Finite Group Databasebeta

← #28374899

Subgroups — #28374899

Order 13952.

420 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 125 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
1Q8×D436C211yesmaximal, characteristic, complemented
2#20295681C211yesmaximal, characteristic
3#20295377C211yesmaximal, characteristic, complemented
4#20295358C212yesmaximal, complemented
5#20295311C211yesmaximal, characteristic, complemented
6#20295254C212yesmaximal, complemented
7#20295231C211yesmaximal, characteristic, complemented
8#20295163C212yesmaximal, complemented
9C8×D436C211yesmaximal, characteristic, complemented
10#103×D109C212yesmaximal, complemented
11C4×SD16×C109C211yesmaximal, characteristic, complemented
12C2×#13182029—21no—
13C2×Q8×D109C2212yes—
14C2×#13182017C2212yes—
15C2×C8×D109C2212yes—
16#16812699—21no—
17#16812699C2211yescharacteristic
18#16812699C2211yescharacteristic
19#16812681—22no—
20#16812679—22no—
21#16812679C2212yes—
22#16812678—21no—
23#16812670C2212yes—
24C4×D436—21no—
25C4×D436C2211yescharacteristic
26C4⋊C4×D109C2212yes—
27#16812630C2212yes—
28C4⋊#11724784C2211yescharacteristic
29#16812624C2212yes—
30#16812624C2212yes—
31#16812623C2212yes—
32#16812622C2212yes—
33#16812611C2212yes—
34C2×SD16×C109C2211yescharacteristic
35C4×Q8×C109C2211yescharacteristic
36C4×D4×C109C2211yescharacteristic
37#106×C109C2211yescharacteristic
38#103×C109C2211yescharacteristic
39#102×C109C2211yescharacteristic
40C4×C8×C109C2211yescharacteristic
41#13182029—44no—
42C2×Dic218—21no—
43C2×Dic218—22no—
44C2×Dic218C2312yes—
45Q8×D109—24no—
46Q8×D109—42no—
47C2×D436D411yescharacteristic
48C2×C22⋊D109—22no—
49C2×C4×D109—22no—
50C2×C4×D109—22no—
51C2×C4×D109C2312yes—
52#13182017—42no—
53C2×#11724784C2312yes—
54C8×D109—42no—
55C4⋊Dic109—21no—
56C4⋊Dic109—21no—
57C4⋊Dic109—21no—
58C4⋊Dic109—22no—
59C4⋊Dic109D411yescharacteristic
60C4⋊Dic109C2312yes—
61#13182008—22no—
62C22⋊Dic109—22no—
63#13182006—22no—
64#13182006D412yes—
65C2×Q8×C109C2311yescharacteristic
66C2×D4×C109C2311yescharacteristic
67C22×C4×C109—21no—
68SD16×C109—22no—
69C2×C8×C109C2311yescharacteristic
70C2×C8×C109C2311yescharacteristic
71C4⋊C4×C109—21no—
72C4⋊C4×C109C2311yescharacteristic
73C4⋊C4×C109C2311yescharacteristic
74C22⋊C4×C109—21no—
75C42×C109—21no—
76C42×C109C2311yescharacteristic
77C23×C109—21no—
78C22×D109C2×D412yes—
79Q8×C109—21no—
80Q8×C109#4112yescomplemented
81D4×C109—21no—
82D4×C109—21no—
83C2×C4×C109—21no—
84C2×C4×C109—21no—
85C2×C4×C109—21no—
86C2×C4×C109—42no—
87C2×C4×C109C2×D411yescharacteristic
88C2×C4×C109C2×D411yescharacteristic
89C2×C4×C109C2411yescharacteristic
90C22⋊D109—44no—
91C2×Dic109—22no—
92C2×Dic109—22no—
93C2×Dic109C2×D412yes—
94D436SD1614yescomplemented
95C4×D109—22no—
96C4×D109—22no—
97C4×D109—44no—
98C4×D109—44no—
99Dic218—22no—
100Dic218—22no—
101Dic218—44no—
102Dic218—42no—
103C8×C109—21no—
104C8×C109—21no—
105#11724784—22no—
106C22×C109—41no—
107C22×C109—41no—
108C22×C109—41no—
109C22×C109C22×D411yescharacteristic
110C2×D109—22no—
111C2×D109—22no—
112C2×D109C2×SD1614yes—
113C4×C109—22no—
114C4×C109—21no—
115C4×C109—41no—
116C4×C109—41no—
117C4×C109C2×SD1612yes—
118C4×C109C2×#4111yescharacteristic
119C4×C109H(5,𝔽2)11yescharacteristic
120Dic109—22no—
121Dic109—42no—
122Dic109—42no—
123C2×C109—81no—
124C2×C109#54611yescharacteristic
125C2×C109C22×SD1611yescharacteristic
126C2×C109#57811yescharacteristic
127D109—24no—
128#3298—1091nomaximal, Sylow
129C109#329811yescharacteristic, complemented, Sylow
