Finite Group Databasebeta

← #13251090

Subgroups — #13251090

Order 1792.

424 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 99 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×#2336C211yesmaximal, characteristic
2#11740466C211yesmaximal, characteristic
3#11729489C211yesmaximal, characteristic
4#11727320C211yesmaximal, characteristic
5#11727247C211yesmaximal, characteristic
6#11725439C211yesmaximal, characteristic
7#11725427C211yesmaximal, characteristic
8C2×C7×#125C2211yescharacteristic
9#90161C2211yescharacteristic
10C7×#389—21no—
11C7×#387—21no—
12C7×#387—21no—
13C7×#387—21no—
14C7×#387—21no—
15C7×#383C2211yescharacteristic
16C4×C7×C4⋊C4—21no—
17C4×C7×C4⋊C4C2211yescharacteristic
18#89841C412yescomplemented
19#89837C412yescomplemented
20#89835—22no—
21#89743—21no—
22#89742—21no—
23#89698C2211yescharacteristic
24#89698C2211yescharacteristic
25C4×#1136C2211yescharacteristic
26#89338—22no—
27#89318—21no—
28#89318—21no—
29#89247—21no—
30#89247—21no—
31#9380—71nomaximal, Sylow
32C7×#125—21no—
33C7×#125—21no—
34C7×#125C2×C412yes—
35C7×#125D412yes—
36C2×C7×C4⋊C4—21no—
37C2×C7×C4⋊C4—21no—
38C2×C7×C4⋊C4—21no—
39C2×C7×C4⋊C4—21no—
40C2×C7×C4⋊C4—21no—
41C2×C7×C4⋊C4D411yescharacteristic
42C2×C7×C4⋊C4D411yescharacteristic
43C2×C42×C7—21no—
44C2×C42×C7—21no—
45C2×C42×C7D411yescharacteristic
46C2×C42×C7D411yescharacteristic
47C2×C42×C7C2311yescharacteristic
48#6489—21no—
49#6487—21no—
50#6487—21no—
51C2×#1136—21no—
52#6433—21no—
53#6432C2×C411yescharacteristic
54#6432C2×C411yescharacteristic
55C4×Dic14—21no—
56C4×Dic14—21no—
57C7×#95—21no—
58C7×#95—21no—
59C7×#95—21no—
60C7×#95—21no—
61C7×#95—21no—
62C7×#95—21no—
63C7×#95—21no—
64C7×#95D411yescharacteristic
65C7×#95D411yescharacteristic
66C22⋊(C7⋊C8)—21no—
67C22⋊(C7⋊C8)—21no—
68#6384—22no—
69#6384—22no—
70#6382—22no—
71#6381—22no—
72C4⋊(C7⋊C8)—21no—
73C4⋊(C7⋊C8)—21no—
74#6377—21no—
75#6377—21no—
76#6377C2×C412yes—
77C4×C7⋊C8—21no—
78#3236—71no—
79#2336D711yescharacteristic
80#1986—71no—
81#1528—71no—
82#1515—71no—
83#1288—71no—
84#1285—71no—
85C22×C4×C7—21no—
86C22×C4×C7—21no—
87C22×C4×C7—21no—
88C22×C4×C7—21no—
89C22×C4×C7—21no—
90C22×C4×C7—21no—
91C22×C4×C7C2×D411yescharacteristic
92C22×C4×C7C2×D411yescharacteristic
93C22×C4×C7C2×D411yescharacteristic
94C2×Dic14—21no—
95C2×Dic14—21no—
96C7×C4⋊C4—22no—
97C7×C4⋊C4—22no—
98C7×C4⋊C4—22no—
99C7×C4⋊C4—22no—
100C7×C4⋊C4—21no—
101C7×C4⋊C4—21no—
102C7×C4⋊C4—42no—
103C7×C4⋊C4—42no—
104C7×C4⋊C4—81no—
105C7×C4⋊C4C22⋊C412yes—
106C7×C4⋊C4C22⋊C412yes—
107C42×C7—41no—
108C42×C7—41no—
109C42×C7—42no—
110C42×C7—81no—
111C42×C7C22×C411yescharacteristic
112C42×C7C2×D411yescharacteristic
113C42×C7#4111yescharacteristic
114C42×C7#4111yescharacteristic
115C22⋊Dic7—41no—
116C4⋊Dic7—21no—
117C4⋊Dic7—21no—
118C4⋊Dic7—21no—
119C4⋊Dic7—21no—
120Dic7⋊C4—41no—
121Dic7⋊C4—41no—
122C4×Dic7—41no—
123#1136—22no—
124#1136—41no—
125C2×C7⋊C8—22no—
126C2×C7⋊C8—41no—
127C2×C7⋊C8—41no—
128#533—71no—
129C2×#125C2×D711yescharacteristic
130#501—72no—
131#491—72no—
132#484—141no—
133#483—141no—
134#479—142no—
135#432—71no—
136#432—71no—
137C4×M16—71no—
138#389—21no—
139#387—21no—
140#387—21no—
141#387—21no—
142#387—21no—
143#383C2×D711yescharacteristic
144C4×C4⋊C4—21no—
145C4×C4⋊C4C2×D711yescharacteristic
146#339—141no—
147#339—141no—
148#330—142no—
149#328—141no—
150#328—141no—
151C23×C7#12011yescharacteristic
152C2×C4×C7—21no—
153C2×C4×C7—21no—
154C2×C4×C7—22no—
155C2×C4×C7—22no—
156C2×C4×C7—22no—
157C2×C4×C7—41no—
158C2×C4×C7—41no—
159C2×C4×C7—41no—
160C2×C4×C7—41no—
161C2×C4×C7—41no—
162C2×C4×C7—41no—
163C2×C4×C7—41no—
164C2×C4×C7—41no—
165C2×C4×C7—41no—
166C2×C4×C7—41no—
167C2×C4×C7—41no—
168C2×C4×C7—41no—
169C2×C4×C7—41no—
170C2×C4×C7—41no—
171C2×C4×C7—41no—
172C2×C4×C7—41no—
173C2×C4×C7—41no—
