Finite Group Databasebeta

← #13250733

Subgroups — #13250733

Order 1792.

530 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 530 conjugacy classes of subgroups — too many to draw. Only its 103 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
1C7×#2319C211yesmaximal, characteristic, complemented
2#11739755C211yesmaximal, characteristic
3#11729353C211yesmaximal, characteristic, complemented
4#11727246C211yesmaximal, characteristic, complemented
5#11727186C211yesmaximal, characteristic, complemented
6#11725427C211yesmaximal, characteristic, complemented
7#11725421C211yesmaximal, characteristic, complemented
8C2×C4×C7×D4C2211yescharacteristic
9#90168C2211yescharacteristic
10C7×#390—21no—
11C7×#389—21no—
12C7×#388—22no—
13C7×#381—21no—
14C7×#380—21no—
15C4×C7×C4⋊C4C2211yescharacteristic
16C7×#376C2211yescharacteristic
17#89989—21no—
18#89787C412yescomplemented
19#89786—22no—
20#89780C412yescomplemented
21#89740—21no—
22C2×#6379—22no—
23#89698C2211yescharacteristic
24#89698C2211yescharacteristic
25C4×#1136C2211yescharacteristic
26D4⋊(C7⋊C8)—22no—
27#89246—21no—
28#89246—21no—
29#9329—71nomaximal, Sylow
30C22×C7×D4D411yescharacteristic
31C23×C4×C7—21no—
32C2×#1157—21no—
33C4×C7×D4—21no—
34C4×C7×D4—21no—
35C4×C7×D4C2×C412yes—
36C4×C7×D4D412yes—
37C2×C7×C4⋊C4—21no—
38C2×C7×C4⋊C4—22no—
39C2×C7×C4⋊C4D411yescharacteristic
40C2×C7×C22⋊C4—21no—
41C2×C7×C22⋊C4—22no—
42C2×C7×C22⋊C4—22no—
43C2×C42×C7—21no—
44C2×C42×C7D412yes—
45C2×C42×C7C2311yescharacteristic
46C2×#1141—22no—
47#6493—21no—
48#6493—21no—
49C2×#1136—21no—
50#6438—21no—
51#6437C2×C411yescharacteristic
52C4×D28—21no—
53C4×D28—21no—
54#6432C2×C411yescharacteristic
55C7×#95—22no—
56C7×#95—21no—
57C7×#95—21no—
58C7×#95D411yescharacteristic
59C7×#95D411yescharacteristic
60#6406—22no—
61C22⋊(C7⋊C8)—21no—
62C22⋊(C7⋊C8)—21no—
63#6383—22no—
64#6382—22no—
65#6379—44no—
66C4⋊(C7⋊C8)—21no—
67C4⋊(C7⋊C8)C2×C412yes—
68#6377—21no—
69#6377—21no—
70#6377—21no—
71C4×C7⋊C8—21no—
72#3109—71no—
73#2319D711yescharacteristic, complemented
74#1980—71no—
75#1505—71no—
76#1490—71no—
77#1300—71no—
78#1285—71no—
79C24×C7—21no—
80C2×C7×D4—21no—
81C2×C7×D4—42no—
82C2×C7×D4C22⋊C412yes—
83C22×C4×C7—21no—
84C22×C4×C7—22no—
85C22×C4×C7—22no—
86C22×C4×C7—21no—
87C22×C4×C7—22no—
88C22×C4×C7—21no—
89C22×C4×C7—82no—
90C22×C4×C7C2×D411yescharacteristic
91C22×C4×C7C2×D411yescharacteristic
92C22×C4×C7C2×D411yescharacteristic
93C2×C22⋊D7—41no—
94#1157—42no—
95#1157—42no—
96C2×D28—21no—
97C2×D28—21no—
98C2×D28—21no—
99C2×C4×D7—41no—
100C2×Dic14—21no—
101C7×C4⋊C4—21no—
102C7×C4⋊C4—42no—
103C7×C4⋊C4—82no—
104C7×C4⋊C4C22⋊C412yes—
105C7×C22⋊C4—22no—
106C7×C22⋊C4—22no—
107C7×C22⋊C4—82no—
108C7×C22⋊C4—82no—
109C42×C7—42no—
110C42×C7—42no—
111C42×C7C22×C411yescharacteristic
112C42×C7C2×D411yescharacteristic
113C42×C7#4111yescharacteristic
114C42×C7#4111yescharacteristic
