Finite Group Databasebeta

← C2×#11735934 #14217540

Subgroups — C2×#11735934 #14217540

Order 1792.

2134 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#1157C211yesmaximal, characteristic
2C22×C7×#97C211yesmaximal, characteristic, complemented
3C2×#89882C212yesmaximal
4C2×#89874C212yesmaximal, complemented
5C2×C4×#1136C211yesmaximal, characteristic, complemented
6#11735934C2116yesmaximal, direct product factor
7C2×#89481C214yesmaximal, complemented
8C2×#89477C214yesmaximal, complemented
9C22×#1157C2211yescharacteristic
10C23×C7×C8C2212yes—
11C22×C42×C7C2211yescharacteristic
12C2×C4×C22⋊D7C414yes—
13C2×#6489C2211yescharacteristic
14C22×#1136C2212yes—
15C4×#1157C418yes—
16C4×#1157C2218yes—
17C2×C4×D28C412yes—
18C2×#6435C2212yes—
19C2×C42×D7C2212yes—
20C2×C4×Dic14C412yes—
21C2×C7×#97C2212yes—
22C2×C7×#97C2212yes—
23C2×C7×#97C2218yescomplemented
24#89882C22116yes—
25C2×(C2×D7)⋊C8C2218yes—
26#89874C22116yescomplemented
27C2×Dic7⋊C8C2218yes—
28C4×#1136C2218yescomplemented
29C2×#6377C2212yes—
30C2×C4×C7⋊C8C2212yes—
31#89481C22132yescomplemented
32#89477C22132yescomplemented
33C2×#2923—71nomaximal, Sylow
34C23×C4×C7C2×C412yes—
35C23×C4×C7C2311yescharacteristic
36C22×C22⋊D7C2×C412yes—
37C2×#1157—28no—
38C2×#1157C2×C414yes—
39C2×#1157C2314yes—
40C22×D28C2×C411yescharacteristic
41C22×C4×D7C2×C414yes—
42C22×C4×D7C2312yes—
43C22×Dic14C2×C411yescharacteristic
44C22×C7×C8—28no—
45C22×C7×C8C2314yes—
46C22×C7×C8C2318yes—
47C2×C42×C7C2×C412yes—
48C2×C42×C7C2×C414yes—
49C2×C42×C7C2311yescharacteristic
50C2×C42×C7C2311yescharacteristic
51C2×C42×C7C2314yes—
52C2×C22⋊Dic7C2×C412yes—
53C4×C22⋊D7C2×C4132yes—
54C2×(C2×D7)⋊C4C2×C414yes—
55#6489C2×C414yes—
56#6489C2314yes—
57C2×C4⋊Dic7C2×C412yes—
58C2×Dic7⋊C4C2×C414yes—
59C2×C4×Dic7C2×C414yes—
60C2×C4×Dic7C2312yes—
61C2×#1136—28no—
62C2×#1136C2318yes—
63C22×C7⋊C8C2314yes—
64C4×D28C2×C4116yes—
65#6435C2×C418yes—
66#6435C2318yes—
67C42×D7C2×C418yes—
68C42×D7C2318yes—
69C4×Dic14C2×C4116yes—
70C7×#97C2318yes—
71C7×#97C2318yes—
72(C2×D7)⋊C8C23132yes—
73Dic7⋊C8C23132yes—
74#6377C2318yes—
75C4×C7⋊C8C2318yes—
76C2×C4×#41—71no—
77#2923—716no—
78C2×#433—74no—
79C2×#432—74no—
80C2×#424—72no—
81C2×#408—72no—
82C2×C4×M16—71no—
83C22×#97D711yescharacteristic, complemented
84C24×C7C22×C411yescharacteristic
85C23×D7C22×C412yes—
86C22×C4×C7—44no—
87C22×C4×C7C22×C412yes—
88C22×C4×C7C22×C412yes—
89C22×C4×C7C22×C418yes—
90C22×C4×C7C22×C411yescharacteristic
91C22×C4×C7C22×C412yes—
92C22×C4×C7#4118yes—
93C22×C4×C7C2412yes—
94C22×C4×C7C2411yescharacteristic
95C2×C22⋊D7—28no—
96C2×C22⋊D7C22×C418yes—
97C22×Dic7C22×C412yes—
98C22×Dic7C22×C414yes—
99#1157—216no—
100#1157—216no—
101C2×D28—24no—
102C2×D28C22×C414yes—
103C2×C4×D7—28no—
104C2×C4×D7—48no—
105C2×C4×D7C22×C414yes—
