Finite Group Databasebeta

← C101×#13058 #38795209

Subgroups — C101×#13058 #38795209

Order 25856.

668 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 668 conjugacy classes of subgroups — too many to draw. Only its 118 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
1C2×#456×C101C211yesmaximal, characteristic
2C101×#2369C211yesmaximal, characteristic, complemented
3C2×#356×C101C211yesmaximal, characteristic, complemented
4C101×#1793C212yesmaximal
5C101×#1394C212yesmaximal, complemented
6C2×2−1+4×C101C2211yescharacteristic
7C2×#137×C101—21no—
8C2×C4⋊Q8×C101C2211yescharacteristic
9#456×C101—22no—
10#456×C101C414yescomplemented
11C2×#104×C101—21no—
12C2×#104×C101—22no—
13#417×C101—22no—
14C2×#101×C101C2212yes—
15C2×#101×C101C2211yescharacteristic
16#391×C101—21no—
17#391×C101C2212yes—
18#389×C101—22no—
19C4×C4⋊C4×C101—22no—
20C4×C4⋊C4×C101—21no—
21#356×C101—22no—
222−1+4×C101—21no—
232−1+4×C101—22no—
242−1+4×C101—42no—
252−1+4×C101C2×C412yes—
26C2×#41×C101—22no—
27C2×#41×C101—21no—
28C2×#41×C101—41no—
29C22×Q8×C101—41no—
30C22×Q8×C101D411yescharacteristic
31C22×Q8×C101C2311yescharacteristic
32#137×C101—24no—
33#137×C101—42no—
34C2×Q16×C101—41no—
35C2×SD16×C101—41no—
36C2×M16×C101—21no—
37C2×M16×C101—22no—
38C4⋊Q8×C101—22no—
39C4⋊Q8×C101C2×C412yes—
40C4⋊Q8×C101D412yes—
41C2×C4⋊C4×C101—21no—
42C2×C4⋊C4×C101—22no—
43C2×C4⋊C4×C101—21no—
44C2×C4⋊C4×C101—21no—
45C2×C4⋊C4×C101D412yes—
46C2×C42×C101—22no—
47C2×C42×C101—21no—
48C2×C42×C101D411yescharacteristic
49C2×C42×C101D412yes—
50#104×C101—24no—
51#104×C101—44no—
52#103×C101—42no—
53#102×C101—42no—
54#101×C101—21no—
55#101×C101—22no—
56#101×C101—22no—
57#101×C101C2×C412yes—
58#95×C101—22no—
59#95×C101—22no—
60#95×C101—22no—
61#41×C101—24no—
62#41×C101—22no—
63#41×C101—22no—
64#41×C101—24no—
65#41×C101—42no—
66#41×C101—41no—
67#41×C101—42no—
68#41×C101—42no—
69#41×C101—44no—
70#41×C101—42no—
71#41×C101—41no—
72C2×Q8×C101—21no—
73C2×Q8×C101—22no—
74C2×Q8×C101—21no—
75C2×Q8×C101—41no—
76C2×Q8×C101—41no—
77C2×Q8×C101—41no—
78C2×Q8×C101—41no—
79C2×Q8×C101—42no—
80C2×Q8×C101—42no—
81C2×Q8×C101—44no—
82C2×Q8×C101C22⋊C412yes—
83C2×Q8×C101C22×C411yescharacteristic
84C2×Q8×C101C2×D411yescharacteristic
85C2×Q8×C101#4112yes—
86C2×D4×C101—22no—
87C2×D4×C101—21no—
88C2×D4×C101—41no—
89C22×C4×C101—21no—
90C22×C4×C101—22no—
91C22×C4×C101—21no—
92C22×C4×C101—22no—
93C22×C4×C101—42no—
94C22×C4×C101—41no—
95C22×C4×C101C2×D411yescharacteristic
96C22×C4×C101C2×D412yes—
97Q16×C101—44no—
98SD16×C101—44no—
99M16×C101—22no—
100M16×C101—41no—
101M16×C101—42no—
102M16×C101—42no—
103C2×C8×C101—41no—
104C2×C8×C101—42no—
105C4⋊C4×C101—24no—
106C4⋊C4×C101—24no—
107C4⋊C4×C101—24no—
108C4⋊C4×C101—42no—
109C4⋊C4×C101—42no—
110C4⋊C4×C101—81no—
111C42×C101—21no—
112C42×C101—42no—
