Finite Group Databasebeta

← #28371361

Subgroups — #28371361

Order 13952.

688 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 688 conjugacy classes of subgroups — too many to draw. Only its 169 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×#16812682C211yesmaximal, characteristic, complemented
2C2×#16812671C211yesmaximal, characteristic
3C22×C4×Dic109C211yesmaximal, characteristic, complemented
4#20294926C212yesmaximal, complemented
5#20294921C212yesmaximal, complemented
6#20294920C212yesmaximal, complemented
7#20294916C212yesmaximal, complemented
8#20294906C211yesmaximal, characteristic, complemented
9#20294903C211yesmaximal, characteristic, complemented
10#20294884C211yesmaximal, characteristic, complemented
11C2×#123×C109C211yesmaximal, characteristic, complemented
12C22×C22⋊D109C2211yescharacteristic
13C22×C22⋊D109C2211yescharacteristic
14C23×Dic109C2211yescharacteristic
15C23×Dic109C2211yescharacteristic
16#16812682—24no—
17#16812671—24no—
18C2×#13182008C2212yes—
19C2×#13182008C2211yescharacteristic
20C2×#13182008C2212yes—
21C2×C22⋊Dic109C2211yescharacteristic
22C2×C22⋊Dic109C2212yes—
23C2×C22⋊Dic109C2211yescharacteristic
24C2×#13182006C2211yescharacteristic
25C2×#13182006C2212yes—
26C2×#13182006C2212yes—
27C2×C4×Dic109—24no—
28C2×C4×Dic109C2212yes—
29C2×C4×Dic109C2212yes—
30#16812606C2212yes—
31#16812606C2212yes—
32#16812606C2212yes—
33C22×D4×C109C2211yescharacteristic
34C23×C4×C109C2211yescharacteristic
35#123×C109—24no—
36C2×C4⋊C4×C109C2212yes—
37C2×C22⋊C4×C109C2212yes—
38C2×C22⋊C4×C109C2211yescharacteristic
39C23×D109C2311yescharacteristic
40C2×C22⋊D109—22no—
41C2×C22⋊D109—22no—
42C2×C22⋊D109—22no—
43C2×C22⋊D109—22no—
44C2×C22⋊D109—24no—
45C22×Dic109—22no—
46C22×Dic109—24no—
47C22×Dic109—24no—
48C22×Dic109D414yes—
49C22×Dic109C2312yes—
50C22×Dic109C2311yescharacteristic
51C22×Dic109C2311yescharacteristic
52C22×Dic109C2312yes—
53C22×Dic109C2311yescharacteristic
54#13182008—24no—
55#13182008—22no—
56#13182008—24no—
57C22⋊Dic109—22no—
58C22⋊Dic109—22no—
59C22⋊Dic109—24no—
60#13182006—24no—
61#13182006—22no—
62#13182006—24no—
63C4×Dic109—24no—
64C4×Dic109—24no—
65C24×C109C2311yescharacteristic
66C24×C109C2311yescharacteristic
67C2×D4×C109—22no—
68C2×D4×C109—22no—
69C2×D4×C109—22no—
70C22×C4×C109—22no—
71C22×C4×C109—22no—
72C22×C4×C109D414yescomplemented
73C22×C4×C109C2312yes—
74C22×C4×C109C2311yescharacteristic
75C22×C4×C109C2312yes—
76C4⋊C4×C109—24no—
77C22⋊C4×C109—22no—
78C22⋊C4×C109—24no—
79C23×C109—22no—
80C23×C109—22no—
81C23×C109—21no—
82C23×C109—21no—
83C23×C109—21no—
84C23×C109—22no—
85C23×C109—22no—
86C23×C109C2×D412yes—
87C23×C109C2×D412yes—
88C23×C109#4112yes—
