Finite Group Databasebeta

← #28374913

Subgroups — #28374913

Order 13952.

386 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20295685C211yesmaximal, characteristic
2#20295383C211yesmaximal, characteristic, complemented
3#20295367C211yesmaximal, characteristic, complemented
4#20295339C211yesmaximal, characteristic, complemented
5#20295330C211yesmaximal, characteristic, complemented
6#20295265C211yesmaximal, characteristic, complemented
7#20295257C211yesmaximal, characteristic, complemented
8#20295248C211yesmaximal, characteristic, complemented
9#20295186C211yesmaximal, characteristic, complemented
10#20295181C211yesmaximal, characteristic, complemented
11C4×#13182019C211yesmaximal, characteristic, complemented
12D4×#11724784C211yesmaximal, characteristic, complemented
13#20295020C211yesmaximal, characteristic, complemented
14C2×#16812622C211yesmaximal, characteristic, complemented
15#552×C109C211yesmaximal, characteristic, complemented
16C2×#13182029C2211yescharacteristic
17C22×Dic218C2211yescharacteristic
18C2×#13182019C2211yescharacteristic
19C2×#13182019C2211yescharacteristic
20C2×#13182016C2211yescharacteristic
21C22×#11724784C2211yescharacteristic
22C22×#11724784C2211yescharacteristic
23#16812687C2211yescharacteristic
24#16812685—21no—
25#16812678C2211yescharacteristic
26#16812677—21no—
27#16812673C2211yescharacteristic
28#16812671—21no—
29C4×Dic218C2211yescharacteristic
30#16812659—21no—
31#16812632C2211yescharacteristic
32#16812632C2211yescharacteristic
33#16812630C2211yescharacteristic
34C4⋊#11724784C2211yescharacteristic
35#16812623C2211yescharacteristic
36#16812622C2211yescharacteristic
37#16812622C2211yescharacteristic
38#16812622C2211yescharacteristic
39#16812622C2211yescharacteristic
40#16812622C2211yescharacteristic
41#16812622C2211yescharacteristic
42#16812619C2211yescharacteristic
43#16812619C2211yescharacteristic
44#16812619C2211yescharacteristic
45C22⋊#11724784C2211yescharacteristic
46C22⋊#11724784C2211yescharacteristic
47C4×#11724784C2211yescharacteristic
48#126×C109—21no—
49#125×C109—21no—
50#125×C109C2211yescharacteristic
51#123×C109—21no—
52C22⋊Q8×C109C2211yescharacteristic
53C22⋊Q8×C109C2211yescharacteristic
54C4×Q8×C109C2211yescharacteristic
55C4×D4×C109C2211yescharacteristic
56#117×C109—21no—
57#117×C109C2211yescharacteristic
58C2×C4⋊C4×C109C2211yescharacteristic
59#13182029—22no—
60#13182029—41no—
61C2×Dic218—22no—
62C2×Dic218—21no—
63C2×Dic218—41no—
64C2×Dic218C2311yescharacteristic
65C2×Dic218C2311yescharacteristic
66C2×D436C2311yescharacteristic
67C2×C22⋊D109—21no—
68C2×C22⋊D109—21no—
69C22×Dic109—21no—
70C2×C4×D109—21no—
71#13182019—22no—
72#13182019—41no—
73#13182016—41no—
74C2×#11724784—41no—
75C2×#11724784—41no—
76C2×#11724784C2311yescharacteristic
77C2×#11724784C2311yescharacteristic
78C2×#11724784C2311yescharacteristic
79C2×#11724784C2311yescharacteristic
80C4⋊Dic109C2311yescharacteristic
81#13182008—21no—
82#13182008—21no—
83C22⋊Dic109—21no—
84#13182006—21no—
85#13182006—21no—
86#13182006—21no—
87C4×Dic109—21no—
88C2×Q8×C109C2311yescharacteristic
89C2×D4×C109D411yescharacteristic
90C22×C4×C109—21no—
91C22×C4×C109C2311yescharacteristic
