Finite Group Databasebeta

← #46838016

Subgroups — #46838016

Order 34688.

234 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 55 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#31774469C211yesmaximal, characteristic
2#31773878C211yesmaximal, characteristic
3#31773869C211yesmaximal, characteristic
4#31773688C211yesmaximal, characteristic, complemented
5#31773687C211yesmaximal, characteristic, complemented
6#31773686C211yesmaximal, characteristic, complemented
7#352×C271C211yesmaximal, characteristic, complemented
8C2×#16864653C2211yescharacteristic
9C22×Dic542C411yescharacteristic, complemented
10M16×D271—21no—
11#22576159—21no—
12#22576146—21no—
13#22576144C411yescharacteristic, complemented
14#22576143C411yescharacteristic, complemented
15#22576141C2211yescharacteristic
16#22576138—21no—
17#22576137—21no—
18#22576116—21no—
19#22576116C411yescharacteristic, complemented
20#22576078—21no—
21#22576077C2211yescharacteristic
22#22576075C2211yescharacteristic
23#22576074—21no—
24#123×C271C2211yescharacteristic
25#100×C271C2211yescharacteristic
26#99×C271C2211yescharacteristic
27#16864653—41no—
28#16864653—41no—
29C2×Dic542—21no—
30C2×Dic542—21no—
31C2×Dic542—41no—
32C2×Dic542—41no—
33C2×Dic542D411yescharacteristic
34C2×C22⋊D271C2×C411yescharacteristic
35C2×C22⋊D271D411yescharacteristic
36C22×Dic271C2×C411yescharacteristic
37C22×Dic271C2×C411yescharacteristic
38C22×Dic271D411yescharacteristic
39C2×C4×D271D411yescharacteristic
40#16864638—21no—
41#16864637—41no—
42C8×D271—41no—
43C4⋊Dic271—21no—
44Dic271⋊C4—21no—
45Dic271⋊C4—21no—
46Dic271⋊C4—21no—
47Dic271⋊C4—41no—
48C22⋊Dic271—21no—
49C22⋊Dic271—21no—
50C22⋊Dic271C2×C411yescharacteristic
51(C2×D271)⋊C4—21no—
52(C2×D271)⋊C4—21no—
53C4×Dic271—21no—
54C4×Dic271—41no—
55C2×D4×C271C2311yescharacteristic
56C22×C4×C271C2×C411yescharacteristic
57M16×C271—21no—
58C4⋊C4×C271—21no—
59C22⋊C4×C271—21no—
60C22⋊C4×C271—21no—
61C22⋊C4×C271C2×C411yescharacteristic
62C22×D271C22⋊C411yescharacteristic
63Dic542—41no—
64Dic542—42no—
65Dic542—41no—
66Dic542—41no—
67C22⋊D271—41no—
68C22⋊D271—41no—
69C2×Dic271—21no—
70C2×Dic271—21no—
71C2×Dic271—21no—
72C2×Dic271—21no—
73C2×Dic271—21no—
74C2×Dic271—21no—
75C2×Dic271—41no—
76C2×Dic271—41no—
77C2×Dic271C22⋊C411yescharacteristic
78C2×Dic271C22⋊C411yescharacteristic
79C2×Dic271C22⋊C411yescharacteristic
80C4×D271—21no—
81C4×D271—21no—
82C271⋊C8—41no—
83C23×C271C22×C411yescharacteristic
84C23×C271C2×D411yescharacteristic
85D4×C271—41no—
86C2×C4×C271—21no—
87C2×C4×C271—21no—
88C2×C4×C271—41no—
89C2×C4×C271—41no—
90C2×C4×C271C2×D411yescharacteristic
91C8×C271—41no—
92C22×C271—21no—
93C22×C271—21no—
94C22×C271—41no—
95C22×C271—41no—
96C22×C271C2×C22⋊C411yescharacteristic
97C2×D271—21no—
98C2×D271—41no—
99C4×C271—21no—
100C4×C271—41no—
101C4×C271—41no—
102C4×C271—81no—
103Dic271—21no—
104Dic271—21no—
105Dic271—41no—
106Dic271—41no—
107Dic271—81no—
108Dic271#9911yescharacteristic
109Dic271#9911yescharacteristic
110C2×C271—21no—
111C2×C271—41no—
