Finite Group Databasebeta

← #11740912

Subgroups — #11740912

Order 896.

370 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
1C7×#542C211yesmaximal, characteristic
2#90207C211yesmaximal, characteristic, complemented
3#89866C211yesmaximal, characteristic, complemented
4#89863C211yesmaximal, characteristic, complemented
5#89826C211yesmaximal, characteristic, complemented
6#89826C211yesmaximal, characteristic, complemented
7#89822C211yesmaximal, characteristic, complemented
8#89817C211yesmaximal, characteristic, complemented
9#89812C211yesmaximal, characteristic
10#89807C211yesmaximal, characteristic
11#89802C211yesmaximal, characteristic, complemented
12C4×#1144C211yesmaximal, characteristic, complemented
13#89790C211yesmaximal, characteristic, complemented
14C4×#1142C211yesmaximal, characteristic, complemented
15#89772C211yesmaximal, characteristic, complemented
16C22×Dic14C2211yescharacteristic
17C7×C4⋊Q8C2211yescharacteristic
18C7×#126—21no—
19C7×#124—21no—
20C7×#124C2211yescharacteristic
21C7×#123—21no—
22C7×C22⋊Q8—21no—
23C7×C22⋊Q8C2211yescharacteristic
24C7×#121C2211yescharacteristic
25C4×C7×Q8C2211yescharacteristic
26C4×C7×D4C2211yescharacteristic
27C7×#117C2211yescharacteristic
28C2×#1144C2211yescharacteristic
29C2×#1144C2211yescharacteristic
30C2×#1142C2211yescharacteristic
31C2×#1142C2211yescharacteristic
32#6489C2211yescharacteristic
33#6450—21no—
34#6450—21no—
35#6440—21no—
36#6440—21no—
37C4×Dic14C2211yescharacteristic
38C4×Dic14C2211yescharacteristic
39C4×Dic14C2211yescharacteristic
40C4×Dic14C2211yescharacteristic
41#6409C2211yescharacteristic
42#6409C2211yescharacteristic
43#6406C2211yescharacteristic
44#6406C2211yescharacteristic
45C22⋊(C7⋊C8)C2211yescharacteristic
46C22⋊(C7⋊C8)C2211yescharacteristic
47#6384C2211yescharacteristic
48#6384C2211yescharacteristic
49#6384C2211yescharacteristic
50#6384C2211yescharacteristic
51#6382C2211yescharacteristic
52#6382C2211yescharacteristic
53#6381C2211yescharacteristic
54#6381C2211yescharacteristic
55C4⋊(C7⋊C8)C2211yescharacteristic
56C4⋊(C7⋊C8)C2211yescharacteristic
57#6377C2211yescharacteristic
58C4×C7⋊C8C2211yescharacteristic
59#3121—71nomaximal, Sylow
60C2×C7×Q8—21no—
61C2×C7×Q8C2311yescharacteristic
62C2×C7×D4—21no—
63C2×C7×D4C2311yescharacteristic
64C22×C4×C7—21no—
65C22×C4×C7C2311yescharacteristic
66C22×Dic7—21no—
67C2×Dic14—21no—
68C2×Dic14—22no—
69C2×Dic14—22no—
70C2×Dic14C2311yescharacteristic
71C2×Dic14C2311yescharacteristic
72C7×C4⋊C4—21no—
73C7×C4⋊C4—21no—
74C7×C4⋊C4—21no—
75C7×C4⋊C4D411yescharacteristic
76C7×C4⋊C4D411yescharacteristic
77C7×C4⋊C4C2311yescharacteristic
78C7×C4⋊C4C2311yescharacteristic
79C7×C22⋊C4—21no—
80C7×C22⋊C4—21no—
81C7×C22⋊C4—21no—
82C7×C22⋊C4—21no—
83C7×C22⋊C4—21no—
84C7×C22⋊C4D411yescharacteristic
