Finite Group Databasebeta

← #49167331

Subgroups — #49167331

Order 37504.

738 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34036965C211yesmaximal, characteristic
2#34036895C211yesmaximal, characteristic, complemented
3#34036597C211yesmaximal, characteristic, complemented
4#34036147C211yesmaximal, characteristic, complemented
5#34036144C214yesmaximal, complemented
6C2×#22627551C211yesmaximal, characteristic, complemented
7C2×#22627550C211yesmaximal, characteristic, complemented
8#104×D293C214yesmaximal, complemented
9#421×C293C211yesmaximal, characteristic, complemented
10#22627672C412yescomplemented
11#22627671—21no—
12#22627671C412yescomplemented
13#22627670—21no—
14#22627670C412yescomplemented
15#22627669—22no—
16#22627669C412yescomplemented
17C2×#17989802—22no—
18C2×#17989802C2211yescharacteristic
19C2×#17989802C2211yescharacteristic
20#41×D293—22no—
21#41×D293C2214yes—
22#22627632—22no—
23#22627630—22no—
24C2×#17989786C2211yescharacteristic, complemented
25M16×D293C2214yescomplemented
26#22627589—21no—
27#22627586—21no—
28#22627584—22no—
29#22627581C2212yescomplemented
30C2×C4×Dic293C2211yescharacteristic, complemented
31C42×D293C2212yescomplemented
32#22627552C2214yescomplemented
33#22627551C2212yescomplemented
34#22627551C2212yescomplemented
35#22627550C2212yescomplemented
36#22627550C2212yescomplemented
37C2×#41×C293C2211yescharacteristic
38C2×M16×C293C2211yescharacteristic, complemented
39#117×C293C2211yescharacteristic, complemented
40#104×C293C2214yescomplemented
41#17989803—22no—
42#17989803C2×C414yescomplemented
43#17989802—22no—
44#17989802—22no—
45#17989802—24no—
46#17989802—24no—
47#17989802—21no—
48#17989802—21no—
49#17989802C2×C412yes—
50#17989802C2×C412yes—
51#17989802C2×C412yescomplemented
52#17989802D412yes—
53#17989802D412yes—
54#17989802C2312yes—
55#17989801—22no—
56#17989801—24no—
57#17989801—24no—
58#17989801C2×C414yescomplemented
59C2×Dic586—22no—
60C2×Dic586C2×C411yescharacteristic, complemented
61C2×Dic586D411yescharacteristic
62Q8×D293—22no—
63Q8×D293C2×C414yescomplemented
64C2×D1172—22no—
65C2×D1172C2×C411yescharacteristic, complemented
66C2×D1172D411yescharacteristic
67C2×C22⋊D293—24no—
68C2×C22⋊D293—21no—
69C2×C22⋊D293D412yes—
70D4×D293—24no—
71D4×D293—22no—
72D4×D293—24no—
73D4×D293C2×C414yescomplemented
74C22×Dic293—21no—
75C2×C4×D293—24no—
76C2×C4×D293—22no—
77C2×C4×D293—21no—
78C2×C4×D293C2312yes—
79#17989786—21no—
80#17989786C2312yes—
81#17989785—24no—
82C2×#14556873—21no—
83C8×D293—24no—
84#17989781—22no—
85#17989779—22no—
86C4×Dic293—22no—
87C4×Dic293—21no—
88C4×Dic293C2311yescharacteristic
89C4×Dic293C2311yescharacteristic
90#41×C293—21no—
91#41×C293—21no—
92#41×C293C2×C412yescomplemented
93#41×C293C2312yes—
94C2×Q8×C293C2×C411yescharacteristic, complemented
95C2×D4×C293—21no—
96C2×D4×C293C2×C411yescharacteristic, complemented
97C22×C4×C293—21no—
98C22×C4×C293C2311yescharacteristic
99M16×C293—21no—
100M16×C293C2312yes—
101C2×C8×C293—21no—
102C4⋊C4×C293—21no—
103C22⋊C4×C293—21no—
104C42×C293C2312yes—
105C22×D293—21no—
106C22×D293—24no—
107C22×D293C2×D412yes—
108Dic586—21no—
109Dic586—24no—
110Dic586C22⋊C412yes—
111Dic586C22⋊C412yes—
112Dic586C22×C412yes—
113D1172—24no—
114D1172—21no—
115D1172C22⋊C412yes—
116D1172C22⋊C412yes—
117D1172C22×C412yes—
118C22⋊D293—22no—
119C22⋊D293—22no—
120C22⋊D293—24no—
121C22⋊D293—24no—
122C22⋊D293C22⋊C414yes—
123C22⋊D293C22⋊C414yes—
124C2×Dic293—21no—
125C2×Dic293—21no—
126C2×Dic293—24no—
127C2×Dic293—21no—
128C2×Dic293—42no—
129C2×Dic293C2×D412yes—
130C4×D293—24no—
131C4×D293—22no—
132C4×D293—22no—
133C4×D293—24no—
134C4×D293—44no—
135C4×D293C22×C414yes—
136C4×D293C2×D414yes—
137#14556873—22no—
138C23×C293—21no—
139C23×C293C2×D411yescharacteristic
140Q8×C293—21no—
141Q8×C293C22×C412yes—
142D4×C293—22no—
143D4×C293—22no—
144D4×C293—21no—
145D4×C293C22×C412yes—
146C2×C4×C293—21no—
147C2×C4×C293—22no—
148C2×C4×C293—22no—
149C2×C4×C293—22no—
150C2×C4×C293C22×C411yescharacteristic
151C2×C4×C293C22×C411yescharacteristic
152C2×C4×C293C2×D411yescharacteristic
153C2×C4×C293C2×D411yescharacteristic
154C2×C4×C293C2×D411yescharacteristic
155C2×C4×C293C2411yescharacteristic
156C8×C293—22no—
157C22×C293—22no—
158C22×C293—41no—
159C22×C293—81no—
160C22×C293C2×C22⋊C411yescharacteristic
161C22×C293C2×C22⋊C411yescharacteristic
162C22×C293C22×D411yescharacteristic