174C2×C4×C7C2×C22⋊C411yescharacteristic
175C2×C4×C7C4×D411yescharacteristic
176C2×C4×C7C4×D411yescharacteristic
177C2×C4×C7#12111yescharacteristic
178C2×C4×C7#12111yescharacteristic
179C2×C4×C7#12311yescharacteristic
180C2×Dic7—41no—
181C2×Dic7—41no—
182Dic14—42no—
183Dic14—81no—
184C7⋊C8—42no—
185C7⋊C8—42no—
186C7⋊C8—81no—
187C2×M16—141no—
188C4⋊Q8—71no—
189C4⋊Q8—71no—
190#125—21no—
191#125—21no—
192#125—141no—
193#125C4×D712yes—
194#125D2812yes—
195C22⋊Q8—141no—
196C22⋊Q8—141no—
197C4×Q8—141no—
198C4×Q8—141no—
199#117—141no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4C22⋊D711yescharacteristic
206C2×C4⋊C4C22⋊D711yescharacteristic
207C2×C42—21no—
208C2×C42—21no—
209C2×C42C22⋊D711yescharacteristic
210C2×C42C22⋊D711yescharacteristic
211C2×C42C22×D711yescharacteristic
212#107—142no—
213#106—142no—
214C4⋊C8—141no—
215C4⋊C8—141no—
216#103—142no—
217#103—142no—
218C22⋊C8—141no—
219C22⋊C8—141no—
220#97—72no—
221#97—141no—
222#97—141no—
223C4×C8—141no—
224#95—21no—
225#95—21no—
226#95—21no—
227#95—21no—
228#95—21no—
229#95—21no—
230#95—21no—
231#95C22⋊D711yescharacteristic
232#95C22⋊D711yescharacteristic
233C22×C7—21no—
234C22×C7—21no—
235C22×C7#38611yescharacteristic
236C22×C7#45511yescharacteristic
237C22×C7#45611yescharacteristic
238C4×C7—21no—
239C4×C7—21no—
240C4×C7—21no—
241C4×C7—21no—
242C4×C7—21no—
243C4×C7—21no—
244C4×C7—42no—
245C4×C7—42no—
246C4×C7—42no—
247C4×C7—81no—
248C4×C7—81no—
249C4×C7—81no—
250Dic7—81no—
251Dic7—81no—
252C2×Q8—141no—
253C2×Q8—141no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4C2×C22⋊D711yescharacteristic
261C22×C4C2×C22⋊D711yescharacteristic
262C22×C4C2×C22⋊D711yescharacteristic
263M16—142no—
264M16—281no—
265C2×C8—142no—
266C2×C8—281no—
267C2×C8—281no—
268C4⋊C4—22no—
269C4⋊C4—22no—
270C4⋊C4—21no—
271C4⋊C4—21no—
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—81no—
277C4⋊C4—141no—
278C4⋊C4—141no—
279C4⋊C4—141no—
280C4⋊C4—141no—
281C4⋊C4—281no—
282C4⋊C4—281no—
283C4⋊C4(C2×D7)⋊C412yes—
284C4⋊C4(C2×D7)⋊C412yes—
285C22⋊C4—281no—
286C42—41no—
287C42—41no—
288C42—42no—
289C42—81no—
290C42—281no—
291C42C2×C4×D711yescharacteristic
292C42C2×D2811yescharacteristic
293C42#115711yescharacteristic
294C42#115711yescharacteristic
295C2×C7—21no—
296C2×C7—21no—
297C2×C7#188211yescharacteristic
298C2×C7#190611yescharacteristic
299C2×C7#191211yescharacteristic
300C23#651611yescharacteristic
301Q8—282no—
302Q8—561no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—281no—
326C2×C4—281no—
327C2×C4C2×(C2×D7)⋊C411yescharacteristic
328C2×C4C4×C22⋊D711yescharacteristic
329C2×C4C4×C22⋊D711yescharacteristic
330C2×C4#649211yescharacteristic
331C2×C4#649311yescharacteristic
332C2×C4#649311yescharacteristic
333C8—282no—
334C8—282no—
335C8—561no—
336C7#938011yescharacteristic, complemented, Sylow
337C22—21no—
338C22—21no—
339C22#8998511yescharacteristic
340C22#9001411yescharacteristic
341C22#9001811yescharacteristic
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—42no—
349C4—42no—
350C4—42no—
351C4—81no—
352C4—81no—
353C4—81no—
354C4—561no—
355C4—561no—
356C2—21no—
357C2—21no—
358C2#1172939911yescharacteristic
359C2#1172941711yescharacteristic
360C2#1172942011yescharacteristic
subgroupG/H
centerC22#89985
commutatorC42×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#2336C2
socleC22×C7#386
prime radicalC23×C7#120
cosocleC2×C42×C7C23
hypercenter#2336D7
hypocenterC7#9380
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2336

Complements

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

NH ≅ G/Ntype
18. #89841C4semidirect
19. #89837C4semidirect
336. C7#9380semidirect

Maximal subgroups

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

subgroupindex
C7×#23362
#117404662
#117294892
#117273202
#117272472
#117254392
#117254272
#93807