Finite Group Databasebeta

← #13249249

Subgroups — #13249249

Order 1792.

536 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2975C211yesmaximal, characteristic, complemented
2C2×#89786C211yesmaximal, characteristic
3#11729351C211yesmaximal, characteristic, complemented
4#11728537C211yesmaximal, characteristic, complemented
5#11728511C211yesmaximal, characteristic, complemented
6#11725547C211yesmaximal, characteristic, complemented
7#11725546C211yesmaximal, characteristic, complemented
8C2×C4×C7×D4C2211yescharacteristic
9C22×#1141C411yescharacteristic
10C2×C4×D28C2211yescharacteristic
11C7×#436—22no—
12C7×#436—22no—
13C7×#431C2211yescharacteristic
14C7×#431C2211yescharacteristic
15C7×#407—22no—
16C2×#6406C411yescharacteristic
17#89786—22no—
18#89786—22no—
19C2×#6383C411yescharacteristic
20C2×#6382C411yescharacteristic
21#89697C2211yescharacteristic
22#89697C2211yescharacteristic
23C2×#6377C2211yescharacteristic
24D4⋊(C7⋊C8)—22no—
25#89263—22no—
26#89254—22no—
27#9093—71nomaximal, Sylow
28C22×C7×D4C2×C411yescharacteristic
29C23×C4×C7—21no—
30C22×D28C2×C411yescharacteristic
31C22×C4×D7—21no—
32C2×C7×M16—24no—
33C4×C7×D4—21no—
34C4×C7×D4—21no—
35C4×C7×D4D412yes—
36C4×C7×D4D412yes—
37C2×C7×C4⋊C4C2×C411yescharacteristic
38C2×C7×C22⋊C4—21no—
39C2×C42×C7C2311yescharacteristic
40C2×#1141—22no—
41C2×#1141—22no—
42C2×#1141—22no—
43C2×(C2×D7)⋊C4—21no—
44C2×C4⋊Dic7C2×C411yescharacteristic
45C22×C7⋊C8C2×C411yescharacteristic
46C22×C7⋊C8C2×C411yescharacteristic
47C4×D28—21no—
48C4×D28—21no—
49C4×D28—21no—
50C4×D28—21no—
51C7×C4⋊C8—21no—
52C7×C4⋊C8—21no—
53C7×C22⋊C8—24no—
54C7×C22⋊C8—21no—
55C7×C22⋊C8D412yes—
56C4×C7×C8—21no—
57C4×C7×C8D412yes—
58#6406—22no—
59C22⋊(C7⋊C8)—21no—
60C22⋊(C7⋊C8)—21no—
61#6383—22no—
62#6382—22no—
63C4⋊(C7⋊C8)—21no—
64C4⋊(C7⋊C8)—21no—
65#6377—21no—
66#6377—21no—
67C4×C7⋊C8—21no—
68C4×C7⋊C8—21no—
69#2975D711yescharacteristic, complemented
70C2×#442—71no—
71#1478—71no—
72#1478—71no—
73#1460—71no—
74#1281—71no—
75#1275—71no—
76C24×C7—21no—
77C23×D7—21no—
78C2×C7×D4—21no—
79C2×C7×D4—21no—
80C2×C7×D4—21no—
81C2×C7×D4—21no—
82C2×C7×D4—21no—
83C2×C7×D4C22⋊C412yes—
84C22×C4×C7—21no—
85C22×C4×C7—22no—
86C22×C4×C7—22no—
87C22×C4×C7—82no—
88C22×C4×C7C22×C411yescharacteristic
89C22×C4×C7C2×D411yescharacteristic
90C22×C4×C7#4111yescharacteristic
91C22×Dic7—21no—
92C2×D28—21no—
93C2×D28—21no—
94C2×D28—21no—
95C2×D28—21no—
96C2×D28—21no—
97C2×D28M1612yescomplemented
98C2×C4×D7—41no—
99C2×C4×D7—41no—
100C2×C4×D7—82no—
101C7×M16—84no—
102C2×C7×C8—22no—
103C2×C7×C8—22no—
104C7×C4⋊C4—21no—
105C7×C4⋊C4C22⋊C412yes—
106C7×C22⋊C4—22no—
107C7×C22⋊C4—22no—
108C42×C7C2×D411yescharacteristic
109C42×C7C2×D411yescharacteristic
110C42×C7C2×D411yescharacteristic
111C42×C7#4111yescharacteristic
112#1141—28no—
113(C2×D7)⋊C4—41no—
114(C2×D7)⋊C4—41no—
115C4⋊Dic7—21no—
116C4⋊Dic7—21no—
117C2×C7⋊C8—21no—
118C2×C7⋊C8—21no—
119C2×C7⋊C8—21no—
120C2×C7⋊C8—21no—
121C2×C7⋊C8—21no—
122C2×C7⋊C8—41no—
123C2×C7⋊C8—41no—
