Finite Group Databasebeta

← #13362964

Subgroups — #13362964

Order 1792.

1270 conjugacy classes of subgroups; only the 231 normal subgroups are shown.

normal subgroups only

This group has 1270 conjugacy classes of subgroups — too many to draw. Only its 231 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11744341C211yesmaximal, characteristic
2#11736142C212yesmaximal, complemented
3C7×#1785C211yesmaximal, characteristic, complemented
4#11729891C211yesmaximal, characteristic, complemented
5#11727878C214yesmaximal, complemented
6#11727821C212yesmaximal, complemented
7#11727746C214yesmaximal, complemented
8C22×C7×#41C2211yescharacteristic
9C2×#6548C2211yescharacteristic
10#90520—24no—
11#90515—24no—
12#90507C2212yes—
13#90495C414yes—
14#90489C2211yescharacteristic
15#90482C2212yes—
16C2×#6492C414yes—
17#90436C2214yes—
18C2×C7×#101C2212yescomplemented
19C2×C7×#100C2212yes—
20C7×#407C2212yescomplemented
21C2×#6410C2212yescomplemented
22C2×#6407C2212yes—
23#89979C2212yescomplemented
24#89904C2214yes—
25#89517—24no—
26D7×#100—24no—
27#89354C2214yescomplemented
28#89351C2214yescomplemented
29#21779—71nomaximal, Sylow
30C2×C7×#41—24no—
31C2×C7×#41—42no—
32C2×C7×#41—42no—
33C22×C7×Q8C2311yescharacteristic
34C22×C7×D4C2×C412yes—
35C22×C7×D4C2311yescharacteristic
36C23×C4×C7C2×C412yes—
37C23×C4×C7C2311yescharacteristic
38C22×C22⋊D7C2×C414yes—
39#6548—42no—
40#6548—42no—
41C2×#1159D414yes—
42C2×D4×D7D414yes—
43C2×#1157C2312yes—
44C2×C7×M16C2314yes—
45#6516C2×C414yes—
46C2×C22⋊Dic7C2×C414yes—
47#6510—22no—
48#6509—24no—
49#6507—22no—
50#6502—28no—
51#6500—24no—
52#6498—24no—
53D4×Dic7—24no—
54#6493—22no—
55#6492—28no—
56#6492C2×C414yes—
57#6492C2×C418yes—
58C4×C22⋊D7—24no—
59C2×(C2×D7)⋊C4—24no—
60#6489C2312yes—
61#6487—22no—
62C2×Dic7⋊C4—24no—
63C2×#1136C2314yes—
64#6470—44no—
65D7×M16—28no—
66#6464—44no—
67C7×#101—42no—
68C7×#100—24no—
69C7×C22⋊C8—42no—
70#6410—42no—
71#6407—24no—
72C22⋊(C7⋊C8)—42no—
73#6398—24no—
74#6397—24no—
75#3485—71no—
76#2888—72no—
77#1915—74no—
78#1891—74no—
79#1829—72no—
80#1785—71no—
81#1785D711yescharacteristic, complemented
82C24×C7C22×C411yescharacteristic
83C24×C7C22×C412yes—
84C23×D7C22⋊C414yes—
85C7×#41—416no—
86C2×C7×Q8—22no—
87C2×C7×Q8—22no—
88C2×C7×Q8—22no—
89C2×C7×D4—24no—
90C2×C7×D4—22no—
91C2×C7×D4—22no—
92C2×C7×D4—42no—
93C2×C7×D4—42no—
94C2×C7×D4C2×D414yes—
95C22×C4×C7—24no—
96C22×C4×C7—22no—
97C22×C4×C7—22no—
98C22×C4×C7—22no—
99C22×C4×C7—42no—
100C22×C4×C7—42no—
101C22×C4×C7C22×C411yescharacteristic
102C22×C4×C7C22×C412yes—
103C22×C4×C7C2411yescharacteristic
104C2×C22⋊D7—24no—
105C2×C22⋊D7—44no—
106C2×C22⋊D7—44no—
107C2×C22⋊D7C22⋊C418yes—
108C2×C22⋊D7C22×C414yes—
109C22×Dic7—24no—
110C22×Dic7C22⋊C414yes—
111#1159—48no—
112D4×D7—48no—
113#1157—24no—
114#1157—24no—
115C2×D28C2×D412yes—
