Finite Group Databasebeta

← #28372866

Subgroups — #28372866

Order 13952.

398 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 133 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#20295654C211yesmaximal, characteristic, complemented
2#20295500C211yesmaximal, characteristic
3#20295329C212yesmaximal, complemented
4#20295325C212yesmaximal, complemented
5#20295315C212yesmaximal, complemented
6#20295305C212yesmaximal, complemented
7#20295069C211yesmaximal, characteristic, complemented
8#20295052C211yesmaximal, characteristic, complemented
9#20295030C211yesmaximal, characteristic
10C2×#16812621C211yesmaximal, characteristic
11#427×C109C211yesmaximal, characteristic, complemented
12C22×D436C2211yescharacteristic
13C2×#13182013C2211yescharacteristic
14#16812699—21no—
15#16812698C2212yescomplemented
16#16812691—21no—
17#16812690C2212yescomplemented
18#16812676—22no—
19C4×D436—21no—
20C4×D436—21no—
21#16812661C2212yes—
22#16812661C2212yes—
23#16812658C2211yescharacteristic
24C2×C4⋊Dic109C2211yescharacteristic
25C22⋊C4×D109—22no—
26#16812632C2212yescomplemented
27#16812630C2212yescomplemented
28#16812628C2212yescomplemented
29#16812628C2212yescomplemented
30#16812621C2212yes—
31#16812621C2212yes—
32#16812619C2212yes—
33#16812619C2212yes—
34C22×C8×C109C2211yescharacteristic
35#117×C109C2211yescharacteristic
36#117×C109C2211yescharacteristic
37#107×C109C2211yescharacteristic, complemented
38#107×C109C2211yescharacteristic, complemented
39#106×C109C2211yescharacteristic, complemented
40#106×C109C2211yescharacteristic, complemented
41C23×D109—21no—
42C2×Dic218—22no—
43C2×D436—21no—
44C2×D436—22no—
45C2×D436Q814yescomplemented
46C2×D436C2312yes—
47C22×Dic109—21no—
48C2×C4×D109—22no—
49C2×C4×D109—22no—
50#13182013—22no—
51C2×#11724784C2312yes—
52C4⋊Dic109—21no—
53C4⋊Dic109—22no—
54C4⋊Dic109D411yescharacteristic
55C4⋊Dic109D411yescharacteristic
56C4⋊Dic109C2312yes—
57#13182008—22no—
58#13182008—22no—
59C22⋊Dic109D412yes—
60#13182006—22no—
61#13182006—22no—
62C4×Dic109C2312yes—
63C22×C4×C109C2311yescharacteristic
64C2×C8×C109—22no—
65C2×C8×C109C2311yescharacteristic
66C2×C8×C109C2311yescharacteristic
67C4⋊C4×C109C2311yescharacteristic
68C4⋊C4×C109C2311yescharacteristic
69C4⋊C4×C109C2311yescharacteristic
70C4⋊C4×C109C2311yescharacteristic
71C22⋊C4×C109—21no—
72C22⋊C4×C109—21no—
73C42×C109—21no—
74C42×C109—21no—
75C23×C109C2×D411yescharacteristic
76C22×D109—22no—
77C22×D109—24no—
78C22×D109—81no—
79C22×D109—81no—
80C2×C4×C109—21no—
81C2×C4×C109—21no—
82C2×C4×C109—21no—
83C2×C4×C109—21no—
84C2×C4×C109C2×D411yescharacteristic
85C2×C4×C109C2×Q812yes—
86C2×C4×C109#4112yes—
87C2×C4×C109C2411yescharacteristic
88C2×Dic109—22no—
89C2×Dic109—81no—
90C2×Dic109C2×D412yes—
91C2×Dic109C2×D412yes—
92D436—24no—
93D436—22no—
94D436C2×Q814yes—
95C4×D109—82no—
96C4×D109—82no—
97Dic218—82no—
98C8×C109—21no—
99C8×C109—21no—
100#11724784—22no—
101C22×C109—21no—
102C22×C109—21no—
103C22×C109C22⋊Q812yes—
104C22×C109C22×D411yescharacteristic
105C2×D109—24no—
106C2×D109—42no—
107C2×D109—42no—
108C2×D109—84no—
109C4×C109—41no—
110C4×C109—41no—
111C4×C109—41no—
112C4×C109—41no—
113C4×C109C22⋊Q812yes—
114C4×C109C22×Q811yescharacteristic
115C4×C109C2×#4111yescharacteristic
116Dic109—22no—
117Dic109—22no—
118Dic109—42no—
119C2×C109—22no—
120C2×C109C2×C22⋊Q811yescharacteristic
121C2×C109#57611yescharacteristic
122C2×C109#57711yescharacteristic
123D109—44no—
124#3081—1091nomaximal, Sylow
125C109#308111yescharacteristic, complemented, Sylow
126#544—1091no—
127#518—1091no—
128#478—1092no—
129#478—1092no—
130#476—1092no—
131#474—1092no—
132#428—1091no—