89C23×C109C2411yescharacteristic
90C22×D109—21no—
91C22×D109—21no—
92C22×D109—22no—
93C22×D109—22no—
94C22×D109—21no—
95D4×C109—44no—
96C2×C4×C109—28no—
97C2×C4×C109—22no—
98C2×C4×C109—22no—
99C2×C4×C109—22no—
100C2×C4×C109—21no—
101C2×C4×C109—21no—
102C2×C4×C109—21no—
103C2×C4×C109—22no—
104C2×C4×C109—22no—
105C2×C4×C109C2×D414yes—
106C22⋊D109—44no—
107C22⋊D109—44no—
108C2×Dic109—28no—
109C2×Dic109—22no—
110C2×Dic109—22no—
111C2×Dic109—22no—
112C2×Dic109—28no—
113C2×Dic109—22no—
114C2×Dic109—21no—
115C2×Dic109—21no—
116C2×Dic109—22no—
117C2×Dic109—22no—
118C2×Dic109—21no—
119C2×Dic109C2×D412yes—
120C2×Dic109C2×D412yes—
121C2×Dic109#4114yes—
122C2×Dic109#4114yes—
123C22×C109—22no—
124C22×C109—22no—
125C22×C109—24no—
126C22×C109—24no—
127C22×C109—41no—
128C22×C109—41no—
129C22×C109—41no—
130C22×C109—42no—
131C22×C109—42no—
132C22×C109#12114yes—
133C22×C109C22×D411yescharacteristic
134C22×C109C22×D411yescharacteristic
135C22×C109C2×#4112yes—
136C22×C109C2×#4112yes—
137C22×C109C2×#4111yescharacteristic
138C2×D109—41no—
139C2×D109—41no—
140C2×D109—41no—
141C2×D109—42no—
142C2×D109—42no—
143C4×C109—24no—
144C4×C109—41no—
145C4×C109—42no—
146Dic109—22no—
147Dic109—24no—
148Dic109—41no—
149Dic109—42no—
150Dic109#12314yescomplemented
151C2×C109—24no—
152C2×C109—81no—
153C2×C109C2×#12111yescharacteristic
154C2×C109C2×#12311yescharacteristic
155C2×C109#52511yescharacteristic
156C2×C109#52912yes—
157C2×C109#53212yes—
158D109—81no—
159#2854—1091nomaximal, Sylow
160C109#285411yescharacteristic, complemented, Sylow
161C2×#123—1091no—
162C2×#123D10911yescharacteristic, complemented
163C2×#121—1091no—
164C22×C42—1091no—
165#390—1092no—
166#389—1092no—
167#388—1092no—
168#388—1092no—
169#386—1091no—
170#386—1091no—
171#380—1091no—
172C22×D4—1091no—
173C22×D4—1091no—
174C22×D4C2×D10911yescharacteristic
175C23×C4—1091no—
176C23×C4—1091no—
177C23×C4C2×D10911yescharacteristic
178#123—24no—
179#123—2184no—
180#121—2184no—
181C2×C4⋊C4—1091no—
182C2×C4⋊C4—1092no—
183C2×C4⋊C4—1092no—
184C2×C4⋊C4C2×D10912yes—
185C2×C22⋊C4—1091no—
186C2×C22⋊C4—1091no—
187C2×C22⋊C4—1092no—
188C2×C22⋊C4—1091no—
189C2×C22⋊C4—1092no—
190C2×C22⋊C4—1092no—
191C2×C22⋊C4C2×D10911yescharacteristic
192C2×C22⋊C4C2×D10912yes—
193C2×C42—1092no—
194C2×C42—1092no—
195C2×C42—2184no—
196#95—1092no—
197#95—1092no—
198#95—1092no—
199C24—1091no—
200C24C22×D10911yescharacteristic
201C24C22×D10911yescharacteristic
202C2×D4—22no—
203C2×D4—22no—
204C2×D4—22no—
205C2×D4—2182no—
206C2×D4—2184no—
207C2×D4—2182no—
208C2×D4—2182no—
209C2×D4—2182no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—1091no—
213C22×C4—1094no—
214C22×C4—1091no—
215C22×C4—1092no—
