Finite Group Databasebeta

← C131×#13359 #45699243

Subgroups — C131×#13359 #45699243

Order 33536.

704 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 704 conjugacy classes of subgroups — too many to draw. Only its 98 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×C131C212yesmaximal
2#2734×C131C211yesmaximal, characteristic, complemented
3C2×#356×C131C212yesmaximal, complemented
4#1793×C131C211yesmaximal, characteristic
5#1394×C131C211yesmaximal, characteristic, complemented
6C2×2−1+4×C131C2211yescharacteristic
7C2×#137×C131—22no—
8C2×C4⋊Q8×C131C2212yes—
9#456×C131—24no—
10#456×C131—24no—
11C2×#104×C131—22no—
12C2×#104×C131—21no—
13#417×C131—21no—
14C2×#101×C131C2211yescharacteristic
15C2×#101×C131C2212yes—
16#398×C131—21no—
17#395×C131—22no—
18#391×C131C2211yescharacteristic
19#389×C131—21no—
20#387×C131—22no—
21#356×C131—24no—
222−1+4×C131—22no—
232−1+4×C131—21no—
242−1+4×C131—42no—
252−1+4×C131D412yes—
26C2×#41×C131—21no—
27C2×#41×C131—22no—
28C2×#41×C131—41no—
29C22×Q8×C131—41no—
30C22×Q8×C131D412yes—
31C22×Q8×C131C2311yescharacteristic
32#137×C131—28no—
33#137×C131—44no—
34C2×Q16×C131—42no—
35C2×SD16×C131—42no—
36C2×M16×C131—22no—
37C2×M16×C131—21no—
38C4⋊Q8×C131—22no—
39C4⋊Q8×C131—22no—
40C4⋊Q8×C131—22no—
41C4⋊Q8×C131—22no—
42C2×C4⋊C4×C131—21no—
43C2×C4⋊C4×C131—22no—
44C2×C4⋊C4×C131—22no—
45C2×C4⋊C4×C131—21no—
46C2×C4⋊C4×C131—21no—
47C2×C4⋊C4×C131D411yescharacteristic
48C2×C42×C131D412yes—
49C2×C42×C131D411yescharacteristic
50#104×C131—28no—
51#104×C131—42no—
52#103×C131—41no—
53#102×C131—41no—
54#101×C131—22no—
55#101×C131—22no—
56#101×C131D414yes—
57#95×C131—22no—
58#95×C131—22no—
59#95×C131—21no—
60#95×C131—22no—
61#41×C131—22no—
62#41×C131—24no—
63#41×C131—24no—
64#41×C131—22no—
65#41×C131—41no—
66#41×C131—42no—
67#41×C131—42no—
68#41×C131—44no—
69#41×C131—42no—
70#41×C131—42no—
71#41×C131—41no—
72C2×Q8×C131—21no—
73C2×Q8×C131—22no—
74C2×Q8×C131—22no—
75C2×Q8×C131—22no—
76C2×Q8×C131—41no—
77C2×Q8×C131—41no—
78C2×Q8×C131—42no—
79C2×Q8×C131—42no—
80C2×Q8×C131—42no—
81C2×Q8×C131—44no—
82C2×Q8×C131—42no—
83C2×Q8×C131C2×D411yescharacteristic
84C2×Q8×C131C2×D412yes—
85C2×Q8×C131#4111yescharacteristic
86C2×D4×C131—21no—
87C2×D4×C131—22no—
88C2×D4×C131—41no—
89C22×C4×C131—21no—
90C22×C4×C131—22no—
91C22×C4×C131—21no—
92C22×C4×C131—22no—
93C22×C4×C131—41no—
94C22×C4×C131—42no—
95C22×C4×C131C2×D412yes—
96C22×C4×C131C2×D411yescharacteristic
97Q16×C131—48no—
98SD16×C131—48no—
99M16×C131—24no—
100M16×C131—42no—
101M16×C131—41no—
102M16×C131—41no—
103C2×C8×C131—42no—
104C2×C8×C131—41no—
105C4⋊C4×C131—24no—
106C4⋊C4×C131—41no—
107C4⋊C4×C131—42no—
108C4⋊C4×C131—42no—
109C4⋊C4×C131—42no—
110C4⋊C4×C131—42no—
111C4⋊C4×C131—81no—