133#427D10911yescharacteristic, complemented
134#424—1091no—
135#418—1091no—
136C2×#102—1091no—
137C22×D4—1091no—
138C2×M16—1091no—
139C22×C8C2×D10911yescharacteristic
140C4⋊Q8—1092no—
141C4⋊Q8—2181no—
142#125—1092no—
143#125—2181no—
144C22⋊Q8—2182no—
145C4×D4—1092no—
146C4×D4—1092no—
147C4×D4—2181no—
148C4×D4—2181no—
149#117—1091no—
150#117C2×D10911yescharacteristic
151#117C2×D10911yescharacteristic
152C2×C4⋊C4—1091no—
153C2×C22⋊C4—2182no—
154#107—1092no—
155#107C2×D10911yescharacteristic, complemented
156#107C2×D10911yescharacteristic, complemented
157#106—1092no—
158#106C2×D10911yescharacteristic, complemented
159#106C2×D10911yescharacteristic, complemented
160C4⋊C8—1092no—
161C4⋊C8—1092no—
162#102—1092no—
163#102—1092no—
164#102—1092no—
165#102—1092no—
166C24—2181no—
167C2×Q8—2182no—
168C2×D4—1094no—
169C2×D4—1092no—
170C2×D4—2181no—
171C2×D4—2182no—
172C22×C4—2182no—
173C22×C4—2182no—
174C22×C4—2181no—
175C22×C4C22×D10911yescharacteristic
176M16—2182no—
177C2×C8—22no—
178C2×C8—1092no—
179C2×C8C22×D10911yescharacteristic
180C2×C8C22×D10911yescharacteristic
181C4⋊C4—1091no—
182C4⋊C4—1091no—
183C4⋊C4—1092no—
184C4⋊C4—2182no—
185C4⋊C4—2182no—
186C4⋊C4—2182no—
187C4⋊C4—2181no—
188C4⋊C4C22×D10911yescharacteristic
189C4⋊C4C22×D10911yescharacteristic
190C4⋊C4C22×D10911yescharacteristic
191C4⋊C4C22×D10911yescharacteristic
192C22⋊C4—21no—
193C22⋊C4—21no—
194C22⋊C4—1092no—
195C22⋊C4—2182no—
196C22⋊C4—2182no—
197C42—21no—
198C42—21no—
199C42—1092no—
200C23—8721no—
201C23—8721no—
202C23—2184no—
203C23—2182no—
204C23D4×D10911yescharacteristic
205Q8—8722no—
206D4—1094no—
207D4—2184no—
208D4—2182no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—8722no—
214C2×C4—8722no—
215C2×C4—8721no—
216C2×C4—1092no—
217C2×C4—1092no—
218C2×C4—2182no—
219C2×C4D4×D10911yescharacteristic
220C2×C4Q8×D10912yes—
221C2×C4#1318202912yes—
222C2×C4C23×D10911yescharacteristic
223C8—21no—
224C8—21no—
225C8—2182no—
226C22—21no—
227C22—21no—
228C22—8724no—
229C22—4362no—
230C22—4362no—
231C22—2184no—
232C22#1681267712yes—
233C22C2×D4×D10911yescharacteristic
234C4—41no—
235C4—41no—
236C4—41no—
237C4—41no—
238C4—4362no—
239C4—2182no—
240C4—2182no—
241C4#1681267712yes—
242C4C2×Q8×D10911yescharacteristic
243C4C2×#1318202911yescharacteristic
244C2—22no—
245C2—4364no—
246C2C2×#1681267711yescharacteristic
247C2#2029582811yescharacteristic
248C2#2029583311yescharacteristic
subgroupG/H
centerC22C2×D4×D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
Fitting#427×C109C2
socleC22×C109C22×D4
prime radicalC22×D436C22
hypercenter#427D109
hypocenterC109#3081
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C22×C4 ◁ #427

Complements

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

NH ≅ G/Ntype
1. #20295654C2semidirect
3. #20295329C2semidirect
4. #20295325C2semidirect
5. #20295315C2semidirect
6. #20295305C2semidirect
7. #20295069C2semidirect
8. #20295052C2semidirect
11. #427×C109C2semidirect
15. #16812698C22semidirect
17. #16812690C22semidirect
26. #16812632C22semidirect
27. #16812630C22semidirect
28. #16812628C22semidirect
29. #16812628C22semidirect
37. #107×C109C22semidirect
38. #107×C109C22semidirect
39. #106×C109C22semidirect
40. #106×C109C22semidirect
45. C2×D436Q8semidirect
125. C109#3081semidirect
133. #427D109semidirect
155. #107C2×D109semidirect
156. #107C2×D109semidirect
158. #106C2×D109semidirect
159. #106C2×D109semidirect

Maximal subgroups

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

subgroupindex
#202956542
#202955002
#202953292
#202953252
#202953152
#202953052
#202950692
#202950522
#202950302
C2×#168126212
#427×C1092
#3081109