115#1141—44no—
116(C2×D7)⋊C4—41no—
117(C2×D7)⋊C4—41no—
118C4⋊Dic7—21no—
119C4⋊Dic7—21no—
120#1136—22no—
121#1136—41no—
122C2×C7⋊C8—22no—
123C2×C7⋊C8—41no—
124C2×C7⋊C8—41no—
125#532—71no—
126C2×C4×D4C2×D711yescharacteristic
127#481—141no—
128#476—72no—
129#459—72no—
130#447—141no—
131#442—142no—
132#432—71no—
133#432—71no—
134C2×#104—142no—
135C4×M16—71no—
136#390—21no—
137#389—21no—
138#388—22no—
139#381—21no—
140#380—21no—
141C4×C4⋊C4C2×D711yescharacteristic
142#376C2×D711yescharacteristic
143#327—141no—
144#327—141no—
145D4⋊C8—142no—
146C23×C7—22no—
147C23×C7—42no—
148C23×C7—42no—
149C23×C7—41no—
150C23×C7—41no—
151C23×C7#12011yescharacteristic
152C22×D7—41no—
153C7×D4—22no—
154C7×D4—22no—
155C7×D4—42no—
156C2×C4×C7—21no—
157C2×C4×C7—21no—
158C2×C4×C7—22no—
159C2×C4×C7—22no—
160C2×C4×C7—41no—
161C2×C4×C7—42no—
162C2×C4×C7—41no—
163C2×C4×C7—41no—
164C2×C4×C7—41no—
165C2×C4×C7—42no—
166C2×C4×C7—42no—
167C2×C4×C7—41no—
168C2×C4×C7—42no—
169C2×C4×C7—41no—
170C2×C4×C7—41no—
171C2×C4×C7—82no—
172C2×C4×C7—82no—
173C2×C4×C7C2×C22⋊C411yescharacteristic
174C2×C4×C7C4×D411yescharacteristic
175C2×C4×C7C4×D411yescharacteristic
176C2×C4×C7#12111yescharacteristic
177C2×C4×C7#12111yescharacteristic
178C2×C4×C7#12311yescharacteristic
179C22⋊D7—81no—
180C22⋊D7—81no—
181C2×Dic7—41no—
182D28—42no—
183D28—42no—
184D28—81no—
185C4×D7—82no—
186Dic14—42no—
187C7⋊C8—42no—
188C7⋊C8—42no—
189C7⋊C8—81no—
190C2×#41—141no—
191C22×D4C22⋊D711yescharacteristic
192C23×C4—21no—
193C2×D8—142no—
194C2×M16—141no—
195C4⋊Q8—71no—
196#127—71no—
197#124—141no—
198#121—141no—
199#121—141no—
200C4×D4—21no—
201C4×D4—21no—
202C4×D4—141no—
203C4×D4—141no—
204C4×D4C4×D712yes—
205C4×D4D2812yes—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4C22⋊D711yescharacteristic
209C2×C22⋊C4—21no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C42—21no—
213C2×C42C22⋊D712yes—
214C2×C42C22×D711yescharacteristic
215#106—142no—
216C4⋊C8—72no—
217C4⋊C8—141no—
218#104—284no—
219#102—142no—
220#102—142no—
221C22⋊C8—141no—
222C22⋊C8—141no—
223#97—141no—
224#97—141no—
225#97—141no—
226C4×C8—141no—
227#95—22no—
228#95—21no—
229#95—21no—
230#95C22⋊D711yescharacteristic
231#95C22⋊D711yescharacteristic
232C22×C7—21no—
233C22×C7—21no—
234C22×C7—42no—
235C22×C7—42no—
236C22×C7—42no—
237C22×C7—82no—
238C22×C7—82no—
239C22×C7#38611yescharacteristic
240C22×C7#45311yescharacteristic
241C22×C7#45411yescharacteristic
242C2×D7—81no—
243C2×D7—81no—
244C2×D7—81no—
245C4×C7—21no—
246C4×C7—21no—
247C4×C7—21no—
248C4×C7—21no—
249C4×C7—21no—
250C4×C7—42no—
251C4×C7—81no—
252C4×C7—82no—
253C4×C7—81no—
254C4×C7#42112yes—
255Dic7—81no—
256C24—21no—
257#41—282no—
258#41—282no—
259C2×Q8—141no—
260C2×D4—21no—
261C2×D4—42no—
262C2×D4—141no—
263C2×D4—141no—
264C2×D4—141no—
265C2×D4—281no—
266C2×D4(C2×D7)⋊C412yes—
267C22×C4—21no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—21no—
