Finite Group Databasebeta

← #26104907

Subgroups — #26104907

Order 12032.

506 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 506 conjugacy classes of subgroups — too many to draw. Only its 103 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#20251719C211yesmaximal, characteristic
2#20243456C211yesmaximal, characteristic, complemented
3#20241212C212yesmaximal, complemented
4#20241208C212yesmaximal, complemented
5C47×#1817C211yesmaximal, characteristic, complemented
6C2×#13094007—21no—
7C2×#13093998C411yescharacteristic
8C2×C22⋊(C47⋊C8)—21no—
9#15689025C411yescharacteristic
10#15689025C2211yescharacteristic
11#15689021C411yescharacteristic
12#15689021C2211yescharacteristic
13#15689014—21no—
14#15689014—21no—
15#15689002—21no—
16#15688983C411yescharacteristic
17#15688878—21no—
18#15688878C2212yes—
19C22×C42×C47C2211yescharacteristic
20C2×C22⋊C8×C47—21no—
21C47×#336—21no—
22C47×#336C2212yes—
23C22×D188C2×C411yescharacteristic
24C22×C22⋊D47—21no—
25C22×C47⋊C8—22no—
26#13094007—44no—
27#13093998—42no—
28C2×C4⋊Dic47C2×C411yescharacteristic
29C2×C4⋊Dic47C2×C411yescharacteristic
30C2×C4⋊Dic47C2×C411yescharacteristic
31C2×Dic47⋊C4—21no—
32C2×Dic47⋊C4—21no—
33C2×C22⋊Dic47—21no—
34C2×(C2×D47)⋊C4—21no—
35C2×(C2×D47)⋊C4—21no—
36C2×(C2×D47)⋊C4—21no—
37C2×(C2×D47)⋊C4—21no—
38C2×(C2×D47)⋊C4—21no—
39C22⋊(C47⋊C8)—42no—
40C47⋊#95—21no—
41C47⋊#95—21no—
42C47⋊#95—21no—
43C23×C4×C47C2×C411yescharacteristic
44C23×C4×C47D411yescharacteristic
45C23×C4×C47D411yescharacteristic
46C22×C8×C47—22no—
47C2×C42×C47—42no—
48C2×C42×C47C2×C411yescharacteristic
49C2×C42×C47D411yescharacteristic
50C2×C42×C47D411yescharacteristic
51C2×C42×C47C2311yescharacteristic
52C22⋊C8×C47—42no—
53C24×C47C22⋊C411yescharacteristic
54C23×D47—21no—
55C22×C4×C47—22no—
56C22×C4×C47—22no—
57C22×C4×C47—22no—
58C22×C4×C47—22no—
59C22×C4×C47—22no—
60C22×C4×C47—22no—
61C22×C4×C47—41no—
62C22×C4×C47—41no—
63C22×C4×C47—41no—
64C22×C4×C47C22⋊C411yescharacteristic
65C22×C4×C47C22⋊C411yescharacteristic
66C22×C4×C47C22⋊C411yescharacteristic
67C22×C4×C47C22×C411yescharacteristic
68C22×C4×C47C2×D411yescharacteristic
69C22×C4×C47C2×D411yescharacteristic
70C2×C22⋊D47—42no—
71C2×C22⋊D47—42no—
72C2×C22⋊D47—42no—
73C22×Dic47—21no—
74C22×Dic47—21no—
75C22×Dic47—21no—
76C2×D188—22no—
77C2×D188—41no—
78C2×D188—41no—
79C2×C8×C47—42no—
80C2×C8×C47—42no—
81C2×C8×C47—42no—
82C42×C47—41no—
83C42×C47—41no—
84C42×C47—41no—
85C42×C47—41no—
86C22⋊Dic47—42no—
87(C2×D47)⋊C4—42no—
88(C2×D47)⋊C4—42no—
89(C2×D47)⋊C4—42no—
90(C2×D47)⋊C4—42no—
91(C2×D47)⋊C4—42no—
92C4⋊Dic47—22no—
93C4⋊Dic47—41no—
94C4⋊Dic47—41no—
95Dic47⋊C4—42no—
96Dic47⋊C4—42no—
97C2×C47⋊C8—42no—
98C2×C47⋊C8—42no—
99C2×C47⋊C8—42no—
100C23×C47—21no—
101C23×C47—21no—
102C23×C47—21no—
103C23×C47—21no—
104C23×C47—21no—
105C23×C47—21no—
106C23×C47#9912yes—
107C23×C47C2×C22⋊C411yescharacteristic
