Finite Group Databasebeta

← #31790441

Subgroups — #31790441

Order 17536.

460 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 141 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
1C22×#14555500C211yesmaximal, characteristic, complemented
2C2×#17984095C211yesmaximal, characteristic
3#22577974C211yesmaximal, characteristic
4#22577659C214yesmaximal, complemented
5#22577642C214yesmaximal, complemented
6#22577439C211yesmaximal, characteristic, complemented
7#22577412C211yesmaximal, characteristic, complemented
8#22577411C211yesmaximal, characteristic, complemented
9#536×C137C211yesmaximal, characteristic
10C22×Dic274C2211yescharacteristic
11C2×#14555500—24no—
12C2×#14555500C2212yes—
13C2×#14555500C2212yes—
14#17984095—24no—
15C2×#14555494C2211yescharacteristic
16C2×#14555494C2212yes—
17C22×#11764040C2212yes—
18C22×#11764040C2211yescharacteristic
19#17984086—21no—
20#17984075—21no—
21#17984069—22no—
22#17984067C2214yes—
23#17984047C2211yescharacteristic
24#17984021C2212yes—
25#17984019C2212yes—
26#17984012C2214yes—
27#17984011C2214yes—
28C2×#41×C137C2211yescharacteristic
29C22×Q8×C137C2211yescharacteristic
30C4⋊Q8×C137—21no—
31#124×C137—21no—
32#123×C137—22no—
33C22⋊Q8×C137C2214yes—
34#117×C137C2211yescharacteristic
35C2×Dic274—21no—
36C2×Dic274—22no—
37C2×Dic274—22no—
38C2×Dic274C2312yes—
39C22×Dic137—21no—
40#14555500—24no—
41#14555500—24no—
42#14555494—24no—
43#14555494—22no—
44C2×#11764040—24no—
45C2×#11764040D414yes—
46C2×#11764040C2312yes—
47C2×#11764040C2311yescharacteristic
48C2×#11764040C2311yescharacteristic
49C4⋊Dic137C2312yes—
50#14555489—24no—
51C22⋊Dic137—21no—
52C22⋊Dic137—24no—
53C4×Dic137—21no—
54#41×C137—22no—
55#41×C137—22no—
56C2×Q8×C137—21no—
57C2×Q8×C137—22no—
58C2×Q8×C137—22no—
59C2×Q8×C137D411yescharacteristic
60C2×Q8×C137C2312yes—
61C2×D4×C137D412yes—
62C2×D4×C137D411yescharacteristic
63C22×C4×C137—21no—
64C22×C4×C137C2312yes—
65C22×C4×C137C2311yescharacteristic
66C4⋊C4×C137—24no—
67C4⋊C4×C137C2312yes—
68C22⋊C4×C137—21no—
69C22⋊C4×C137—24no—
70C42×C137—21no—
71Dic274—22no—
72Dic274—22no—
73Dic274—42no—
74C2×Dic137—22no—
75C2×Dic137—22no—
76C2×Dic137—42no—
77#11764040—22no—
78#11764040C2×D412yes—
79#11764040C2×D412yes—
80C23×C137C2×D412yes—
81C23×C137C2×D411yescharacteristic
82Q8×C137—22no—
83Q8×C137—22no—
84Q8×C137—22no—
85Q8×C137—42no—
86D4×C137—24no—
87D4×C137—22no—
88C2×C4×C137—22no—
89C2×C4×C137—22no—
90C2×C4×C137—22no—
91C2×C4×C137—22no—
92C2×C4×C137—42no—
93C2×C4×C137C2×D412yes—
94C2×C4×C137C2×D412yes—
95C2×C4×C137C2×D411yescharacteristic
96C2×C4×C137#4112yes—
97C2×C4×C137C2411yescharacteristic
98C22×C137—22no—
99C22×C137—22no—
100C22×C137—24no—
101C22×C137#12112yes—
102C22×C137C22×D411yescharacteristic
103C4×C137—22no—
104C4×C137—42no—
105C4×C137—42no—
106C4×C137#12112yes—
107C4×C137C22×D411yescharacteristic
108C4×C137C2×#4111yescharacteristic
109Dic137—42no—
110Dic137—42no—
111C2×C137—22no—
112C2×C137—42no—
113C2×C137C2×#12111yescharacteristic
