Finite Group Databasebeta

← C2×#20293331 #28349989

Subgroups — C2×#20293331 #28349989

Order 13696.

1200 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20293331C2116yesmaximal, direct product factor
2C2×#16812107C211yesmaximal, characteristic, complemented
3C2×#16812099C211yesmaximal, characteristic, complemented
4C2×#16812099C211yesmaximal, characteristic, complemented
5C2×#16812087C211yesmaximal, characteristic, complemented
6C2×#16812087C211yesmaximal, characteristic, complemented
7C2×#16812086C211yesmaximal, characteristic, complemented
8C2×#16812086C211yesmaximal, characteristic, complemented
9C2×#16812085C211yesmaximal, characteristic, complemented
10C2×#16812074C211yesmaximal, characteristic, complemented
11C2×C4×C22⋊D107C211yesmaximal, characteristic
12C2×(C22⋊D107)⋊C4C211yesmaximal, characteristic
13C2×(C22⋊D107)⋊C4C211yesmaximal, characteristic
14C22×Dic107⋊C4C211yesmaximal, characteristic, complemented
15C2×C4⋊C4×D107C211yesmaximal, characteristic, complemented
16C2×C22⋊Q8×C107C211yesmaximal, characteristic, complemented
17C22×Dic214C2211yescharacteristic
18C22×Dic214C2211yescharacteristic
19C22×C22⋊D107C2211yescharacteristic
20C23×Dic107C2211yescharacteristic
21C23×Dic107C2211yescharacteristic
22C22×C4×D107C2211yescharacteristic
23C22×C4×D107C2211yescharacteristic
24C22×C4×D107C2211yescharacteristic
25#16812107C2218yescomplemented
26#16812099C2218yescomplemented
27#16812099C2218yescomplemented
28#16812087C2218yescomplemented
29#16812087C2218yescomplemented
30#16812086C2218yescomplemented
31#16812086C2218yescomplemented
32#16812085C2218yescomplemented
33#16812074C2218yescomplemented
34C4×C22⋊D107C2218yes—
35(C22⋊D107)⋊C4C2218yes—
36(C22⋊D107)⋊C4C2218yes—
37C2×C4⋊Dic107C2211yescharacteristic
38C2×C4⋊Dic107C2211yescharacteristic
39C2×Dic107⋊C4C2211yescharacteristic
40C2×Dic107⋊C4C2211yescharacteristic
41C2×Dic107⋊C4C2211yescharacteristic
42C2×Dic107⋊C4C2211yescharacteristic
43C2×Dic107⋊C4C2211yescharacteristic
44C2×Dic107⋊C4C2211yescharacteristic
45C2×Dic107⋊C4C2211yescharacteristic
46C2×Dic107⋊C4C2211yescharacteristic
47C2×Dic107⋊C4C2211yescharacteristic
48C2×Dic107⋊C4C2211yescharacteristic
49C2×Dic107⋊C4C2211yescharacteristic
50C2×Dic107⋊C4C2218yescomplemented
51C4⋊C4×D107C2218yescomplemented
52C2×C22⋊Dic107C2211yescharacteristic
53C2×(C2×D107)⋊C4C2211yescharacteristic
54C2×(C2×D107)⋊C4C2211yescharacteristic
55C2×(C2×D107)⋊C4C2211yescharacteristic
56C2×C4×Dic107C2211yescharacteristic
57C2×C4×Dic107C2211yescharacteristic
58C2×C4×Dic107C2211yescharacteristic
59C22×Q8×C107C2211yescharacteristic
60C23×C4×C107C2211yescharacteristic
61C22⋊Q8×C107C2218yescomplemented
62C2×C4⋊C4×C107C2211yescharacteristic
63C2×C4⋊C4×C107C2211yescharacteristic
64C2×C4⋊C4×C107C2211yescharacteristic
65C2×C22⋊C4×C107C2211yescharacteristic
66C2×C22⋊C4×C107C2211yescharacteristic
67C23×D107C2311yescharacteristic