92C22×C4×C109C2311yescharacteristic
93C4⋊C4×C109—21no—
94C4⋊C4×C109—21no—
95C4⋊C4×C109—21no—
96C4⋊C4×C109—21no—
97C4⋊C4×C109—21no—
98C4⋊C4×C109—21no—
99C4⋊C4×C109D411yescharacteristic
100C4⋊C4×C109C2311yescharacteristic
101C4⋊C4×C109C2311yescharacteristic
102C4⋊C4×C109C2311yescharacteristic
103C22⋊C4×C109—21no—
104C22⋊C4×C109—21no—
105C22⋊C4×C109—21no—
106C22⋊C4×C109D411yescharacteristic
107C22⋊C4×C109D411yescharacteristic
108C42×C109—21no—
109C42×C109—21no—
110C42×C109C2311yescharacteristic
111C23×C109C2×D411yescharacteristic
112C23×C109C2×D411yescharacteristic
113C22×D109—21no—
114Q8×C109—21no—
115Q8×C109—21no—
116D4×C109—22no—
117C2×C4×C109—21no—
118C2×C4×C109—21no—
119C2×C4×C109—21no—
120C2×C4×C109—21no—
121C2×C4×C109—21no—
122C2×C4×C109—21no—
123C2×C4×C109—21no—
124C2×C4×C109—21no—
125C2×C4×C109—42no—
126C2×C4×C109C2×D411yescharacteristic
127C2×C4×C109C2×D411yescharacteristic
128C2×C4×C109C2×D411yescharacteristic
129C2×C4×C109C2×D411yescharacteristic
130C2×C4×C109C2411yescharacteristic
131C22⋊D109—42no—
132C22⋊D109—42no—
133C2×Dic109—21no—
134C2×Dic109—21no—
135C2×Dic109—21no—
136C2×Dic109—42no—
137D436—21no—
138D436—21no—
139C4×D109—42no—
140Dic218—21no—
141Dic218—21no—
142Dic218—21no—
143Dic218—42no—
144Dic218#4112yes—
145#11724784—21no—
146#11724784—21no—
147#11724784—21no—
148#11724784—21no—
149C22×C109—21no—
150C22×C109—21no—
151C22×C109—21no—
152C22×C109—21no—
153C22×C109—21no—
154C22×C109#13512yes—
155C22×C109C22×D411yescharacteristic
156C2×D109—41no—
157C2×D109—41no—
158C2×D109—41no—
159C4×C109—21no—
160C4×C109—21no—
161C4×C109—21no—
162C4×C109—21no—
163C4×C109—41no—
164C4×C109—41no—
165C4×C109—41no—
166C4×C109—41no—
167C4×C109C2×#4111yescharacteristic
168C4×C109H(5,𝔽2)11yescharacteristic
169Dic109—22no—
170Dic109—41no—
171Dic109—41no—
172C2×C109—22no—
173C2×C109—41no—
174C2×C109#54611yescharacteristic
175C2×C109C2×#13511yescharacteristic
176C2×C109#57811yescharacteristic
177D109—81no—
178#3300—1091nomaximal, Sylow
179C109#330011yescharacteristic, complemented, Sylow
180#552D10911yescharacteristic, complemented
181#548—1091no—
182#488—1091no—
183#485—1091no—
184#480—1091no—
185#479—1091no—
186#467—1091no—
187#465—1091no—
188#464—1091no—
189#452—1091no—
190#451—1091no—
191C4×Q16—1091no—
192C8×D4—1091no—
193#416—1091no—
194C2×#103—1091no—
195C2×#41—1091no—
196C22×Q8—1091no—
197C2×Q16—1091no—
198C2×Q16—1091no—
199C2×SD16—1091no—
200C22×C8—1091no—
201C22×C8—1091no—
202#126—21no—
203#125—21no—
204#125C2×D10911yescharacteristic
205#124—1091no—
206#124—2181no—
207#123—21no—
208C22⋊Q8—1091no—
209C22⋊Q8—2181no—
210C22⋊Q8C2×D10911yescharacteristic
211C22⋊Q8C2×D10911yescharacteristic
212#121—1091no—
213#121—2181no—
214C4×Q8—1091no—
215C4×Q8C2×D10911yescharacteristic
216C4×D4—2181no—
217C4×D4C2×D10911yescharacteristic
218#117—21no—
219#117C2×D10911yescharacteristic
220C2×C4⋊C4C2×D10911yescharacteristic
221#107—1091no—
222#107—1091no—
223#106—1091no—
224C4⋊C8—1091no—
225#103—1091no—
226#103—1091no—
227#103—1091no—
228#103—1091no—
229#103—1091no—
230#103—1091no—
231#103—1091no—
232#102—1091no—
233#102—1091no—
234#102—1091no—
235C22⋊C8—1091no—
236C22⋊C8—1091no—
237C4×C8—1091no—