271C22×C4—22no—
272C22×C4—21no—
273C22×C4—82no—
274C22×C4—281no—
275C22×C4C2×C22⋊D711yescharacteristic
276C22×C4C2×C22⋊D711yescharacteristic
277C22×C4C2×C22⋊D711yescharacteristic
278D8—284no—
279M16—142no—
280M16—281no—
281C2×C8—142no—
282C2×C8—281no—
283C2×C8—281no—
284C4⋊C4—21no—
285C4⋊C4—42no—
286C4⋊C4—82no—
287C4⋊C4—141no—
288C4⋊C4—141no—
289C4⋊C4(C2×D7)⋊C412yes—
290C22⋊C4—22no—
291C22⋊C4—22no—
292C22⋊C4—82no—
293C22⋊C4—82no—
294C22⋊C4—281no—
295C22⋊C4—281no—
296C42—42no—
297C42—42no—
298C42C2×C4×D711yescharacteristic
299C42C2×D2811yescharacteristic
300C42#115711yescharacteristic
301C42#115711yescharacteristic
302C2×C7—21no—
303C2×C7—21no—
304C2×C7—82no—
305C2×C7#188311yescharacteristic
306C2×C7#189711yescharacteristic
307C2×C7#190411yescharacteristic
308D7—161no—
309C23—22no—
310C23—42no—
311C23—42no—
312C23—41no—
313C23—41no—
314C23—281no—
315C23#651611yescharacteristic
316Q8—282no—
317D4—22no—
318D4—22no—
319D4—42no—
320D4—282no—
321D4—282no—
322D4—561no—
323D4—561no—
324D4—561no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—41no—
330C2×C4—42no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—41no—
337C2×C4—42no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—82no—
341C2×C4—82no—
342C2×C4—281no—
343C2×C4—562no—
344C2×C4C2×(C2×D7)⋊C411yescharacteristic
345C2×C4C4×C22⋊D711yescharacteristic
346C2×C4C4×C22⋊D711yescharacteristic
347C2×C4#649211yescharacteristic
348C2×C4#649311yescharacteristic
349C2×C4#649311yescharacteristic
350C8—282no—
351C8—282no—
352C8—561no—
353C7#932911yescharacteristic, complemented, Sylow
354C22—21no—
355C22—21no—
356C22—42no—
357C22—42no—
358C22—42no—
359C22—82no—
360C22—82no—
361C22—561no—
362C22—561no—
363C22—561no—
364C22#8998511yescharacteristic
365C22#9001311yescharacteristic
366C22#9001711yescharacteristic
367C4—21no—
368C4—21no—
369C4—21no—
370C4—21no—
371C4—21no—
372C4—42no—
373C4—81no—
374C4—82no—
375C4—81no—
376C4—561no—
377C4#8977712yes—
378C2—21no—
379C2—21no—
380C2—82no—
381C2—1121no—
382C2#1172934511yescharacteristic
383C2#1172940511yescharacteristic
384C2#1172942111yescharacteristic
subgroupG/H
centerC22#89985
commutatorC42×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#2319C2
socleC22×C7#386
prime radical#11739755C2
cosocleC2×C42×C7C23
hypercenter#2319D7
hypocenterC7#9329
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2319

Complements

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

NH ≅ G/Ntype
1. C7×#2319C2semidirect
3. #11729353C2semidirect
4. #11727246C2semidirect
5. #11727186C2semidirect
6. #11725427C2semidirect
7. #11725421C2semidirect
18. #89787C4semidirect
20. #89780C4semidirect
73. #2319D7semidirect
353. C7#9329semidirect

Maximal subgroups

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

subgroupindex
C7×#23192
#117397552
#117293532
#117272462
#117271862
#117254272
#117254212
#93297