112C2×C271—41no—
113C2×C271C2×#9911yescharacteristic
114D271—41no—
115C271#212311yescharacteristic, complemented, Sylow
116#2123—2711nomaximal, Sylow
117#536—2711no—
118#413—2711no—
119#410—2711no—
120#352—2711no—
121#352—2711no—
122#352—2711no—
123#352D27111yescharacteristic, complemented
124C2×#41—2711no—
125C22×Q8—2711no—
126C2×M16—5421no—
127C4⋊Q8—5421no—
128#124—5421no—
129#123—2711no—
130#123—2711no—
131#123—2711no—
132#123C2×D27111yescharacteristic
133C22⋊Q8—5421no—
134C22⋊Q8—5421no—
135#117—2711no—
136#117—5421no—
137#101—5421no—
138#100—2711no—
139#100C2×D27111yescharacteristic
140#99—2711no—
141#99—5421no—
142#99C2×D27111yescharacteristic
143#41—10841no—
144#41—10841no—
145C2×Q8—2711no—
146C2×Q8—5421no—
147C2×Q8—5421no—
148C2×Q8—10841no—
149C2×Q8—10841no—
150C2×D4—2711no—
151C2×D4—2711no—
152C2×D4C22×D27111yescharacteristic
153C22×C4—2711no—
154C22×C4—2711no—
155C22×C4—2711no—
156C22×C4—2711no—
157C22×C4C4×D27111yescharacteristic
158M16—21no—
159M16—5421no—
160M16—10841no—
161C2×C8—10841no—
162C4⋊C4—21no—
163C4⋊C4—5421no—
164C4⋊C4—5421no—
165C4⋊C4—5421no—
166C4⋊C4—5421no—
167C4⋊C4—10841no—
168C22⋊C4—21no—
169C22⋊C4—21no—
170C22⋊C4—2711no—
171C22⋊C4—5421no—
172C22⋊C4—5421no—
173C22⋊C4—5421no—
174C22⋊C4—5421no—
175C22⋊C4C4×D27111yescharacteristic
176C42—5421no—
177C42—10841no—
178C23—2711no—
179C23C2×C4×D27111yescharacteristic
180C23D4×D27111yescharacteristic
181Q8—10841no—
182Q8—10841no—
183Q8—10841no—
184Q8—10842no—
185D4—41no—
186D4—10841no—
187D4—10841no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—2711no—
193C2×C4—2711no—
194C2×C4—2711no—
195C2×C4—5421no—
196C2×C4—5421no—
197C2×C4—5421no—
198C2×C4—5421no—
199C2×C4—5421no—
200C2×C4—5421no—
201C2×C4—5421no—
202C2×C4—5421no—
203C2×C4—10841no—
204C2×C4—10841no—
205C2×C4D4×D27111yescharacteristic
206C8—41no—
207C8—10841no—
208C22—21no—
209C22—21no—
210C22—41no—
211C22—41no—
212C22—5421no—
213C22—10841no—
214C22C22⋊C4×D27111yescharacteristic
215C4—21no—
216C4—41no—
217C4—41no—
218C4—81no—
219C4—2711no—
220C4—2711no—
221C4—5421no—
222C4—5421no—
223C4—10841no—
224C4—10841no—
225C4—21681no—
226C2—21no—
227C2—41no—
228C2—41no—
229C2—10841no—
230C2#99×D27111yescharacteristic
subgroupG/H
centerC2#99×D271
commutatorC23×C271C22×C4
FrattiniC2×D4C22×D271
Fitting#352×C271C2
socleC2×C271C2×#99
prime radicalC2×#16864653C22
cosocleC2×D4×C271C23
hypercenter#352D271
hypocenterC271#2123
LCSG ▷ C23×C271 ▷ C22×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #352

Complements

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

NH ≅ G/Ntype
4. #31773688C2semidirect
5. #31773687C2semidirect
6. #31773686C2semidirect
7. #352×C271C2semidirect
9. C22×Dic542C4semidirect
13. #22576144C4semidirect
14. #22576143C4semidirect
19. #22576116C4semidirect
115. C271#2123semidirect
123. #352D271semidirect

Maximal subgroups

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

subgroupindex
#317744692
#317738782
#317738692
#317736882
#317736872
#317736862
#352×C2712
#2123271