108C22×D47—41no—
109C22×D47—41no—
110C22×D47—41no—
111C22×D47—41no—
112C22×D47—41no—
113C22×D47—41no—
114C22×D47—41no—
115C2×C4×C47—21no—
116C2×C4×C47—21no—
117C2×C4×C47—21no—
118C2×C4×C47—21no—
119C2×C4×C47—21no—
120C2×C4×C47—21no—
121C2×C4×C47—21no—
122C2×C4×C47—21no—
123C2×C4×C47—21no—
124C2×C4×C47—21no—
125C2×C4×C47—21no—
126C2×C4×C47—21no—
127C2×C4×C47—21no—
128C2×C4×C47—21no—
129C2×C4×C47—21no—
130C2×C4×C47—44no—
131C2×C4×C47—44no—
132C2×C4×C47—44no—
133C2×C4×C47#13112yes—
134C2×C4×C47#13512yes—
135C2×C4×C47AGL(1,ℤ/8)12yes—
136C22⋊D47—84no—
137C2×Dic47—41no—
138C2×Dic47—41no—
139C2×Dic47—41no—
140C2×Dic47—41no—
141C2×Dic47—41no—
142C2×Dic47—41no—
143C2×Dic47—41no—
144C2×Dic47—41no—
145C2×Dic47—41no—
146D188—82no—
147C8×C47—82no—
148C47⋊C8—82no—
149#7718—471nomaximal, Sylow
150C22×C47—22no—
151C22×C47—22no—
152C22×C47—22no—
153C22×C47—41no—
154C22×C47—41no—
155C22×C47—41no—
156C22×C47—41no—
157C22×C47#40811yescharacteristic
158C22×C47C2×#9911yescharacteristic
159C22×C47#41311yescharacteristic
160C22×C47#41611yescharacteristic
161C22×C47#41811yescharacteristic
162C22×C47#42111yescharacteristic
163C22×C47#42111yescharacteristic
164C2×D47—81no—
165C2×D47—81no—
166C2×D47—81no—
167C2×D47—81no—
168C2×D47—81no—
169C2×D47—81no—
170C2×D47—81no—
171C4×C47—22no—
172C4×C47—22no—
173C4×C47—22no—
174C4×C47—41no—
175C4×C47—41no—
176C4×C47—41no—
177Dic47—81no—
178Dic47—81no—
179Dic47—81no—
180#2854—471no—
181#1817—471no—
182#1817D4711yescharacteristic, complemented
183#1271—472no—
184#1270—472no—
185C2×C47—42no—
186C2×C47#146811yescharacteristic
187C2×C47#148211yescharacteristic
188C2×C47#149611yescharacteristic
189C2×C47#151011yescharacteristic
190C2×C47#151611yescharacteristic
191C2×C47#153511yescharacteristic
192C2×C47#155311yescharacteristic
193D47—161no—
194C2×#123—941no—
195C2×#121—471no—
196C22×C42C2×D4711yescharacteristic
197C2×C22⋊C8—21no—
198C2×C22⋊C8—941no—
199#390—471no—
200#390—471no—
201#389—471no—
202#389—471no—
203#388—941no—
204#388—941no—
205#386—941no—
206#380—471no—
207#336—21no—
208#336—472no—
209#336—941no—
210#336C2×D4712yes—
211C47#771811yescharacteristic, complemented, Sylow
212C22×D4—471no—
213C22×D4—941no—
214C23×C4C4×D4711yescharacteristic
215C23×C4D18811yescharacteristic
216C23×C4C22⋊D4711yescharacteristic
217C22×C8—22no—
218C22×C8—942no—
219#123—1884no—
220#121—1882no—
221C2×C4⋊C4—471no—
222C2×C4⋊C4—471no—
223C2×C4⋊C4—471no—
224C2×C4⋊C4—941no—
225C2×C4⋊C4—941no—
226C2×C22⋊C4—941no—
227C2×C22⋊C4—941no—
228C2×C22⋊C4—941no—
229C2×C22⋊C4—941no—
230C2×C22⋊C4—941no—
231C2×C22⋊C4—941no—
232C2×C42—42no—
233C2×C42C4×D4711yescharacteristic
234C2×C42D18811yescharacteristic
235C2×C42C22⋊D4711yescharacteristic
236C2×C42C22×D4711yescharacteristic
237C22⋊C8—42no—
238C22⋊C8—1882no—
239#95—941no—
240#95—941no—
241#95—941no—
242C24—941no—
243C24(C2×D47)⋊C411yescharacteristic