112C4⋊C4×C131—81no—
113C42×C131—22no—
114C42×C131—22no—
115C42×C131—41no—
116C23×C131—41no—
117C23×C131#12011yescharacteristic
118Q8×C131—24no—
119Q8×C131—22no—
120Q8×C131—41no—
121Q8×C131—42no—
122Q8×C131—41no—
123Q8×C131—41no—
124Q8×C131—42no—
125Q8×C131—44no—
126Q8×C131—82no—
127D4×C131—24no—
128D4×C131—22no—
129D4×C131—41no—
130D4×C131—42no—
131D4×C131—42no—
132D4×C131—42no—
133C2×C4×C131—21no—
134C2×C4×C131—22no—
135C2×C4×C131—21no—
136C2×C4×C131—22no—
137C2×C4×C131—22no—
138C2×C4×C131—24no—
139C2×C4×C131—44no—
140C2×C4×C131—42no—
141C2×C4×C131—42no—
142C2×C4×C131—44no—
143C2×C4×C131—41no—
144C2×C4×C131—42no—
145C2×C4×C131—42no—
146C2×C4×C131—41no—
147C2×C4×C131—41no—
148C2×C4×C131—42no—
149C2×C4×C131—42no—
150C2×C4×C131—42no—
151C2×C4×C131—41no—
152C2×C4×C131—42no—
153C2×C4×C131#12012yes—
154C2×C4×C131#12112yes—
155C2×C4×C131#12111yescharacteristic
156C2×C4×C131#12711yescharacteristic
157C8×C131—44no—
158C8×C131—81no—
159C22×C131—21no—
160C22×C131—21no—
161C22×C131—42no—
162C22×C131—81no—
163C22×C131—81no—
164C22×C131#39211yescharacteristic
165C22×C131B(4,𝔽2)11yescharacteristic
166C22×C131B(4,𝔽2)11yescharacteristic
167C4×C131—22no—
168C4×C131—21no—
169C4×C131—21no—
170C4×C131—22no—
171C4×C131—44no—
172C4×C131—42no—
173C4×C131—42no—
174C4×C131—81no—
175C4×C131—82no—
176C2×C131—21no—
177C2×C131—21no—
178C2×C131—82no—
179C2×C131#202111yescharacteristic
180C2×C131#220311yescharacteristic
181C2×C131#220511yescharacteristic
182#13359C13111yesmaximal, characteristic, direct product factor, Sylow
183C131#1335911yescharacteristic, direct product factor, Sylow
184C2×#456C2×C13112yes—
185#2734C2×C13111yescharacteristic, complemented
186C2×#356C2×C13112yescomplemented
187#1793C2×C13111yescharacteristic
188#1394C2×C13111yescharacteristic, complemented
189C2×2−1+4C22×C13111yescharacteristic
190C2×#137—22no—
191C2×C4⋊Q8C22×C13112yes—
192#456—24no—
193#456—24no—
194C2×#104—22no—
195C2×#104—21no—
196#417—21no—
197C2×#101C22×C13111yescharacteristic
198C2×#101C22×C13112yes—
199#398—21no—
200#395—22no—
201#391C22×C13111yescharacteristic
202#389—21no—
203#387—22no—
204#356—24no—
2052−1+4—22no—
2062−1+4—21no—
2072−1+4—42no—
2082−1+4D4×C13112yes—
209C2×#41—22no—
210C2×#41—21no—
211C2×#41—41no—
212C22×Q8—41no—
213C22×Q8D4×C13112yes—
214C22×Q8C23×C13111yescharacteristic
215#137—28no—
216#137—44no—
217C2×Q16—42no—
218C2×SD16—42no—
219C2×M16—22no—
220C2×M16—21no—
221C4⋊Q8—22no—
222C4⋊Q8—22no—
223C4⋊Q8—22no—
224C4⋊Q8—22no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4D4×C13111yescharacteristic
231C2×C42D4×C13112yes—
232C2×C42D4×C13111yescharacteristic
233#104—28no—
234#104—42no—
235#103—41no—
236#102—41no—
237#101—22no—
238#101—22no—