124C2×C7⋊C8M1612yescomplemented
125C22×D8—71no—
126C2×C4×D4—71no—
127C2×C4×D4C2×D711yescharacteristic
128#442—142no—
129#442—142no—
130#436—22no—
131#436—22no—
132#431—71no—
133#431—71no—
134#431C2×D711yescharacteristic
135#431C2×D711yescharacteristic
136C2×#106—71no—
137C2×#102—71no—
138C2×#102—71no—
139#407—22no—
140C2×#97—71no—
141D4⋊C8—142no—
142D4⋊C8—142no—
143#323—142no—
144C23×C7—22no—
145C23×C7—41no—
146C23×C7—41no—
147C23×C7—81no—
148C23×C7—81no—
149C23×C7C4×D411yescharacteristic
150C22×D7—22no—
151C22×D7—41no—
152C22×D7—41no—
153C22×D7—81no—
154C22×D7—81no—
155C7×D4—22no—
156C7×D4—22no—
157C7×D4—41no—
158C7×D4—41no—
159C2×C4×C7—22no—
160C2×C4×C7—41no—
161C2×C4×C7—41no—
162C2×C4×C7—44no—
163C2×C4×C7—82no—
164C2×C4×C7—81no—
165C2×C4×C7C2×C22⋊C411yescharacteristic
166C2×C4×C7C4×D411yescharacteristic
167C2×C4×C7#12011yescharacteristic
168C2×C4×C7#12111yescharacteristic
169C2×C4×C7#12111yescharacteristic
170C2×C4×C7#12311yescharacteristic
171C2×C4×C7C2×M1611yescharacteristic
172C2×C4×C7C2×M1611yescharacteristic
173C2×C4×C7#13111yescharacteristic
174C2×C4×C7#13111yescharacteristic
175C2×Dic7—41no—
176C2×Dic7—81no—
177D28—22no—
178D28—22no—
179D28—41no—
180D28—41no—
181C4×D7—82no—
182C4×D7—82no—
183C7×C8—42no—
184C7×C8—42no—
185C7⋊C8—22no—
186C7⋊C8—41no—
187C7⋊C8—81no—
188C7⋊C8—81no—
189C22×D4—71no—
190C22×D4C4×D711yescharacteristic
191C23×C4—21no—
192C23×C4—141no—
193C2×D8—142no—
194C2×D8—142no—
195C2×D8—142no—
196C2×M16—24no—
197C22×C8—71no—
198C22×C8—71no—
199C4×D4—21no—
200C4×D4—21no—
201C4×D4—141no—
202C4×D4—141no—
203C4×D4—141no—
204C4×D4—141no—
205C4×D4D2812yes—
206C4×D4C22⋊D712yes—
207C2×C4⋊C4—71no—
208C2×C4⋊C4C4×D711yescharacteristic
209C2×C22⋊C4—21no—
210C2×C22⋊C4—141no—
211C2×C42C22×D711yescharacteristic
212#106—142no—
213C4⋊C8—21no—
214C4⋊C8—21no—
215C4⋊C8—141no—
216C4⋊C8—141no—
217#102—142no—
218#102—142no—
219C22⋊C8—24no—
220C22⋊C8—21no—
221C22⋊C8—141no—
222C22⋊C8—141no—
223C22⋊C8C22⋊D712yes—
224#97—141no—
225#97—141no—
226C4×C8—21no—
227C4×C8—141no—
228C4×C8—141no—
229C4×C8C22⋊D712yes—
230C22×C7—21no—
231C22×C7—21no—
232C22×C7—42no—
233C22×C7—42no—
234C22×C7—42no—
235C22×C7—82no—
236C22×C7—82no—
237C22×C7#38611yescharacteristic
238C22×C7#43511yescharacteristic
239C22×C7#43611yescharacteristic
240C2×D7—42no—
241C2×D7—82no—
242C2×D7—81no—
243C2×D7—81no—
244C2×D7—82no—
245C4×C7—21no—
246C4×C7—21no—
247C4×C7—21no—
248C4×C7—21no—
249C4×C7—42no—
250C4×C7#40711yescharacteristic
251C4×C7#40811yescharacteristic
252C4×C7#43511yescharacteristic
253C4×C7#43611yescharacteristic
254Dic7—81no—
255C24—21no—
256C24—141no—
257C2×D4—21no—
258C2×D4—21no—
259C2×D4—21no—
260C2×D4—21no—
261C2×D4—21no—
262C2×D4—72no—
263C2×D4—141no—
264C2×D4—141no—
265C2×D4—141no—
266C2×D4—141no—
267C2×D4—141no—
268C2×D4(C2×D7)⋊C412yes—
269C22×C4—21no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—82no—
273C22×C4—141no—
274C22×C4—281no—
275C22×C4—281no—
276C22×C4—562no—
277C22×C4C2×C4×D711yescharacteristic
