70#11764040—41no—
71C23×C137—21no—
72D4×C137—21no—
73D4×C137—21no—
74C2×C4×C137—21no—
75C2×C4×C137—41no—
76C2×C4×C137—41no—
77C2×C4×C137C22×C411yescharacteristic
78C2×C4×C137C2×D411yescharacteristic
79C2×C4×C137C2×D411yescharacteristic
80C8×C137—41no—
81C8×C137—41no—
82C22×C137—41no—
83C22×C137—41no—
84C22×C137—41no—
85C22×C137C2×C22⋊C411yescharacteristic
86C2×D137—21no—
87C2×D137—21no—
88C2×D137—21no—
89C4×C137—21no—
90C4×C137—41no—
91C4×C137#13111yescharacteristic
92C4×C137#13111yescharacteristic
93C4×C137AGL(1,ℤ/8)11yescharacteristic
94C4×C137#13711yescharacteristic
95Dic137—21no—
96Dic137—21no—
97Dic137—21no—
98Dic137—41no—
99Dic137—41no—
100C2×C137—81no—
101C2×C137#40811yescharacteristic
102C2×C137#41711yescharacteristic
103C2×C137#42111yescharacteristic
104D137—41no—
105C137#149511yescharacteristic, complemented, Sylow
106#1495—1371nomaximal, Sylow
107#529—1371no—
108#431—1371no—
109#423—1371no—
110Q8⋊C8—1371no—
111Q8⋊C8—1371no—
112D4⋊C8—1371no—
113D4⋊C8D13711yescharacteristic, complemented
114C2×M16—2741no—
115#126—2741no—
116#125—1371no—
117#124—1371no—
118#123—2741no—
119C22⋊Q8—1371no—
120#121—1371no—
121C4×Q8—1371no—
122C4×D4—2741no—
123C4×D4C2×D13711yescharacteristic
124#117—2741no—
125C2×C42—1371no—
126C4⋊C8—2741no—
127C4⋊C8—2741no—
128C4⋊C8—1371no—
129C4⋊C8C2×D13711yescharacteristic
130C22⋊C8—2741no—
131C4×C8—1371no—
132C4×C8C2×D13711yescharacteristic
133C2×Q8—1371no—
134C2×D4—2741no—
135C2×D4C4×D13711yescharacteristic
136C22×C4—21no—
137C22×C4—1371no—
138C22×C4—1371no—
139C22×C4—1371no—
140M16—5482no—
141C2×C8—21no—
142C2×C8—21no—
143C2×C8—2741no—
144C2×C8—2741no—
145C4⋊C4—2741no—
146C4⋊C4—2741no—
147C4⋊C4—2741no—
148C4⋊C4—2741no—
149C4⋊C4—1371no—
150C4⋊C4C4×D13711yescharacteristic
151C22⋊C4—21no—
152C22⋊C4—2741no—
153C22⋊C4—2741no—
154C22⋊C4—2741no—
155C22⋊C4—2741no—
156C42—2741no—
157C42—1371no—
158C42—1371no—
159C42—1371no—
160C42C22×D13711yescharacteristic
161C23—21no—
162C23—1371no—
163Q8—2741no—
164Q8—2741no—
165D4—21no—
166D4—21no—
167D4—5482no—
168C2×C4—21no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—2741no—
172C2×C4—2741no—
173C2×C4—2741no—
174C2×C4—2741no—
175C2×C4—2741no—
176C2×C4—2741no—
177C2×C4—5481no—
178C2×C4—1371no—
179C2×C4—1371no—
180C2×C4—1371no—
181C2×C4C2×C4×D13711yescharacteristic
182C2×C4D4×D13711yescharacteristic
183C2×C4D4×D13711yescharacteristic
184C8—41no—
185C8—41no—
186C8—5481no—
187C8—5481no—
188C22—41no—
189C22—41no—
190C22—41no—
191C22—2741no—
192C22—2741no—
193C22—2741no—
194C22C22⋊C4×D13711yescharacteristic
195C4—21no—
196C4—41no—
197C4—2741no—
198C4—2741no—
199C4—2741no—
200C4—5481no—
201C4—5481no—
202C4#1798409411yescharacteristic
203C4#1798409611yescharacteristic
204C4#1798411211yescharacteristic
205C4#1798411711yescharacteristic
206C2—81no—
207C2—5481no—
208C2#2257736811yescharacteristic
209C2#2257740111yescharacteristic
210C2#2257741811yescharacteristic
subgroupG/H
centerC2×C4D4×D137
commutatorC2×C4×C137C22×C4
FrattiniC42C22×D137
FittingD4⋊C8×C137C2
socleC22×C137C2×C22⋊C4
prime radical#17984059C4
cosocleC42×C137C23
hypercenterD4⋊C8D137
hypocenterC137#1495
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2×C4 ◁ C42 ◁ D4⋊C8

Complements

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

NH ≅ G/Ntype
2. #22577454C2semidirect
3. #22577425C2semidirect
4. #22577135C2semidirect
5. #22577133C2semidirect
6. D4⋊#11764040C2semidirect
7. D4⋊C8×C137C2semidirect
105. C137#1495semidirect
113. D4⋊C8D137semidirect

Maximal subgroups

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

subgroupindex
#225779352
#225774542
#225774252
#225771352
#225771332
D4⋊#117640402
D4⋊C8×C1372
#1495137