Finite Group Databasebeta

← C79×#13677 #34125366

Subgroups — C79×#13677 #34125366

Order 20224.

420 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 58 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
1C79×#3288C211yesmaximal, characteristic, complemented
2C79×#2202C211yesmaximal, characteristic, complemented
3C79×#2199C211yesmaximal, characteristic, complemented
4C79×#1655C211yesmaximal, characteristic, complemented
5C79×#1406C211yesmaximal, characteristic, complemented
6C79×#1404C211yesmaximal, characteristic, complemented
7C79×#1403C211yesmaximal, characteristic, complemented
8#577×C79—21no—
9#576×C79—21no—
10H(3,𝔽4)×C79C2211yescharacteristic, complemented
11#496×C79—21no—
12#496×C79C2211yescharacteristic
13#472×C79—21no—
14#471×C79—21no—
15B(4,𝔽2)×C79—21no—
16#455×C79C2211yescharacteristic
17#454×C79—21no—
18#454×C79C2211yescharacteristic
19#453×C79—21no—
20#444×C79—21no—
21#355×C79—21no—
22#355×C79C2211yescharacteristic, complemented
23#354×C79C2211yescharacteristic, complemented
24AΣL(1,𝔽16)×C79—21no—
25#329×C79—21no—
26#329×C79C2211yescharacteristic, complemented
272−1+4×C79D411yescharacteristic, complemented
28H(5,𝔽2)×C79—21no—
29H(5,𝔽2)×C79D411yescharacteristic
30#137×C79—21no—
31#137×C79—41no—
32AGL(1,ℤ/8)×C79—21no—
33AGL(1,ℤ/8)×C79—21no—
34AGL(1,ℤ/8)×C79—21no—
35AGL(1,ℤ/8)×C79—41no—
36AGL(1,ℤ/8)×C79—41no—
37AGL(1,ℤ/8)×C79—41no—
38AGL(1,ℤ/8)×C79—41no—
39#135×C79—21no—
40#135×C79—41no—
41#135×C79—41no—
42C2×SD16×C79—21no—
43C2×SD16×C79—41no—
44C2×SD16×C79—41no—
45C2×D8×C79—21no—
46C2×D8×C79—41no—
47C2×D8×C79—41no—
48#131×C79—21no—
49#131×C79—21no—
50#127×C79D411yescharacteristic, complemented
51#124×C79—21no—
52#124×C79—21no—
53#124×C79—41no—
54#124×C79C2311yescharacteristic
55#120×C79—21no—
56#120×C79—41no—
57#120×C79—41no—
58#108×C79—21no—
59#104×C79—21no—
60#104×C79—21no—
61#104×C79—41no—
62#101×C79—21no—
63#101×C79D411yescharacteristic, complemented
64#100×C79—21no—
65#100×C79D411yescharacteristic
66#99×C79—21no—
67#99×C79—41no—
68#97×C79—21no—
69#97×C79D411yescharacteristic, complemented
70C24×C79—21no—
71#41×C79—21no—
72#41×C79—21no—
73#41×C79—41no—
74#41×C79—41no—
75#41×C79—41no—
76#41×C79—41no—
77#41×C79—41no—
78C2×Q8×C79—21no—
79C2×Q8×C79—41no—
80C2×Q8×C79C2×D411yescharacteristic
81C2×D4×C79—21no—
82C2×D4×C79—21no—
83C2×D4×C79—21no—
84C2×D4×C79—41no—
85C2×D4×C79—41no—
86C2×D4×C79—41no—
87C2×D4×C79—41no—
88C2×D4×C79—41no—
89C2×D4×C79—41no—
90C2×D4×C79—81no—
91C2×D4×C79—81no—
92C2×D4×C79C2×D411yescharacteristic
93Q16×C79—21no—
94Q16×C79—41no—
95SD16×C79—21no—
96SD16×C79—21no—
97SD16×C79—41no—
98SD16×C79—41no—
99SD16×C79—41no—
100SD16×C79—41no—
101SD16×C79—41no—
102SD16×C79—41no—
103SD16×C79—81no—
104SD16×C79—82no—
105D8×C79—21no—
106D8×C79—41no—
107D8×C79—41no—
108D8×C79—41no—
109D8×C79—41no—
110D8×C79—41no—
111D8×C79—81no—
112D8×C79—82no—
113M16×C79—21no—
114M16×C79—21no—
115M16×C79—41no—
116M16×C79—41no—
117M16×C79—41no—
118C2×C8×C79—21no—
119C2×C8×C79—41no—
120C2×C8×C79—41no—
121C2×C8×C79—41no—
122C22⋊C4×C79—21no—
123C22⋊C4×C79—41no—
124C22⋊C4×C79—41no—
125C22⋊C4×C79—41no—
126C22⋊C4×C79—41no—
127C22⋊C4×C79—41no—
128C22⋊C4×C79—81no—
129C42×C79—21no—
130C42×C79C2×D411yescharacteristic
131C23×C79—21no—
132C23×C79—41no—
133C23×C79—41no—
134C23×C79—41no—
135C23×C79—81no—
136C23×C79—81no—
137C23×C79—161no—
138Q8×C79—21no—
139Q8×C79—21no—
140Q8×C79—41no—
141Q8×C79—41no—
142Q8×C79—81no—
143D4×C79—21no—
144D4×C79—21no—
145D4×C79—41no—
146D4×C79—41no—
147D4×C79—41no—
148D4×C79—41no—
149D4×C79—41no—
150D4×C79—41no—
151D4×C79—81no—
152D4×C79—81no—
153D4×C79—81no—
154D4×C79—81no—
155D4×C79—81no—
156D4×C79—81no—
157D4×C79—81no—
158D4×C79—161no—
159D4×C79—162no—
160C2×C4×C79—21no—
161C2×C4×C79—21no—
162C2×C4×C79—41no—
163C2×C4×C79—41no—
164C2×C4×C79—41no—
165C2×C4×C79—41no—
166C2×C4×C79—81no—
167C2×C4×C79—81no—
168C2×C4×C79—81no—
169C2×C4×C79#12011yescharacteristic
170C8×C79—21no—
171C8×C79—21no—
172C8×C79—41no—
173C8×C79—41no—
174C8×C79—42no—
175C8×C79—81no—
176C22×C79—41no—
177C22×C79—41no—
178C22×C79—41no—
179C22×C79—81no—
180C22×C79—81no—
181C22×C79—81no—
182C22×C79—82no—
183C22×C79—81no—
184C22×C79—81no—
185C22×C79—161no—
186C22×C79—161no—
187C22×C79—161no—
