Finite Group Databasebeta

← C113×#12300 #43280320

Subgroups — C113×#12300 #43280320

Order 28928.

328 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 328 conjugacy classes of subgroups — too many to draw. Only its 70 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
1C113×#3197C211yesmaximal, characteristic
2C113×#2239C211yesmaximal, characteristic, complemented
3C113×#2238C211yesmaximal, characteristic, complemented
4C113×#2091C211yesmaximal, characteristic, complemented
5C113×#1342C211yesmaximal, characteristic, complemented
6C113×#1341C211yesmaximal, characteristic, complemented
7C113×#1320C211yesmaximal, characteristic, complemented
8#525×C113—21no—
9#485×C113—21no—
10#482×C113—21no—
11#470×C113C2211yescharacteristic
12#469×C113C2211yescharacteristic
13#468×C113—21no—
14#452×C113—21no—
15#449×C113—21no—
16#430×C113—21no—
17#430×C113C2211yescharacteristic
18#407×C113—21no—
19#407×C113C2211yescharacteristic
20#365×C113—21no—
21#365×C113—21no—
22#362×C113—21no—
23#361×C113—21no—
24#349×C113C2211yescharacteristic
25#349×C113C2211yescharacteristic
26#338×C113C2211yescharacteristic
27C2×#41×C113—21no—
28C23×C4×C113D411yescharacteristic
29C2×Q16×C113D411yescharacteristic
30C2×SD16×C113—21no—
31C2×D8×C113D411yescharacteristic
32C2×M16×C113—21no—
33C2×M16×C113—21no—
34C2×M16×C113D411yescharacteristic
35C2×M16×C113C2311yescharacteristic
36#123×C113—41no—
37C22⋊Q8×C113—21no—
38C22⋊Q8×C113—21no—
39#121×C113—21no—
40#121×C113—21no—
41#120×C113—41no—
42C4×D4×C113—41no—
43C4×D4×C113—41no—
44#117×C113—21no—
45M32×C113—21no—
46M32×C113—21no—
47#108×C113—21no—
48#108×C113—21no—
49#108×C113—42no—
50#107×C113—21no—
51#106×C113D411yescharacteristic
52#106×C113D411yescharacteristic
53#103×C113—21no—
54#103×C113—21no—
55#102×C113—21no—
56#102×C113—21no—
57C22⋊C8×C113—21no—
58C22⋊C8×C113—21no—
59C22⋊C8×C113—42no—
60C24×C113SD1611yescharacteristic
61#41×C113—81no—
62#41×C113—81no—
63C2×Q8×C113—21no—
64C2×D4×C113—21no—
65C2×D4×C113—41no—
66C2×D4×C113—41no—
67C2×D4×C113—41no—
68C22×C4×C113—21no—
69C22×C4×C113—21no—
70C22×C4×C113—41no—
71C22×C4×C113—41no—
72C22×C4×C113—41no—
73C22×C4×C113SD1611yescharacteristic
74C22×C4×C113C2×D411yescharacteristic
75Q16×C113—41no—
76SD16×C113—81no—
77D8×C113—41no—
78M16×C113—41no—
79M16×C113—41no—
80M16×C113—41no—
81M16×C113—41no—
82M16×C113—41no—
83M16×C113—41no—
84C2×C8×C113—22no—
85C2×C8×C113—21no—
86C2×C8×C113—41no—
87C2×C8×C113C2×D411yescharacteristic
88C2×C8×C113C2×D411yescharacteristic
89C4⋊C4×C113—21no—
90C4⋊C4×C113—21no—
91C4⋊C4×C113—41no—
92C4⋊C4×C113—41no—
93C22⋊C4×C113—41no—
94C22⋊C4×C113—41no—
95C22⋊C4×C113—41no—
96C22⋊C4×C113—41no—
97C22⋊C4×C113—41no—
98C42×C113—41no—
99C16×C113—41no—
100C16×C113—41no—
101C23×C113—21no—
102C23×C113—41no—
103C23×C113—41no—
104C23×C113—81no—
105C23×C113C2×SD1611yescharacteristic
106Q8×C113—41no—
107Q8×C113—41no—
108D4×C113—41no—
109D4×C113—41no—
110D4×C113—82no—
111D4×C113—82no—
112D4×C113—82no—
113C2×C4×C113—21no—
114C2×C4×C113—21no—
115C2×C4×C113—21no—
116C2×C4×C113—41no—
117C2×C4×C113—41no—
118C2×C4×C113—41no—
119C2×C4×C113—41no—
120C2×C4×C113—41no—
121C2×C4×C113—41no—
122C2×C4×C113—81no—
123C2×C4×C113—82no—
124C2×C4×C113#12011yescharacteristic
125C2×C4×C113AGL(1,ℤ/8)11yescharacteristic
126C8×C113—21no—
127C8×C113—21no—
128C8×C113—42no—
129C8×C113—41no—
130C8×C113—81no—
131C22×C113—21no—
132C22×C113—41no—
133C22×C113—41no—
134C22×C113—41no—
135C22×C113—42no—
136C22×C113—41no—
137C22×C113—81no—
138C22×C113—81no—
139C22×C113—81no—
140C22×C113—81no—
141C22×C113#45011yescharacteristic
142C4×C113—21no—
143C4×C113—41no—
144C4×C113—81no—
145C4×C113—81no—
146C4×C113—81no—
147C4×C113#45311yescharacteristic
148C4×C113#45811yescharacteristic
149#12300C11311yesmaximal, characteristic, direct product factor, Sylow
150C2×C113—21no—
151C2×C113—41no—
