Finite Group Databasebeta

← C2×#11729530 #13322315

Subgroups — C2×#11729530 #13322315

Order 1792.

1182 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C7×C4⋊Q8C211yesmaximal, characteristic
2C2×#89698C211yesmaximal, characteristic
3C2×#89698C211yesmaximal, characteristic
4#11729530C218yesmaximal, direct product factor
5C2×#89342C214yesmaximal, complemented
6C23×C7×Q8—21no—
7C2×C7×C4⋊Q8—21no—
8C2×C7×C4⋊Q8—21no—
9C2×C7×C4⋊Q8—22no—
10C2×C7×C4⋊Q8—22no—
11C2×C7×C4⋊Q8C412yes—
12C2×C7×C4⋊Q8C414yes—
13C2×C7×C4⋊Q8C2212yes—
14C2×C7×C4⋊Q8C2214yes—
15C22×C7×C4⋊C4—21no—
16C22×C7×C4⋊C4C412yes—
17C22×C42×C7C2211yescharacteristic
18C2×C22⋊(C7⋊C8)—21no—
19C2×C22⋊(C7⋊C8)—21no—
20#89698—22no—
21#89698—22no—
22#89698C2214yes—
23#89698C2214yes—
24C2×C4⋊(C7⋊C8)C2212yes—
25C2×C4⋊(C7⋊C8)C2212yes—
26C2×#6377—21no—
27C2×#6377—21no—
28#89342C22116yescomplemented
29C2×#1556—71nomaximal, Sylow
30C22×C7×Q8—24no—
31C22×C7×Q8—22no—
32C22×C7×Q8—24no—
33C22×C7×Q8—22no—
34C22×C7×Q8—48no—
35C23×C4×C7—22no—
36C23×C4×C7C2×C411yescharacteristic
37C23×C4×C7D411yescharacteristic
38C23×C4×C7D411yescharacteristic
39C7×C4⋊Q8—24no—
40C7×C4⋊Q8—24no—
41C7×C4⋊Q8—24no—
42C7×C4⋊Q8—24no—
43C7×C4⋊Q8—24no—
44C7×C4⋊Q8—22no—
45C7×C4⋊Q8—22no—
46C7×C4⋊Q8C2×C414yes—
47C7×C4⋊Q8C2×C418yes—
48C7×C4⋊Q8C2314yes—
49C2×C7×C4⋊C4—24no—
50C2×C7×C4⋊C4—22no—
51C2×C7×C4⋊C4—24no—
52C2×C7×C4⋊C4—22no—
53C2×C7×C4⋊C4—24no—
54C2×C7×C4⋊C4—22no—
55C2×C7×C4⋊C4C2×C414yes—
56C2×C7×C4⋊C4C2×C418yes—
57C2×C42×C7C2×C412yes—
58C2×C42×C7C2×C411yescharacteristic
59C2×C42×C7D411yescharacteristic
60C2×C42×C7D411yescharacteristic
61C2×C42×C7D412yes—
62C2×C42×C7D412yes—
63C2×C42×C7C2311yescharacteristic
64C2×C42×C7C2312yes—
65C22×C7⋊C8—22no—
66C22×C7⋊C8—22no—
67C22⋊(C7⋊C8)—24no—
68C22⋊(C7⋊C8)—24no—
69C4⋊(C7⋊C8)—22no—
70C4⋊(C7⋊C8)—22no—
71C4⋊(C7⋊C8)C2314yes—
72C4⋊(C7⋊C8)C2314yes—
73#6377—24no—
74#6377—24no—
75C22×C4⋊Q8D711yescharacteristic
76C2×#432—71no—
77C2×#432—71no—
78#1556—78no—
79C2×#332—74no—
80C24×C7C22⋊C411yescharacteristic
81C2×C7×Q8—28no—
82C2×C7×Q8—24no—
83C2×C7×Q8—24no—
84C2×C7×Q8—48no—
85C2×C7×Q8—416no—
86C2×C7×Q8—416no—
87C2×C7×Q8—48no—
88C22×C4×C7—24no—
89C22×C4×C7—24no—
90C22×C4×C7—24no—
91C22×C4×C7—28no—
92C22×C4×C7—24no—
93C22×C4×C7—48no—
94C22×C4×C7C22⋊C411yescharacteristic
95C22×C4×C7C22⋊C412yes—
96C22×C4×C7C22⋊C411yescharacteristic
97C22×C4×C7C22⋊C412yes—
98C22×C4×C7C22⋊C412yes—
99C22×C4×C7C22⋊C411yescharacteristic
100C22×C4×C7C22×C412yes—
101C22×C4×C7C22×C411yescharacteristic
102C22×C4×C7C2×D412yes—
103C22×C4×C7C2×D412yes—
104C22×C4×C7C2×D411yescharacteristic
105C22×C4×C7C2×D411yescharacteristic
106C7×C4⋊C4—28no—
107C7×C4⋊C4—24no—
108C7×C4⋊C4—28no—
109C7×C4⋊C4—24no—
110C7×C4⋊C4—24no—
111C7×C4⋊C4C22×C418yes—
112C42×C7C22⋊C412yes—
113C42×C7C22⋊C412yes—
114C42×C7C22⋊C412yes—
115C42×C7C22×C412yes—
116C42×C7C22×C411yescharacteristic
117C42×C7C2×D412yes—