188C22×C79—321no—
189C22×C79B(4,𝔽2)11yescharacteristic
190C4×C79—21no—
191C4×C79—41no—
192C4×C79—41no—
193C4×C79—41no—
194C4×C79—41no—
195C4×C79—81no—
196C4×C79—161no—
197#13677C7911yesmaximal, characteristic, direct product factor, Sylow
198C2×C79—21no—
199C2×C79—81no—
200C2×C79—81no—
201C2×C79—81no—
202C2×C79—161no—
203C2×C79—161no—
204C2×C79#219611yescharacteristic
205#3288C2×C7911yescharacteristic, complemented
206#2202C2×C7911yescharacteristic, complemented
207#2199C2×C7911yescharacteristic, complemented
208#1655C2×C7911yescharacteristic, complemented
209#1406C2×C7911yescharacteristic, complemented
210#1404C2×C7911yescharacteristic, complemented
211#1403C2×C7911yescharacteristic, complemented
212C79#1367711yescharacteristic, direct product factor, Sylow
213#577—21no—
214#576—21no—
215H(3,𝔽4)C22×C7911yescharacteristic, complemented
216#496—21no—
217#496C22×C7911yescharacteristic
218#472—21no—
219#471—21no—
220B(4,𝔽2)—21no—
221#455C22×C7911yescharacteristic
222#454—21no—
223#454C22×C7911yescharacteristic
224#453—21no—
225#444—21no—
226#355—21no—
227#355C22×C7911yescharacteristic, complemented
228#354C22×C7911yescharacteristic, complemented
229AΣL(1,𝔽16)—21no—
230#329—21no—
231#329C22×C7911yescharacteristic, complemented
2322−1+4D4×C7911yescharacteristic, complemented
233H(5,𝔽2)—21no—
234H(5,𝔽2)D4×C7911yescharacteristic
235#137—21no—
236#137—41no—
237AGL(1,ℤ/8)—21no—
238AGL(1,ℤ/8)—21no—
239AGL(1,ℤ/8)—21no—
240AGL(1,ℤ/8)—41no—
241AGL(1,ℤ/8)—41no—
242AGL(1,ℤ/8)—41no—
243AGL(1,ℤ/8)—41no—
244#135—21no—
245#135—41no—
246#135—41no—
247C2×SD16—21no—
248C2×SD16—41no—
249C2×SD16—41no—
250C2×D8—21no—
251C2×D8—41no—
252C2×D8—41no—
253#131—21no—
254#131—21no—
255#127D4×C7911yescharacteristic, complemented
256#124—21no—
257#124—21no—
258#124—41no—
259#124C23×C7911yescharacteristic
260#120—21no—
261#120—41no—
262#120—41no—
263#108—21no—
264#104—21no—
265#104—21no—
266#104—41no—
267#101—21no—
268#101D4×C7911yescharacteristic, complemented
269#100—21no—
270#100D4×C7911yescharacteristic
271#99—21no—
272#99—41no—
273#97—21no—
274#97D4×C7911yescharacteristic, complemented
275C24—21no—
276#41—21no—
277#41—21no—
278#41—41no—
279#41—41no—
280#41—41no—
281#41—41no—
282#41—41no—
283C2×Q8—21no—
284C2×Q8—41no—
285C2×Q8C2×D4×C7911yescharacteristic
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4—41no—
290C2×D4—41no—
291C2×D4—41no—
292C2×D4—41no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—81no—
296C2×D4—81no—
297C2×D4C2×D4×C7911yescharacteristic
298Q16—21no—
299Q16—41no—
300SD16—21no—
301SD16—21no—
302SD16—41no—
303SD16—41no—
304SD16—41no—
305SD16—41no—
306SD16—41no—
307SD16—41no—
308SD16—82no—
309SD16—81no—
310D8—21no—
311D8—41no—
312D8—41no—
313D8—41no—
314D8—41no—
315D8—41no—
316D8—82no—
317D8—81no—
318M16—21no—
319M16—21no—
320M16—41no—
321M16—41no—
322M16—41no—
323C2×C8—21no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—41no—
327C22⋊C4—21no—
328C22⋊C4—41no—
329C22⋊C4—41no—
330C22⋊C4—41no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—81no—
334C42—21no—
335C42C2×D4×C7911yescharacteristic
336C23—21no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23—81no—
341C23—81no—
342C23—161no—
343Q8—21no—
344Q8—21no—
345Q8—41no—
346Q8—41no—
347Q8—81no—
348D4—21no—
349D4—21no—
350D4—41no—
351D4—41no—
352D4—41no—
353D4—41no—
354D4—41no—
355D4—41no—
356D4—81no—
357D4—81no—
358D4—81no—
359D4—81no—
360D4—81no—
361D4—81no—
362D4—81no—
363D4—161no—
364D4—162no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4#120×C7911yescharacteristic
375C8—21no—
376C8—21no—
377C8—41no—
378C8—41no—
379C8—42no—
380C8—81no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—81no—
385C22—81no—
386C22—81no—
387C22—82no—
388C22—81no—
389C22—81no—
390C22—161no—
391C22—161no—
392C22—161no—
393C22—321no—
394C22B(4,𝔽2)×C7911yescharacteristic
395C4—21no—
396C4—41no—
397C4—41no—
398C4—41no—
399C4—41no—
400C4—81no—
401C4—161no—
402C2—21no—
403C2—81no—
404C2—81no—
405C2—81no—
406C2—161no—
407C2—161no—
408C2C79×#219611yescharacteristic
subgroupG/H
center, socleC2×C79#2196
commutator, Frattini, cosocle#124C23×C79
LCSG ▷ #124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C79 ◁ C22×C79 ◁ C2×C4×C79 ◁ #124×C79 ◁ G

