Finite Group Databasebeta

← C2×#11735990 #14223420

Subgroups — C2×#11735990 #14223420

Order 1792.

1854 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90210C212yesmaximal
2C2×C7×#405C211yesmaximal, characteristic
3C2×#89893C212yesmaximal
4C2×#89878C211yesmaximal, characteristic
5C22×#6391C211yesmaximal, characteristic
6#11735990C2116yesmaximal, direct product factor
7C2×#89474C214yesmaximal, complemented
8C2×#89455C212yesmaximal, complemented
9C2×#89452C212yesmaximal, complemented
10C22×C7×M16C2211yescharacteristic
11C23×C7×C8C2211yescharacteristic
12C2×C7×#117C2211yescharacteristic
13C2×#6489C2212yes—
14C22×C4⋊Dic7C2212yes—
15#90210—28no—
16#90210C22116yes—
17C2×#6453—22no—
18C2×#6451—22no—
19C2×#6441—24no—
20C2×#6433C2214yes—
21C2×#6432C2214yes—
22C7×#405C2218yes—
23C2×C7×#97C2212yes—
24C2×C4×C7×C8C2212yes—
25#89893C22116yes—
26#89878C2218yes—
27C2×#6392C2214yes—
28C2×#6392C2212yes—
29C2×#6391C2214yes—
30C2×#6391C2211yescharacteristic
31C2×#6391C2212yes—
32C2×#6391C2211yescharacteristic
33C2×#6391C2212yes—
34C2×#6391C2218yes—
35#89474C22132yescomplemented
36#89455C22116yescomplemented
37#89452C22116yescomplemented
38C2×#3163—71nomaximal, Sylow
39C23×C4×C7C2311yescharacteristic
40C23×Dic7—22no—
41C22×Dic14—24no—
42C2×C7×M16Q818yescomplemented
43C2×C7×M16C2314yes—
44C22×C7×C8Q818yescomplemented
45C22×C7×C8C2311yescharacteristic
46C22×C7×C8C2312yes—
47C22×C7×C8C2311yescharacteristic
48C22×C7×C8C2314yes—
49C7×#117D414yes—
50C7×#117C2314yes—
51C2×C7×C4⋊C4D412yes—
52C2×C7×C22⋊C4D412yes—
53C2×C42×C7C2312yes—
54C2×C22⋊Dic7—22no—
55#6489—24no—
56#6489C2318yes—
57C2×C4⋊Dic7—24no—
58C2×C4⋊Dic7—22no—
59C2×C4⋊Dic7—24no—
60C2×C4⋊Dic7C2312yes—
61C2×C4⋊Dic7C2314yes—
62C2×C4⋊Dic7C2318yes—
63C2×C4⋊Dic7C2312yes—
64C2×Dic7⋊C4—24no—
65C2×Dic7⋊C4—24no—
66C2×C4×Dic7—22no—
67#6453—216no—
68#6451—216no—
69#6441—232no—
70#6433—28no—
71#6433C23116yes—
72#6432—28no—
73#6432C23116yes—
74C7×#97C2318yes—
75C4×C7×C8C2318yes—
76#6392C2318yes—
77#6392C23116yes—
78#6391C2318yes—
79#6391C2318yes—
80#6391C2314yes—
81#6391C2314yes—
82#6391C23116yes—
83C2×#544—72no—
84#3163—716no—
85C2×#501—74no—
86C2×#499—72no—
87C2×#498—72no—
88C2×#428—72no—
89C2×#427—71no—
90C22×#106—71no—
91C2×#405D711yescharacteristic
92C24×C7C2×D411yescharacteristic
93C22×C4×C7C2×D414yes—
94C22×C4×C7C2×D411yescharacteristic
95C22×C4×C7C2×D412yes—
96C22×C4×C7C2×Q814yes—
97C22×C4×C7C2×Q814yes—
98C22×C4×C7C2411yescharacteristic
99C22×C4×C7C2412yes—
100C22×Dic7—22no—
101C22×Dic7—22no—
102C22×Dic7—24no—
103C22×Dic7—28no—
104C22×Dic7—84no—
105C2×Dic14—216no—
106C2×Dic14—88no—
107C7×M16C2×Q8116yescomplemented
108C2×C7×C8C2×Q814yes—
109C2×C7×C8C2×Q814yes—
110C2×C7×C8C2×Q818yes—
111C2×C7×C8C2×Q8116yescomplemented
112C2×C7×C8C2412yes—
