Finite Group Databasebeta

← C2×#11729118 #13341543

Subgroups — C2×#11729118 #13341543

Order 1792.

734 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×C7×#500C211yesmaximal, characteristic
2C2×#89851C211yesmaximal, characteristic
3C2×#89865C211yesmaximal, characteristic
4C2×#89853C211yesmaximal, characteristic
5C2×#89853C211yesmaximal, characteristic
6C2×#89453C211yesmaximal, characteristic
7#11729118C218yesmaximal, direct product factor
8C2×#89249C211yesmaximal, characteristic
9C2×C7×C4⋊Q8C2211yescharacteristic
10C2×C7×C4⋊Q8C2211yescharacteristic
11C2×#6432C2211yescharacteristic
12C2×#6432C2211yescharacteristic
13C7×#500—22no—
14C7×#500C2214yes—
15C2×C7×#107—22no—
16C2×C4×C7×C8C2211yescharacteristic
17C2×#6409—22no—
18C2×#6409—22no—
19C2×#6390—22no—
20#89865—22no—
21#89865C2214yes—
22#89853—22no—
23#89853—22no—
24#89853C2214yes—
25#89853C2214yes—
26#89851—22no—
27#89851C2214yes—
28C2×#6384—22no—
29C2×#6382—22no—
30C2×C4×C7⋊C8C2211yescharacteristic
31C2×C4×C7⋊C8C2211yescharacteristic
32#89453—22no—
33#89453C2214yes—
34#89249C2214yes—
35C2×#1718—71nomaximal, Sylow
36C22×C7×Q8—21no—
37C22×C7×Q8—21no—
38C22×Dic14—21no—
39C22×Dic14—21no—
40C22×C7×C8D412yes—
41C7×C4⋊Q8—21no—
42C7×C4⋊Q8—21no—
43C7×C4⋊Q8—21no—
44C7×C4⋊Q8—21no—
45C7×C4⋊Q8—21no—
46C7×C4⋊Q8—21no—
47C7×C4⋊Q8C2312yes—
48C7×C4⋊Q8C2312yes—
49C2×C7×C4⋊C4—21no—
50C2×C7×C4⋊C4—21no—
51C2×C7×C4⋊C4—21no—
52C2×C7×C4⋊C4—21no—
53C2×C42×C7C2311yescharacteristic
54C2×C4⋊Dic7—21no—
55C2×C4⋊Dic7—21no—
56C2×C4⋊Dic7—21no—
57C2×C4⋊Dic7—21no—
58C22×C7⋊C8D412yes—
59C22×C7⋊C8D412yes—
60#6432—21no—
61#6432—21no—
62#6432—21no—
63#6432—21no—
64#6432—21no—
65#6432—21no—
66#6432C2312yes—
67#6432C2312yes—
68C7×#107—28no—
69C4×C7×C8D412yes—
70C4×C7×C8C2312yes—
71#6409—28no—
72#6409—28no—
73#6390—28no—
74#6384—28no—
75#6382—28no—
76C4×C7⋊C8D412yes—
77C4×C7⋊C8D412yes—
78C4×C7⋊C8C2312yes—
79C4×C7⋊C8C2312yes—
80C2×#500D711yescharacteristic
81C2×#498—71no—
82C2×#487—71no—
83C2×#487—71no—
84C2×#487—71no—
85C2×#487—71no—
86#1718—78no—
87C2×#322—71no—
88C2×C7×Q8—22no—
89C2×C7×Q8—22no—
90C2×C7×Q8—41no—
91C2×C7×Q8—41no—
92C2×C7×Q8—44no—
93C2×C7×Q8—44no—
94C22×C4×C7—41no—
95C22×C4×C7—41no—
96C22×C4×C7C2×D411yescharacteristic
97C22×C4×C7C2×D411yescharacteristic
98C22×C4×C7C2×D411yescharacteristic
99C22×Dic7—41no—
100C22×Dic7—41no—
101C2×Dic14—22no—
102C2×Dic14—22no—
103C2×Dic14—44no—
104C2×Dic14—44no—
105C2×Dic14—41no—
106C2×Dic14—41no—
107C2×C7×C8—24no—
108C2×C7×C8C2×D414yes—
109C7×C4⋊C4—22no—
110C7×C4⋊C4—22no—
111C7×C4⋊C4—22no—
112C7×C4⋊C4—22no—
113C7×C4⋊C4—41no—
