Finite Group Databasebeta

← #11736830

Subgroups — #11736830

Order 896.

372 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90214C211yesmaximal, characteristic
2#90209C211yesmaximal, characteristic, complemented
3C7×#433C211yesmaximal, characteristic, complemented
4#89621C211yesmaximal, characteristic, complemented
5#89619C211yesmaximal, characteristic, complemented
6#89613C211yesmaximal, characteristic, complemented
7#89601C211yesmaximal, characteristic, complemented
8#89505C211yesmaximal, characteristic
9#89505C211yesmaximal, characteristic
10#89494C211yesmaximal, characteristic, complemented
11#89493C211yesmaximal, characteristic, complemented
12#89488C211yesmaximal, characteristic, complemented
13#89482C211yesmaximal, characteristic, complemented
14C4×Dic28C211yesmaximal, characteristic, complemented
15C4×#1132C211yesmaximal, characteristic, complemented
16C2×#1157C2211yescharacteristic
17C7×#117C2211yescharacteristic
18C2×C4⋊Dic7C2211yescharacteristic
19#6487C2211yescharacteristic
20#6487C2211yescharacteristic
21C2×Dic28C2211yescharacteristic
22C2×Dic28C2211yescharacteristic
23C2×#1132C2211yescharacteristic
24C2×#1132C2211yescharacteristic
25#6455—21no—
26D7×C4⋊C4—21no—
27#6452—21no—
28#6452—21no—
29(C22⋊D7)⋊C4—21no—
30(C22⋊D7)⋊C4—21no—
31#6442—21no—
32#6442—21no—
33C4×D28C2211yescharacteristic
34C4×D28C2211yescharacteristic
35#6433C2211yescharacteristic
36#6432C2211yescharacteristic
37C4×Dic14C2211yescharacteristic
38C4×Dic14C2211yescharacteristic
39C4×Dic14C2211yescharacteristic
40C4×Dic14C2211yescharacteristic
41C7×C4⋊C8C2211yescharacteristic
42C7×C4⋊C8C2211yescharacteristic
43C7×C22⋊C8C2211yescharacteristic
44C7×C22⋊C8C2211yescharacteristic
45C7×#97C2211yescharacteristic
46C4×C7×C8C2211yescharacteristic
47#6395C2211yescharacteristic
48#6395C2211yescharacteristic
49#6392C2211yescharacteristic
50#6392C2211yescharacteristic
51#6391C2211yescharacteristic
52#6391C2211yescharacteristic
53#6390C2211yescharacteristic
54#6390C2211yescharacteristic
55#6390C2211yescharacteristic
56#6390C2211yescharacteristic
57#6390C2211yescharacteristic
58#6390C2211yescharacteristic
59#3120—71nomaximal, Sylow
60C22×C4×C7C2311yescharacteristic
61C2×C22⋊D7—21no—
62C22×Dic7—21no—
63#1157—22no—
64#1157—22no—
65C2×D28C2311yescharacteristic
66C2×C4×D7—21no—
67C2×C4×D7—21no—
68C2×C4×D7—21no—
69C2×Dic14C2311yescharacteristic
70C2×Dic14C2311yescharacteristic
71C2×Dic14C2311yescharacteristic
72C2×C7×C8C2311yescharacteristic
73C2×C7×C8C2311yescharacteristic
74C2×C7×C8C2311yescharacteristic
75C2×C7×C8C2311yescharacteristic
76C7×C4⋊C4D411yescharacteristic
77C7×C4⋊C4D411yescharacteristic
78C7×C22⋊C4D411yescharacteristic
79C7×C22⋊C4D411yescharacteristic
80C42×C7C2311yescharacteristic
81C42×C7C2311yescharacteristic
82C22⋊Dic7—21no—
83C22⋊Dic7—21no—
84(C2×D7)⋊C4—21no—
