Finite Group Databasebeta

← #28364160

Subgroups — #28364160

Order 13952.

430 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 75 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#20295585C211yesmaximal, characteristic
2C2×#16812682C211yesmaximal, characteristic, complemented
3C2×#16812614C211yesmaximal, characteristic, complemented
4C2×#16812613C211yesmaximal, characteristic, complemented
5#20294889C211yesmaximal, characteristic, complemented
6#20294792C211yesmaximal, characteristic, complemented
7C2×#123×C109C211yesmaximal, characteristic, complemented
8C22×C22⋊D109C2211yescharacteristic
9#16812682—22no—
10#16812682—21no—
11#16812682—22no—
12#16812680—21no—
13#16812671—22no—
14#16812668—21no—
15#16812667—21no—
16C2×#13182008—21no—
17C2×C22⋊Dic109C2211yescharacteristic
18C2×C22⋊Dic109C2211yescharacteristic
19C2×C22⋊Dic109C2211yescharacteristic
20C2×#13182006—21no—
21#16812614—22no—
22#16812613—22no—
23#16812606—22no—
24C22×D4×C109C2211yescharacteristic
25C23×C4×C109C2211yescharacteristic
26#123×C109—22no—
27#123×C109—22no—
28C2×C4⋊C4×C109—21no—
29C2×C22⋊C4×C109—21no—
30C2×C22⋊C4×C109C2211yescharacteristic
31C23×D109D411yescharacteristic
32C2×C22⋊D109—21no—
33C2×C22⋊D109—21no—
34C2×C22⋊D109—21no—
35C2×C22⋊D109—21no—
36C2×C22⋊D109—21no—
37C2×C22⋊D109—21no—
38C2×C22⋊D109—22no—
39C22×Dic109—21no—
40C22×Dic109—21no—
41C22×Dic109D411yescharacteristic
42C22×Dic109D412yes—
43#13182008—24no—
44#13182008—22no—
45C22⋊Dic109—22no—
46C22⋊Dic109—21no—
47C22⋊Dic109—21no—
48C22⋊Dic109—21no—
49C22⋊Dic109—21no—
50C22⋊Dic109—21no—
51C22⋊Dic109D412yescomplemented
52#13182006—24no—
53#13182006—22no—
54C24×C109D411yescharacteristic
55C24×C109C2311yescharacteristic
56C2×D4×C109—21no—
57C2×D4×C109—21no—
58C2×D4×C109—21no—
59C2×D4×C109—21no—
60C2×D4×C109—21no—
61C2×D4×C109—21no—
62C22×C4×C109—21no—
63C22×C4×C109—21no—
64C22×C4×C109—21no—
65C22×C4×C109—21no—
66C22×C4×C109—21no—
67C22×C4×C109—21no—
68C22×C4×C109—21no—
69C22×C4×C109D411yescharacteristic
70C22×C4×C109D412yescomplemented
71C4⋊C4×C109—24no—
72C22⋊C4×C109—24no—
73C22⋊C4×C109—21no—
74C22⋊C4×C109D412yescomplemented
75C23×C109—21no—
76C23×C109—21no—
77C23×C109—21no—
78C23×C109—42no—
79C23×C109—42no—
80C23×C109C2×D411yescharacteristic
81C23×C109C2×D411yescharacteristic
82C23×C109C2×D411yescharacteristic
83C23×C109C2×D411yescharacteristic
84C23×C109C2×D411yescharacteristic
85C23×C109C2×D411yescharacteristic
86C23×C109#4111yescharacteristic
87C22×D109—21no—
88C22×D109—21no—
89C22×D109—21no—
90C22×D109—42no—
91D4×C109—41no—
92D4×C109—41no—
93D4×C109—41no—
94D4×C109—41no—
95C2×C4×C109—22no—
96C2×C4×C109—22no—
97C2×C4×C109—21no—
98C2×C4×C109—22no—
99C2×C4×C109—21no—
100C2×C4×C109—22no—
101C2×C4×C109—21no—
102C2×C4×C109—42no—
103C2×C4×C109—41no—
104C2×C4×C109—41no—
105C2×C4×C109—41no—
106C2×C4×C109—41no—
107C22⋊D109—41no—
108C22⋊D109—41no—
109C22⋊D109—41no—
110C22⋊D109—41no—
111C22⋊D109—82no—
112C2×Dic109—22no—
113C2×Dic109—22no—
114C2×Dic109—21no—
115C2×Dic109—21no—
116C2×Dic109—21no—
117C2×Dic109—42no—
118C2×Dic109—42no—
119C2×Dic109—42no—
120C2×Dic109—41no—
121C22×C109—22no—
122C22×C109—22no—
123C22×C109—22no—
124C22×C109—41no—
125C22×C109—41no—
126C22×C109—41no—
127C22×C109—41no—
128C22×C109—42no—
129C22×C109—41no—
130C22×C109—42no—
131C22×C109—81no—
132C22×C109#12011yescharacteristic
133C22×C109#12011yescharacteristic
134C22×C109#12011yescharacteristic
135C22×C109#12111yescharacteristic
136C22×C109#12111yescharacteristic
137C22×C109#12111yescharacteristic
138C22×C109#12711yescharacteristic
139C2×D109—42no—
140C2×D109—41no—
141C2×D109—42no—
142C2×D109—81no—
143C4×C109—42no—
144C4×C109—42no—
145C4×C109—41no—
146Dic109—42no—
147Dic109—41no—
148Dic109—42no—
149Dic109—81no—
150C2×C109—21no—
151C2×C109—21no—
152C2×C109—21no—
153C2×C109—21no—
154C2×C109—21no—
155C2×C109—21no—
156C2×C109—81no—
157C2×C109#39211yescharacteristic