163C2×D293—22no—
164C2×D293—42no—
165C2×D293—84no—
166C2×D293—81no—
167C2×D293C2×C22⋊C414yes—
168C4×C293—22no—
169C4×C293—42no—
170C4×C293C2×C22⋊C411yescharacteristic
171C4×C293C2×C22⋊C411yescharacteristic
172C4×C293C23×C411yescharacteristic
173C4×C293C22×D411yescharacteristic
174Dic293—22no—
175Dic293—22no—
176Dic293—41no—
177Dic293C2×C22⋊C414yes—
178C2×C293—21no—
179C2×C293—21no—
180C2×C293—21no—
181C2×C293—42no—
182C2×C293C22×C22⋊C411yescharacteristic
183D293—24no—
184D293—42no—
185C293#290111yescharacteristic, complemented, Sylow
186#2901—2931nomaximal, Sylow
187#585—2931no—
188#568—2931no—
189C4×#41—2931no—
190#421—2931no—
191#421—2934no—
192#421D29311yescharacteristic, complemented
193C2×#104—2931no—
194C2×#104—2931no—
195C2×#104—2934no—
1962−1+4—2932no—
1972−1+4—2932no—
1982−1+4—5861no—
199H(5,𝔽2)—2932no—
200H(5,𝔽2)—2932no—
201H(5,𝔽2)—5862no—
202H(5,𝔽2)—5861no—
203C2×#41—2931no—
204C2×#41—2934no—
205C2×#41—2931no—
206C2×#41—5862no—
207C2×#41—5862no—
208C2×#41C2×D29311yescharacteristic
209#131—5862no—
210#131—5862no—
211C2×M16—2931no—
212C2×M16—2934no—
213C2×M16C2×D29311yescharacteristic, complemented
214C4×Q8—5861no—
215C4×D4—5862no—
216C4×D4—5861no—
217#117—2932no—
218#117C2×D29311yescharacteristic, complemented
219C2×C42—2931no—
220C2×C42—2932no—
221#104—2932no—
222#104—2932no—
223#104—2932no—
224#104—2932no—
225#104—2934no—
226#104C2×D29314yescomplemented
227#41—21no—
228#41—21no—
229#41—2932no—
230#41—2932no—
231#41—2932no—
232#41—2932no—
233#41—2934no—
234#41—2934no—
235#41—2932no—
236#41—2932no—
237#41—5864no—
238#41—5864no—
239#41—5862no—
240#41—5862no—
241#41—5864no—
242#41—5864no—
243#41—5862no—
244#41—5862no—
245#41—5861no—
246#41—5861no—
247#41C4×D29312yescomplemented
248#41C22×D29312yes—
249C2×Q8—2931no—
250C2×Q8—2934no—
251C2×Q8—2931no—
252C2×Q8—5862no—
253C2×Q8—5862no—
254C2×Q8C4×D29311yescharacteristic, complemented
255C2×D4—21no—
256C2×D4—2932no—
257C2×D4—2934no—
258C2×D4—2931no—
259C2×D4—2931no—
260C2×D4—5862no—
261C2×D4—5864no—
262C2×D4—5862no—
263C2×D4—5864no—
264C2×D4—5864no—
265C2×D4—5861no—
266C2×D4C4×D29311yescharacteristic, complemented
267C22×C4—21no—
268C22×C4—2932no—
269C22×C4—5864no—
270C22×C4—5862no—
271C22×C4—5861no—
272C22×C4—5861no—
273C22×C4C22×D29311yescharacteristic
274M16—21no—
275M16—2932no—
276M16—5864no—
277M16—5861no—
278M16C22×D29312yes—
279C2×C8—21no—
280C2×C8—5864no—
281C2×C8—5861no—
282C4⋊C4—21no—
283C4⋊C4—5862no—
284C22⋊C4—21no—
285C22⋊C4—5862no—
286C42—2931no—
287C42—2931no—
288C42—5862no—
289C42—5861no—
290C42C22×D29312yes—
291C23—21no—
292C23—2932no—
293C23—5864no—
294C23—5861no—
295C23D4×D29311yescharacteristic
296Q8—21no—
297Q8—2932no—
298Q8—2932no—
299Q8—2932no—
300Q8—5864no—
301Q8—5861no—
302Q8C2×C4×D29312yes—
303D4—22no—
304D4—22no—
305D4—21no—
306D4—2934no—
307D4—2934no—
308D4—2932no—
309D4—2932no—
310D4—2932no—
311D4—5864no—
312D4—5864no—
313D4—5864no—
314D4—5862no—
315D4—5862no—
316D4—5861no—
317D4C2×C4×D29312yes—
318C2×C4—21no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—2932no—
323C2×C4—2934no—
324C2×C4—2934no—
325C2×C4—5864no—
326C2×C4—5864no—
327C2×C4—5864no—
328C2×C4—5861no—
329C2×C4—5861no—
330C2×C4—5862no—
331C2×C4—5862no—
332C2×C4—5861no—
333C2×C4—11724no—
334C2×C4—11722no—
335C2×C4C2×C4×D29311yescharacteristic
336C2×C4C2×C4×D29311yescharacteristic
337C2×C4D4×D29311yescharacteristic
338C2×C4D4×D29311yescharacteristic
339C2×C4D4×D29311yescharacteristic
340C2×C4C23×D29311yescharacteristic
341C8—22no—
342C8—5862no—
343C22—22no—
344C22—41no—
345C22—81no—
346C22—2934no—
347C22—23441no—
348C22—23444no—
349C22—5862no—
350C22—11722no—
351C22C22⋊C4×D29311yescharacteristic
352C22C22⋊C4×D29311yescharacteristic
353C22C2×D4×D29311yescharacteristic
354C4—22no—
355C4—42no—
356C4—2934no—
357C4—5862no—
358C4—5862no—
359C4—11721no—
360C4C22⋊C4×D29311yescharacteristic
361C4C22⋊C4×D29311yescharacteristic
362C4C22×C4×D29311yescharacteristic
363C4C2×D4×D29311yescharacteristic
364C2—21no—
365C2—21no—
366C2—21no—
367C2—42no—
368C2—5864no—
369C2—11722no—
370C2C2×C22⋊C4×D29311yescharacteristic
subgroupG/H
centerC4C2×D4×D293
commutatorC4×C293C23×C4
FrattiniC2×C4C23×D293
Fitting#421×C293C2
socleC2×C293C22×C22⋊C4
prime radical#34036965C2
cosocleC2×C4×C293C24
hypercenter#421D293
hypocenterC293#2901
LCSG ▷ C4×C293 ▷ C2×C293 ▷ C293
UCSC1 ◁ C4 ◁ C22×C4 ◁ #421