239#101D4×C13114yes—
240#95—21no—
241#95—22no—
242#95—22no—
243#95—22no—
244#41—22no—
245#41—24no—
246#41—24no—
247#41—22no—
248#41—41no—
249#41—42no—
250#41—42no—
251#41—44no—
252#41—42no—
253#41—42no—
254#41—41no—
255C2×Q8—21no—
256C2×Q8—22no—
257C2×Q8—22no—
258C2×Q8—22no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×Q8—42no—
262C2×Q8—42no—
263C2×Q8—42no—
264C2×Q8—44no—
265C2×Q8—42no—
266C2×Q8C2×D4×C13111yescharacteristic
267C2×Q8C2×D4×C13112yes—
268C2×Q8#41×C13111yescharacteristic
269C2×D4—21no—
270C2×D4—22no—
271C2×D4—41no—
272C22×C4—21no—
273C22×C4—22no—
274C22×C4—21no—
275C22×C4—22no—
276C22×C4—41no—
277C22×C4—42no—
278C22×C4C2×D4×C13112yes—
279C22×C4C2×D4×C13111yescharacteristic
280Q16—48no—
281SD16—48no—
282M16—24no—
283M16—42no—
284M16—41no—
285M16—41no—
286C2×C8—42no—
287C2×C8—41no—
288C4⋊C4—24no—
289C4⋊C4—41no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—81no—
295C4⋊C4—81no—
296C42—22no—
297C42—22no—
298C42—41no—
299C23—41no—
300C23#120×C13111yescharacteristic
301Q8—24no—
302Q8—22no—
303Q8—41no—
304Q8—42no—
305Q8—41no—
306Q8—41no—
307Q8—42no—
308Q8—44no—
309Q8—82no—
310D4—24no—
311D4—22no—
312D4—41no—
313D4—42no—
314D4—42no—
315D4—42no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—24no—
322C2×C4—44no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—44no—
326C2×C4—41no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—41no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4#120×C13112yes—
337C2×C4#121×C13112yes—
338C2×C4#121×C13111yescharacteristic
339C2×C4#127×C13111yescharacteristic
340C8—44no—
341C8—81no—
342C22—21no—
343C22—21no—
344C22—42no—
345C22—81no—
346C22—81no—
347C22#392×C13111yescharacteristic
348C22B(4,𝔽2)×C13111yescharacteristic
349C22B(4,𝔽2)×C13111yescharacteristic
350C4—22no—
351C4—21no—
352C4—21no—
353C4—22no—
354C4—44no—
355C4—42no—
356C4—42no—
357C4—82no—
358C4—81no—
359C2—21no—
360C2—21no—
361C2—82no—
362C2#2021×C13111yescharacteristic
363C2#2203×C13111yescharacteristic
364C2#2205×C13111yescharacteristic
subgroupG/H
center, socleC22×C131B(4,𝔽2)
commutator, Frattini, cosocleC22×Q8C23×C131
prime radicalC2×2−1+4×C131C22
LCSG ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C131 ◁ C23×C131 ◁ C22×Q8×C131 ◁ G

Complements

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

NH ≅ G/Ntype
2. #2734×C131C2semidirect
3. C2×#356×C131C2semidirect
5. #1394×C131C2semidirect
182. #13359C131direct (N × H)
183. C131#13359direct (N × H)
185. #2734C2×C131semidirect
186. C2×#356C2×C131semidirect
188. #1394C2×C131semidirect

Maximal subgroups

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

subgroupindex
C2×#456×C1312
#2734×C1312
C2×#356×C1312
#1793×C1312
#1394×C1312
#13359131