Finite Group Databasebeta

← #13508505

Subgroups — #13508505

Order 1792.

534 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2951C211yesmaximal, characteristic
2#11735991C211yesmaximal, characteristic, complemented
3#11729832C211yesmaximal, characteristic
4#11729831C211yesmaximal, characteristic
5#11728666C212yesmaximal, complemented
6#11728665C212yesmaximal, complemented
7#11728642C212yesmaximal, complemented
8#11728640C212yesmaximal, complemented
9#11728350C211yesmaximal, characteristic, complemented
10C2×#89284C211yesmaximal, characteristic, complemented
11#11727636C211yesmaximal, characteristic, complemented
12C2×C7×#135C2211yescharacteristic
13C7×#520—21no—
14#90210—21no—
15C7×#440—22no—
16C4×C7×Q16C2212yes—
17C4×C7×D8C2212yes—
18C2×C7×#107C2211yescharacteristic
19C7×#417—21no—
20C7×#405C2211yescharacteristic
21#89893—21no—
22#89878C2211yescharacteristic
23C2×#6392C2211yescharacteristic
24C2×#6386C2211yescharacteristic
25C22×C7⋊C16C2211yescharacteristic
26#89474—22no—
27#89455C2212yes—
28#89454C2212yes—
29#89359C2211yescharacteristic
30#89359C2212yes—
31#89359C2211yescharacteristic
32#89357C2211yescharacteristic
33#89357C2212yes—
34#89357C2211yescharacteristic
35#89285C2212yes—
36#89284C2212yes—
37#89284C2212yes—
38#89284C2212yes—
39C4⋊(C7⋊C16)C2212yes—
40C4⋊(C7⋊C16)C2212yes—
41#33633—71nomaximal, Sylow
42C2×C7×#41—21no—
43C7×#135—22no—
44C7×#135Q814yescomplemented
45C2×C7×Q16C2311yescharacteristic
46C2×C7×SD16—21no—
47C2×C7×D8C2311yescharacteristic
48C2×C7×M16D411yescharacteristic
49C22×C7×C8C2311yescharacteristic
50C4×C7×Q8—22no—
51C4×C7×D4—22no—
52C4×C7×D4—42no—
53C7×#117—41no—
54C7×#117D411yescharacteristic
55C2×C7×C4⋊C4—21no—
56C2×C7×C4⋊C4—41no—
57#6489—21no—
58C2×C4⋊Dic7—21no—
59#6453—41no—
60#6451—41no—
61#6441—42no—
62#6433—22no—
63#6432—22no—
64C7×#107—21no—
65C7×#107C2312yes—
66C7×#103—22no—
67C7×#102—22no—
68C7×#97D412yes—
69C4×C7×C8C2312yes—
70#6392—21no—
71#6392—22no—
72#6392C2312yes—
73#6392C2311yescharacteristic
74#6392C2311yescharacteristic
75#6391—21no—
76#6391—22no—
77#6386—22no—
78C2×C7⋊C16—22no—
79C2×C7⋊C16C2311yescharacteristic
80C2×C7⋊C16C2311yescharacteristic
81C2×C7⋊C16C2312yes—
82#3164—71no—
83#2951D711yescharacteristic
84#2229—72no—
85#2228—72no—
86#2227—72no—
87#2224—72no—
88#2159—71no—
89C2×#366—71no—
90#2151—71no—
91#2139—71no—
92#2138—71no—
93C7×#41—24no—
94C7×#41—42no—
95C2×C7×Q8—21no—
96C2×C7×D4—21no—
97C2×C7×D4—41no—
98C22×C4×C7—41no—
99C22×C4×C7—41no—
100C22×C4×C7—42no—
101C22×C4×C7C2×D411yescharacteristic
102C22×Dic7—41no—
103C2×Dic14—42no—
104C7×Q16—21no—
105C7×Q16C2×Q812yes—
106C7×SD16—24no—
107C7×D8—21no—
108C7×D8C2×Q812yes—
109C7×M16—22no—
110C2×C7×C8C2×D412yes—
111C2×C7×C8C2×D411yescharacteristic