Complements

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

NH ≅ G/Ntype
2. #34036895C2semidirect
3. #34036597C2semidirect
4. #34036147C2semidirect
5. #34036144C2semidirect
6. C2×#22627551C2semidirect
7. C2×#22627550C2semidirect
8. #104×D293C2semidirect
9. #421×C293C2semidirect
10. #22627672C4semidirect
12. #22627671C4semidirect
14. #22627670C4semidirect
16. #22627669C4semidirect
24. C2×#17989786C22semidirect
25. M16×D293C22semidirect
29. #22627581C22semidirect
30. C2×C4×Dic293C22semidirect
31. C42×D293C22semidirect
32. #22627552C22semidirect
33. #22627551C22semidirect
34. #22627551C22semidirect
35. #22627550C22semidirect
36. #22627550C22semidirect
38. C2×M16×C293C22semidirect
39. #117×C293C22semidirect
40. #104×C293C22semidirect
42. #17989803C2×C4semidirect
51. #17989802C2×C4semidirect
58. #17989801C2×C4semidirect
60. C2×Dic586C2×C4semidirect
63. Q8×D293C2×C4semidirect
65. C2×D1172C2×C4semidirect
73. D4×D293C2×C4semidirect
92. #41×C293C2×C4semidirect
94. C2×Q8×C293C2×C4semidirect
96. C2×D4×C293C2×C4semidirect
185. C293#2901semidirect
192. #421D293semidirect
213. C2×M16C2×D293semidirect
218. #117C2×D293semidirect
226. #104C2×D293semidirect
247. #41C4×D293semidirect
254. C2×Q8C4×D293semidirect
266. C2×D4C4×D293semidirect

Maximal subgroups

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

subgroupindex
#340369652
#340368952
#340365972
#340361472
#340361442
C2×#226275512
C2×#226275502
#104×D2932
#421×C2932
#2901293