Finite Group Databasebeta

← C2×#11730963 #13523089

Subgroups — C2×#11730963 #13523089

Order 1792.

2918 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6487C211yesmaximal, characteristic
2C23×C4⋊Dic7C211yesmaximal, characteristic
3C2×C4×C7×C22⋊C4C211yesmaximal, characteristic
4C2×#89715C214yesmaximal
5C2×#89701C212yesmaximal, complemented
6C2×#89708C212yesmaximal, complemented
7#11730963C2116yesmaximal, direct product factor
8C2×#89431C214yesmaximal, complemented
9C24×C4×C7C2211yescharacteristic
10C24×Dic7C2212yes—
11C23×Dic14C2211yescharacteristic
12C22×C7×C22⋊C4C2212yes—
13C22×C42×C7C2212yes—
14C22×C22⋊Dic7C2212yes—
15C22×C4⋊Dic7—28no—
16C22×C4⋊Dic7C2218yes—
17C22×C4⋊Dic7C2212yes—
18C22×C4⋊Dic7C2212yes—
19C22×C4⋊Dic7C2211yescharacteristic
20C2×#6487—28no—
21C2×#6487C2218yes—
22C22×Dic7⋊C4C2212yes—
23C4×C7×C22⋊C4C2218yes—
24C2×C7×#95C2212yes—
25C2×C7⋊#95C2218yes—
26C2×C7⋊#95C2218yes—
27#89715C22132yes—
28#89708C22116yescomplemented
29#89701C22116yescomplemented
30#89431C22132yescomplemented
31C2×#2503—71nomaximal, Sylow
32C25×C7C2311yescharacteristic
33C23×C4×C7—28no—
34C23×C4×C7D418yes—
35C23×C4×C7C2314yes—
36C23×C4×C7C2311yescharacteristic
37C23×C4×C7C2314yes—
38C23×C4×C7C2311yescharacteristic
39C23×Dic7—28no—
40C23×Dic7—216no—
41C23×Dic7C2312yes—
42C23×Dic7C2312yes—
43C23×Dic7C2318yes—
44C23×Dic7C2314yes—
45C22×Dic14—24no—
46C22×Dic14—28no—
47C22×Dic14C2314yes—
48C2×C7×C22⋊C4—28no—
49C2×C7×C22⋊C4C2318yes—
50C2×C42×C7—28no—
51C2×C42×C7C2318yes—
52C2×C22⋊Dic7—28no—
53C2×C22⋊Dic7C2318yes—
54C2×C4⋊Dic7—24no—
55C2×C4⋊Dic7—28no—
56C2×C4⋊Dic7—232no—
57C2×C4⋊Dic7—28no—
58C2×C4⋊Dic7C2318yes—
59C2×C4⋊Dic7C2318yes—
60C2×C4⋊Dic7C2314yes—
61#6487—232no—
62C2×Dic7⋊C4—28no—
63C2×Dic7⋊C4C2318yes—
64C7×#95C2318yes—
65C7⋊#95C23132yes—
66C7⋊#95C23132yes—
67C22×C22⋊Q8—71no—
68C23×C4⋊C4—71no—
69#2503—716no—
70C2×#400—74no—
71C2×#391—72no—
72C2×#382—72no—
73C2×#381—74no—
74C2×C4×C22⋊C4D711yescharacteristic
75C24×C7—24no—
76C24×C7—24no—
77C24×C7C2×D414yes—
78C24×C7#4118yes—
79C24×C7C2412yes—
80C24×C7C2411yescharacteristic
81C22×C4×C7—24no—
82C22×C4×C7—232no—
83C22×C4×C7—216no—
84C22×C4×C7—28no—
85C22×C4×C7—28no—
86C22×C4×C7—28no—
87C22×C4×C7C2×D414yes—
88C22×C4×C7C2×D4116yes—
89C22×C4×C7C2×D414yes—
90C22×C4×C7#4118yes—
91C22×C4×C7C2418yes—
92C22×C4×C7C2412yes—
93C22×C4×C7C2412yes—
94C22×Dic7—24no—
95C22×Dic7—24no—
96C22×Dic7—216no—
97C22×Dic7—28no—
98C22×Dic7—28no—
99C22×Dic7—232no—
100C22×Dic7—216no—
101C22×Dic7—28no—
102C22×Dic7—432no—
103C22×Dic7C2414yes—
104C22×Dic7C2414yes—
105C22×Dic7C2418yes—