114C7×C4⋊C4—41no—
115C7×C4⋊C4—41no—
116C7×C4⋊C4—41no—
117C42×C7C2×D411yescharacteristic
118C42×C7C2×D411yescharacteristic
119C42×C7C2×D411yescharacteristic
120C42×C7C2411yescharacteristic
121C4⋊Dic7—22no—
122C4⋊Dic7—22no—
123C4⋊Dic7—22no—
124C4⋊Dic7—22no—
125C4⋊Dic7—41no—
126C4⋊Dic7—41no—
127C4⋊Dic7—41no—
128C4⋊Dic7—41no—
129C2×C7⋊C8—24no—
130C2×C7⋊C8—24no—
131C2×C7⋊C8C2×D414yes—
132C2×C7⋊C8C2×D414yes—
133C2×C4⋊Q8—71no—
134C2×C4⋊Q8—71no—
135C2×C4⋊Q8C2×D711yescharacteristic
136C2×C4⋊Q8C2×D711yescharacteristic
137#500—22no—
138#500C2×D714yes—
139#498—74no—
140#498—142no—
141#487—74no—
142#487—74no—
143#487—74no—
144#487—74no—
145#487—142no—
146#487—142no—
147#487—142no—
148#487—142no—
149C2×#107—22no—
150C2×#106—142no—
151C2×#103—142no—
152C2×#103—142no—
153C2×#103—142no—
154C2×#103—142no—
155C2×C4×C8—71no—
156C2×C4×C8—71no—
157C2×C4×C8C2×D711yescharacteristic
158#322—74no—
159C23×C7#12711yescharacteristic
160C7×Q8—44no—
161C7×Q8—44no—
162C7×Q8—82no—
163C7×Q8—82no—
164C2×C4×C7—42no—
165C2×C4×C7—42no—
166C2×C4×C7—82no—
167C2×C4×C7—82no—
168C2×C4×C7#12711yescharacteristic
169C2×C4×C7#12711yescharacteristic
170C2×C4×C7#12711yescharacteristic
171C2×C4×C7#13514yes—
172C2×C4×C7#13714yes—
173C2×C4×C7#13714yes—
174C2×C4×C7C22×D411yescharacteristic
175C2×C4×C7C22×D411yescharacteristic
176C2×C4×C7C22×D411yescharacteristic
177C2×Dic7—42no—
178C2×Dic7—42no—
179C2×Dic7—82no—
180C2×Dic7—82no—
181Dic14—44no—
182Dic14—44no—
183Dic14—82no—
184Dic14—82no—
185C7×C8—24no—
186C7⋊C8—24no—
187C7⋊C8—24no—
188C22×Q8—21no—
189C22×Q8—21no—
190C22×Q8—141no—
191C22×Q8—141no—
192C22×C8—72no—
193C22×C8—72no—
194C22×C8C22⋊D712yes—
195C4⋊Q8—21no—
196C4⋊Q8—21no—
197C4⋊Q8—21no—
198C4⋊Q8—21no—
199C4⋊Q8—21no—
200C4⋊Q8—21no—
201C4⋊Q8—72no—
202C4⋊Q8—72no—
203C4⋊Q8—141no—
204C4⋊Q8—141no—
205C4⋊Q8—141no—
206C4⋊Q8—141no—
207C4⋊Q8—141no—
208C4⋊Q8—141no—
209C4⋊Q8C22×D712yes—
210C4⋊Q8C22×D712yes—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—141no—
216C2×C4⋊C4—141no—
217C2×C4⋊C4—141no—
218C2×C4⋊C4—141no—
219C2×C42C22×D711yescharacteristic
220#107—28no—
221#106—148no—
222#103—148no—
223#103—148no—
224#103—148no—
225#103—148no—
226C4×C8—72no—
227C4×C8—72no—
228C4×C8—72no—
229C4×C8—72no—
230C4×C8C22⋊D712yes—
231C4×C8C22×D712yes—
232C22×C7#49512yes—
233C22×C7#49712yes—
234C22×C7#49712yes—
235C22×C7C2×#12711yescharacteristic
236C4×C7—82no—
237C4×C7—82no—
238C4×C7#49512yes—
239C4×C7#49712yes—
240C4×C7#49712yes—
241C4×C7C2×#13512yes—
242C4×C7C2×#13712yes—
243C4×C7C2×#13712yes—
244Dic7—82no—
245Dic7—82no—
246C2×Q8—22no—
