Finite Group Databasebeta

← #28374773

Subgroups — #28374773

Order 13952.

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

normal subgroups only

This group has 482 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
1C2×#16812722C211yesmaximal, characteristic, complemented
2#20295687C212yesmaximal, complemented
3#20295411C211yesmaximal, characteristic, complemented
4#20295264C211yesmaximal, characteristic, complemented
5#20295262C211yesmaximal, characteristic, complemented
6#20295250C212yesmaximal, complemented
7#20295187C212yesmaximal, complemented
8#20295165C212yesmaximal, complemented
9Q16×Dic109C211yesmaximal, characteristic, complemented
10C8×C22⋊D109C211yesmaximal, characteristic, complemented
11C22×Q16×C109C211yesmaximal, characteristic, complemented
12C2×#13182030C2212yes—
13#16812722—24no—
14C2×#13182016C2212yes—
15C2×#13182016C2212yes—
16C2×D872C2211yescharacteristic
17C2×C8×D109C2211yescharacteristic
18#16812688—22no—
19#16812688C2212yes—
20#16812675—22no—
21#16812675C2212yes—
22#16812673—22no—
23#16812673C2212yes—
24Q8×Dic109C2212yes—
25C4×C22⋊D109—22no—
26C4×C22⋊D109C2211yescharacteristic
27#16812633C2211yescharacteristic
28#16812628C2211yescharacteristic
29#16812623C2212yes—
30#16812623C2212yes—
31#16812621C2212yes—
32C22⋊#11724784C2211yescharacteristic
33#16812611C2211yescharacteristic
34C8×Dic109C2211yescharacteristic
35C22×Q8×C109C2212yes—
36C2×Q16×C109—24no—
37C2×Q16×C109C2212yes—
38C2×Q16×C109C2211yescharacteristic
39C2×Q16×C109C2211yescharacteristic
40C22×C8×C109C2211yescharacteristic
41#13182030—24no—
42#13182030—42no—
43C2×D436—22no—
44C2×D436C2312yes—
45C2×C22⋊D109—22no—
46C2×C22⋊D109D411yescharacteristic
47C2×C4×D109—22no—
48C2×C4×D109C2311yescharacteristic
49#13182016—24no—
50#13182016—42no—
51D872—22no—
52C2×#11724784C2311yescharacteristic
53C2×#11724784C2311yescharacteristic
54C8×D109—22no—
55C4⋊Dic109—22no—
56C4⋊Dic109C2312yes—
57#13182008—22no—
58#13182008D411yescharacteristic
59C22⋊Dic109D411yescharacteristic
60#13182006—22no—
61#13182006—22no—
62#13182006—22no—
63#13182006—22no—
64#13182006D411yescharacteristic
65C4×Dic109—22no—
66C4×Dic109C2311yescharacteristic
67C2×Q8×C109—22no—
68C2×Q8×C109—24no—
69C2×Q8×C109—42no—
70C2×Q8×C109C2312yes—
71C2×Q8×C109C2312yes—
72C22×C4×C109—22no—
73C22×C4×C109C2311yescharacteristic
74Q16×C109—24no—
75Q16×C109—22no—
76Q16×C109D414yescomplemented
77C2×C8×C109—22no—
78C2×C8×C109C2311yescharacteristic
79C2×C8×C109C2311yescharacteristic
80C23×C109C2×D411yescharacteristic
81C22×D109—22no—
82C22×D109C2×D411yescharacteristic
83Q8×C109—22no—
84Q8×C109—22no—
85Q8×C109—22no—
86Q8×C109—44no—
87Q8×C109C2×D414yes—
88C2×C4×C109—22no—
89C2×C4×C109—21no—
90C2×C4×C109—21no—
91C2×C4×C109—22no—
92C2×C4×C109—44no—
93C2×C4×C109C2×D411yescharacteristic
94C2×C4×C109C2411yescharacteristic
95C22⋊D109—22no—
96C22⋊D109—44no—
97C2×Dic109—22no—
98C2×Dic109C2×D411yescharacteristic
99C2×Dic109C2×D411yescharacteristic
100C2×Dic109C2×D411yescharacteristic
101D436—22no—
102D436—22no—
103D436—44no—
104C4×D109—21no—
105C4×D109—21no—
106C4×D109—44no—
107C8×C109—21no—
108C8×C109C2×D412yes—
109#11724784—21no—
110#11724784—21no—
111C22×C109—21no—
112C22×C109—21no—
113C22×C109#13512yescomplemented
114C22×C109C22×D411yescharacteristic
115C2×D109—21no—
116C2×D109—21no—
117C2×D109—21no—
118C2×D109—42no—
119C2×D109—42no—
120C2×D109—42no—
121C4×C109—24no—
122C4×C109—21no—
123C4×C109—42no—
124C4×C109C22×D411yescharacteristic
125C4×C109H(5,𝔽2)11yescharacteristic
126Dic109—21no—
127Dic109—21no—
128Dic109—21no—
129Dic109—42no—
130C2×C109—22no—
131C2×C109D4211yescharacteristic
132C2×C109C2×#13511yescharacteristic
133C2×C109#57811yescharacteristic