112C2×C7×C8C2×Q812yes—
113C2×C7×C8#4112yes—
114C2×C7×C8C2411yescharacteristic
115C7×C4⋊C4—22no—
116C7×C4⋊C4—41no—
117C7×C4⋊C4—42no—
118C7×C4⋊C4D811yescharacteristic
119C7×C4⋊C4D811yescharacteristic
120C7×C22⋊C4—42no—
121C7×C22⋊C4D812yes—
122C42×C7—42no—
123C42×C7C2×D412yes—
124C22⋊Dic7—41no—
125C4⋊Dic7—21no—
126C4⋊Dic7—22no—
127C4⋊Dic7—21no—
128C4⋊Dic7—42no—
129C4⋊Dic7—41no—
130Dic7⋊C4—42no—
131Dic7⋊C4—42no—
132C4×Dic7—41no—
133C7⋊C16—21no—
134C7⋊C16—21no—
135C7⋊C16—22no—
136C2×#135C2×D711yescharacteristic
137#544—141no—
138#520—21no—
139C2×M32—71no—
140C22×C16—71no—
141#501—142no—
142#500—72no—
143#499—72no—
144#440—22no—
145C4×Q16C2×D712yes—
146C4×D8C2×D712yes—
147#428—141no—
148#427—71no—
149C2×#107—71no—
150C2×#107C2×D711yescharacteristic
151#417—21no—
152#405C2×D711yescharacteristic
153#367—72no—
154#366—72no—
155#366—72no—
156#366—72no—
157C4⋊C16—72no—
158C4⋊C16—72no—
159#358—71no—
160#358—71no—
161#358—72no—
162#357—71no—
163#357—71no—
164#357—72no—
165C23×C7—41no—
166C23×C7C2×D811yescharacteristic
167C7×Q8—22no—
168C7×Q8—41no—
169C7×D4—22no—
170C7×D4—44no—
171C7×D4—41no—
172C2×C4×C7—41no—
173C2×C4×C7—44no—
174C2×C4×C7—42no—
175C2×C4×C7—162no—
176C2×C4×C7—161no—
177C2×C4×C7C22⋊Q812yes—
178C2×C4×C7C2×D812yes—
179C2×C4×C7C2×D812yes—
180C2×C4×C7C2×D811yescharacteristic
181C2×C4×C7#13712yes—
182C2×C4×C7C22×D411yescharacteristic
183C2×Dic7—41no—
184C2×Dic7—41no—
185C2×Dic7—42no—
186C2×Dic7—161no—
187Dic14—162no—
188C7×C8—22no—
189C7×C8C22⋊Q812yes—
190C7×C8C22×Q811yescharacteristic
191C7×C8C2×#4111yescharacteristic
192C2×#41—21no—
193#135—22no—
194#135Dic1414yescomplemented
195C2×Q16C22×D711yescharacteristic
196C2×SD16—21no—
197C2×D8C22×D711yescharacteristic
198C2×M16C22⋊D711yescharacteristic
199C22×C8C22×D711yescharacteristic
200C4⋊Q8—142no—
201C4⋊Q8—281no—
202#125—142no—
203#125—281no—
204C22⋊Q8—282no—
205C4×Q8—22no—
206C4×D4—22no—
207C4×D4—42no—
208#117—41no—
209#117—141no—
210#117C22⋊D711yescharacteristic
211C2×C4⋊C4—21no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—141no—
214M32—142no—
215C2×C16—71no—
216C2×C16—71no—
217C2×C16—72no—
218C2×C16—142no—
219#107—21no—
220#107—72no—
221#107—71no—
222#107—71no—
223#107—141no—
224#107—142no—
225#107C22×D712yes—
226#106—141no—
227#106—142no—
228#103—22no—
229#102—22no—
230#97C22⋊D712yes—
231C4×C8C22×D712yes—
232C22×C7—21no—
233C22×C7—21no—
234C22×C7—42no—
235C22×C7—81no—
236C22×C7—81no—
237C22×C7#47412yes—
238C22×C7C22×D811yescharacteristic
239C4×C7—22no—
240C4×C7—22no—
241C4×C7—42no—
242C4×C7—82no—
243C4×C7#47412yes—
244C4×C7C2×C22⋊Q811yescharacteristic
245C4×C7C2×#13711yescharacteristic
246Dic7—81no—
247Dic7—82no—
248Dic7—81no—
249#41—24no—
250#41—42no—
251C2×Q8—21no—
252C2×Q8—282no—