106C2×C4×D7C22×C4116yes—
107C2×C4×D7C2414yes—
108C2×Dic14—24no—
109C2×Dic14C22×C414yes—
110C2×C7×C8—28no—
111C2×C7×C8—216no—
112C2×C7×C8C2418yes—
113C42×C7C22×C418yes—
114C42×C7C22×C412yes—
115C42×C7C22×C412yes—
116C42×C7C2412yes—
117C42×C7C2412yes—
118C22⋊Dic7C22×C418yes—
119(C2×D7)⋊C4C22×C4116yes—
120C4⋊Dic7C22×C418yes—
121Dic7⋊C4C22×C4116yes—
122C4×Dic7C22×C4116yes—
123C4×Dic7C22×C414yes—
124C4×Dic7C2414yes—
125#1136—216no—
126C2×C7⋊C8—28no—
127C2×C7⋊C8C2418yes—
128C22×#41—71no—
129C22×M16—72no—
130C23×C8C2×D712yes—
131C4×#41—78no—
132C4×#41—78no—
133C2×C4×Q8—72no—
134C2×C4×D4—74no—
135C2×C4×D4—72no—
136C2×#117—71no—
137C2×#117—72no—
138C22×C42—72no—
139C22×C42C2×D711yescharacteristic
140#433—732no—
141#432—732no—
142#424—716no—
143C2×C4⋊C8—78no—
144#408—716no—
145C2×C22⋊C8—78no—
146C4×M16—78no—
147C2×#97—72no—
148C2×#97C2×D712yes—
149C2×#97C2×D712yes—
150C2×#97C2×D718yescomplemented
151C2×C4×C8—72no—
152C23×C7—24no—
153C23×C7#11714yes—
154C23×C7C23×C412yes—
155C23×C7C23×C411yescharacteristic
156C22×D7—24no—
157C22×D7—28no—
158C22×D7C23×C414yes—
159C2×C4×C7—24no—
160C2×C4×C7—24no—
161C2×C4×C7—48no—
162C2×C4×C7#11714yes—
163C2×C4×C7#11718yes—
164C2×C4×C7C23×C414yes—
165C2×C4×C7C23×C414yes—
166C2×C4×C7C23×C411yescharacteristic
167C2×C4×C7C23×C412yes—
168C2×C4×C7C2×#4114yes—
169C2×C4×C7C2×#4118yes—
170C2×C4×C7C2511yescharacteristic
171C22⋊D7—216no—
172C2×Dic7—24no—
173C2×Dic7—28no—
174C2×Dic7C23×C414yes—
175C2×Dic7C23×C418yes—
176D28—28no—
177C4×D7—28no—
178C4×D7—28no—
179C4×D7—416no—
180Dic14—28no—
181C7×C8—28no—
182C7⋊C8—28no—
183C2×#41—74no—
184C2×#41—74no—
185C2×#41—148no—
186C22×Q8—71no—
187C22×D4—72no—
188C22×D4—71no—
189C23×C4—72no—
190C23×C4—74no—
191C23×C4C4×D712yes—
192C23×C4C22×D711yescharacteristic
193C2×M16—78no—
194C2×M16—148no—
195C22×C8—28no—
196C22×C8—74no—
197C22×C8C22×D718yes—
198C22×C8C22×D714yes—
199C4×Q8—716no—
200C4×D4—732no—
201C4×D4—716no—
202#117—74no—
203#117—74no—
204#117—78no—
205#117—78no—
206C2×C4⋊C4—74no—
207C2×C4⋊C4—72no—
208C2×C22⋊C4—72no—
209C2×C22⋊C4—74no—
210C2×C42—74no—
211C2×C42—72no—
212C2×C42—78no—
213C2×C42—78no—
214C2×C42C4×D712yes—
215C2×C42C4×D714yes—
216C2×C42C22×D714yes—
217C2×C42C22×D711yescharacteristic
218C2×C42C22×D711yescharacteristic
219C4⋊C8—732no—
220C22⋊C8—732no—
221#97—78no—
222#97C22×D718yes—
223#97C22×D718yes—
224C4×C8—78no—
225C22×C7—28no—
226C22×C7—24no—
227C22×C7C2×#11714yes—
228C22×C7C2×#11712yes—
229C22×C7#56814yes—
230C22×C7C24×C411yescharacteristic
231C2×D7—24no—
232C2×D7—28no—
233C2×D7—48no—
234C4×C7—24no—
235C4×C7—24no—
236C4×C7C2×#11714yes—
237C4×C7C2×#11712yes—
238C4×C7C22×#4112yes—
239Dic7—24no—
240Dic7—28no—
241C24—72no—
242C24C2×C4×D711yescharacteristic
243#41—1416no—
244#41—1416no—
245C2×Q8—74no—
246C2×Q8—144no—
247C2×D4—78no—