134D109—41no—
135D109—82no—
136#3285—1091nomaximal, Sylow
137C109#328511yescharacteristic, complemented, Sylow
138C2×#135—1091no—
139C22×Q16D10911yescharacteristic, complemented
140#548—1092no—
141#495—1091no—
142#467—1091no—
143#466—1091no—
144#464—1092no—
145#452—1092no—
146#448—1092no—
147C4×Q16—1091no—
148C8×D4—1091no—
149C2×#41—1092no—
150C22×Q8C2×D10912yes—
151#135—2184no—
152C2×Q16—24no—
153C2×Q16C2×D10912yes—
154C2×Q16C2×D10911yescharacteristic
155C2×Q16C2×D10911yescharacteristic
156C2×SD16—1092no—
157C2×SD16—1092no—
158C2×D8—1091no—
159C22×C8—1091no—
160C22×C8C2×D10911yescharacteristic
161#124—1092no—
162#124—2182no—
163C22⋊Q8—1092no—
164C22⋊Q8—2182no—
165#121—1092no—
166#121—2182no—
167C4×Q8—1092no—
168C4×D4—1091no—
169C4×D4—2182no—
170#107—1091no—
171C4⋊C8—1091no—
172#103—1092no—
173#103—1092no—
174#102—1092no—
175C22⋊C8—1091no—
176C22⋊C8—1091no—
177C4×C8—1091no—
178#41—4362no—
179#41—2184no—
180C2×Q8—22no—
181C2×Q8—24no—
182C2×Q8—42no—
183C2×Q8C22×D10912yes—
184C2×Q8C22×D10912yes—
185C2×D4—1091no—
186C2×D4—1092no—
187C2×D4—2182no—
188C2×D4—2182no—
189C22×C4—22no—
190C22×C4—1091no—
191C22×C4—2182no—
192C22×C4C22×D10911yescharacteristic
193Q16—24no—
194Q16—22no—
195Q16C22⋊D10914yescomplemented
196SD16—4362no—
197SD16—2184no—
198D8—2182no—
199C2×C8—22no—
200C2×C8—1091no—
201C2×C8—1091no—
202C2×C8—2182no—
203C2×C8C22×D10911yescharacteristic
204C2×C8C22×D10911yescharacteristic
205C4⋊C4—1091no—
206C4⋊C4—1092no—
207C4⋊C4—2182no—
208C4⋊C4—2182no—
209C22⋊C4—1091no—
210C22⋊C4—1091no—
211C22⋊C4—2182no—
212C22⋊C4—2182no—
213C22⋊C4—2182no—
214C22⋊C4—2182no—
215C42—1091no—
216C42—2182no—
217C23—1091no—
218C23—2182no—
219C23D4×D10911yescharacteristic
220Q8—22no—
221Q8—22no—
222Q8—22no—
223Q8—44no—
224Q8C2×C22⋊D10914yes—
225D4—4364no—
226D4—4364no—
227D4—2182no—
228D4—2182no—
229D4—2182no—
230C2×C4—22no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—44no—
235C2×C4—1091no—
236C2×C4—1091no—
237C2×C4—1091no—
238C2×C4—4364no—
239C2×C4—2182no—
240C2×C4—2181no—
241C2×C4—2181no—
242C2×C4D4×D10911yescharacteristic
243C2×C4C23×D10911yescharacteristic
244C8—21no—
245C8—2181no—
246C8—2181no—
247C8C2×C22⋊D10912yes—
248C22—21no—
249C22—21no—
250C22—4362no—
251C22—4362no—
252C22—4362no—
253C22—2181no—
254C22—2181no—
255C22—2181no—
256C22#1681272212yescomplemented
257C22C2×D4×D10911yescharacteristic
258C4—21no—
259C4—24no—
260C4—42no—
261C4—4362no—
262C4—2181no—
263C4—2181no—
264C4—2181no—
265C4C22×C22⋊D10911yescharacteristic
266C4#1681274411yescharacteristic
267C2—22no—
268C2—8722no—
269C2—4361no—
270C2D4×C22⋊D10911yescharacteristic
271C2C2×#1681272211yescharacteristic
272C2#2029583911yescharacteristic
subgroupG/H
centerC22C2×D4×D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
FittingC22×Q16×C109C2
socleC22×C109C22×D4
hypercenterC22×Q16D109
hypocenterC109#3285
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×Q16

Complements

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

NH ≅ G/Ntype
1. C2×#16812722C2semidirect
2. #20295687C2semidirect
3. #20295411C2semidirect
4. #20295264C2semidirect
5. #20295262C2semidirect
6. #20295250C2semidirect
7. #20295187C2semidirect
8. #20295165C2semidirect
9. Q16×Dic109C2semidirect
10. C8×C22⋊D109C2semidirect
11. C22×Q16×C109C2semidirect
76. Q16×C109D4semidirect
113. C22×C109#135semidirect
137. C109#3285semidirect
139. C22×Q16D109semidirect
195. Q16C22⋊D109semidirect
256. C22#16812722semidirect

Maximal subgroups

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

subgroupindex
C2×#168127222
#202956872
#202954112
#202952642
#202952622
#202952502
#202951872
#202951652
Q16×Dic1092
C8×C22⋊D1092
C22×Q16×C1092
#3285109