85(C2×D7)⋊C4—21no—
86C4⋊Dic7—21no—
87C4⋊Dic7—21no—
88C4⋊Dic7—21no—
89C4⋊Dic7C2311yescharacteristic
90C4⋊Dic7C2311yescharacteristic
91C4⋊Dic7C2311yescharacteristic
92C4⋊Dic7C2311yescharacteristic
93Dic7⋊C4—21no—
94Dic7⋊C4—21no—
95Dic7⋊C4—21no—
96Dic7⋊C4—21no—
97Dic7⋊C4—21no—
98Dic7⋊C4—21no—
99Dic7⋊C4—21no—
100Dic7⋊C4—21no—
101C4×Dic7—21no—
102C4×Dic7—21no—
103C4×Dic7—21no—
104C4×Dic7—21no—
105Dic28—22no—
106Dic28—22no—
107#1132—22no—
108#1132—22no—
109#541—71no—
110#520—71no—
111#484—71no—
112#483—71no—
113#479—71no—
114#478—71no—
115#462—71no—
116#461—71no—
117#452—71no—
118#452—71no—
119#441—71no—
120#440—71no—
121C4×Q16—71no—
122C4×SD16—71no—
123#433D711yescharacteristic, complemented
124C23×C7C2×D411yescharacteristic
125C22×D7—21no—
126C2×C4×C7—21no—
127C2×C4×C7—21no—
128C2×C4×C7C2×D411yescharacteristic
129C2×C4×C7C2×D411yescharacteristic
130C2×C4×C7C2×D411yescharacteristic
131C2×C4×C7C2×D411yescharacteristic
132C2×C4×C7C2×D411yescharacteristic
133C2×C4×C7C2411yescharacteristic
134C22⋊D7—24no—
135C2×Dic7—21no—
136C2×Dic7—21no—
137C2×Dic7—21no—
138C2×Dic7—21no—
139C2×Dic7—21no—
140C2×Dic7—21no—
141C2×Dic7—21no—
142C2×Dic7—81no—
143D28—21no—
144D28#4112yes—
145C4×D7—24no—
146C4×D7—81no—
147C4×D7—81no—
148Dic14—21no—
149Dic14—21no—
150Dic14—21no—
151Dic14—21no—
152Dic14—21no—
153Dic14#4112yes—
154C7×C8—21no—
155C7×C8—21no—
156C7×C8—21no—
157C7×C8—21no—
158C2×#41—71no—
159C2×Q16—71no—
160C2×Q16—71no—
161C2×SD16—71no—
162C2×SD16—71no—
163C4⋊Q8—71no—
164#126—141no—
165#126—141no—
166#125—71no—
167#125—141no—
168#125—141no—
169C22⋊Q8—71no—
170C22⋊Q8—71no—
171C4×Q8—71no—
172C4×Q8—71no—
173C4×Q8—71no—
174C4×Q8—71no—
175C4×D4—71no—
176C4×D4—71no—
177C4×D4—141no—
178C4×D4—141no—
179#117—141no—
180#117C2×D711yescharacteristic
181C2×C4⋊C4—71no—
182C2×C4⋊C4—141no—
183#107—71no—
184#107—71no—
185#106—71no—
186#106—71no—
187C4⋊C8C2×D711yescharacteristic
188C4⋊C8C2×D711yescharacteristic
189#103—71no—
190#103—71no—
191#103—71no—
192#103—71no—
193#103—71no—
194#103—71no—
195#102—71no—
196#102—71no—
197C22⋊C8C2×D711yescharacteristic
198C22⋊C8C2×D711yescharacteristic
199#97C2×D711yescharacteristic
200C4×C8C2×D711yescharacteristic
201C22×C7—21no—
202C22×C7—21no—
203C22×C7—21no—
204C22×C7C22×D411yescharacteristic
205C2×D7—22no—
206C2×D7—41no—
207C2×D7—41no—
208C4×C7—21no—
209C4×C7—21no—
210C4×C7—21no—
211C4×C7—21no—
212C4×C7—21no—
213C4×C7C2×#4111yescharacteristic
214C4×C7C2×#4111yescharacteristic
215Dic7—22no—
216Dic7—41no—
217Dic7—41no—
218Dic7—41no—
219Dic7—41no—
220Dic7—41no—
221Dic7—41no—
222#41—142no—
223#41—142no—
224C2×Q8—71no—
225C2×Q8—71no—
226C2×Q8—71no—
227C2×D4—71no—
228C2×D4—141no—
229C22×C4—141no—
230C22×C4—141no—
231C22×C4—141no—
232C22×C4—141no—
233C22×C4C22×D711yescharacteristic
234Q16—142no—
235Q16—142no—
236SD16—142no—
237SD16—142no—