113C42×C101—44no—
114C42×C101—81no—
115C42×C101C22⋊C412yes—
116C23×C101—41no—
117C23×C101#12011yescharacteristic
118Q8×C101—22no—
119Q8×C101—24no—
120Q8×C101—41no—
121Q8×C101—41no—
122Q8×C101—42no—
123Q8×C101—42no—
124Q8×C101—42no—
125Q8×C101—41no—
126Q8×C101—81no—
127D4×C101—22no—
128D4×C101—24no—
129D4×C101—42no—
130D4×C101—41no—
131D4×C101—42no—
132D4×C101—42no—
133C2×C4×C101—22no—
134C2×C4×C101—22no—
135C2×C4×C101—22no—
136C2×C4×C101—22no—
137C2×C4×C101—21no—
138C2×C4×C101—22no—
139C2×C4×C101—21no—
140C2×C4×C101—24no—
141C2×C4×C101—22no—
142C2×C4×C101—42no—
143C2×C4×C101—44no—
144C2×C4×C101—44no—
145C2×C4×C101—42no—
146C2×C4×C101—42no—
147C2×C4×C101—42no—
148C2×C4×C101—41no—
149C2×C4×C101—41no—
150C2×C4×C101—41no—
151C2×C4×C101—42no—
152C2×C4×C101—42no—
153C2×C4×C101—42no—
154C2×C4×C101C2×C22⋊C411yescharacteristic
155C2×C4×C101C4×D412yes—
156C2×C4×C101#12112yes—
157C2×C4×C101#12311yescharacteristic
158C8×C101—42no—
159C8×C101—82no—
160C22×C101—21no—
161C22×C101—21no—
162C22×C101—42no—
163C22×C101—81no—
164C22×C101—81no—
165C22×C101#38611yescharacteristic
166C22×C101B(4,𝔽2)11yescharacteristic
167C22×C101B(4,𝔽2)11yescharacteristic
168C4×C101—24no—
169C4×C101—21no—
170C4×C101—22no—
171C4×C101—22no—
172C4×C101—21no—
173C4×C101—44no—
174C4×C101—42no—
175C4×C101—42no—
176C4×C101—82no—
177#13058C10111yesmaximal, characteristic, direct product factor, Sylow
178C2×C101—21no—
179C2×C101—21no—
180C2×C101—82no—
181C2×C101#188911yescharacteristic
182C2×C101#220411yescharacteristic
183C2×C101#220511yescharacteristic
184C2×#456C2×C10111yescharacteristic
185#2369C2×C10111yescharacteristic, complemented
186C2×#356C2×C10111yescharacteristic, complemented
187#1793C2×C10112yes—
188#1394C2×C10112yescomplemented
189C101#1305811yescharacteristic, direct product factor, Sylow
190C2×2−1+4C22×C10111yescharacteristic
191C2×#137—21no—
192C2×C4⋊Q8C22×C10111yescharacteristic
193#456—22no—
194#456C4×C10114yescomplemented
195C2×#104—21no—
196C2×#104—22no—
197#417—22no—
198C2×#101C22×C10112yes—
199C2×#101C22×C10111yescharacteristic
200#391—21no—
201#391C22×C10112yes—
202#389—22no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—21no—
205#356—22no—
2062−1+4—21no—
2072−1+4—22no—
2082−1+4—42no—
2092−1+4C2×C4×C10112yes—
210C2×#41—21no—
211C2×#41—22no—
212C2×#41—41no—
213C22×Q8—41no—
214C22×Q8D4×C10111yescharacteristic
215C22×Q8C23×C10111yescharacteristic
216#137—24no—
217#137—42no—
218C2×Q16—41no—
219C2×SD16—41no—
220C2×M16—21no—
221C2×M16—22no—
222C4⋊Q8—22no—
223C4⋊Q8C2×C4×C10112yes—
224C4⋊Q8D4×C10112yes—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4D4×C10112yes—
230C2×C42—22no—
231C2×C42—21no—
232C2×C42D4×C10111yescharacteristic
233C2×C42D4×C10112yes—
234#104—24no—
235#104—44no—
236#103—42no—
237#102—42no—
238#101—21no—
