Finite Group Databasebeta

← #28346178

Subgroups — #28346178

Order 13696.

452 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20293425C211yesmaximal, characteristic, complemented
2C2×#16812129C211yesmaximal, characteristic, complemented
3C2×#16812116C211yesmaximal, characteristic, complemented
4#20292900C212yesmaximal, complemented
5#20292895C212yesmaximal, complemented
6#20292892C212yesmaximal, complemented
7#20292888C212yesmaximal, complemented
8C2×#16812044C211yesmaximal, characteristic, complemented
9#20292611C211yesmaximal, characteristic, complemented
10#20292609C211yesmaximal, characteristic, complemented
11#575×C107C211yesmaximal, characteristic, complemented
12C2×#13133818C2211yescharacteristic
13#16812140—21no—
14#16812139C2212yes—
15#16812135—21no—
16#16812134C2212yes—
17#16812129—21no—
18#16812129—21no—
19#16812129—21no—
20#16812129—21no—
21#16812129—22no—
22C2×#13133808C2211yescharacteristic, complemented
23C2×#13133806C2211yescharacteristic, complemented
24C2×#13133805C2211yescharacteristic, complemented
25C2×#13133803C2211yescharacteristic, complemented
26#16812116—21no—
27#16812116—21no—
28#16812116C2212yes—
29#16812116C2212yes—
30C2×#13133802—21no—
31C2×#13133802C2211yescharacteristic
32C2×#13133802C2211yescharacteristic
33C22×C107⋊C8—21no—
34C22×C107⋊C8C2211yescharacteristic
35#16812044C2212yescomplemented
36#16812044C2212yescomplemented
37#16812028C2212yescomplemented
38#16812027C2212yescomplemented
39#16812026C2212yescomplemented
40#16812025C2212yescomplemented
41C2×#41×C107C2211yescharacteristic
42C2×#41×C107C2211yescharacteristic
43#137×C107—21no—
44#137×C107C2212yes—
45AGL(1,ℤ/8)×C107—21no—
46AGL(1,ℤ/8)×C107C2212yes—
47#135×C107—22no—
48C2×M16×C107C2211yescharacteristic
49#13133818—21no—
50#13133818—21no—
51#13133818—21no—
52#13133818—21no—
53C2×Dic214C2311yescharacteristic
54C2×D428C2311yescharacteristic
55C2×C22⋊D107—21no—
56C2×C4×D107—21no—
57#13133808—22no—
58#13133808—21no—
59#13133808—21no—
60#13133806—22no—
61#13133806—21no—
62#13133806—21no—
63#13133805—22no—
64#13133805—21no—
65#13133805—21no—
66#13133803—22no—
67#13133803—21no—
68#13133803—21no—
69#13133802—21no—
70#13133802—21no—
71#13133802—22no—
72#13133802—22no—
73#13133802C2312yes—
74#13133802C2312yes—
75C2×C107⋊C8—21no—
76C2×C107⋊C8—21no—
77C2×C107⋊C8—22no—
78C2×C107⋊C8—22no—
79C2×C107⋊C8D411yescharacteristic
80C2×C107⋊C8D411yescharacteristic
81C2×C107⋊C8D411yescharacteristic
82C2×C107⋊C8D411yescharacteristic
83C2×C107⋊C8C2311yescharacteristic
84C2×C107⋊C8C2311yescharacteristic
85#41×C107—21no—
86#41×C107—21no—
87#41×C107—21no—
88#41×C107—21no—
89#41×C107—21no—
90#41×C107—21no—
91#41×C107D412yes—
92#41×C107C2312yes—
93C2×Q8×C107D411yescharacteristic
94C2×Q8×C107C2311yescharacteristic
95C2×D4×C107—21no—
96C2×D4×C107—21no—
97C2×D4×C107D411yescharacteristic
98C2×D4×C107C2311yescharacteristic
99C22×C4×C107—21no—
100C22×C4×C107—21no—
101C22×C4×C107C2311yescharacteristic
102Q16×C107—22no—
103SD16×C107—22no—
104SD16×C107—22no—
105D8×C107—22no—
106M16×C107—21no—
107M16×C107C2312yes—