152C2×C113—81no—
153C2×C113—161no—
154C2×C113#161311yescharacteristic
155#3197C2×C11311yescharacteristic
156#2239C2×C11311yescharacteristic, complemented
157#2238C2×C11311yescharacteristic, complemented
158#2091C2×C11311yescharacteristic, complemented
159#1342C2×C11311yescharacteristic, complemented
160#1341C2×C11311yescharacteristic, complemented
161#1320C2×C11311yescharacteristic, complemented
162C113#1230011yescharacteristic, direct product factor, Sylow
163#525—21no—
164#485—21no—
165#482—21no—
166#470C22×C11311yescharacteristic
167#469C22×C11311yescharacteristic
168#468—21no—
169#452—21no—
170#449—21no—
171#430—21no—
172#430C22×C11311yescharacteristic
173#407—21no—
174#407C22×C11311yescharacteristic
175#365—21no—
176#365—21no—
177#362—21no—
178#361—21no—
179#349C22×C11311yescharacteristic
180#349C22×C11311yescharacteristic
181#338C22×C11311yescharacteristic
182C2×#41—21no—
183C23×C4D4×C11311yescharacteristic
184C2×Q16D4×C11311yescharacteristic
185C2×SD16—21no—
186C2×D8D4×C11311yescharacteristic
187C2×M16—21no—
188C2×M16—21no—
189C2×M16D4×C11311yescharacteristic
190C2×M16C23×C11311yescharacteristic
191#123—41no—
192C22⋊Q8—21no—
193C22⋊Q8—21no—
194#121—21no—
195#121—21no—
196#120—41no—
197C4×D4—41no—
198C4×D4—41no—
199#117—21no—
200M32—21no—
201M32—21no—
202#108—21no—
203#108—21no—
204#108—42no—
205#107—21no—
206#106D4×C11311yescharacteristic
207#106D4×C11311yescharacteristic
208#103—21no—
209#103—21no—
210#102—21no—
211#102—21no—
212C22⋊C8—21no—
213C22⋊C8—21no—
214C22⋊C8—42no—
215C24SD16×C11311yescharacteristic
216#41—81no—
217#41—81no—
218C2×Q8—21no—
219C2×D4—21no—
220C2×D4—41no—
221C2×D4—41no—
222C2×D4—41no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4SD16×C11311yescharacteristic
229C22×C4C2×D4×C11311yescharacteristic
230Q16—41no—
231SD16—81no—
232D8—41no—
233M16—41no—
234M16—41no—
235M16—41no—
236M16—41no—
237M16—41no—
238M16—41no—
239C2×C8—22no—
240C2×C8—21no—
241C2×C8—41no—
242C2×C8C2×D4×C11311yescharacteristic
243C2×C8C2×D4×C11311yescharacteristic
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C22⋊C4—41no—
249C22⋊C4—41no—
250C22⋊C4—41no—
251C22⋊C4—41no—
252C22⋊C4—41no—
253C42—41no—
254C16—41no—
255C16—41no—
256C23—21no—
257C23—41no—
258C23—41no—
259C23—81no—
260C23C2×SD16×C11311yescharacteristic
261Q8—41no—
262Q8—41no—
263D4—41no—
264D4—41no—
265D4—82no—
266D4—82no—
267D4—82no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—81no—
278C2×C4—82no—
279C2×C4#120×C11311yescharacteristic
280C2×C4AGL(1,ℤ/8)×C11311yescharacteristic
281C8—21no—
282C8—21no—
283C8—42no—
284C8—41no—
285C8—81no—
286C22—21no—
287C22—41no—
288C22—41no—
289C22—41no—
290C22—42no—
291C22—41no—
292C22—81no—
293C22—81no—
294C22—81no—
295C22—81no—
296C22#450×C11311yescharacteristic
297C4—21no—
298C4—41no—
299C4—81no—
300C4—81no—
301C4—81no—
302C4#453×C11311yescharacteristic
303C4#458×C11311yescharacteristic
304C2—21no—
305C2—41no—
306C2—81no—
307C2—161no—
308C2C113×#161311yescharacteristic
subgroupG/H
center, socleC2×C113#1613
commutator, Frattini, cosocleC2×M16C23×C113
prime radicalC113×#3197C2
LCSG ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C113 ◁ C2×C4×C113 ◁ C2×M16×C113 ◁ G

Complements

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

NH ≅ G/Ntype
2. C113×#2239C2semidirect
3. C113×#2238C2semidirect
4. C113×#2091C2semidirect
5. C113×#1342C2semidirect
6. C113×#1341C2semidirect
7. C113×#1320C2semidirect
149. #12300C113direct (N × H)
156. #2239C2×C113semidirect
157. #2238C2×C113semidirect
158. #2091C2×C113semidirect
159. #1342C2×C113semidirect
160. #1341C2×C113semidirect
161. #1320C2×C113semidirect
162. C113#12300direct (N × H)

Maximal subgroups

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

subgroupindex
C113×#31972
C113×#22392
C113×#22382
C113×#20912
C113×#13422
C113×#13412
C113×#13202
#12300113