216C22×C4—1092no—
217C22×C4—1091no—
218C22×C4—2184no—
219C22×C4—2184no—
220C22×C4—2182no—
221C22×C4C22⋊D10914yescomplemented
222C22×C4C22×D10912yes—
223C22×C4C22×D10911yescharacteristic
224C22×C4C22×D10912yes—
225C4⋊C4—24no—
226C4⋊C4—2184no—
227C4⋊C4—2182no—
228C4⋊C4—2184no—
229C22⋊C4—22no—
230C22⋊C4—24no—
231C22⋊C4—2184no—
232C22⋊C4—2182no—
233C22⋊C4—2184no—
234C22⋊C4—2182no—
235C22⋊C4—2182no—
236C22⋊C4—2184no—
237C42—2184no—
238C42—2184no—
239C23—22no—
240C23—22no—
241C23—21no—
242C23—21no—
243C23—22no—
244C23—22no—
245C23—21no—
246C23—2181no—
247C23—2181no—
248C23—2182no—
249C23—2182no—
250C23—2181no—
251C23D4×D10912yes—
252C23C2×C22⋊D10912yes—
253C23#1318202812yes—
254C23C23×D10911yescharacteristic
255D4—44no—
256D4—4364no—
257D4—4364no—
258C2×C4—28no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—1092no—
268C2×C4—1094no—
269C2×C4—1092no—
270C2×C4—1094no—
271C2×C4—2188no—
272C2×C4—2181no—
273C2×C4—2182no—
274C2×C4—2181no—
275C2×C4—2182no—
276C2×C4—2182no—
277C2×C4—2181no—
278C2×C4—2188no—
279C2×C4—2182no—
280C2×C4—2182no—
281C2×C4—2182no—
282C2×C4C2×C22⋊D10914yes—
283C22—22no—
284C22—22no—
285C22—24no—
286C22—24no—
287C22—41no—
288C22—41no—
289C22—41no—
290C22—42no—
291C22—42no—
292C22—4361no—
293C22—4361no—
294C22—4361no—
295C22—4362no—
296C22—4362no—
297C22#1681267114yes—
298C22C2×D4×D10911yescharacteristic
299C22C22×C22⋊D10911yescharacteristic
300C22#41×D10912yes—
301C22#41×D10912yes—
302C22C2×#1318202811yescharacteristic
303C4—24no—
304C4—41no—
305C4—42no—
306C4—1094no—
307C4—4361no—
308C4—4362no—
309C4—2182no—
310C4—2184no—
311C2—24no—
312C2—81no—
313C2—8721no—
314C2C2×#1681267111yescharacteristic
315C2#123×D10911yescharacteristic
316C2#2029553411yescharacteristic
317C2#2029555312yes—
318C2#2029557212yes—
subgroupG/H
center, FrattiniC23C23×D109
commutator, socle, cosocleC23×C109C24
FittingC2×#123×C109C2
prime radicalC2×#16812671C2
hypercenterC2×#123D109
hypocenterC109#2854
LCSG ▷ C23×C109 ▷ C109
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. C2×#16812682C2semidirect
3. C22×C4×Dic109C2semidirect
4. #20294926C2semidirect
5. #20294921C2semidirect
6. #20294920C2semidirect
7. #20294916C2semidirect
8. #20294906C2semidirect
9. #20294903C2semidirect
10. #20294884C2semidirect
11. C2×#123×C109C2semidirect
72. C22×C4×C109D4semidirect
150. Dic109#123semidirect
160. C109#2854semidirect
162. C2×#123D109semidirect
221. C22×C4C22⋊D109semidirect

Maximal subgroups

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

subgroupindex
C2×#168126822
C2×#168126712
C22×C4×Dic1092
#202949262
#202949212
#202949202
#202949162
#202949062
#202949032
#202948842
C2×#123×C1092
#2854109