108C2×C8×C107—21no—
109C23×C107—21no—
110C23×C107—21no—
111C23×C107#4111yescharacteristic
112C22×D107—21no—
113Q8×C107—21no—
114Q8×C107—21no—
115Q8×C107—21no—
116Q8×C107C2×D412yes—
117D4×C107—22no—
118D4×C107—22no—
119D4×C107—21no—
120D4×C107—21no—
121D4×C107—21no—
122D4×C107—41no—
123D4×C107—41no—
124D4×C107C2×D412yes—
125C2×C4×C107—21no—
126C2×C4×C107—22no—
127C2×C4×C107—22no—
128C2×C4×C107—21no—
129C2×C4×C107—41no—
130C2×C4×C107—41no—
131C2×C4×C107C2×D411yescharacteristic
132C2×C4×C107C2×D411yescharacteristic
133C2×C4×C107C2×D411yescharacteristic
134C2×C4×C107C2×D411yescharacteristic
135C2×C4×C107#4111yescharacteristic
136C2×C4×C107C2411yescharacteristic
137C22⋊D107—41no—
138C22⋊D107—41no—
139C2×Dic107—21no—
140D428—21no—
141D428—21no—
142C4×D107—41no—
143C4×D107—41no—
144Dic214—21no—
145Dic214—21no—
146C8×C107—22no—
147C107⋊C8—22no—
148C107⋊C8—22no—
149C107⋊C8C2×D411yescharacteristic
150C107⋊C8C2×D411yescharacteristic
151C107⋊C8C2×D411yescharacteristic
152C107⋊C8C2×D411yescharacteristic
153C22×C107—22no—
154C22×C107—41no—
155C22×C107—42no—
156C22×C107—41no—
157C22×C107—81no—
158C22×C107#12111yescharacteristic
159C22×C107#12111yescharacteristic
160C22×C107C2×#4111yescharacteristic
161C2×D107—41no—
162C2×D107—81no—
163C4×C107—22no—
164C4×C107—41no—
165C4×C107#12111yescharacteristic
166C4×C107#12111yescharacteristic
167C4×C107C22×D411yescharacteristic
168C4×C107C22×D411yescharacteristic
169Dic107—41no—
170C2×C107—21no—
171C2×C107—21no—
172C2×C107—21no—
173C2×C107—42no—
174C2×C107—81no—
175C2×C107C2×#12111yescharacteristic
176D107—81no—
177#3067—1071nomaximal, Sylow
178C107#306711yescharacteristic, complemented, Sylow
179#575—1071no—
180#575D10711yescharacteristic, complemented
181C2×#135—1071no—
182C2×#131—1071no—
183#473—1072no—
184#472—1072no—
185#471—1072no—
186#471—1072no—
187C2×#108—1071no—
188#413—1071no—
189#413—1071no—
190C2×#41—1071no—
191C2×#41C2×D10711yescharacteristic
192C2×#41C2×D10711yescharacteristic
193#137—21no—
194#137—1072no—
195#137—2141no—
196#137C2×D10712yes—
197AGL(1,ℤ/8)—21no—
198AGL(1,ℤ/8)—1072no—
199AGL(1,ℤ/8)—2141no—
200AGL(1,ℤ/8)C2×D10712yes—
201#135—22no—
202#135—2141no—
203#135—2141no—
204#135—2141no—
205#135—2141no—
206#135—2142no—
207C2×Q16—1071no—
208C2×SD16—1071no—
209C2×SD16—1071no—
210C2×D8—1071no—
211#131—1072no—
212#131—1072no—
213#131—2141no—
214#131—2141no—
215C2×M16—1071no—
216C2×M16—1071no—
217C2×M16—2141no—
218C2×M16C2×D10711yescharacteristic
219C22×C8—1071no—
220C22×C8—2141no—
221#108—1072no—
222#108—1072no—
223#101—1072no—
224#101—1072no—
225#100—1072no—
226#100—1072no—
227#41—21no—
228#41—21no—
229#41—21no—
230#41—21no—
231#41—21no—
232#41—21no—
233#41—2141no—
234#41—2141no—
235#41—2141no—
236#41—2141no—
237#41C22⋊D10712yes—
238#41C22×D10712yes—
239C2×Q8—1071no—
240C2×Q8C22⋊D10711yescharacteristic
241C2×Q8C22×D10711yescharacteristic
242C2×D4—21no—
243C2×D4—21no—
244C2×D4—1071no—
245C2×D4—2141no—
246C2×D4C22⋊D10711yescharacteristic
247C2×D4C22×D10711yescharacteristic
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—2141no—
