Finite Group Databasebeta

← #11740913

Subgroups — #11740913

Order 896.

334 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#542C211yesmaximal, characteristic
2#90209C211yesmaximal, characteristic, complemented
3#89866C211yesmaximal, characteristic, complemented
4#89863C211yesmaximal, characteristic, complemented
5#89826C211yesmaximal, characteristic, complemented
6#89826C211yesmaximal, characteristic, complemented
7#89824C211yesmaximal, characteristic, complemented
8#89823C211yesmaximal, characteristic, complemented
9#89814C211yesmaximal, characteristic
10#89813C211yesmaximal, characteristic
11#89803C211yesmaximal, characteristic, complemented
12#89794C211yesmaximal, characteristic, complemented
13#89791C211yesmaximal, characteristic, complemented
14#89781C211yesmaximal, characteristic, complemented
15#89771C211yesmaximal, characteristic, complemented
16C7×C4⋊Q8C2211yescharacteristic
17C7×#126—21no—
18C7×#124—21no—
19C7×#124C2211yescharacteristic
20C7×#123—21no—
21C7×C22⋊Q8—21no—
22C7×C22⋊Q8C2211yescharacteristic
23C7×#121C2211yescharacteristic
24C4×C7×Q8C2211yescharacteristic
25C4×C7×D4C2211yescharacteristic
26C7×#117C2211yescharacteristic
27C2×#1144C2211yescharacteristic
28C2×#1144C2211yescharacteristic
29C2×#1142C2211yescharacteristic
30C2×#1142C2211yescharacteristic
31#6487C2211yescharacteristic
32#6487C2211yescharacteristic
33C2×#1136C2211yescharacteristic
34C22×C7⋊C8C2211yescharacteristic
35#6452—21no—
36#6452—21no—
37#6442—21no—
38#6442—21no—
39#6433C2211yescharacteristic
40#6432C2211yescharacteristic
41C4×Dic14C2211yescharacteristic
42C4×Dic14C2211yescharacteristic
43#6409C2211yescharacteristic
44#6409C2211yescharacteristic
45#6406C2211yescharacteristic
46#6406C2211yescharacteristic
47#6384C2211yescharacteristic
48#6384C2211yescharacteristic
49#6384C2211yescharacteristic
50#6384C2211yescharacteristic
51#6382C2211yescharacteristic
52#6382C2211yescharacteristic
53#6381C2211yescharacteristic
54#6381C2211yescharacteristic
55C4⋊(C7⋊C8)C2211yescharacteristic
56C4⋊(C7⋊C8)C2211yescharacteristic
57C4⋊(C7⋊C8)C2211yescharacteristic
58C4⋊(C7⋊C8)C2211yescharacteristic
59#3258—71nomaximal, Sylow
60C2×C7×Q8—21no—
61C2×C7×Q8C2311yescharacteristic
62C2×C7×D4—21no—
63C2×C7×D4C2311yescharacteristic
64C22×C4×C7—21no—
65C22×C4×C7C2311yescharacteristic
66C2×Dic14C2311yescharacteristic
67C2×Dic14C2311yescharacteristic
68C7×C4⋊C4—21no—
69C7×C4⋊C4—21no—
70C7×C4⋊C4—21no—
71C7×C4⋊C4D411yescharacteristic
72C7×C4⋊C4D411yescharacteristic
73C7×C4⋊C4C2311yescharacteristic
74C7×C4⋊C4C2311yescharacteristic
75C7×C22⋊C4—21no—
76C7×C22⋊C4—21no—