106C2×Dic14—28no—
107C2×Dic14—216no—
108C2×Dic14—416no—
109C7×C22⋊C4—216no—
110C42×C7—216no—
111C22⋊Dic7—216no—
112C4⋊Dic7—28no—
113C4⋊Dic7—216no—
114C4⋊Dic7—216no—
115Dic7⋊C4—216no—
116C23×Q8—71no—
117C24×C4—72no—
118C24×C4C2×D711yescharacteristic
119C2×C22⋊Q8—78no—
120C2×C22⋊Q8—148no—
121C22×C4⋊C4—72no—
122C22×C4⋊C4—72no—
123C22×C4⋊C4—78no—
124C22×C4⋊C4—72no—
125C22×C4⋊C4—71no—
126C22×C4⋊C4—148no—
127C22×C22⋊C4—72no—
128C22×C22⋊C4C2×D712yes—
129C22×C42C2×D712yes—
130#400—732no—
131#391—716no—
132#382—716no—
133#381—732no—
134C4×C22⋊C4C2×D718yes—
135C2×#95—78no—
136C2×#95—78no—
137C2×#95C2×D712yes—
138C23×C7—28no—
139C23×C7—28no—
140C23×C7—232no—
141C23×C7—24no—
142C23×C7—24no—
143C23×C7#124116yes—
144C23×C7C22×D414yes—
145C23×C7C22×D412yes—
146C23×C7C2×#4118yes—
147C23×C7C2×#4114yes—
148C23×C7C2×#4114yes—
149C23×C72−1+414yes—
150C23×C7C2511yescharacteristic
151C2×C4×C7—28no—
152C2×C4×C7—24no—
153C2×C4×C7—232no—
154C2×C4×C7—28no—
155C2×C4×C7—28no—
156C2×C4×C7—28no—
157C2×C4×C7—416no—
158C2×C4×C7#123116yes—
159C2×C4×C7C22×D414yes—
160C2×C4×C7C22×D414yes—
161C2×C4×C7C2×#4118yes—
162C2×Dic7—24no—
163C2×Dic7—24no—
164C2×Dic7—28no—
165C2×Dic7—28no—
166C2×Dic7—216no—
167C2×Dic7—28no—
168C2×Dic7—48no—
169C2×Dic7—432no—
170C2×Dic7—416no—
171C2×Dic7—48no—
172Dic14—416no—
173C25C22×D711yescharacteristic
174C22×Q8—74no—
175C22×Q8—144no—
176C22×Q8—148no—
177C23×C4—28no—
178C23×C4—78no—
179C23×C4—72no—
180C23×C4—72no—
181C23×C4—74no—
182C23×C4—148no—
183C23×C4—1416no—
184C23×C4D2818yes—
185C23×C4C22×D714yes—
186C23×C4C22×D714yes—
187C23×C4C22×D711yescharacteristic
188C23×C4C22×D711yescharacteristic
189C22⋊Q8—1432no—
190C2×C4⋊C4—78no—
191C2×C4⋊C4—78no—
192C2×C4⋊C4—78no—
193C2×C4⋊C4—74no—
194C2×C4⋊C4—148no—
195C2×C4⋊C4—1432no—
196C2×C4⋊C4—148no—
197C2×C4⋊C4—148no—
198C2×C4⋊C4—144no—
199C2×C22⋊C4—28no—
200C2×C22⋊C4—78no—
201C2×C22⋊C4—148no—
202C2×C22⋊C4C22×D718yes—
203C2×C42—28no—
204C2×C42C22×D718yes—
205#95—732no—
206#95—732no—
207#95C22×D718yes—
208C22×C7—24no—
209C22×C7—24no—
210C22×C7—216no—
211C22×C7—232no—
212C22×C7C2×#12314yes—
213C22×C7C2×#12418yes—
214C22×C7C2×#12414yes—
215C22×C7#53614yes—
216C22×C7#552116yes—
217C22×C7C23×D411yescharacteristic
218C22×C7C22×#4112yes—
219C22×C7C22×#4112yes—
220C22×C7C2×2−1+412yes—
221C4×C7—24no—
222C4×C7—48no—
223C4×C7C2×#12318yes—
224Dic7—44no—
225Dic7—48no—
226Dic7—44no—
227C24—24no—
228C24—24no—
229C24C2×D2814yes—
230C24#115718yes—
231C24C23×D712yes—
232C24C23×D711yescharacteristic
233C2×Q8—148no—
234C2×Q8—1416no—
235C2×Q8—2816no—
236C22×C4—24no—
237C22×C4—232no—