Complements

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

NH ≅ G/Ntype
1. C79×#3288C2semidirect
2. C79×#2202C2semidirect
3. C79×#2199C2semidirect
4. C79×#1655C2semidirect
5. C79×#1406C2semidirect
6. C79×#1404C2semidirect
7. C79×#1403C2semidirect
10. H(3,𝔽4)×C79C22semidirect
22. #355×C79C22semidirect
23. #354×C79C22semidirect
26. #329×C79C22semidirect
27. 2−1+4×C79D4semidirect
50. #127×C79D4semidirect
63. #101×C79D4semidirect
69. #97×C79D4semidirect
197. #13677C79direct (N × H)
205. #3288C2×C79semidirect
206. #2202C2×C79semidirect
207. #2199C2×C79semidirect
208. #1655C2×C79semidirect
209. #1406C2×C79semidirect
210. #1404C2×C79semidirect
211. #1403C2×C79semidirect
212. C79#13677direct (N × H)
215. H(3,𝔽4)C22×C79semidirect
227. #355C22×C79semidirect
228. #354C22×C79semidirect
231. #329C22×C79semidirect
232. 2−1+4D4×C79semidirect
255. #127D4×C79semidirect
268. #101D4×C79semidirect
274. #97D4×C79semidirect

Maximal subgroups

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

subgroupindex
C79×#32882
C79×#22022
C79×#21992
C79×#16552
C79×#14062
C79×#14042
C79×#14032
#1367779