116C2×C4×D7C2×D414yes—
117C2×Dic14C2×D412yes—
118C7×M16—24no—
119C7×M16—24no—
120C2×C7×C8—24no—
121C22⋊Dic7—24no—
122C22⋊Dic7—24no—
123C22⋊Dic7C22×C414yes—
124(C2×D7)⋊C4—24no—
125(C2×D7)⋊C4—28no—
126C4⋊Dic7—22no—
127Dic7⋊C4—24no—
128Dic7⋊C4—28no—
129C4×Dic7—22no—
130#1136—24no—
131#1136—24no—
132C2×C7⋊C8—24no—
133#1131—48no—
134C8×D7—48no—
135C2×H(5,𝔽2)—71no—
136C22×#41C2×D711yescharacteristic
137#568—74no—
138#547—144no—
139#546—144no—
140#536—72no—
141#534—74no—
142#525—72no—
143C2×#123—74no—
144#518—71no—
145#413—74no—
146#413—144no—
147C2×#101—72no—
148C2×#101C2×D712yescomplemented
149C2×#100—72no—
150C2×#100—144no—
151C2×#100C2×D712yes—
152#407—72no—
153#407C2×D712yescomplemented
154#344—74no—
155#344—74no—
156C23×C7—22no—
157C23×C7—22no—
158C23×C7—21no—
159C23×C7—21no—
160C23×C7—21no—
161C23×C7—81no—
162C23×C7—82no—
163C23×C7C2×C22⋊C414yes—
164C23×C7C23×C411yescharacteristic
165C22×D7—44no—
166C22×D7—84no—
167C22×D7C2×C22⋊C414yes—
168C7×Q8—44no—
169C7×D4—28no—
170C7×D4—48no—
171C2×C4×C7—21no—
172C2×C4×C7—22no—
173C2×C4×C7—21no—
174C2×C4×C7—21no—
175C2×C4×C7—22no—
176C2×C4×C7—22no—
177C2×C4×C7—48no—
178C2×C4×C7—44no—
179C2×C4×C7C22×D412yes—
180C2×C4×C7H(5,𝔽2)12yes—
181C2×C4×C7H(5,𝔽2)12yes—
182C22⋊D7—24no—
183C22⋊D7—24no—
184C22⋊D7—48no—
185C22⋊D7—84no—
186C2×Dic7—24no—
187C2×Dic7—44no—
188C2×Dic7—84no—
189C2×Dic7C2×C22⋊C414yes—
190D28—24no—
191C4×D7—28no—
192Dic14—24no—
193C7×C8—44no—
194C7⋊C8—44no—
195H(5,𝔽2)—282no—
196H(5,𝔽2)—282no—
197C2×#41—24no—
198C2×#41—42no—
199C2×#41—42no—
200C2×#41—74no—
201C2×#41—72no—
202C22×Q8C22×D711yescharacteristic
203C22×D4—74no—
204C22×D4—74no—
205C22×D4C4×D712yes—
206C22×D4C22×D711yescharacteristic
207C23×C4C4×D712yes—
208C23×C4C22×D711yescharacteristic
209#131—284no—
210#131—284no—
211C2×M16—74no—
212C2×M16—148no—
213C2×M16C22×D714yes—
214C4⋊Q8—142no—
215#124—142no—
216#123—74no—
217#123—78no—
218#123—148no—
219#123—144no—
220C22⋊Q8—142no—
221C22⋊Q8—144no—
222#121—142no—
223#121—148no—
224#120—74no—
225#120—144no—
226C4×D4—144no—
227C4×D4—144no—
228#117—72no—
229C2×C4⋊C4—144no—
230C2×C22⋊C4—74no—
231C2×C22⋊C4—144no—
232#101—42no—
233#101—144no—
234#101—282no—
235#100—24no—
236#100—144no—
237#100—144no—
238C22⋊C8—42no—
239C22⋊C8—282no—
240C22×C7—24no—
241C22×C7—41no—
242C22×C7—44no—
243C22×C7—42no—
244C22×C7—42no—
245C22×C7—42no—
246C22×C7—41no—
247C22×C7—82no—
248C22×C7—82no—
249C22×C7C22×C22⋊C411yescharacteristic
250C22×C7#51811yescharacteristic
251C22×C7#51811yescharacteristic
252C2×D7—24no—
253C2×D7—44no—
254C2×D7—164no—
255C4×C7—41no—
256C4×C7—42no—
257C4×C7#53414yes—
258Dic7—24no—
259Dic7—44no—
260C24—74no—
261C24C2×C4×D711yescharacteristic
262C24C2×C4×D712yes—
263#41—416no—
264#41—144no—
265#41—144no—
266#41—288no—
267C2×Q8—22no—
268C2×Q8—22no—
269C2×Q8—22no—
270C2×Q8—72no—
271C2×D4—24no—
