Finite Group Databasebeta

← C127×#13427 #44577100

Subgroups — C127×#13427 #44577100

Order 32512.

462 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 90 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
1C127×#2731C211yesmaximal, characteristic
2C2×#356×C127C211yesmaximal, characteristic, complemented
3C2×#355×C127C211yesmaximal, characteristic
4C127×#1888C211yesmaximal, characteristic
5C127×#1886C211yesmaximal, characteristic, complemented
6C127×#1857C211yesmaximal, characteristic, complemented
7C127×#1394C211yesmaximal, characteristic, complemented
8C2×C4⋊Q8×C127C2211yescharacteristic
9C2×#124×C127C2211yescharacteristic
10#519×C127C2211yescharacteristic
11#428×C127—21no—
12#419×C127—21no—
13#418×C127—21no—
14C2×#101×C127C2211yescharacteristic
15C2×#101×C127C2211yescharacteristic
16C2×#101×C127C2211yescharacteristic
17#396×C127—21no—
18#394×C127—21no—
19#393×C127—21no—
20#391×C127C2211yescharacteristic
21#388×C127—21no—
22#388×C127—21no—
23#387×C127—21no—
24#356×C127—22no—
25#355×C127—22no—
26#339×C127—21no—
27#339×C127—21no—
28#339×C127—21no—
29C22×Q8×C127D411yescharacteristic
30C22×Q8×C127C2311yescharacteristic
31C22×D4×C127D411yescharacteristic
32C2×M16×C127—21no—
33C2×M16×C127—21no—
34C2×M16×C127—21no—
35C4⋊Q8×C127—21no—
36C4⋊Q8×C127—21no—
37C4⋊Q8×C127—21no—
38C4⋊Q8×C127—21no—
39#124×C127—21no—
40#124×C127—21no—
41#124×C127—21no—
42#124×C127—21no—
43C4×Q8×C127—41no—
44C4×Q8×C127—41no—
45#117×C127—21no—
46#117×C127—21no—
47#117×C127—21no—
48C2×C4⋊C4×C127—21no—
49C2×C4⋊C4×C127—21no—
50C2×C4⋊C4×C127—21no—
51C2×C4⋊C4×C127D411yescharacteristic
52C2×C22⋊C4×C127—21no—
53C2×C22⋊C4×C127—21no—
54C2×C22⋊C4×C127—21no—
55C2×C22⋊C4×C127—21no—
56C2×C22⋊C4×C127—21no—
57C2×C22⋊C4×C127—21no—
58C2×C42×C127D411yescharacteristic
59C2×C42×C127D411yescharacteristic
60C2×C42×C127D411yescharacteristic
61#107×C127—41no—
62#106×C127—41no—
63#103×C127—41no—
64#103×C127—41no—
65#102×C127—41no—
66#102×C127—41no—
67#101×C127—21no—
68#101×C127—21no—
69#101×C127—21no—
70#101×C127—21no—
71#101×C127—21no—
72#101×C127D412yes—
73#95×C127—21no—
74#95×C127—21no—
75#95×C127—21no—
76#95×C127—21no—
77#95×C127—21no—
78C24×C127—21no—
79C2×Q8×C127—21no—
80C2×Q8×C127—21no—
81C2×Q8×C127—41no—
82C2×Q8×C127—41no—
83C2×Q8×C127—41no—
84C2×Q8×C127—41no—
85C2×Q8×C127C2×D411yescharacteristic
86C2×Q8×C127#4111yescharacteristic
87C2×Q8×C127#4111yescharacteristic
88C2×Q8×C127#4111yescharacteristic
89C2×D4×C127—21no—
90C2×D4×C127—21no—
91C2×D4×C127—41no—
92C2×D4×C127—41no—
93C22×C4×C127—21no—
94C22×C4×C127—21no—
95C22×C4×C127—21no—
96C22×C4×C127—21no—
97C22×C4×C127—21no—
98C22×C4×C127C2×D411yescharacteristic
99C22×C4×C127C2×D411yescharacteristic
100C22×C4×C127C2×D411yescharacteristic
101M16×C127—22no—
102M16×C127—41no—
103M16×C127—41no—
104M16×C127—41no—
105M16×C127—41no—
106M16×C127—41no—
107C2×C8×C127—41no—
108C2×C8×C127—41no—
109C2×C8×C127—41no—
110C4⋊C4×C127—21no—
111C4⋊C4×C127—21no—
112C4⋊C4×C127—21no—
113C4⋊C4×C127—21no—
114C4⋊C4×C127—21no—
115C4⋊C4×C127—21no—
116C4⋊C4×C127—41no—
117C4⋊C4×C127—42no—
118C4⋊C4×C127—41no—
119C4⋊C4×C127—41no—
120C4⋊C4×C127—41no—
121C4⋊C4×C127—41no—
122C22⋊C4×C127—42no—
123C22⋊C4×C127—42no—
124C22⋊C4×C127—41no—
125C22⋊C4×C127—41no—