158C2×C109#45812yes—
159D109—81no—
160#2022—1091nomaximal, Sylow
161C109#202211yescharacteristic, complemented, Sylow
162#534—1091no—
163C2×#123—1091no—
164C2×#123D10911yescharacteristic, complemented
165C2×#99—1091no—
166C2×#99—1091no—
167#381—1091no—
168#342—1091no—
169C22×D4—1091no—
170C22×D4C2×D10911yescharacteristic
171C23×C4C2×D10911yescharacteristic
172#123—22no—
173#123—22no—
174#123—2182no—
175#123—2181no—
176#123—2181no—
177#123—2182no—
178#121—2182no—
179#120—2181no—
180#120—2181no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—2181no—
183C2×C22⋊C4—21no—
184C2×C22⋊C4—1091no—
185C2×C22⋊C4—1091no—
186C2×C22⋊C4—1091no—
187C2×C22⋊C4—2181no—
188C2×C22⋊C4C2×D10911yescharacteristic
189#99—2182no—
190#99—2182no—
191#95—2182no—
192C24—1091no—
193C24C22⋊D10911yescharacteristic
194C24C22×D10911yescharacteristic
195C2×D4—21no—
196C2×D4—21no—
197C2×D4—21no—
198C2×D4—21no—
199C2×D4—21no—
200C2×D4—21no—
201C2×D4—2182no—
202C2×D4—2181no—
203C2×D4—2181no—
204C2×D4—2181no—
205C2×D4—2181no—
206C2×D4—2181no—
207C2×D4—2181no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—1092no—
216C22×C4—1091no—
217C22×C4—2181no—
218C22×C4—2181no—
219C22×C4C22⋊D10911yescharacteristic
220C22×C4C22⋊D10912yescomplemented
221C4⋊C4—24no—
222C4⋊C4—2184no—
223C4⋊C4—2182no—
224C22⋊C4—24no—
225C22⋊C4—21no—
226C22⋊C4—1092no—
227C22⋊C4—2184no—
228C22⋊C4—2182no—
229C22⋊C4—2181no—
230C22⋊C4—2181no—
231C22⋊C4—2182no—
232C22⋊C4—2181no—
233C22⋊C4—2181no—
234C22⋊C4—2181no—
235C22⋊C4C22⋊D10912yescomplemented
236C23—21no—
237C23—21no—
238C23—21no—
239C23—42no—
240C23—42no—
241C23—4362no—
242C23—2181no—
243C23—2181no—
244C23—2181no—
245C23D4×D10911yescharacteristic
246C23D4×D10911yescharacteristic
247C23D4×D10911yescharacteristic
248C23C2×C22⋊D10911yescharacteristic
249C23C2×C22⋊D10911yescharacteristic
250C23C2×C22⋊D10911yescharacteristic
251C23#1318202811yescharacteristic
252D4—41no—
253D4—41no—
254D4—41no—
255D4—41no—
256D4—8722no—
257D4—4361no—
258D4—4361no—
259D4—4361no—
260D4—4361no—
261C2×C4—22no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—21no—
266C2×C4—22no—
267C2×C4—21no—
268C2×C4—42no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—4362no—
274C2×C4—4361no—
275C2×C4—4362no—
276C2×C4—4362no—
277C2×C4—2182no—
278C2×C4—2181no—
279C2×C4—2182no—
280C2×C4—2181no—
281C2×C4—2181no—
282C22—22no—
283C22—22no—
284C22—22no—
285C22—41no—
286C22—41no—
287C22—41no—
288C22—41no—
289C22—42no—
290C22—41no—
291C22—42no—
292C22—81no—
293C22—8721no—
294C22—4362no—
295C22—4361no—
296C22—4362no—
297C22#1681266711yescharacteristic
298C22#1681266711yescharacteristic
299C22#1681266811yescharacteristic
300C22#1681267011yescharacteristic
301C22#1681267111yescharacteristic
302C22#1681267111yescharacteristic
303C22#1681269511yescharacteristic
304C4—42no—
305C4—42no—
306C4—41no—
307C4—8721no—
308C4—4362no—
309C4—4361no—
310C4—4362no—
311C2—21no—
312C2—21no—
313C2—21no—
314C2—21no—
315C2—21no—
316C2—21no—
317C2—81no—
318C2—8721no—
319C2#2029493411yescharacteristic
320C2#2029521512yes—
subgroupG/H
centerC22#16812667
commutator, cosocleC24×C109C23
FrattiniC24C22×D109
FittingC2×#123×C109C2
socleC22×C109#120
prime radical#20295585C2
hypercenterC2×#123D109
hypocenterC109#2022
LCSG ▷ C24×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C24 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
2. C2×#16812682C2semidirect
3. C2×#16812614C2semidirect
4. C2×#16812613C2semidirect
5. #20294889C2semidirect
6. #20294792C2semidirect
7. C2×#123×C109C2semidirect
51. C22⋊Dic109D4semidirect
70. C22×C4×C109D4semidirect
74. C22⋊C4×C109D4semidirect
161. C109#2022semidirect
164. C2×#123D109semidirect
220. C22×C4C22⋊D109semidirect
235. C22⋊C4C22⋊D109semidirect

Maximal subgroups

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

subgroupindex
#202955852
C2×#168126822
C2×#168126142
C2×#168126132
#202948892
#202947922
C2×#123×C1092
#2022109