130D4×Q8—1091no—
131#547—1091no—
132#487—1091no—
133#483—1092no—
134#475—1091no—
135#465—1092no—
136#461—1091no—
137#448—1092no—
138C4×SD16D10911yescharacteristic, complemented
139C8×D4—1091no—
140C2×#103—1092no—
141C2×#41—2181no—
142C22×Q8—1092no—
143C2×SD16—1092no—
144C2×SD16C2×D10911yescharacteristic
145C22×C8—1092no—
146C4⋊Q8—1091no—
147C4⋊Q8—1091no—
148C4⋊Q8—2181no—
149#123—2182no—
150C22⋊Q8—1092no—
151C22⋊Q8—2182no—
152C22⋊Q8—2181no—
153#121—1092no—
154C4×Q8C2×D10911yescharacteristic
155C4×D4—1091no—
156C4×D4—2181no—
157C4×D4C2×D10911yescharacteristic
158C2×C4⋊C4—1092no—
159#106—1092no—
160#106C2×D10911yescharacteristic
161C4⋊C8—1091no—
162#103—1092no—
163#103—1092no—
164#103—1092no—
165#103—1092no—
166#103C2×D10911yescharacteristic
167#102C2×D10911yescharacteristic
168C22⋊C8—1092no—
169C4×C8C2×D10911yescharacteristic
170#41—4364no—
171C2×Q8—1092no—
172C2×Q8—4362no—
173C2×Q8—2182no—
174C2×Q8—2184no—
175C2×Q8—2181no—
176C2×Q8C22×D10911yescharacteristic
177C2×D4—1091no—
178C2×D4—2182no—
179C2×D4C22×D10911yescharacteristic
180C22×C4—21no—
181C22×C4—1092no—
182C22×C4—2182no—
183C22×C4—2182no—
184SD16—22no—
185SD16—4362no—
186C2×C8—1092no—
187C2×C8—4362no—
188C2×C8C22×D10911yescharacteristic
189C2×C8C22×D10911yescharacteristic
190C4⋊C4—21no—
191C4⋊C4—1091no—
192C4⋊C4—1092no—
193C4⋊C4—2182no—
194C4⋊C4—2182no—
195C4⋊C4—2181no—
196C4⋊C4—2181no—
197C4⋊C4—2181no—
198C4⋊C4C22×D10911yescharacteristic
199C4⋊C4C22×D10911yescharacteristic
200C22⋊C4—21no—
201C22⋊C4—1092no—
202C22⋊C4—2182no—
203C22⋊C4—2182no—
204C42—21no—
205C42C22×D10911yescharacteristic
206C23—21no—
207C23—1092no—
208Q8—21no—
209Q8—4364no—
210Q8—4362no—
211Q8—2182no—
212Q8—2182no—
213Q8#1318202912yescomplemented
214D4—21no—
215D4—21no—
216D4—1094no—
217D4—4364no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—42no—
222C2×C4—1092no—
223C2×C4—4364no—
224C2×C4—4364no—
225C2×C4—2182no—
226C2×C4—2182no—
227C2×C4—2182no—
228C2×C4—2182no—
229C2×C4D4×D10911yescharacteristic
230C2×C4D4×D10911yescharacteristic
231C2×C4C23×D10911yescharacteristic
232C8—21no—
233C8—21no—
234C8—2182no—
235C22—41no—
236C22—41no—
237C22—41no—
238C22—1094no—
239C22—2182no—
240C22—2182no—
241C22C2×D4×D10911yescharacteristic
242C4—21no—
243C4—22no—
244C4—41no—
245C4—41no—
246C4—4362no—
247C4—4362no—
248C4—2182no—
249C4SD16×D10912yes—
250C4C2×#1318202911yescharacteristic
251C4#1681274511yescharacteristic
252C2—81no—
253C2—2184no—
254C2#2029567211yescharacteristic
255C2C2×SD16×D10911yescharacteristic
256C2#2029584011yescharacteristic
subgroupG/H
centerC22C2×D4×D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
FittingC4×SD16×C109C2
socleC22×C109C22×D4
prime radical#20295681C2
hypercenterC4×SD16D109
hypocenterC109#3298
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C42 ◁ C4×SD16

Complements

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

NH ≅ G/Ntype
1. Q8×D436C2semidirect
3. #20295377C2semidirect
4. #20295358C2semidirect
5. #20295311C2semidirect
6. #20295254C2semidirect
7. #20295231C2semidirect
8. #20295163C2semidirect
9. C8×D436C2semidirect
10. #103×D109C2semidirect
11. C4×SD16×C109C2semidirect
80. Q8×C109#41semidirect
94. D436SD16semidirect
129. C109#3298semidirect
138. C4×SD16D109semidirect
213. Q8#13182029semidirect

Maximal subgroups

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

subgroupindex
Q8×D4362
#202956812
#202953772
#202953582
#202953112
#202952542
#202952312
#202951632
C8×D4362
#103×D1092
C4×SD16×C1092
#3298109