126C22⋊C4×C127—41no—
127C22⋊C4×C127—41no—
128C22⋊C4×C127—41no—
129C22⋊C4×C127—81no—
130C22⋊C4×C127—81no—
131C42×C127—21no—
132C42×C127—21no—
133C42×C127—21no—
134C42×C127—21no—
135C42×C127—41no—
136C42×C127—41no—
137C42×C127—41no—
138C42×C127—41no—
139C23×C127—41no—
140C23×C127—41no—
141C23×C127—41no—
142C23×C127—41no—
143C23×C127—41no—
144C23×C127—41no—
145C23×C127—41no—
146C23×C127#12011yescharacteristic
147Q8×C127—41no—
148Q8×C127—41no—
149Q8×C127—41no—
150Q8×C127—41no—
151Q8×C127—42no—
152Q8×C127—81no—
153D4×C127—42no—
154D4×C127—81no—
155C2×C4×C127—21no—
156C2×C4×C127—21no—
157C2×C4×C127—21no—
158C2×C4×C127—21no—
159C2×C4×C127—21no—
160C2×C4×C127—21no—
161C2×C4×C127—22no—
162C2×C4×C127—41no—
163C2×C4×C127—41no—
164C2×C4×C127—41no—
165C2×C4×C127—41no—
166C2×C4×C127—41no—
167C2×C4×C127—41no—
168C2×C4×C127—41no—
169C2×C4×C127—41no—
170C2×C4×C127—41no—
171C2×C4×C127—41no—
172C2×C4×C127—41no—
173C2×C4×C127—41no—
174C2×C4×C127—41no—
175C2×C4×C127—41no—
176C2×C4×C127—41no—
177C2×C4×C127—41no—
178C2×C4×C127#12111yescharacteristic
179C2×C4×C127#12111yescharacteristic
180C2×C4×C127#12111yescharacteristic
181C2×C4×C127#12311yescharacteristic
182C2×C4×C127#12311yescharacteristic
183C2×C4×C127#12411yescharacteristic
184C8×C127—42no—
185C8×C127—81no—
186C8×C127—81no—
187C22×C127—21no—
188C22×C127—21no—
189C22×C127—81no—
190C22×C127—81no—
191C22×C127—81no—
192C22×C127—81no—
193C22×C127—81no—
194C22×C127—81no—
195C22×C127—81no—
196C22×C127#39411yescharacteristic
197C22×C127B(4,𝔽2)11yescharacteristic
198C22×C127#45811yescharacteristic
199C4×C127—21no—
200C4×C127—21no—
201C4×C127—21no—
202C4×C127—21no—
203C4×C127—21no—
204C4×C127—21no—
205C4×C127—42no—
206C4×C127—81no—
207C4×C127—81no—
208C4×C127—81no—
209C4×C127—81no—
210C4×C127—81no—
211C4×C127—81no—
212#13427C12711yesmaximal, characteristic, direct product factor, Sylow
213C2×C127—21no—
214C2×C127—21no—
215C2×C127—161no—
216C2×C127#204411yescharacteristic
217C2×C127#220311yescharacteristic
218C2×C127#220411yescharacteristic
219#2731C2×C12711yescharacteristic
220C2×#356C2×C12711yescharacteristic, complemented
221C2×#355C2×C12711yescharacteristic
222#1888C2×C12711yescharacteristic
223#1886C2×C12711yescharacteristic, complemented
224#1857C2×C12711yescharacteristic, complemented
225#1394C2×C12711yescharacteristic, complemented
226C127#1342711yescharacteristic, direct product factor, Sylow
227C2×C4⋊Q8C22×C12711yescharacteristic
228C2×#124C22×C12711yescharacteristic
229#519C22×C12711yescharacteristic
230#428—21no—
231#419—21no—
232#418—21no—
233C2×#101C22×C12711yescharacteristic
234C2×#101C22×C12711yescharacteristic
235C2×#101C22×C12711yescharacteristic
236#396—21no—
237#394—21no—
238#393—21no—
239#391C22×C12711yescharacteristic
240#388—21no—
241#388—21no—
242#387—21no—
243#356—22no—
244#355—22no—
245#339—21no—
246#339—21no—
247#339—21no—
248C22×Q8D4×C12711yescharacteristic
249C22×Q8C23×C12711yescharacteristic
250C22×D4D4×C12711yescharacteristic
251C2×M16—21no—
252C2×M16—21no—
253C2×M16—21no—
254C4⋊Q8—21no—
255C4⋊Q8—21no—
256C4⋊Q8—21no—
257C4⋊Q8—21no—
258#124—21no—
259#124—21no—
260#124—21no—
261#124—21no—
262C4×Q8—41no—
263C4×Q8—41no—
264#117—21no—
265#117—21no—