68C2×Dic214—24no—
69C2×Dic214—42no—
70C2×Dic214C2314yes—
71C2×Dic214C2314yes—
72C2×C22⋊D107Q818yescomplemented
73C2×C22⋊D107C2314yes—
74C22×Dic107—24no—
75C22×Dic107—42no—
76C22×Dic107C2311yescharacteristic
77C22×Dic107C2311yescharacteristic
78C22×Dic107C2311yescharacteristic
79C22×Dic107C2311yescharacteristic
80C22×Dic107C2311yescharacteristic
81C22×Dic107C2314yes—
82C22×Dic107C2314yes—
83C22×Dic107C2311yescharacteristic
84C22×Dic107C2311yescharacteristic
85C2×C4×D107—24no—
86C2×C4×D107—42no—
87C2×C4×D107—42no—
88C2×C4×D107C2314yes—
89C2×C4×D107C2314yes—
90C2×C4×D107C2314yes—
91C4⋊Dic107C2314yes—
92C4⋊Dic107C2314yes—
93Dic107⋊C4C2314yes—
94Dic107⋊C4C2314yes—
95Dic107⋊C4C2314yes—
96Dic107⋊C4C2314yes—
97Dic107⋊C4C2314yes—
98Dic107⋊C4C2314yes—
99Dic107⋊C4C2314yes—
100Dic107⋊C4C2314yes—
101Dic107⋊C4C2314yes—
102Dic107⋊C4C2314yes—
103Dic107⋊C4C2314yes—
104C22⋊Dic107C2314yes—
105(C2×D107)⋊C4C2314yes—
106(C2×D107)⋊C4C2314yes—
107(C2×D107)⋊C4C2314yes—
108C4×Dic107C2314yes—
109C4×Dic107C2314yes—
110C4×Dic107C2314yes—
111C24×C107C2311yescharacteristic
112C2×Q8×C107—42no—
113C2×Q8×C107C2314yes—
114C22×C4×C107—42no—
115C22×C4×C107C2311yescharacteristic
116C22×C4×C107C2311yescharacteristic
117C22×C4×C107C2311yescharacteristic
118C22×C4×C107C2311yescharacteristic
119C22×C4×C107C2311yescharacteristic
120C22×C4×C107C2311yescharacteristic
121C22×C4×C107C2314yes—
122C4⋊C4×C107C2314yes—
123C4⋊C4×C107C2314yes—
124C4⋊C4×C107C2314yes—
125C22⋊C4×C107C2314yes—
126C22⋊C4×C107C2314yes—
127C23×C107—22no—
128C23×C107—22no—
129C23×C107C2×Q814yes—
130C23×C107C2412yes—
131C23×C107C2411yescharacteristic
132C22×D107—22no—
133C22×D107—22no—
134C22×D107C2×Q814yes—
135C22×D107C2412yes—
136Q8×C107—44no—
137C2×C4×C107—22no—
138C2×C4×C107—22no—
139C2×C4×C107—22no—
140C2×C4×C107—22no—
141C2×C4×C107—22no—
142C2×C4×C107—22no—
143C2×C4×C107—44no—
144C2×C4×C107C2412yes—
145C2×C4×C107C2412yes—
146C2×C4×C107C2412yes—
147C2×C4×C107C2412yes—
148C2×C4×C107C2412yes—
149C2×C4×C107C2412yes—
150C22⋊D107C2×Q8116yescomplemented
151C2×Dic107—22no—
152C2×Dic107—22no—
153C2×Dic107—22no—
154C2×Dic107—22no—
155C2×Dic107—22no—
156C2×Dic107—28no—
157C2×Dic107—44no—
158C2×Dic107C2×Q814yes—
159C2×Dic107#4114yes—
160C2×Dic107C2412yes—
161C2×Dic107C2412yes—
162C2×Dic107C2412yes—
163C2×Dic107C2412yes—
164C2×Dic107C2412yes—
165C2×Dic107C2412yes—
166C2×Dic107C2412yes—
167C4×D107—28no—
168C4×D107—44no—
169C4×D107—44no—
170Dic214—28no—
171Dic214—44no—
172C22×C107—28no—
173C22×C107—22no—
174C22×C107—22no—
175C22×C107C22×Q814yes—
176C22×C107C22×Q812yes—
177C22×C107C2×#4112yes—
178C22×C107H(5,𝔽2)12yes—
179C22×C107C2511yescharacteristic
180C2×D107—28no—
181C2×D107—22no—
182C2×D107—22no—
183C2×D107C22×Q814yes—
184C4×C107—22no—
185C4×C107—22no—
186C4×C107—22no—
187C4×C107—22no—
188C4×C107—22no—
189C4×C107—22no—
190Dic107—22no—
191Dic107—22no—
192Dic107—22no—
193Dic107—22no—
194Dic107—22no—