118C42×C7C2×D412yes—
119C42×C7C2×D411yescharacteristic
120C42×C7C2×D411yescharacteristic
121C42×C7C2411yescharacteristic
122C2×C7⋊C8—24no—
123C2×C7⋊C8—24no—
124C2×C7⋊C8—44no—
125C2×C7⋊C8—44no—
126C23×Q8—21no—
127C2×C4⋊Q8—21no—
128C2×C4⋊Q8—21no—
129C2×C4⋊Q8—22no—
130C2×C4⋊Q8—22no—
131C2×C4⋊Q8Dic714yes—
132C2×C4⋊Q8Dic712yes—
133C2×C4⋊Q8C2×D712yes—
134C2×C4⋊Q8C2×D714yes—
135C22×C4⋊C4—21no—
136C22×C4⋊C4Dic712yes—
137C22×C42C2×D711yescharacteristic
138#432—74no—
139#432—74no—
140#432—142no—
141#432—142no—
142C2×C4⋊C8—72no—
143C2×C4⋊C8—72no—
144C2×C22⋊C8—141no—
145C2×C22⋊C8—141no—
146C2×#97—141no—
147C2×#97—141no—
148#332—716no—
149C23×C7—22no—
150C23×C7—22no—
151C23×C7#10114yes—
152C23×C7C2×C22⋊C412yes—
153C23×C7C2×C22⋊C411yescharacteristic
154C7×Q8—416no—
155C7×Q8—48no—
156C7×Q8—48no—
157C2×C4×C7—24no—
158C2×C4×C7—22no—
159C2×C4×C7—24no—
160C2×C4×C7—22no—
161C2×C4×C7—22no—
162C2×C4×C7—22no—
163C2×C4×C7—24no—
164C2×C4×C7—24no—
165C2×C4×C7—24no—
166C2×C4×C7—24no—
167C2×C4×C7—28no—
168C2×C4×C7—416no—
169C2×C4×C7—48no—
170C2×C4×C7C2×C22⋊C412yes—
171C2×C4×C7C2×C22⋊C412yes—
172C2×C4×C7C2×C22⋊C412yes—
173C2×C4×C7C2×C22⋊C411yescharacteristic
174C2×C4×C7C2×C22⋊C411yescharacteristic
175C2×C4×C7C2×C22⋊C411yescharacteristic
176C2×C4×C7AGL(1,ℤ/8)14yes—
177C2×C4×C7#13714yes—
178C2×C4×C7C23×C411yescharacteristic
179C2×C4×C7C22×D411yescharacteristic
180C2×C4×C7C22×D411yescharacteristic
181C7⋊C8—44no—
182C7⋊C8—44no—
183C22×Q8—24no—
184C22×Q8—22no—
185C22×Q8—24no—
186C22×Q8—22no—
187C22×Q8—48no—
188C23×C4—22no—
189C23×C4C2×Dic711yescharacteristic
190C23×C4C22⋊D711yescharacteristic
191C23×C4C22⋊D711yescharacteristic
192C22×C8—142no—
193C22×C8—142no—
194C4⋊Q8—24no—
195C4⋊Q8—24no—
196C4⋊Q8—24no—
197C4⋊Q8—24no—
198C4⋊Q8—24no—
199C4⋊Q8—22no—
200C4⋊Q8—22no—
201C4⋊Q8C2×Dic714yes—
202C4⋊Q8C2×Dic718yes—
203C4⋊Q8C22×D714yes—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4C2×Dic718yes—
211C2×C4⋊C4C2×Dic714yes—
212C2×C42C2×Dic712yes—
213C2×C42C2×Dic711yescharacteristic
214C2×C42C22⋊D711yescharacteristic
215C2×C42C22⋊D711yescharacteristic
216C2×C42C22⋊D712yes—
217C2×C42C22⋊D712yes—
218C2×C42C22×D711yescharacteristic
219C2×C42C22×D712yes—
220C4⋊C8—74no—
221C4⋊C8—74no—
222C4⋊C8—142no—
223C4⋊C8—142no—
224C22⋊C8—144no—
225C22⋊C8—144no—
226#97—144no—
227#97—144no—
228C22×C7—28no—
229C22×C7—22no—
230C22×C7—22no—
231C22×C7C2×#10114yes—
232C22×C7C2×#10112yes—
233C22×C7#41812yes—
234C22×C7#41912yes—
235C22×C7C22×C22⋊C411yescharacteristic
236C4×C7—21no—
237C4×C7—22no—
238C4×C7—21no—
239C4×C7—22no—
240C4×C7—22no—
241C4×C7—48no—
242C4×C7#41812yes—
243C4×C7#41912yes—
244C4×C7C2×AGL(1,ℤ/8)12yes—
245C4×C7C2×#13712yes—
246C24C22⋊Dic711yescharacteristic
247C2×Q8—28no—
248C2×Q8—24no—
249C2×Q8—24no—
250C2×Q8—48no—
251C2×Q8—416no—
252C2×Q8—416no—
253C2×Q8—48no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—28no—