244C2×D4—942no—
245C2×D4—1881no—
246C2×D4—1881no—
247C2×D4—1882no—
248C2×D4—1882no—
249C2×D4—1882no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—941no—
260C22×C4—941no—
261C22×C4—941no—
262C22×C4(C2×D47)⋊C411yescharacteristic
263C22×C4(C2×D47)⋊C411yescharacteristic
264C22×C4(C2×D47)⋊C411yescharacteristic
265C22×C4C2×C4×D4711yescharacteristic
266C22×C4C2×D18811yescharacteristic
267C22×C4C2×C22⋊D4711yescharacteristic
268C2×C8—42no—
269C2×C8—42no—
270C2×C8—42no—
271C2×C8—1882no—
272C2×C8—1882no—
273C2×C8—1882no—
274C4⋊C4—942no—
275C4⋊C4—1881no—
276C4⋊C4—1881no—
277C4⋊C4—1882no—
278C4⋊C4—1882no—
279C22⋊C4—1882no—
280C22⋊C4—1882no—
281C22⋊C4—1882no—
282C22⋊C4—1882no—
283C22⋊C4—1882no—
284C22⋊C4—1882no—
285C42—41no—
286C42—41no—
287C42—41no—
288C42—41no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23—1881no—
296C23—1881no—
297C23—1881no—
298C23—1881no—
299C23—1881no—
300C23—1881no—
301C23—1881no—
302C23#1309393812yes—
303C23C2×(C2×D47)⋊C411yescharacteristic
304D4—3762no—
305D4—3764no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—44no—
322C2×C4—44no—
323C2×C4—44no—
324C2×C4—1881no—
325C2×C4—1881no—
326C2×C4—1881no—
327C2×C4—1881no—
328C2×C4—1881no—
329C2×C4—1881no—
330C2×C4—1881no—
331C2×C4—1881no—
332C2×C4—1881no—
333C2×C4#1309403212yes—
334C2×C4#1309404612yes—
335C2×C4#1309404912yes—
336C8—82no—
337C8—3762no—
338C22—22no—
339C22—22no—
340C22—22no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—3761no—
346C22—3761no—
347C22—3761no—
348C22—3761no—
349C22—3761no—
350C22—3761no—
351C22—3761no—
352C22#1568909111yescharacteristic
353C22C2×#1309393811yescharacteristic
354C22#1568910511yescharacteristic
355C22#1568912011yescharacteristic
356C22#1568912711yescharacteristic
357C22#1568913911yescharacteristic
358C22#1568914111yescharacteristic
359C4—22no—
360C4—22no—
361C4—22no—
362C4—41no—
363C4—41no—
364C4—41no—
365C4—3761no—
366C4—3761no—
367C4—3761no—
368C2—42no—
369C2—7521no—
370C2#2024173711yescharacteristic
371C2#2024179711yescharacteristic
372C2#2024185611yescharacteristic
373C2#2024191611yescharacteristic
374C2#2024194511yescharacteristic
375C2#2024202011yescharacteristic
376C2#2024209111yescharacteristic
subgroupG/H
centerC23C2×(C2×D47)⋊C4
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1817C2
socleC23×C47C2×C22⋊C4
prime radicalC2×#13093998C4
cosocleC2×C42×C47C23
hypercenter#1817D47
hypocenterC47#7718
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1817

Complements

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

NH ≅ G/Ntype
2. #20243456C2semidirect
3. #20241212C2semidirect
4. #20241208C2semidirect
5. C47×#1817C2semidirect
182. #1817D47semidirect
211. C47#7718semidirect

Maximal subgroups

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

subgroupindex
#202517192
#202434562
#202412122
#202412082
C47×#18172
#771847