Finite Group Databasebeta

← #28374954

Subgroups — #28374954

Order 13952.

398 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 123 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#20295681C211yesmaximal, characteristic, complemented
2#20295378C211yesmaximal, characteristic, complemented
3#20295367C212yesmaximal, complemented
4#20295313C211yesmaximal, characteristic, complemented
5#20295253C212yesmaximal, complemented
6#20295231C211yesmaximal, characteristic, complemented
7#20295169C212yesmaximal, complemented
8C4×#13182014C211yesmaximal, characteristic, complemented
9D4×#11724784C211yesmaximal, characteristic, complemented
10#20295020C212yesmaximal, complemented
11#555×C109C211yesmaximal, characteristic, complemented
12C2×#13182029C2212yes—
13C2×#13182017C2212yes—
14C2×#13182014C2211yescharacteristic
15C22×#11724784C2212yes—
16#16812699C2211yescharacteristic
17#16812681—22no—
18#16812678C2212yes—
19#16812670—21no—
20C4×D436C2211yescharacteristic
21C4⋊C4×D109—21no—
22#16812632C2211yescharacteristic
23#16812630C2212yes—
24C4⋊#11724784C2211yescharacteristic
25#16812622C2212yes—
26#16812622C2212yes—
27#16812622C2212yes—
28#16812620C2211yescharacteristic
29#16812619C2212yes—
30#16812619C2211yescharacteristic
31C22⋊#11724784C2212yes—
32C4×#11724784C2211yescharacteristic
33C4⋊Q8×C109C2211yescharacteristic
34#124×C109—22no—
35#121×C109—21no—
36#121×C109C2212yes—
37C4×D4×C109C2211yescharacteristic
38C4×D4×C109C2211yescharacteristic
39#117×C109—21no—
40#117×C109C2212yes—
41#13182029—24no—
42#13182029—42no—
43C2×Dic218C2312yes—
44C2×D436C2311yescharacteristic
45C2×C22⋊D109—22no—
46C2×C4×D109—21no—
47C2×C4×D109—22no—
48#13182017—42no—
49#13182014—22no—
50C2×#11724784—42no—
51C2×#11724784C2311yescharacteristic
52C2×#11724784C2311yescharacteristic
53C2×#11724784C2312yes—
54C4⋊Dic109—21no—
55C4⋊Dic109—21no—
56C4⋊Dic109C2311yescharacteristic
57#13182008—22no—
58C22⋊Dic109—22no—
59#13182006—21no—
60C2×Q8×C109—21no—
61C2×D4×C109—22no—
62C2×D4×C109D411yescharacteristic
63C2×D4×C109C2311yescharacteristic
64C22×C4×C109—21no—
65C22×C4×C109C2312yes—
66C4⋊C4×C109—21no—
67C4⋊C4×C109D411yescharacteristic
68C4⋊C4×C109C2311yescharacteristic
69C4⋊C4×C109C2312yes—
70C22⋊C4×C109—22no—
71C22⋊C4×C109—22no—
72C22⋊C4×C109—21no—
73C22⋊C4×C109D412yes—
74C42×C109—22no—
75C42×C109C2311yescharacteristic
76C23×C109—21no—
77C23×C109C2×D412yes—
78C22×D109—21no—
79Q8×C109—81no—
80D4×C109—22no—
81D4×C109—21no—
82D4×C109—21no—
83D4×C109—82no—
84C2×C4×C109—22no—
85C2×C4×C109—22no—
86C2×C4×C109—21no—
87C2×C4×C109—22no—
88C2×C4×C109—42no—
89C2×C4×C109C2×D412yes—
90C2×C4×C109C2×D411yescharacteristic
91C2×C4×C109C2×D411yescharacteristic
92C2×C4×C109C2411yescharacteristic
93C22⋊D109—44no—
94C2×Dic109—21no—
95C2×Dic109—22no—
96D436—21no—
97D436#4112yes—
98C4×D109—44no—
99C4×D109—42no—
100Dic218—22no—
101Dic218—22no—
102#11724784—21no—
103#11724784—21no—
104#11724784—22no—
105C22×C109—22no—
106C22×C109—22no—
107C22×C109—22no—
108C22×C109—41no—
109C22×C109—41no—
110C22×C109—41no—
111C22×C109C22×D411yescharacteristic
112C2×D109—22no—
113C2×D109—41no—
114C2×D109—41no—
115C4×C109—22no—
116C4×C109—21no—
117C4×C109—41no—
118C4×C109—42no—
119C4×C109#13512yes—
120C4×C109C2×#4111yescharacteristic
121C4×C109H(5,𝔽2)11yescharacteristic
122Dic109—42no—
123Dic109—41no—
124C2×C109—42no—
125C2×C109—81no—
126C2×C109#54611yescharacteristic
127C2×C109C2×#13511yescharacteristic