251C22×C4C22×D10711yescharacteristic
252Q16—22no—
253Q16—2141no—
254Q16—2141no—
255Q16—2142no—
256SD16—22no—
257SD16—22no—
258SD16—2141no—
259SD16—2141no—
260SD16—2141no—
261SD16—2141no—
262SD16—2142no—
263SD16—2142no—
264D8—22no—
265D8—2141no—
266D8—2141no—
267D8—2142no—
268M16—21no—
269M16—1072no—
270M16—1072no—
271M16—2142no—
272M16—2142no—
273M16—2141no—
274M16—2141no—
275M16C22×D10712yes—
276C2×C8—21no—
277C2×C8—1071no—
278C2×C8—1071no—
279C2×C8—1071no—
280C2×C8—1071no—
281C2×C8—1071no—
282C2×C8—1071no—
283C2×C8—2142no—
284C2×C8—2142no—
285C2×C8—2141no—
286C2×C8—2141no—
287C23—21no—
288C23—21no—
289C23—2141no—
290C23#1313381711yescharacteristic
291Q8—21no—
292Q8—21no—
293Q8—21no—
294Q8—2141no—
295Q8—2141no—
296Q8C2×C22⋊D10712yes—
297D4—22no—
298D4—22no—
299D4—21no—
300D4—21no—
301D4—21no—
302D4—41no—
303D4—41no—
304D4—4281no—
305D4—4281no—
306D4—2141no—
307D4—2141no—
308D4C2×C22⋊D10712yes—
309C2×C4—21no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—21no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—4281no—
316C2×C4—4281no—
317C2×C4—2141no—
318C2×C4D4×D10711yescharacteristic
319C2×C4D4×D10711yescharacteristic
320C2×C4C2×C22⋊D10711yescharacteristic
321C2×C4C2×C22⋊D10711yescharacteristic
322C2×C4#1313381711yescharacteristic
323C2×C4C23×D10711yescharacteristic
324C8—22no—
325C8—1071no—
326C8—1071no—
327C8—1071no—
328C8—1071no—
329C8—2142no—
330C8—2142no—
331C22—22no—
332C22—41no—
333C22—42no—
334C22—41no—
335C22—81no—
336C22—8561no—
337C22—4281no—
338C22#1681208111yescharacteristic
339C22#1681208111yescharacteristic
340C22C2×#1313381711yescharacteristic
341C4—22no—
342C4—41no—
343C4—4281no—
344C4#1681208111yescharacteristic
345C4#1681208111yescharacteristic
346C4C2×D4×D10711yescharacteristic
347C4C22×C22⋊D10711yescharacteristic
348C2—21no—
349C2—21no—
350C2—21no—
351C2—42no—
352C2—81no—
353C2—8561no—
354C2C2×#1681208111yescharacteristic
subgroupG/H
centerC4#16812081
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#575×C107C2
socleC2×C107C2×#121
hypercenter#575D107
hypocenterC107#3067
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C4 ◁ C22×C4 ◁ #575

Complements

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

NH ≅ G/Ntype
1. #20293425C2semidirect
2. C2×#16812129C2semidirect
3. C2×#16812116C2semidirect
4. #20292900C2semidirect
5. #20292895C2semidirect
6. #20292892C2semidirect
7. #20292888C2semidirect
8. C2×#16812044C2semidirect
9. #20292611C2semidirect
10. #20292609C2semidirect
11. #575×C107C2semidirect
22. C2×#13133808C22semidirect
23. C2×#13133806C22semidirect
24. C2×#13133805C22semidirect
25. C2×#13133803C22semidirect
35. #16812044C22semidirect
36. #16812044C22semidirect
37. #16812028C22semidirect
38. #16812027C22semidirect
39. #16812026C22semidirect
40. #16812025C22semidirect
178. C107#3067semidirect
180. #575D107semidirect

Maximal subgroups

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

subgroupindex
#202934252
C2×#168121292
C2×#168121162
#202929002
#202928952
#202928922
#202928882
C2×#168120442
#202926112
#202926092
#575×C1072
#3067107