258C22×C4—24no—
259C22×C4—48no—
260C22×C4C22⋊Dic712yes—
261C22×C4C22⋊Dic712yes—
262C22×C4C22⋊Dic712yes—
263C22×C4C22⋊Dic711yescharacteristic
264C22×C4C22⋊Dic711yescharacteristic
265C22×C4C22⋊Dic711yescharacteristic
266C22×C4C22×Dic712yes—
267C22×C4C22×Dic711yescharacteristic
268C22×C4C2×C22⋊D712yes—
269C22×C4C2×C22⋊D712yes—
270C22×C4C2×C22⋊D711yescharacteristic
271C22×C4C2×C22⋊D711yescharacteristic
272C2×C8—144no—
273C2×C8—144no—
274C2×C8—284no—
275C2×C8—284no—
276C4⋊C4—28no—
277C4⋊C4—24no—
278C4⋊C4—28no—
279C4⋊C4—24no—
280C4⋊C4—24no—
281C4⋊C4C22×Dic718yes—
282C42C22⋊Dic712yes—
283C42C22⋊Dic712yes—
284C42C22⋊Dic712yes—
285C42C22×Dic712yes—
286C42C22×Dic711yescharacteristic
287C42C2×C22⋊D712yes—
288C42C2×C22⋊D712yes—
289C42C2×C22⋊D711yescharacteristic
290C42C2×C22⋊D711yescharacteristic
291C42C23×D711yescharacteristic
292C2×C7—24no—
293C2×C7#155614yescomplemented
294C2×C7C22×#10111yescharacteristic
295C2×C7C2×#41811yescharacteristic
296C2×C7C2×#41911yescharacteristic
297C23—22no—
298C23—22no—
299C23#641014yes—
300C23C2×C22⋊Dic712yes—
301C23C2×C22⋊Dic711yescharacteristic
302Q8—416no—
303Q8—48no—
304Q8—48no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—24no—
308C2×C4—22no—
309C2×C4—24no—
310C2×C4—22no—
311C2×C4—24no—
312C2×C4—24no—
313C2×C4—24no—
314C2×C4—24no—
315C2×C4—28no—
316C2×C4—416no—
317C2×C4—48no—
318C2×C4#649514yes—
319C2×C4#650514yes—
320C2×C4C2×C22⋊Dic712yes—
321C2×C4C2×C22⋊Dic712yes—
322C2×C4C2×C22⋊Dic712yes—
323C2×C4C2×C22⋊Dic711yescharacteristic
324C2×C4C2×C22⋊Dic711yescharacteristic
325C2×C4C2×C22⋊Dic711yescharacteristic
326C2×C4C23×Dic711yescharacteristic
327C2×C4C22×C22⋊D711yescharacteristic
328C2×C4C22×C22⋊D711yescharacteristic
329C8—284no—
330C8—284no—
331C7C2×#155611yescharacteristic, complemented, Sylow
332C22—22no—
333C22—22no—
334C22—28no—
335C22#8998712yes—
336C22#8999712yes—
337C22C2×#641014yes—
338C22C2×#641012yes—
339C22C22×C22⋊Dic711yescharacteristic
340C4—22no—
341C4—22no—
342C4—21no—
343C4—22no—
344C4—21no—
345C4—48no—
346C4#8998712yes—
347C4#8999712yes—
348C4C2×#649512yes—
349C4C2×#650512yes—
350C2—24no—
351C2#1172953014yesdirect product factor
352C2C2×#8998711yescharacteristic
353C2C2×#8999711yescharacteristic
354C2C22×#641011yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic7
commutatorC2×C4×C7C23×C4
FrattiniC42C23×D7
FittingC22×C7×C4⋊Q8C2
socleC23×C7C2×C22⋊C4
prime radicalC24×C7C22⋊C4
cosocleC42×C7C24
hypercenterC22×C4⋊Q8D7
hypocenterC7C2×#1556
LCSG ▷ C2×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4⋊Q8

Complements

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

NH ≅ G/Ntype
4. #11729530C2direct (N × H)
5. C2×#89342C2semidirect
28. #89342C22semidirect
293. C2×C7#1556semidirect
331. C7C2×#1556semidirect
351. C2#11729530direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C7×C4⋊Q82
C2×#896982
#117295302
C2×#893422
C2×#15567