113C2×C7×C8C2414yes—
114C2×C7×C8C2412yes—
115C7×C4⋊C4C2×D418yes—
116C7×C22⋊C4C2×D418yes—
117C42×C7C2×D414yes—
118C42×C7C2414yes—
119C22⋊Dic7—28no—
120C4⋊Dic7—22no—
121C4⋊Dic7—24no—
122C4⋊Dic7—216no—
123C4⋊Dic7—22no—
124C4⋊Dic7—28no—
125C4⋊Dic7C2414yes—
126C4⋊Dic7C2418yes—
127C4⋊Dic7C2414yes—
128Dic7⋊C4—216no—
129Dic7⋊C4—216no—
130C4×Dic7—28no—
131C22×M16C2×D711yescharacteristic
132C23×C8C2×D711yescharacteristic
133#544—716no—
134#544—148no—
135C2×C4⋊Q8—74no—
136C2×C4⋊Q8—142no—
137C2×#125—74no—
138C2×#125—142no—
139C2×C22⋊Q8—144no—
140C2×#117—72no—
141C2×#117C2×D711yescharacteristic
142C22×C4⋊C4—72no—
143#501—732no—
144#499—716no—
145#498—716no—
146#428—716no—
147#427—78no—
148C2×#107—74no—
149C2×#107—72no—
150C2×#106—71no—
151C2×#106—72no—
152C2×#106—74no—
153C2×#106—71no—
154C2×#106—72no—
155C2×#106—78no—
156#405C2×D718yes—
157C2×#97C2×D712yes—
158C2×C4×C8C2×D712yes—
159C23×C7—24no—
160C23×C7C4⋊Q814yes—
161C23×C7C22×D412yes—
162C23×C7C22×D411yescharacteristic
163C2×C4×C7—28no—
164C2×C4×C7C4⋊Q818yes—
165C2×C4×C7C4⋊Q814yes—
166C2×C4×C7C22×D418yes—
167C2×C4×C7C22×D412yes—
168C2×C4×C7C22×D411yescharacteristic
169C2×C4×C7C22×Q812yes—
170C2×C4×C7C22×Q812yes—
171C2×C4×C7C22×Q814yes—
172C2×C4×C7C22×Q814yes—
173C2×C4×C7C2511yescharacteristic
174C2×Dic7—24no—
175C2×Dic7—24no—
176C2×Dic7—28no—
177C2×Dic7—44no—
178C2×Dic7—48no—
179C2×Dic7—44no—
180C2×Dic7—88no—
181Dic14—816no—
182C7×C8C22×Q814yes—
183C7×C8C22×Q814yes—
184C7×C8C22×Q818yes—
185C22×Q8—144no—
186C23×C4—142no—
187C23×C4C22×D711yescharacteristic
188C2×M16Dic1418yescomplemented
189C2×M16C22×D714yes—
190C22×C8Dic1418yescomplemented
191C22×C8C22×D711yescharacteristic
192C22×C8C22×D711yescharacteristic
193C22×C8C22×D712yes—
194C22×C8C22×D714yes—
195C4⋊Q8—716no—
196C4⋊Q8—1416no—
197C4⋊Q8—148no—
198#125—716no—
199#125—1416no—
200#125—148no—
201C22⋊Q8—1432no—
202#117—78no—
203#117—144no—
204#117D2814yes—
205#117C22×D714yes—
206C2×C4⋊C4—74no—
207C2×C4⋊C4—78no—
208C2×C4⋊C4—72no—
209C2×C4⋊C4—72no—
210C2×C4⋊C4—142no—
211C2×C4⋊C4—144no—
212C2×C4⋊C4—144no—
213C2×C4⋊C4—144no—
214C2×C4⋊C4—144no—
215C2×C4⋊C4D2812yes—
216C2×C22⋊C4—142no—
217C2×C22⋊C4D2812yes—
218C2×C42—142no—
219C2×C42C22×D712yes—
220#107—78no—
221#107—716no—
222#106—78no—
223#106—78no—
224#106—74no—
225#106—74no—
226#106—716no—
227#97C22×D718yes—
228C4×C8C22×D718yes—
229C22×C7—28no—
230C22×C7—24no—
231C22×C7C2×C4⋊Q814yes—
232C22×C7C2×C4⋊Q812yes—
233C22×C7#57714yes—
234C22×C7C23×D411yescharacteristic
235C4×C7—28no—
236C4×C7C2×C4⋊Q814yes—
237C4×C7C2×C4⋊Q812yes—
238C4×C7C23×Q811yescharacteristic
239C4×C7C23×Q811yescharacteristic
240Dic7—44no—
241Dic7—48no—
242Dic7—44no—
243C24C2×D2811yescharacteristic
244C2×Q8—1416no—
245C2×Q8—568no—
246C22×C4—142no—
247C22×C4—142no—
248C22×C4—144no—