266#117—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4D4×C12711yescharacteristic
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C42D4×C12711yescharacteristic
278C2×C42D4×C12711yescharacteristic
279C2×C42D4×C12711yescharacteristic
280#107—41no—
281#106—41no—
282#103—41no—
283#103—41no—
284#102—41no—
285#102—41no—
286#101—21no—
287#101—21no—
288#101—21no—
289#101—21no—
290#101—21no—
291#101D4×C12712yes—
292#95—21no—
293#95—21no—
294#95—21no—
295#95—21no—
296#95—21no—
297C24—21no—
298C2×Q8—21no—
299C2×Q8—21no—
300C2×Q8—41no—
301C2×Q8—41no—
302C2×Q8—41no—
303C2×Q8—41no—
304C2×Q8C2×D4×C12711yescharacteristic
305C2×Q8#41×C12711yescharacteristic
306C2×Q8#41×C12711yescharacteristic
307C2×Q8#41×C12711yescharacteristic
308C2×D4—21no—
309C2×D4—21no—
310C2×D4—41no—
311C2×D4—41no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4C2×D4×C12711yescharacteristic
318C22×C4C2×D4×C12711yescharacteristic
319C22×C4C2×D4×C12711yescharacteristic
320M16—22no—
321M16—41no—
322M16—41no—
323M16—41no—
324M16—41no—
325M16—41no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—41no—
329C4⋊C4—21no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—41no—
336C4⋊C4—42no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C22⋊C4—42no—
342C22⋊C4—42no—
343C22⋊C4—41no—
344C22⋊C4—41no—
345C22⋊C4—41no—
346C22⋊C4—41no—
347C22⋊C4—41no—
348C22⋊C4—81no—
349C22⋊C4—81no—
350C42—21no—
351C42—21no—
352C42—21no—
353C42—21no—
354C42—41no—
355C42—41no—
356C42—41no—
357C42—41no—
358C23—41no—
359C23—41no—
360C23—41no—
361C23—41no—
362C23—41no—
363C23—41no—
364C23—41no—
365C23#120×C12711yescharacteristic
366Q8—41no—
367Q8—41no—
368Q8—41no—
369Q8—41no—
370Q8—42no—
371Q8—81no—
372D4—42no—
373D4—81no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—22no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4#121×C12711yescharacteristic
398C2×C4#121×C12711yescharacteristic
399C2×C4#121×C12711yescharacteristic
400C2×C4#123×C12711yescharacteristic
401C2×C4#123×C12711yescharacteristic
402C2×C4#124×C12711yescharacteristic
403C8—42no—
404C8—81no—
405C8—81no—
406C22—21no—
407C22—21no—
408C22—81no—
409C22—81no—
410C22—81no—
411C22—81no—
412C22—81no—
413C22—81no—
414C22—81no—
415C22#394×C12711yescharacteristic
416C22B(4,𝔽2)×C12711yescharacteristic
417C22#458×C12711yescharacteristic
418C4—21no—
419C4—21no—
420C4—21no—
421C4—21no—
422C4—21no—
423C4—21no—
424C4—42no—
425C4—81no—
426C4—81no—
427C4—81no—
428C4—81no—
429C4—81no—
430C4—81no—
431C2—21no—
432C2—21no—
433C2—161no—
434C2C127×#204411yescharacteristic
435C2C127×#220311yescharacteristic
436C2C127×#220411yescharacteristic
subgroupG/H
center, socleC22×C127B(4,𝔽2)
commutator, Frattini, cosocleC22×Q8C23×C127
prime radicalC22×D4×C127D4
LCSG ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C23×C127 ◁ C22×Q8×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#356×C127C2semidirect
5. C127×#1886C2semidirect
6. C127×#1857C2semidirect
7. C127×#1394C2semidirect
212. #13427C127direct (N × H)
220. C2×#356C2×C127semidirect
223. #1886C2×C127semidirect
224. #1857C2×C127semidirect
225. #1394C2×C127semidirect
226. C127#13427direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#27312
C2×#356×C1272
C2×#355×C1272
C127×#18882
C127×#18862
C127×#18572
C127×#13942
#13427127