253C2×D4—21no—
254C2×D4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—42no—
258C22×C4—281no—
259C22×C4C2×C22⋊D711yescharacteristic
260Q16—21no—
261Q16C2×Dic1412yes—
262SD16—24no—
263D8—21no—
264D8C2×Dic1412yes—
265M16—22no—
266C2×C8C2×Dic1412yes—
267C2×C8#115712yes—
268C2×C8C2×C22⋊D712yes—
269C2×C8C2×C22⋊D711yescharacteristic
270C2×C8C23×D711yescharacteristic
271C4⋊C4—22no—
272C4⋊C4—41no—
273C4⋊C4—42no—
274C4⋊C4—141no—
275C4⋊C4—142no—
276C4⋊C4—141no—
277C4⋊C4—282no—
278C4⋊C4—282no—
279C4⋊C4—281no—
280C4⋊C4—282no—
281C4⋊C4#114111yescharacteristic
282C4⋊C4#114111yescharacteristic
283C22⋊C4—42no—
284C22⋊C4—281no—
285C22⋊C4#114112yes—
286C42—42no—
287C42—281no—
288C42C2×C22⋊D712yes—
289C16—141no—
290C16—141no—
291C16—142no—
292C2×C7—22no—
293C2×C7—82no—
294C2×C7C2×#47411yescharacteristic
295C2×C7#341511yescharacteristic
296C2×C7#341711yescharacteristic
297C23—41no—
298C23C2×#114111yescharacteristic
299Q8—22no—
300Q8—41no—
301Q8—1122no—
302D4—22no—
303D4—44no—
304D4—41no—
305C2×C4—41no—
306C2×C4—44no—
307C2×C4—42no—
308C2×C4—162no—
309C2×C4—161no—
310C2×C4—281no—
311C2×C4—281no—
312C2×C4—282no—
313C2×C4—1121no—
314C2×C4#648712yes—
315C2×C4C2×#114112yes—
316C2×C4C2×#114112yes—
317C2×C4C2×#114111yescharacteristic
318C2×C4#651412yes—
319C2×C4C22×C22⋊D711yescharacteristic
320C8—22no—
321C8#648712yes—
322C8C22×Dic1411yescharacteristic
323C8C2×#115711yescharacteristic
324C7#3363311yescharacteristic, complemented, Sylow
325C22—21no—
326C22—21no—
327C22—42no—
328C22—81no—
329C22—81no—
330C22#8977912yes—
331C22C22×#114111yescharacteristic
332C4—22no—
333C4—22no—
334C4—42no—
335C4—82no—
336C4—562no—
337C4—561no—
338C4—561no—
339C4#8977912yes—
340C4C2×#648711yescharacteristic
341C4C2×#651411yescharacteristic
342C2—22no—
343C2—82no—
344C2C2×#8977911yescharacteristic
345C2#1174145211yescharacteristic
346C2#1174146211yescharacteristic
subgroupG/H
centerC22C22×#1141
commutator, cosocleC2×C7×C8C24
FrattiniC2×C8C23×D7
FittingC7×#2951C2
socleC22×C7C22×D8
prime radicalC2×C7×#135C22
hypercenter#2951D7
hypocenterC7#33633
LCSG ▷ C2×C7×C8 ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ #117 ◁ #405 ◁ #2951

Complements

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

NH ≅ G/Ntype
2. #11735991C2semidirect
5. #11728666C2semidirect
6. #11728665C2semidirect
7. #11728642C2semidirect
8. #11728640C2semidirect
9. #11728350C2semidirect
10. C2×#89284C2semidirect
11. #11727636C2semidirect
44. C7×#135Q8semidirect
194. #135Dic14semidirect
324. C7#33633semidirect

Maximal subgroups

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

subgroupindex
C7×#29512
#117359912
#117298322
#117298312
#117286662
#117286652
#117286422
#117286402
#117283502
C2×#892842
#117276362
#336337