238#41—4361no—
239#41—2182no—
240C2×Q8—1091no—
241C2×Q8—1091no—
242C2×Q8—4361no—
243C2×Q8—2182no—
244C2×Q8—2181no—
245C2×Q8C22×D10911yescharacteristic
246C2×D4—1091no—
247C2×D4—2181no—
248C2×D4—2181no—
249C2×D4C22⋊D10911yescharacteristic
250C22×C4—21no—
251C22×C4—2181no—
252C22×C4—2181no—
253C22×C4C22×D10911yescharacteristic
254C22×C4C22×D10911yescharacteristic
255Q16—4361no—
256Q16—2182no—
257SD16—4361no—
258C2×C8—1091no—
259C2×C8—1091no—
260C2×C8—1091no—
261C2×C8—1091no—
262C2×C8—4361no—
263C2×C8—4361no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—1091no—
271C4⋊C4—2181no—
272C4⋊C4—2181no—
273C4⋊C4C22⋊D10911yescharacteristic
274C4⋊C4C22×D10911yescharacteristic
275C4⋊C4C22×D10911yescharacteristic
276C4⋊C4C22×D10911yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—21no—
280C22⋊C4—2181no—
281C22⋊C4—2181no—
282C22⋊C4—2181no—
283C22⋊C4—2181no—
284C22⋊C4C22⋊D10911yescharacteristic
285C22⋊C4C22⋊D10911yescharacteristic
286C42—21no—
287C42—21no—
288C42—2181no—
289C42C22×D10911yescharacteristic
290C23—2181no—
291C23C2×C22⋊D10911yescharacteristic
292C23C2×C22⋊D10911yescharacteristic
293Q8—21no—
294Q8—21no—
295Q8—1092no—
296Q8—4362no—
297Q8—2181no—
298Q8—2181no—
299Q8—2181no—
300D4—22no—
301D4—4362no—
302D4—4362no—
303D4—2181no—
304D4—2181no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—42no—
314C2×C4—4362no—
315C2×C4—4362no—
316C2×C4—2181no—
317C2×C4—2181no—
318C2×C4—2181no—
319C2×C4C2×C22⋊D10911yescharacteristic
320C2×C4C2×C22⋊D10911yescharacteristic
321C2×C4C2×C22⋊D10911yescharacteristic
322C2×C4C2×C22⋊D10911yescharacteristic
323C2×C4C23×D10911yescharacteristic
324C8—2181no—
325C8—2181no—
326C8—2181no—
327C8—2181no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—4361no—
334C22—4361no—
335C22—4361no—
336C22#1681271912yes—
337C22C22×C22⋊D10911yescharacteristic
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—41no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—4361no—
347C4—4361no—
348C4—2182no—
349C4#41×D10911yescharacteristic
350C4#1681274511yescharacteristic
351C2—22no—
352C2—41no—
353C2—8721no—
354C2#2029566411yescharacteristic
355C2C2×#1681271911yescharacteristic
356C2#2029584111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
Fitting#552×C109C2
socleC22×C109C22×D4
prime radical#20295685C2
hypercenter#552D109
hypocenterC109#3300
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C4×D4 ◁ #552

Complements

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

NH ≅ G/Ntype
2. #20295383C2semidirect
3. #20295367C2semidirect
4. #20295339C2semidirect
5. #20295330C2semidirect
6. #20295265C2semidirect
7. #20295257C2semidirect
8. #20295248C2semidirect
9. #20295186C2semidirect
10. #20295181C2semidirect
11. C4×#13182019C2semidirect
12. D4×#11724784C2semidirect
13. #20295020C2semidirect
14. C2×#16812622C2semidirect
15. #552×C109C2semidirect
179. C109#3300semidirect
180. #552D109semidirect

Maximal subgroups

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

subgroupindex
#202956852
#202953832
#202953672
#202953392
#202953302
#202952652
#202952572
#202952482
#202951862
#202951812
C4×#131820192
D4×#117247842
#202950202
C2×#168126222
#552×C1092
#3300109