85C7×C22⋊C4D411yescharacteristic
86C42×C7—21no—
87C42×C7C2311yescharacteristic
88C42×C7C2311yescharacteristic
89C22⋊Dic7—21no—
90#1144—22no—
91#1144—22no—
92#1142—22no—
93#1142—22no—
94C4⋊Dic7C2311yescharacteristic
95C4⋊Dic7C2311yescharacteristic
96Dic7⋊C4—21no—
97Dic7⋊C4—21no—
98Dic7⋊C4—21no—
99Dic7⋊C4—21no—
100C4×Dic7—21no—
101C4×Dic7—21no—
102C4×Dic7—21no—
103C4×Dic7—21no—
104C4×Dic7—21no—
105C2×C7⋊C8C2311yescharacteristic
106C2×C7⋊C8C2311yescharacteristic
107C2×C7⋊C8C2311yescharacteristic
108C2×C7⋊C8C2311yescharacteristic
109#542D711yescharacteristic
110#519—71no—
111#485—71no—
112#482—71no—
113#477—71no—
114#475—71no—
115#464—71no—
116#464—71no—
117#451—71no—
118#448—71no—
119#441—71no—
120#440—71no—
121C4×Q16—71no—
122C4×SD16—71no—
123#433—71no—
124C23×C7—21no—
125C23×C7C2×D411yescharacteristic
126C7×Q8—21no—
127C7×Q8—21no—
128C7×Q8—81no—
129C7×D4—21no—
130C7×D4—21no—
131C7×D4—81no—
132C2×C4×C7—21no—
133C2×C4×C7—21no—
134C2×C4×C7—21no—
135C2×C4×C7—21no—
136C2×C4×C7—21no—
137C2×C4×C7—81no—
138C2×C4×C7C2×D411yescharacteristic
139C2×C4×C7C2×D411yescharacteristic
140C2×C4×C7C2×D411yescharacteristic
141C2×C4×C7C2×D411yescharacteristic
142C2×C4×C7C2×D411yescharacteristic
143C2×C4×C7C2411yescharacteristic
144C2×Dic7—24no—
145C2×Dic7—21no—
146C2×Dic7—21no—
147C2×Dic7—21no—
148C2×Dic7—21no—
149Dic14—24no—
150Dic14—21no—
151Dic14—21no—
152Dic14#4112yes—
153Dic14#4112yes—
154C7⋊C8—21no—
155C7⋊C8—21no—
156C7⋊C8—21no—
157C7⋊C8—21no—
158C22×Q8—71no—
159C2×Q16—71no—
160C2×Q16—71no—
161C2×SD16—71no—
162C2×SD16—71no—
163C4⋊Q8C2×D711yescharacteristic
164#126—21no—
165#124—21no—
166#124C2×D711yescharacteristic
167#123—21no—
168C22⋊Q8—21no—
169C22⋊Q8C2×D711yescharacteristic
170#121C2×D711yescharacteristic
171C4×Q8—71no—
172C4×Q8—71no—
173C4×Q8—71no—
174C4×Q8—71no—
175C4×Q8—141no—
176C4×Q8—141no—
177C4×Q8C2×D711yescharacteristic
178C4×D4C2×D711yescharacteristic
179#117—71no—
180#117—141no—
181#117—141no—
182#117C2×D711yescharacteristic
183#107—71no—
184#107—71no—
185#106—71no—
186#106—71no—
187C4⋊C8—71no—
188C4⋊C8—71no—
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⋊C8—71no—
198C22⋊C8—71no—
199#97—71no—
200C4×C8—71no—
201C22×C7—21no—
202C22×C7—21no—
203C22×C7—21no—
204C22×C7—41no—
205C22×C7—41no—
206C22×C7—41no—
207C22×C7C22×D411yescharacteristic
208C4×C7—21no—
209C4×C7—21no—
210C4×C7—21no—
211C4×C7—21no—
212C4×C7—21no—
213C4×C7—41no—
214C4×C7—41no—
215C4×C7—41no—
216C4×C7C2×#4111yescharacteristic
217C4×C7C2×#4111yescharacteristic
218Dic7—22no—
219Dic7—22no—
220Dic7—41no—
221Dic7—41no—
222C2×Q8—21no—
223C2×Q8—71no—
224C2×Q8—71no—
225C2×Q8—142no—
226C2×Q8—142no—
227C2×Q8—141no—
228C2×Q8C22×D711yescharacteristic
229C2×D4—21no—
230C2×D4C22×D711yescharacteristic
231C22×C4—21no—
232C22×C4—141no—