238C22×C4—216no—
239C22×C4—28no—
240C22×C4—28no—
241C22×C4—28no—
242C22×C4—74no—
243C22×C4—74no—
244C22×C4—78no—
245C22×C4—144no—
246C22×C4—144no—
247C22×C4—1416no—
248C22×C4—148no—
249C22×C4—1432no—
250C22×C4—1416no—
251C22×C4—148no—
252C22×C4—148no—
253C22×C4—2832no—
254C22×C4C2×D2814yes—
255C22×C4C2×D28116yes—
256C22×C4C2×D2814yes—
257C22×C4#115918yes—
258C22×C4C23×D718yes—
259C22×C4C23×D712yes—
260C22×C4C23×D712yes—
261C4⋊C4—1416no—
262C4⋊C4—1416no—
263C4⋊C4—1416no—
264C4⋊C4—148no—
265C22⋊C4—216no—
266C22⋊C4—1416no—
267C42—216no—
268C2×C7—28no—
269C2×C7#250318yescomplemented
270C2×C7C22×#12311yescharacteristic
271C2×C7C22×#12411yescharacteristic
272C2×C7C2×#53611yescharacteristic
273C2×C7C2×#55214yes—
274C23—28no—
275C23—28no—
276C23—232no—
277C23—24no—
278C23—24no—
279C23#6438116yes—
280C23C22×D2814yes—
281C23C22×D2812yes—
282C23C2×#115718yes—
283C23C2×#115714yes—
284C23C2×#115914yes—
285C23#655414yes—
286C23C24×D711yescharacteristic
287Q8—2816no—
288C2×C4—28no—
289C2×C4—24no—
290C2×C4—232no—
291C2×C4—28no—
292C2×C4—28no—
293C2×C4—28no—
294C2×C4—416no—
295C2×C4—144no—
296C2×C4—144no—
297C2×C4—148no—
298C2×C4—1416no—
299C2×C4—148no—
300C2×C4—148no—
301C2×C4—2832no—
302C2×C4—2816no—
303C2×C4—288no—
304C2×C4—288no—
305C2×C4#6449116yes—
306C2×C4C22×D2814yes—
307C2×C4C22×D2814yes—
308C2×C4C2×#115918yes—
309C7C2×#250311yescharacteristic, complemented, Sylow
310C22—232no—
311C22—24no—
312C22—24no—
313C22—216no—
314C22C2×#643818yes—
315C22C2×#643814yes—
316C22C2×#644914yes—
317C22#9021714yes—
318C22#90232116yes—
319C22C23×D2811yescharacteristic
320C22C22×#115712yes—
321C22C22×#115912yes—
322C22C2×#655412yes—
323C4—24no—
324C4—48no—
325C4—288no—
326C4—284no—
327C4—284no—
328C4C2×#644918yes—
329C2—28no—
330C2#1173096318yesdirect product factor
331C2C22×#643811yescharacteristic
332C2C22×#644911yescharacteristic
333C2C2×#9021711yescharacteristic
334C2C2×#9023214yes—
subgroupG/H
centerC24C23×D7
commutator, cosocleC23×C7C25
FrattiniC23C24×D7
FittingC2×C4×C7×C22⋊C4C2
socleC24×C7C24
prime radicalC25×C7C23
hypercenterC2×C4×C22⋊C4D7
hypocenterC7C2×#2503
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
5. C2×#89701C2semidirect
6. C2×#89708C2semidirect
7. #11730963C2direct (N × H)
8. C2×#89431C2semidirect
28. #89708C22semidirect
29. #89701C22semidirect
30. #89431C22semidirect
269. C2×C7#2503semidirect
309. C7C2×#2503semidirect
330. C2#11730963direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#64872
C23×C4⋊Dic72
C2×C4×C7×C22⋊C42
C2×#897152
C2×#897012
C2×#897082
#117309632
C2×#894312
C2×#25037