239#101—22no—
240#101—22no—
241#101C2×C4×C10112yes—
242#95—22no—
243#95—22no—
244#95—22no—
245#41—24no—
246#41—22no—
247#41—22no—
248#41—24no—
249#41—42no—
250#41—41no—
251#41—42no—
252#41—42no—
253#41—44no—
254#41—42no—
255#41—41no—
256C2×Q8—21no—
257C2×Q8—22no—
258C2×Q8—21no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×Q8—41no—
262C2×Q8—41no—
263C2×Q8—42no—
264C2×Q8—42no—
265C2×Q8—44no—
266C2×Q8C22⋊C4×C10112yes—
267C2×Q8C22×C4×C10111yescharacteristic
268C2×Q8C2×D4×C10111yescharacteristic
269C2×Q8#41×C10112yes—
270C2×D4—22no—
271C2×D4—21no—
272C2×D4—41no—
273C22×C4—21no—
274C22×C4—22no—
275C22×C4—21no—
276C22×C4—22no—
277C22×C4—42no—
278C22×C4—41no—
279C22×C4C2×D4×C10111yescharacteristic
280C22×C4C2×D4×C10112yes—
281Q16—44no—
282SD16—44no—
283M16—22no—
284M16—41no—
285M16—42no—
286M16—42no—
287C2×C8—41no—
288C2×C8—42no—
289C4⋊C4—24no—
290C4⋊C4—24no—
291C4⋊C4—24no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—81no—
295C42—21no—
296C42—42no—
297C42—44no—
298C42—81no—
299C42C22⋊C4×C10112yes—
300C23—41no—
301C23#120×C10111yescharacteristic
302Q8—22no—
303Q8—24no—
304Q8—41no—
305Q8—41no—
306Q8—42no—
307Q8—42no—
308Q8—42no—
309Q8—41no—
310Q8—81no—
311D4—22no—
312D4—24no—
313D4—42no—
314D4—41no—
315D4—42no—
316D4—42no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—22no—
324C2×C4—24no—
325C2×C4—22no—
326C2×C4—42no—
327C2×C4—44no—
328C2×C4—44no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4C2×C22⋊C4×C10111yescharacteristic
339C2×C4C4×D4×C10112yes—
340C2×C4#121×C10112yes—
341C2×C4#123×C10111yescharacteristic
342C8—42no—
343C8—82no—
344C22—21no—
345C22—21no—
346C22—42no—
347C22—81no—
348C22—81no—
349C22#386×C10111yescharacteristic
350C22B(4,𝔽2)×C10111yescharacteristic
351C22B(4,𝔽2)×C10111yescharacteristic
352C4—24no—
353C4—21no—
354C4—22no—
355C4—22no—
356C4—21no—
357C4—44no—
358C4—42no—
359C4—42no—
360C4—82no—
361C2—21no—
362C2—21no—
363C2—82no—
364C2C101×#188911yescharacteristic
365C2C101×#220411yescharacteristic
366C2C101×#220511yescharacteristic
subgroupG/H
center, socleC22×C101B(4,𝔽2)
commutatorC2×Q8C22×C4×C101
Frattini, cosocleC22×Q8C23×C101
prime radicalC2×2−1+4×C101C22
LCSG ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C101 ◁ C23×C101 ◁ C22×Q8×C101 ◁ G

Complements

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

NH ≅ G/Ntype
2. C101×#2369C2semidirect
3. C2×#356×C101C2semidirect
5. C101×#1394C2semidirect
10. #456×C101C4semidirect
177. #13058C101direct (N × H)
185. #2369C2×C101semidirect
186. C2×#356C2×C101semidirect
188. #1394C2×C101semidirect
189. C101#13058direct (N × H)
194. #456C4×C101semidirect

Maximal subgroups

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

subgroupindex
C2×#456×C1012
C101×#23692
C2×#356×C1012
C101×#17932
C101×#13942
#13058101