195Dic107C22×Q814yes—
196Dic107C2×#4114yes—
197C2×C107—24no—
198C2×C107#55414yescomplemented
199C2×C107C23×Q811yescharacteristic
200C2×C107C22×#4111yescharacteristic
201C2×C107C2×H(5,𝔽2)11yescharacteristic
202D107—24no—
203C2×#554—1071nomaximal, Sylow
204C107C2×#55411yescharacteristic, complemented, Sylow
205#554—10716no—
206C2×C4⋊Q8—1071no—
207C2×#125—1071no—
208C2×#125—1071no—
209C2×C22⋊Q8—1071no—
210C2×C22⋊Q8—1071no—
211C2×C22⋊Q8—1071no—
212C2×C22⋊Q8—1071no—
213C2×C22⋊Q8—1071no—
214C2×C22⋊Q8D10711yescharacteristic, complemented
215C2×C4×Q8—1071no—
216C2×C4×D4—1071no—
217C2×C4×D4—1071no—
218C2×C4×D4—1071no—
219C22×C4⋊C4—1071no—
220C22×C4⋊C4—1071no—
221C22×Q8—1071no—
222C22×Q8—1071no—
223C22×Q8C2×D10711yescharacteristic
224C22×D4—1071no—
225C23×C4—1071no—
226C23×C4—1071no—
227C23×C4—1071no—
228C23×C4—1071no—
229C23×C4—1071no—
230C23×C4C2×D10711yescharacteristic
231C4⋊Q8—1078no—
232#125—1078no—
233#125—1078no—
234C22⋊Q8—1078no—
235C22⋊Q8—1078no—
236C22⋊Q8—1078no—
237C22⋊Q8—1078no—
238C22⋊Q8—1078no—
239C22⋊Q8C2×D10718yescomplemented
240C4×Q8—1078no—
241C4×D4—1078no—
242C4×D4—1078no—
243C4×D4—1078no—
244C2×C4⋊C4—1071no—
245C2×C4⋊C4—1071no—
246C2×C4⋊C4—1071no—
247C2×C4⋊C4—1071no—
248C2×C4⋊C4—1071no—
249C2×C4⋊C4—1071no—
250C2×C4⋊C4—1078no—
251C2×C4⋊C4—1071no—
252C2×C4⋊C4—1071no—
253C2×C4⋊C4—1071no—
254C2×C4⋊C4—1071no—
255C2×C4⋊C4—1071no—
256C2×C4⋊C4—1071no—
257C2×C4⋊C4—1071no—
258C2×C4⋊C4—1078no—
259C2×C4⋊C4C2×D10711yescharacteristic
260C2×C4⋊C4C2×D10711yescharacteristic
261C2×C4⋊C4C2×D10711yescharacteristic
262C2×C22⋊C4—1071no—
263C2×C22⋊C4—1071no—
264C2×C22⋊C4—1071no—
265C2×C22⋊C4—1071no—
266C2×C22⋊C4C2×D10711yescharacteristic
267C2×C22⋊C4C2×D10711yescharacteristic
268C2×C42—1071no—
269C2×C42—1071no—
270C2×C42—1071no—
271C24—1071no—
272C24C22×D10711yescharacteristic
273C2×Q8—42no—
274C2×Q8—1074no—
275C2×Q8—1074no—
276C2×Q8—4282no—
277C2×Q8—2144no—
278C2×Q8C22×D10714yes—
279C2×D4—1078no—
280C2×D4—1074no—
281C22×C4—42no—
282C22×C4—1071no—
283C22×C4—1071no—
284C22×C4—1074no—
285C22×C4—1074no—
286C22×C4—1071no—
287C22×C4—1071no—
288C22×C4—1071no—
289C22×C4—1071no—
290C22×C4—1071no—
291C22×C4—1074no—
292C22×C4—1074no—
293C22×C4—1074no—
294C22×C4—4282no—
295C22×C4—4282no—
296C22×C4—4282no—
297C22×C4—2144no—
298C22×C4—2144no—
299C22×C4C22×D10711yescharacteristic
300C22×C4C22×D10711yescharacteristic
301C22×C4C22×D10711yescharacteristic
302C22×C4C22×D10711yescharacteristic
303C22×C4C22×D10711yescharacteristic
304C22×C4C22×D10711yescharacteristic
305C22×C4C22×D10714yes—
306C4⋊C4—1074no—
307C4⋊C4—1074no—
308C4⋊C4—1074no—
309C4⋊C4—1074no—
310C4⋊C4—1074no—
311C4⋊C4—1074no—
312C4⋊C4—1074no—
313C4⋊C4—1074no—
314C4⋊C4—1074no—
315C4⋊C4—1074no—
316C4⋊C4—1074no—
317C4⋊C4—1074no—
318C4⋊C4—1074no—
319C4⋊C4C22×D10714yes—
320C4⋊C4C22×D10714yes—
321C4⋊C4C22×D10714yes—
322C22⋊C4—1074no—
323C22⋊C4—1074no—
324C22⋊C4—1074no—