238C2×C8C22×D711yescharacteristic
239C2×C8C22×D711yescharacteristic
240C2×C8C22×D711yescharacteristic
241C2×C8C22×D711yescharacteristic
242C4⋊C4—71no—
243C4⋊C4—71no—
244C4⋊C4—71no—
245C4⋊C4—71no—
246C4⋊C4—141no—
247C4⋊C4—141no—
248C4⋊C4—141no—
249C4⋊C4—141no—
250C4⋊C4—141no—
251C4⋊C4—141no—
252C4⋊C4—141no—
253C4⋊C4—141no—
254C4⋊C4—141no—
255C4⋊C4—141no—
256C4⋊C4—141no—
257C4⋊C4D2811yescharacteristic
258C4⋊C4D2811yescharacteristic
259C22⋊C4—141no—
260C22⋊C4—141no—
261C22⋊C4—141no—
262C22⋊C4—141no—
263C22⋊C4D2811yescharacteristic
264C22⋊C4D2811yescharacteristic
265C42—141no—
266C42—141no—
267C42—141no—
268C42—141no—
269C42C22×D711yescharacteristic
270C42C22×D711yescharacteristic
271C2×C7—41no—
272C2×C7#52511yescharacteristic
273C2×C7#57711yescharacteristic
274C2×C7#57811yescharacteristic
275D7—42no—
276C23—141no—
277C23C2×D2811yescharacteristic
278Q8—72no—
279Q8—141no—
280Q8—141no—
281Q8—141no—
282Q8—141no—
283Q8—141no—
284D4—72no—
285D4—144no—
286D4—141no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—144no—
290C2×C4—141no—
291C2×C4—141no—
292C2×C4—141no—
293C2×C4—141no—
294C2×C4—141no—
295C2×C4—141no—
296C2×C4—141no—
297C2×C4—561no—
298C2×C4—561no—
299C2×C4—561no—
300C2×C4C2×D2811yescharacteristic
301C2×C4C2×D2811yescharacteristic
302C2×C4C2×D2811yescharacteristic
303C2×C4C2×D2811yescharacteristic
304C2×C4C2×D2811yescharacteristic
305C2×C4C23×D711yescharacteristic
306C8—21no—
307C8—21no—
308C8—21no—
309C8—21no—
310C7#312011yescharacteristic, complemented, Sylow
311C22—21no—
312C22—21no—
313C22—21no—
314C22—142no—
315C22—281no—
316C22—281no—
317C22C22×D2811yescharacteristic
318C4—21no—
319C4—21no—
320C4—21no—
321C4—21no—
322C4—21no—
323C4—142no—
324C4—281no—
325C4—281no—
326C4—281no—
327C4—281no—
328C4—281no—
329C4—281no—
330C4D7×#4111yescharacteristic
331C4D7×#4111yescharacteristic
332C2—41no—
333C2—282no—
334C2#9021511yescharacteristic
335C2#9044211yescharacteristic
336C2#9044411yescharacteristic
subgroupG/H
centerC22C22×D28
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#433C2
socleC22×C7C22×D4
prime radicalC2×#1157C22
hypercenter#433D7
hypocenterC7#3120
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ #117 ◁ #433

Complements

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

NH ≅ G/Ntype
2. #90209C2semidirect
3. C7×#433C2semidirect
4. #89621C2semidirect
5. #89619C2semidirect
6. #89613C2semidirect
7. #89601C2semidirect
10. #89494C2semidirect
11. #89493C2semidirect
12. #89488C2semidirect
13. #89482C2semidirect
14. C4×Dic28C2semidirect
15. C4×#1132C2semidirect
123. #433D7semidirect
310. C7#3120semidirect

Maximal subgroups

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

subgroupindex
#902142
#902092
C7×#4332
#896212
#896192
#896132
#896012
#895052
#894942
#894932
#894882
#894822
C4×Dic282
C4×#11322
#31207