248C2×D4—74no—
249C2×D4—148no—
250C2×D4—144no—
251C22×C4—44no—
252C22×C4—74no—
253C22×C4—72no—
254C22×C4—74no—
255C22×C4—74no—
256C22×C4—716no—
257C22×C4—148no—
258C22×C4—288no—
259C22×C4C2×C4×D712yes—
260C22×C4C2×C4×D712yes—
261C22×C4C2×C4×D718yes—
262C22×C4C2×C4×D711yescharacteristic
263C22×C4C2×C4×D712yes—
264C22×C4#115718yes—
265C22×C4C23×D712yes—
266C22×C4C23×D711yescharacteristic
267M16—1416no—
268C2×C8—28no—
269C2×C8—216no—
270C2×C8—78no—
271C2×C8—148no—
272C2×C8C23×D718yes—
273C4⋊C4—716no—
274C4⋊C4—78no—
275C22⋊C4—78no—
276C22⋊C4—716no—
277C42—74no—
278C42—716no—
279C42—74no—
280C42C2×C4×D712yes—
281C42C2×C4×D712yes—
282C42C2×C4×D718yes—
283C42C23×D712yes—
284C42C23×D712yes—
285C2×C7—24no—
286C2×C7#292314yescomplemented
287C2×C7C22×#11711yescharacteristic
288C2×C7C2×#56812yes—
289D7—44no—
290C23—24no—
291C23—74no—
292C23—144no—
293C23—148no—
294C23#643514yes—
295C23C22×C4×D712yes—
296C23C22×C4×D711yescharacteristic
297Q8—148no—
298D4—1416no—
299D4—148no—
300C2×C4—24no—
301C2×C4—24no—
302C2×C4—48no—
303C2×C4—78no—
304C2×C4—74no—
305C2×C4—144no—
306C2×C4—148no—
307C2×C4—148no—
308C2×C4—148no—
309C2×C4—2816no—
310C2×C4#643514yes—
311C2×C4#643518yes—
312C2×C4C22×C4×D714yes—
313C2×C4C22×C4×D714yes—
314C2×C4C22×C4×D711yescharacteristic
315C2×C4C22×C4×D712yes—
316C2×C4C2×#115714yes—
317C2×C4C2×#115718yes—
318C2×C4C24×D711yescharacteristic
319C8—28no—
320C8—148no—
321C7C2×#292311yescharacteristic, complemented, Sylow
322C22—28no—
323C22—24no—
324C22—144no—
325C22—148no—
326C22—288no—
327C22C2×#643514yes—
328C22C2×#643512yes—
329C22#9043614yes—
330C22C23×C4×D711yescharacteristic
331C4—24no—
332C4—24no—
333C4—144no—
334C4—148no—
335C4C2×#643512yes—
336C4C2×#643514yes—
337C4C22×#115712yes—
338C2—24no—
339C2—284no—
340C2#1173593414yesdirect product factor
341C2C22×#643511yescharacteristic
342C2C2×#9043612yes—
subgroupG/H
centerC22×C4C23×D7
commutatorC22×C7C24×C4
FrattiniC2×C4C24×D7
FittingC22×C7×#97C2
socleC23×C7C23×C4
prime radicalC22×#1157C22
cosocleC2×C4×C7C25
hypercenterC22×#97D7
hypocenterC7C2×#2923
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C22×C4 ◁ C22×#97

Complements

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

NH ≅ G/Ntype
2. C22×C7×#97C2semidirect
4. C2×#89874C2semidirect
5. C2×C4×#1136C2semidirect
6. #11735934C2direct (N × H)
7. C2×#89481C2semidirect
8. C2×#89477C2semidirect
23. C2×C7×#97C22semidirect
26. #89874C22semidirect
28. C4×#1136C22semidirect
31. #89481C22semidirect
32. #89477C22semidirect
83. C22×#97D7semidirect
150. C2×#97C2×D7semidirect
286. C2×C7#2923semidirect
321. C7C2×#2923semidirect
340. C2#11735934direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4×#11572
C22×C7×#972
C2×#898822
C2×#898742
C2×C4×#11362
#117359342
C2×#894812
C2×#894772
C2×#29237