247C2×Q8—22no—
248C2×Q8—41no—
249C2×Q8—41no—
250C2×Q8—44no—
251C2×Q8—44no—
252C2×Q8—142no—
253C2×Q8—142no—
254C2×Q8—281no—
255C2×Q8—281no—
256C2×Q8—284no—
257C2×Q8—284no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—281no—
261C22×C4—281no—
262C22×C4D4×D711yescharacteristic
263C22×C4D4×D711yescharacteristic
264C22×C4C2×C22⋊D711yescharacteristic
265C2×C8—24no—
266C2×C8—74no—
267C2×C8—74no—
268C2×C8—144no—
269C2×C8—144no—
270C2×C8C2×C22⋊D714yes—
271C4⋊C4—22no—
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—22no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—142no—
280C4⋊C4—142no—
281C4⋊C4—142no—
282C4⋊C4—142no—
283C4⋊C4—281no—
284C4⋊C4—281no—
285C4⋊C4—281no—
286C4⋊C4—281no—
287C42D4×D711yescharacteristic
288C42D4×D711yescharacteristic
289C42C2×C22⋊D711yescharacteristic
290C42C23×D711yescharacteristic
291C2×C7#171814yescomplemented
292C2×C7C2×#49511yescharacteristic
293C2×C7C2×#49711yescharacteristic
294C2×C7C2×#49711yescharacteristic
295C23#650311yescharacteristic
296Q8—44no—
297Q8—44no—
298Q8—82no—
299Q8—82no—
300Q8—284no—
301Q8—284no—
302Q8—562no—
303Q8—562no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—82no—
307C2×C4—82no—
308C2×C4—282no—
309C2×C4—282no—
310C2×C4—562no—
311C2×C4—562no—
312C2×C4#647514yes—
313C2×C4#648114yes—
314C2×C4#650311yescharacteristic
315C2×C4#650311yescharacteristic
316C2×C4#650311yescharacteristic
317C2×C4#650514yes—
318C2×C4C2×D4×D711yescharacteristic
319C2×C4C2×D4×D711yescharacteristic
320C2×C4C22×C22⋊D711yescharacteristic
321C8—24no—
322C8—144no—
323C8—144no—
324C7C2×#171811yescharacteristic, complemented, Sylow
325C22#8985812yes—
326C22#8992712yes—
327C22#8996212yes—
328C22C2×#650311yescharacteristic
329C4—82no—
330C4—82no—
331C4—562no—
332C4—562no—
333C4#8985812yes—
334C4#8992712yes—
335C4#8996212yes—
336C4C2×#647512yes—
337C4C2×#648112yes—
338C4C2×#650512yes—
339C2#1172911814yesdirect product factor
340C2C2×#8985811yescharacteristic
341C2C2×#8992711yescharacteristic
342C2C2×#8996211yescharacteristic
subgroupG/H
centerC23#6503
commutator, cosocleC42×C7C24
FrattiniC42C23×D7
FittingC2×C7×#500C2
socle, prime radicalC23×C7#127
hypercenterC2×#500D7
hypocenterC7C2×#1718
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#500

Complements

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

NH ≅ G/Ntype
7. #11729118C2direct (N × H)
291. C2×C7#1718semidirect
324. C7C2×#1718semidirect
339. C2#11729118direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#5002
C2×#898512
C2×#898652
C2×#898532
C2×#894532
#117291182
C2×#892492
C2×#17187