114C2×C137#57812yes—
115C137#305911yescharacteristic, complemented, Sylow
116#3059—1371nomaximal, Sylow
117C22×Q16—1371no—
118C2×#131—1371no—
119#536—1371no—
120#536D13711yescharacteristic
121#470—1374no—
122#467—1374no—
123#427—1371no—
124#419—1371no—
125#419—1371no—
126C2×#41C2×D13711yescharacteristic
127C22×Q8—1371no—
128C22×Q8C2×D13711yescharacteristic
129C2×Q16—2744no—
130C2×Q16—1372no—
131C2×Q16—1372no—
132#131—2744no—
133C2×M16—1371no—
134C2×M16—1372no—
135C22×C8—1372no—
136C22×C8—1371no—
137C4⋊Q8—21no—
138C4⋊Q8—2741no—
139#124—21no—
140#124—2741no—
141#123—22no—
142#123—2742no—
143C22⋊Q8—1374no—
144C22⋊Q8C2×D13714yes—
145#117—1371no—
146#117C2×D13711yescharacteristic
147#107—1372no—
148#106—1372no—
149#103—1374no—
150#103—1374no—
151#41—22no—
152#41—22no—
153C2×Q8—21no—
154C2×Q8—22no—
155C2×Q8—22no—
156C2×Q8—2741no—
157C2×Q8—2742no—
158C2×Q8—2742no—
159C2×Q8—1372no—
160C2×Q8C22⋊D13711yescharacteristic
161C2×Q8C22×D13712yes—
162C2×D4C22⋊D13712yes—
163C2×D4C22⋊D13711yescharacteristic
164C22×C4—21no—
165C22×C4—2741no—
166C22×C4C22×D13712yes—
167C22×C4C22×D13711yescharacteristic
168Q16—2744no—
169Q16—2744no—
170M16—2742no—
171M16—2744no—
172C2×C8—2744no—
173C2×C8—1372no—
174C2×C8—1374no—
175C2×C8—1371no—
176C2×C8—1371no—
177C4⋊C4—24no—
178C4⋊C4—2744no—
179C4⋊C4—1372no—
180C4⋊C4C22×D13712yes—
181C22⋊C4—21no—
182C22⋊C4—24no—
183C22⋊C4—2741no—
184C22⋊C4—2744no—
185C42—21no—
186C42—2741no—
187C23C2×C22⋊D13712yes—
188C23C2×C22⋊D13711yescharacteristic
189Q8—22no—
190Q8—22no—
191Q8—22no—
192Q8—42no—
193Q8—2742no—
194Q8—2742no—
195Q8—5482no—
196D4—24no—
197D4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—42no—
203C2×C4—2742no—
204C2×C4—2742no—
205C2×C4—5482no—
206C2×C4D4×D13712yes—
207C2×C4C2×C22⋊D13712yes—
208C2×C4C2×C22⋊D13711yescharacteristic
209C2×C4#1455550912yes—
210C2×C4C23×D13711yescharacteristic
211C8—2742no—
212C8—1372no—
213C8—1372no—
214C22—22no—
215C22—24no—
216C22—22no—
217C22#1798406012yes—
218C22C22×C22⋊D13711yescharacteristic
219C4—22no—
220C4—42no—
221C4—42no—
222C4—5482no—
223C4—5482no—
224C4#1798406012yes—
225C4C2×D4×D13711yescharacteristic
226C4C2×#1455550911yescharacteristic
227C2—22no—
228C2—42no—
229C2C2×#1798406011yescharacteristic
230C2#2257821812yes—
subgroupG/H
centerC22C22×C22⋊D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#536×C137C2
socleC22×C137C22×D4
prime radicalC2×#41×C137C22
hypercenter#536D137
hypocenterC137#3059
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C2×#41 ◁ #536

Complements

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

NH ≅ G/Ntype
1. C22×#14555500C2semidirect
4. #22577659C2semidirect
5. #22577642C2semidirect
6. #22577439C2semidirect
7. #22577412C2semidirect
8. #22577411C2semidirect
115. C137#3059semidirect

Maximal subgroups

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

subgroupindex
C22×#145555002
C2×#179840952
#225779742
#225776592
#225776422
#225774392
#225774122
#225774112
#536×C1372
#3059137