278C22×C4#115711yescharacteristic
279C22×C4C2×C22⋊D711yescharacteristic
280D8—148no—
281M16—84no—
282C2×C8—22no—
283C2×C8—22no—
284C2×C8—72no—
285C2×C8—141no—
286C2×C8—141no—
287C2×C8—141no—
288C2×C8—141no—
289C2×C8—141no—
290C2×C8—281no—
291C2×C8—281no—
292C4⋊C4—21no—
293C4⋊C4—141no—
294C4⋊C4—141no—
295C4⋊C4(C2×D7)⋊C412yes—
296C22⋊C4—22no—
297C22⋊C4—22no—
298C22⋊C4—281no—
299C22⋊C4—281no—
300C42C2×D2811yescharacteristic
301C42#115711yescharacteristic
302C42C2×C22⋊D711yescharacteristic
303C42C2×C22⋊D711yescharacteristic
304C2×C7—21no—
305C2×C7—21no—
306C2×C7—82no—
307C2×C7#187011yescharacteristic
308C2×C7#190311yescharacteristic
309C2×C7#190411yescharacteristic
310D7—82no—
311C23—22no—
312C23—41no—
313C23—41no—
314C23—81no—
315C23—81no—
316C23—142no—
317C23—281no—
318C23—281no—
319C23—561no—
320C23—561no—
321C23C4×C22⋊D711yescharacteristic
322D4—22no—
323D4—22no—
324D4—41no—
325D4—41no—
326D4—142no—
327D4—142no—
328D4—281no—
329D4—281no—
330C2×C4—22no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—44no—
334C2×C4—82no—
335C2×C4—81no—
336C2×C4—281no—
337C2×C4—562no—
338C2×C4—562no—
339C2×C4—561no—
340C2×C4D7×M1611yescharacteristic
341C2×C4D7×M1611yescharacteristic
342C2×C4#647011yescharacteristic
343C2×C4#647011yescharacteristic
344C2×C4C2×(C2×D7)⋊C411yescharacteristic
345C2×C4C4×C22⋊D711yescharacteristic
346C2×C4#649211yescharacteristic
347C2×C4#649311yescharacteristic
348C2×C4#649311yescharacteristic
349C2×C4#651611yescharacteristic
350C8—42no—
351C8—42no—
352C8—142no—
353C8—281no—
354C8—561no—
355C8—561no—
356C7#909311yescharacteristic, complemented, Sylow
357C22—21no—
358C22—21no—
359C22—42no—
360C22—42no—
361C22—42no—
362C22—82no—
363C22—82no—
364C22—282no—
365C22—562no—
366C22—562no—
367C22—561no—
368C22—561no—
369C22#8989911yescharacteristic
370C22#8990011yescharacteristic
371C22#8998511yescharacteristic
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—42no—
377C4—561no—
378C4#8989811yescharacteristic
379C4#8989911yescharacteristic
380C4#8990011yescharacteristic
381C4#8990111yescharacteristic
382C2—21no—
383C2—21no—
384C2—82no—
385C2—562no—
386C2#1172940511yescharacteristic
387C2#1172975011yescharacteristic
388C2#1172977711yescharacteristic
subgroupG/H
centerC22#89985
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#2975C2
socleC22×C7#386
prime radicalC22×#1141C4
cosocleC2×C42×C7C23
hypercenter#2975D7
hypocenterC7#9093
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2975

Complements

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

NH ≅ G/Ntype
1. C7×#2975C2semidirect
3. #11729351C2semidirect
4. #11728537C2semidirect
5. #11728511C2semidirect
6. #11725547C2semidirect
7. #11725546C2semidirect
69. #2975D7semidirect
97. C2×D28M16semidirect
124. C2×C7⋊C8M16semidirect
356. C7#9093semidirect

Maximal subgroups

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

subgroupindex
C7×#29752
C2×#897862
#117293512
#117285372
#117285112
#117255472
#117255462
#90937