128C2×C109#57711yescharacteristic
129D109—42no—
130#3303—1091nomaximal, Sylow
131C109#330311yescharacteristic, complemented, Sylow
132#555D10911yescharacteristic, complemented
133#547—1091no—
134#487—1091no—
135#485—1092no—
136#476—1091no—
137#465—1092no—
138#461—1091no—
139#449—1092no—
140C4×D8—1091no—
141C8×D4—1091no—
142#416—1092no—
143C2×#41—1092no—
144C2×SD16—1092no—
145C2×D8—1091no—
146C22×C8—1092no—
147C4⋊Q8—1091no—
148C4⋊Q8C2×D10911yescharacteristic
149#124—22no—
150#123—2182no—
151C22⋊Q8—1092no—
152#121—21no—
153#121—2181no—
154#121C2×D10912yes—
155C4×D4—1091no—
156C4×D4C2×D10911yescharacteristic
157C4×D4C2×D10911yescharacteristic
158#117—21no—
159#117C2×D10912yes—
160C2×C4⋊C4—2181no—
161#107—1091no—
162#106—1092no—
163C4⋊C8—1091no—
164#103—1092no—
165#103—1092no—
166#103—1092no—
167#102—1091no—
168#102—1091no—
169#102—1092no—
170C22⋊C8—1092no—
171C4×C8—1091no—
172#41—4362no—
173#41—2184no—
174C2×Q8—21no—
175C2×Q8—1092no—
176C2×D4—22no—
177C2×D4—1091no—
178C2×D4—2182no—
179C2×D4C22⋊D10911yescharacteristic
180C2×D4C22×D10911yescharacteristic
181C22×C4—21no—
182C22×C4—2181no—
183C22×C4—2182no—
184C22×C4C22×D10912yes—
185SD16—4362no—
186D8—2182no—
187C2×C8—1091no—
188C2×C8—1091no—
189C2×C8—1092no—
190C2×C8—4362no—
191C4⋊C4—21no—
192C4⋊C4—1091no—
193C4⋊C4—2182no—
194C4⋊C4—2181no—
195C4⋊C4—2181no—
196C4⋊C4C22⋊D10911yescharacteristic
197C4⋊C4C22×D10912yes—
198C4⋊C4C22×D10911yescharacteristic
199C22⋊C4—22no—
200C22⋊C4—22no—
201C22⋊C4—21no—
202C22⋊C4—2182no—
203C22⋊C4—2181no—
204C22⋊C4C22⋊D10912yes—
205C42—22no—
206C42C22×D10911yescharacteristic
207C23—21no—
208C23—2181no—
209C23C2×C22⋊D10912yes—
210Q8—81no—
211Q8—2182no—
212Q8—2182no—
213D4—22no—
214D4—21no—
215D4—21no—
216D4—82no—
217D4—1092no—
218D4—4364no—
219D4—2181no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—42no—
225C2×C4—4364no—
226C2×C4—4362no—
227C2×C4—2181no—
228C2×C4—2182no—
229C2×C4C2×C22⋊D10912yes—
230C2×C4C2×C22⋊D10911yescharacteristic
231C2×C4C2×C22⋊D10911yescharacteristic
232C2×C4C23×D10911yescharacteristic
233C8—2181no—
234C8—2181no—
235C8—2182no—
236C22—22no—
237C22—22no—
238C22—22no—
239C22—41no—
240C22—41no—
241C22—41no—
242C22—4361no—
243C22—4361no—
244C22—2182no—
245C22C22×C22⋊D10911yescharacteristic
246C4—21no—
247C4—22no—
248C4—41no—
249C4—42no—
250C4—4361no—
251C4—4362no—
252C4#1681271912yes—
253C4#41×D10911yescharacteristic
254C4#1681274511yescharacteristic
255C2—42no—
256C2—81no—
257C2—4362no—
258C2#2029566411yescharacteristic
259C2C2×#1681271911yescharacteristic
260C2#2029583711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
Fitting#555×C109C2
socleC22×C109C22×D4
hypercenter#555D109
hypocenterC109#3303
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C4×D4 ◁ #555

Complements

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

NH ≅ G/Ntype
1. #20295681C2semidirect
2. #20295378C2semidirect
3. #20295367C2semidirect
4. #20295313C2semidirect
5. #20295253C2semidirect
6. #20295231C2semidirect
7. #20295169C2semidirect
8. C4×#13182014C2semidirect
9. D4×#11724784C2semidirect
10. #20295020C2semidirect
11. #555×C109C2semidirect
131. C109#3303semidirect
132. #555D109semidirect

Maximal subgroups

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

subgroupindex
#202956812
#202953782
#202953672
#202953132
#202952532
#202952312
#202951692
C4×#131820142
D4×#117247842
#202950202
#555×C1092
#3303109