272C2×D4—22no—
273C2×D4—22no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—74no—
277C2×D4—78no—
278C2×D4—72no—
279C2×D4—144no—
280C2×D4—288no—
281C2×D4—284no—
282C2×D4—284no—
283C2×D4D4×D714yes—
284C22×C4—24no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—74no—
291C22×C4—74no—
292C22×C4—144no—
293C22×C4C2×C4×D711yescharacteristic
294C22×C4C2×C4×D712yes—
295C22×C4C23×D711yescharacteristic
296M16—24no—
297M16—24no—
298M16—144no—
299M16—144no—
300M16—288no—
301C2×C8—24no—
302C2×C8—144no—
303C2×C8—288no—
304C4⋊C4—144no—
305C4⋊C4—148no—
306C4⋊C4—142no—
307C22⋊C4—74no—
308C22⋊C4—144no—
309C22⋊C4—148no—
310C22⋊C4—144no—
311C22⋊C4—144no—
312C42—142no—
313C2×C7—21no—
314C2×C7—21no—
315C2×C7—21no—
316C2×C7—44no—
317C2×C7—81no—
318C2×C7#230811yescharacteristic
319D7—44no—
320C23—22no—
321C23—22no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—74no—
326C23—81no—
327C23—82no—
328C23—284no—
329C23—564no—
330C23D7×C22⋊C414yes—
331C23C22×C4×D711yescharacteristic
332Q8—44no—
333Q8—144no—
334D4—28no—
335D4—48no—
336D4—144no—
337D4—144no—
338D4—144no—
339D4—288no—
340D4—564no—
341C2×C4—21no—
342C2×C4—22no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—48no—
348C2×C4—44no—
349C2×C4—74no—
350C2×C4—148no—
351C2×C4—144no—
352C2×C4—284no—
353C2×C4—564no—
354C2×C4C2×D4×D712yes—
355C2×C4#654812yes—
356C2×C4#654812yes—
357C8—44no—
358C8—284no—
359C7#2177911yescharacteristic, complemented, Sylow
360C22—24no—
361C22—41no—
362C22—44no—
363C22—42no—
364C22—42no—
365C22—42no—
366C22—41no—
367C22—82no—
368C22—82no—
369C22—144no—
370C22—284no—
371C22—1124no—
372C22C2×D7×C22⋊C411yescharacteristic
373C22#9017711yescharacteristic
374C22#9017711yescharacteristic
375C4—41no—
376C4—42no—
377C4—144no—
378C4—284no—
379C4#9028014yes—
380C2—21no—
381C2—21no—
382C2—21no—
383C2—44no—
384C2—81no—
385C2—284no—
386C2#1173129411yescharacteristic
subgroupG/H
centerC2#11731294
commutatorC23×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1785C2
socleC2×C7#2308
prime radical#11744341C2
cosocleC22×C4×C7C24
hypercenter#1785D7
hypocenterC7#21779
LCSG ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2 ◁ C22×C4 ◁ #1785

Complements

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

NH ≅ G/Ntype
2. #11736142C2semidirect
3. C7×#1785C2semidirect
4. #11729891C2semidirect
5. #11727878C2semidirect
6. #11727821C2semidirect
7. #11727746C2semidirect
18. C2×C7×#101C22semidirect
20. C7×#407C22semidirect
21. C2×#6410C22semidirect
23. #89979C22semidirect
27. #89354C22semidirect
28. #89351C22semidirect
81. #1785D7semidirect
148. C2×#101C2×D7semidirect
153. #407C2×D7semidirect
359. C7#21779semidirect

Maximal subgroups

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

subgroupindex
#117443412
#117361422
C7×#17852
#117298912
#117278782
#117278212
#117277462
#217797