249C22×C4—148no—
250C22×C4—564no—
251C22×C4C2×Dic1414yes—
252C22×C4C2×Dic1414yes—
253C22×C4C2×D2814yes—
254C22×C4C2×D2812yes—
255C22×C4C2×D2811yescharacteristic
256C22×C4C23×D712yes—
257C22×C4C23×D711yescharacteristic
258M16C2×Dic14116yescomplemented
259C2×C8C2×Dic1414yes—
260C2×C8C2×Dic1414yes—
261C2×C8C2×Dic1418yes—
262C2×C8C2×Dic14116yescomplemented
263C2×C8C23×D712yes—
264C2×C8C23×D714yes—
265C2×C8C23×D712yes—
266C4⋊C4—78no—
267C4⋊C4—74no—
268C4⋊C4—74no—
269C4⋊C4—1416no—
270C4⋊C4—1416no—
271C4⋊C4—148no—
272C4⋊C4—144no—
273C4⋊C4—142no—
274C4⋊C4—142no—
275C4⋊C4—1416no—
276C4⋊C4C2×D2818yes—
277C22⋊C4—148no—
278C22⋊C4C2×D2818yes—
279C42—148no—
280C42C2×D2814yes—
281C42C23×D714yes—
282C2×C7—24no—
283C2×C7#316314yescomplemented
284C2×C7C22×C4⋊Q811yescharacteristic
285C2×C7C2×#57712yes—
286C23—24no—
287C23#643214yes—
288C23C22×D2812yes—
289C23C22×D2811yescharacteristic
290Q8—5616no—
291C2×C4—28no—
292C2×C4—144no—
293C2×C4—144no—
294C2×C4—148no—
295C2×C4—284no—
296C2×C4—288no—
297C2×C4—284no—
298C2×C4—568no—
299C2×C4#643218yes—
300C2×C4#643214yes—
301C2×C4C22×Dic1412yes—
302C2×C4C22×Dic1412yes—
303C2×C4C22×Dic1414yes—
304C2×C4C22×Dic1414yes—
305C2×C4C22×D2818yes—
306C2×C4C22×D2812yes—
307C2×C4C22×D2811yescharacteristic
308C2×C4C24×D711yescharacteristic
309C8C22×Dic1414yes—
310C8C22×Dic1414yes—
311C8C22×Dic1418yes—
312C7C2×#316311yescharacteristic, complemented, Sylow
313C22—28no—
314C22—24no—
315C22C2×#643214yes—
316C22C2×#643212yes—
317C22#9044214yes—
318C22C23×D2811yescharacteristic
319C4—28no—
320C4—288no—
321C4—284no—
322C4—284no—
323C4C2×#643212yes—
324C4C2×#643214yes—
325C4C23×Dic1411yescharacteristic
326C4C23×Dic1411yescharacteristic
327C2—24no—
328C2#1173599014yesdirect product factor
329C2C22×#643211yescharacteristic
330C2C2×#9044212yes—
subgroupG/H
centerC23C22×D28
commutator, cosocleC2×C4×C7C25
FrattiniC2×C4C24×D7
FittingC2×C7×#405C2
socleC23×C7C22×D4
prime radicalC24×C7C2×D4
hypercenterC2×#405D7
hypocenterC7C2×#3163
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#405

Complements

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

NH ≅ G/Ntype
6. #11735990C2direct (N × H)
7. C2×#89474C2semidirect
8. C2×#89455C2semidirect
9. C2×#89452C2semidirect
35. #89474C22semidirect
36. #89455C22semidirect
37. #89452C22semidirect
42. C2×C7×M16Q8semidirect
44. C22×C7×C8Q8semidirect
107. C7×M16C2×Q8semidirect
111. C2×C7×C8C2×Q8semidirect
188. C2×M16Dic14semidirect
190. C22×C8Dic14semidirect
258. M16C2×Dic14semidirect
262. C2×C8C2×Dic14semidirect
283. C2×C7#3163semidirect
312. C7C2×#3163semidirect
328. C2#11735990direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#902102
C2×C7×#4052
C2×#898932
C2×#898782
C22×#63912
#117359902
C2×#894742
C2×#894552
C2×#894522
C2×#31637