325C22⋊C4—1074no—
326C22⋊C4C22×D10714yes—
327C22⋊C4C22×D10714yes—
328C42—1074no—
329C42—1074no—
330C42—1074no—
331C23—22no—
332C23—22no—
333C23—1074no—
334C23—1072no—
335C23—2142no—
336C23—2142no—
337C23Q8×D10714yes—
338C23C23×D10712yes—
339C23C23×D10711yescharacteristic
340Q8—44no—
341Q8—4284no—
342Q8—2148no—
343D4—10716no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—44no—
351C2×C4—1074no—
352C2×C4—1074no—
353C2×C4—1072no—
354C2×C4—1072no—
355C2×C4—1072no—
356C2×C4—1072no—
357C2×C4—1072no—
358C2×C4—1072no—
359C2×C4—1072no—
360C2×C4—4284no—
361C2×C4—4284no—
362C2×C4—4284no—
363C2×C4—2148no—
364C2×C4—2148no—
365C2×C4—2142no—
366C2×C4—2142no—
367C2×C4—2142no—
368C2×C4—2142no—
369C2×C4—2142no—
370C2×C4C23×D10712yes—
371C2×C4C23×D10712yes—
372C2×C4C23×D10712yes—
373C2×C4C23×D10712yes—
374C2×C4C23×D10712yes—
375C2×C4C23×D10712yes—
376C22—22no—
377C22—22no—
378C22—28no—
379C22—1074no—
380C22—2142no—
381C22—2142no—
382C22—2148no—
383C22C2×Q8×D10714yes—
384C22C2×Q8×D10712yes—
385C22#41×D10712yes—
386C22#1681215412yes—
387C22C24×D10711yescharacteristic
388C4—22no—
389C4—22no—
390C4—22no—
391C4—22no—
392C4—22no—
393C4—22no—
394C4—1074no—
395C4—1074no—
396C4—2142no—
397C4—2142no—
398C4—2142no—
399C4—2142no—
400C4—2142no—
401C2—24no—
402C2—2144no—
403C2#2029333114yesdirect product factor
404C2C22×Q8×D10711yescharacteristic
405C2C2×#41×D10711yescharacteristic
406C2C2×#1681215411yescharacteristic
subgroupG/H
centerC23C23×D107
commutator, cosocleC22×C107C25
FrattiniC22C24×D107
FittingC2×C22⋊Q8×C107C2
socleC23×C107C24
prime radicalC22×C22⋊D107C22
hypercenterC2×C22⋊Q8D107
hypocenterC107C2×#554
LCSG ▷ C22×C107 ▷ C107
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. #20293331C2direct (N × H)
2. C2×#16812107C2semidirect
3. C2×#16812099C2semidirect
4. C2×#16812099C2semidirect
5. C2×#16812087C2semidirect
6. C2×#16812087C2semidirect
7. C2×#16812086C2semidirect
8. C2×#16812086C2semidirect
9. C2×#16812085C2semidirect
10. C2×#16812074C2semidirect
14. C22×Dic107⋊C4C2semidirect
15. C2×C4⋊C4×D107C2semidirect
16. C2×C22⋊Q8×C107C2semidirect
25. #16812107C22semidirect
26. #16812099C22semidirect
27. #16812099C22semidirect
28. #16812087C22semidirect
29. #16812087C22semidirect
30. #16812086C22semidirect
31. #16812086C22semidirect
32. #16812085C22semidirect
33. #16812074C22semidirect
50. C2×Dic107⋊C4C22semidirect
51. C4⋊C4×D107C22semidirect
61. C22⋊Q8×C107C22semidirect
72. C2×C22⋊D107Q8semidirect
150. C22⋊D107C2×Q8semidirect
198. C2×C107#554semidirect
204. C107C2×#554semidirect
214. C2×C22⋊Q8D107semidirect
239. C22⋊Q8C2×D107semidirect
403. C2#20293331direct (N × H)

Maximal subgroups

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

subgroupindex
#202933312
C2×#168121072
C2×#168120992
C2×#168120872
C2×#168120862
C2×#168120852
C2×#168120742
C2×C4×C22⋊D1072
C2×(C22⋊D107)⋊C42
C22×Dic107⋊C42
C2×C4⋊C4×D1072
C2×C22⋊Q8×C1072
C2×#554107