233C22×C4C22×D711yescharacteristic
234Q16—142no—
235Q16—142no—
236SD16—142no—
237SD16—142no—
238C2×C8—71no—
239C2×C8—71no—
240C2×C8—71no—
241C2×C8—71no—
242C4⋊C4—21no—
243C4⋊C4—21no—
244C4⋊C4—21no—
245C4⋊C4—71no—
246C4⋊C4—71no—
247C4⋊C4—141no—
248C4⋊C4—141no—
249C4⋊C4—141no—
250C4⋊C4—141no—
251C4⋊C4C22⋊D711yescharacteristic
252C4⋊C4C22⋊D711yescharacteristic
253C4⋊C4C22×D711yescharacteristic
254C4⋊C4C22×D711yescharacteristic
255C22⋊C4—21no—
256C22⋊C4—21no—
257C22⋊C4—21no—
258C22⋊C4—21no—
259C22⋊C4—21no—
260C22⋊C4—141no—
261C22⋊C4C22⋊D711yescharacteristic
262C22⋊C4C22⋊D711yescharacteristic
263C42—21no—
264C42—141no—
265C42—141no—
266C42—141no—
267C42—141no—
268C42—141no—
269C42C22×D711yescharacteristic
270C42C22×D711yescharacteristic
271C2×C7—41no—
272C2×C7—81no—
273C2×C7#52511yescharacteristic
274C2×C7#57711yescharacteristic
275C2×C7#57811yescharacteristic
276C23—21no—
277C23C2×C22⋊D711yescharacteristic
278Q8—21no—
279Q8—21no—
280Q8—72no—
281Q8—72no—
282Q8—81no—
283Q8—144no—
284Q8—141no—
285Q8—141no—
286D4—21no—
287D4—21no—
288D4—81no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—81no—
295C2×C4—144no—
296C2×C4—141no—
297C2×C4—141no—
298C2×C4—141no—
299C2×C4—141no—
300C2×C4C2×C22⋊D711yescharacteristic
301C2×C4C2×C22⋊D711yescharacteristic
302C2×C4C2×C22⋊D711yescharacteristic
303C2×C4C2×C22⋊D711yescharacteristic
304C2×C4C2×C22⋊D711yescharacteristic
305C2×C4C23×D711yescharacteristic
306C8—141no—
307C8—141no—
308C8—141no—
309C8—141no—
310C7#312111yescharacteristic, complemented, Sylow
311C22—21no—
312C22—21no—
313C22—21no—
314C22—41no—
315C22—41no—
316C22—41no—
317C22C22×C22⋊D711yescharacteristic
318C4—21no—
319C4—21no—
320C4—21no—
321C4—21no—
322C4—21no—
323C4—41no—
324C4—41no—
325C4—41no—
326C4—142no—
327C4—142no—
328C4—281no—
329C4—281no—
330C4D7×#4111yescharacteristic
331C4D7×#4111yescharacteristic
332C2—41no—
333C2—81no—
334C2#9051911yescharacteristic
335C2#9052711yescharacteristic
336C2#9052911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#542C2
socleC22×C7C22×D4
prime radicalC7×#121C22
hypercenter#542D7
hypocenterC7#3121
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ #117 ◁ #542

Complements

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

NH ≅ G/Ntype
2. #90207C2semidirect
3. #89866C2semidirect
4. #89863C2semidirect
5. #89826C2semidirect
6. #89826C2semidirect
7. #89822C2semidirect
8. #89817C2semidirect
11. #89802C2semidirect
12. C4×#1144C2semidirect
13. #89790C2semidirect
14. C4×#1142C2semidirect
15. #89772C2semidirect
310. C7#3121semidirect

Maximal subgroups

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

subgroupindex
C7×#5422
#902072
#898662
#898632
#898262
#898222
#898172
#898122
#898072
#898022
C4×#11442
#897902
C4×#11422
#897722
#31217