Finite Group Databasebeta

← #46835342

Subgroups — #46835342

Order 34688.

260 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 49 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#31774485C211yesmaximal, characteristic
2#31774053C211yesmaximal, characteristic, complemented
3#31774036C211yesmaximal, characteristic, complemented
4#31773633C211yesmaximal, characteristic, complemented
5#31773630C211yesmaximal, characteristic, complemented
6#31773621C211yesmaximal, characteristic, complemented
7#393×C271C211yesmaximal, characteristic, complemented
8C2×#16864644—21no—
9C2×#16864641—21no—
10#22576153—21no—
11#22576146—21no—
12#22576142—21no—
13#22576139C2211yescharacteristic
14#22576134C2211yescharacteristic
15#22576133—21no—
16#22576129—21no—
17C4×C22⋊D271—21no—
18#22576084—21no—
19#22576084—21no—
20C22⋊(C271⋊C8)C2211yescharacteristic
21C22⋊(C271⋊C8)C2211yescharacteristic
22C22×Q8×C271C2211yescharacteristic
23C2×C22⋊C4×C271—21no—
24C2×C22⋊C4×C271C2211yescharacteristic
25#95×C271—21no—
26#95×C271C2211yescharacteristic
27C2×Dic542D411yescharacteristic
28C2×D1084D411yescharacteristic
29C2×C22⋊D271—21no—
30C2×C22⋊D271—21no—
31C2×C22⋊D271—21no—
32C2×C4×D271—21no—
33#16864644—42no—
34#16864641—42no—
35C2×C271⋊C8—21no—
36C2×C271⋊C8—21no—
37C4⋊Dic271D411yescharacteristic
38C4⋊Dic271D411yescharacteristic
39Dic271⋊C4—21no—
40Dic271⋊C4—21no—
41C22⋊Dic271—21no—
42C22⋊Dic271—21no—
43C22⋊Dic271—21no—
44(C2×D271)⋊C4—21no—
45(C2×D271)⋊C4—21no—
46C4×Dic271—21no—
47C24×C271D411yescharacteristic
48C2×Q8×C271—21no—
49C2×Q8×C271—21no—
50C2×Q8×C271—42no—
51C22×C4×C271—21no—
52C22×C4×C271—21no—
53C22×C4×C271D411yescharacteristic
54C22×C4×C271C2311yescharacteristic
55C22⋊C4×C271—21no—
56C22⋊C4×C271—21no—
57C22⋊C4×C271—42no—
58C22×D271—21no—
59Dic542—41no—
60D1084—41no—
61C22⋊D271—41no—
62C22⋊D271—41no—
63C22⋊D271—81no—
64C22⋊D271—81no—
65C2×Dic271—21no—
66C2×Dic271—21no—
67C2×Dic271—21no—
68C4×D271—81no—
69C271⋊C8—41no—
70C271⋊C8—41no—
71C23×C271—21no—
72C23×C271—21no—
73C23×C271—21no—
74C23×C271—41no—
75C23×C271—41no—
76C23×C271C2×D411yescharacteristic
77Q8×C271—22no—
78Q8×C271—22no—
79Q8×C271—41no—
80Q8×C271—41no—
81C2×C4×C271—22no—
82C2×C4×C271—21no—
83C2×C4×C271—41no—
84C2×C4×C271—41no—
85C2×C4×C271—41no—
86C2×C4×C271—41no—
87C2×C4×C271—41no—
88C2×C4×C271—41no—
89C2×C4×C271—41no—
90C2×C4×C271C2×D411yescharacteristic
91C2×C4×C271C2×D411yescharacteristic
92C22×C271—21no—
93C22×C271—21no—
94C22×C271—42no—
95C22×C271—41no—
96C22×C271—41no—
97C22×C271—41no—
98C22×C271—81no—
99C22×C271#12011yescharacteristic
100C22×C271#13511yescharacteristic
101C22×C271#13711yescharacteristic
102C2×D271—41no—
103C2×D271—41no—
104C2×D271—41no—
105C4×C271—21no—
106C4×C271—21no—
107C4×C271—41no—
108C4×C271—42no—
109C4×C271—81no—
110Dic271—41no—
111Dic271—41no—
112Dic271—41no—
113C2×C271—21no—
114C2×C271—21no—
115C2×C271—81no—
116C2×C271#45211yescharacteristic
117C2×C271#45511yescharacteristic
118C2×C271B(4,𝔽2)11yescharacteristic
119D271—81no—
120C271#160611yescharacteristic, complemented, Sylow
121#1606—2711nomaximal, Sylow
122#538—2711no—
123#451—2711no—
124#448—2711no—
125#393D27111yescharacteristic, complemented
126#328—2711no—
127#327—2711no—
128#323—2711no—
129C22×Q8C2×D27111yescharacteristic
130C2×Q16—5421no—
131C2×SD16—5421no—
132#126—5421no—
133#124—5421no—
134#123—5421no—
135C22⋊Q8—2711no—
136#121—2711no—
137#121—5421no—
138#120—5421no—
139C4×D4—5421no—
140C2×C22⋊C4—21no—
141C2×C22⋊C4C2×D27111yescharacteristic
142#103—5421no—
143#103—5421no—
144C22⋊C8—2711no—
145C22⋊C8—2711no—
146#95—21no—
147#95C2×D27111yescharacteristic
148C24C22⋊D27111yescharacteristic
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—42no—
152C2×Q8—2711no—
153C2×D4—2711no—
154C2×D4—5421no—
155C2×D4—5421no—
156C2×D4—5421no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—5421no—
160C22×C4C22⋊D27111yescharacteristic
161C22×C4C22×D27111yescharacteristic
162Q16—10842no—
163SD16—10842no—
164C2×C8—5421no—
165C2×C8—5421no—
166C4⋊C4—2711no—
167C4⋊C4—2711no—
168C4⋊C4—5421no—
169C4⋊C4—5421no—
170C22⋊C4—21no—
171C22⋊C4—21no—
172C22⋊C4—42no—
173C22⋊C4—5421no—
174C22⋊C4—5421no—
175C22⋊C4—5421no—
176C22⋊C4—5421no—
177C22⋊C4—5421no—
178C42—5421no—
179C23—21no—
180C23—21no—
181C23—21no—
182C23—41no—
183C23—41no—
184C23—5421no—
185C23C2×C22⋊D27111yescharacteristic
186Q8—22no—
187Q8—22no—
188Q8—41no—
189Q8—41no—
190Q8—10841no—
191D4—10841no—
192D4—10841no—
193D4—10841no—
194D4—21681no—
195D4—21681no—
196C2×C4—22no—
197C2×C4—21no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—5421no—
206C2×C4—5421no—
207C2×C4—5421no—
208C2×C4—21681no—
209C2×C4D4×D27111yescharacteristic
210C2×C4D4×D27111yescharacteristic
211C8—10841no—
212C8—10841no—
213C22—21no—
214C22—21no—
215C22—42no—
216C22—41no—
217C22—41no—
218C22—41no—
219C22—81no—
220C22—10841no—
221C22—10841no—
222C22—10841no—
223C22#2257612811yescharacteristic
224C22#2257618011yescharacteristic
225C22#2257619111yescharacteristic
226C4—21no—
227C4—21no—
228C4—41no—
229C4—42no—
230C4—81no—
231C4—10841no—
232C4—10841no—
233C4—10841no—
234C2—21no—
235C2—21no—
236C2—81no—
237C2—21681no—
238C2#3177405711yescharacteristic
239C2#3177407411yescharacteristic
240C2#3177408311yescharacteristic
subgroupG/H
centerC22#22576128
commutator, cosocleC22×C4×C271C23
FrattiniC22×C4C22×D271
Fitting#393×C271C2
socleC22×C271#120
prime radical#31774485C2
hypercenter#393D271
hypocenterC271#1606
LCSG ▷ C22×C4×C271 ▷ C22×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #393

Complements

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

NH ≅ G/Ntype
2. #31774053C2semidirect
3. #31774036C2semidirect
4. #31773633C2semidirect
5. #31773630C2semidirect
6. #31773621C2semidirect
7. #393×C271C2semidirect
120. C271#1606semidirect
125. #393D271semidirect

Maximal subgroups

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

subgroupindex
#317744852
#317740532
#317740362
#317736332
#317736302
#317736212
#393×C2712
#1606271