77C7×C22⋊C4—21no—
78C7×C22⋊C4—21no—
79C7×C22⋊C4—21no—
80C7×C22⋊C4D411yescharacteristic
81C7×C22⋊C4D411yescharacteristic
82C42×C7—21no—
83C42×C7C2311yescharacteristic
84C42×C7C2311yescharacteristic
85C22⋊Dic7—21no—
86C22⋊Dic7—21no—
87#1144—41no—
88#1144—41no—
89#1142—41no—
90#1142—41no—
91C4⋊Dic7—21no—
92C4⋊Dic7—21no—
93C4⋊Dic7C2311yescharacteristic
94C4⋊Dic7C2311yescharacteristic
95Dic7⋊C4—21no—
96Dic7⋊C4—21no—
97Dic7⋊C4—21no—
98Dic7⋊C4—21no—
99Dic7⋊C4—21no—
100Dic7⋊C4—21no—
101C4×Dic7—21no—
102C4×Dic7—21no—
103#1136—41no—
104C2×C7⋊C8—41no—
105C2×C7⋊C8C2311yescharacteristic
106C2×C7⋊C8C2311yescharacteristic
107C2×C7⋊C8C2311yescharacteristic
108C2×C7⋊C8C2311yescharacteristic
109#542D711yescharacteristic
110#541—71no—
111#479—71no—
112#478—71no—
113#477—71no—
114#475—71no—
115#470—71no—
116#468—71no—
117#467—71no—
118#465—71no—
119#464—71no—
120#464—71no—
121#462—71no—
122#461—71no—
123#424—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—21no—
145C2×Dic7—21no—
146C2×Dic7—21no—
147C2×Dic7—21no—
148Dic14—21no—
149Dic14—21no—
150Dic14—21no—
151Dic14—21no—
152C7⋊C8—21no—
153C7⋊C8—21no—
154C7⋊C8—21no—
155C7⋊C8—21no—
156C2×Q16—71no—
157C2×Q16—71no—
158C2×SD16—71no—
159C2×SD16—71no—
160C2×M16—71no—
161C22×C8—71no—
162C4⋊Q8—71no—
163C4⋊Q8C2×D711yescharacteristic
164#126—21no—
165#126—141no—
166#126—141no—
167#125—71no—
168#125—141no—
169#125—141no—
170#124—21no—
171#124C2×D711yescharacteristic
172#123—21no—
173C22⋊Q8—21no—
174C22⋊Q8—71no—
175C22⋊Q8—71no—
176C22⋊Q8C2×D711yescharacteristic
177#121C2×D711yescharacteristic
178C4×Q8—71no—
179C4×Q8—71no—
180C4×Q8C2×D711yescharacteristic
181C4×D4C2×D711yescharacteristic
182#117C2×D711yescharacteristic
183#107—71no—
184#107—71no—
185#106—71no—
186#106—71no—
187C4⋊C8—71no—
188C4⋊C8—71no—
189C4⋊C8—71no—
190C4⋊C8—71no—
191#103—71no—
192#103—71no—
193#103—71no—
194#103—71no—
195#103—71no—
196#103—71no—
197#102—71no—
198#102—71no—
199C22×C7—21no—
200C22×C7—21no—
201C22×C7—21no—
202C22×C7—41no—
203C22×C7—41no—
204C22×C7—41no—
205C22×C7C22×D411yescharacteristic
206C4×C7—21no—
207C4×C7—21no—
208C4×C7—21no—
209C4×C7—21no—
210C4×C7—21no—
211C4×C7—41no—
212C4×C7—41no—
213C4×C7—41no—
214C4×C7H(5,𝔽2)11yescharacteristic
215C4×C72−1+411yescharacteristic
216Dic7—41no—
217Dic7—41no—
218Dic7—41no—
219Dic7—41no—
220C2×Q8—21no—
221C2×Q8—71no—
222C2×Q8—71no—
223C2×Q8C22×D711yescharacteristic
224C2×D4—21no—
225C2×D4C22×D711yescharacteristic
226C22×C4—21no—
227C22×C4C22×D711yescharacteristic
228Q16—281no—
229Q16—281no—
230SD16—281no—
231SD16—281no—
232M16—281no—
233C2×C8—71no—
234C2×C8—71no—
235C2×C8—71no—
236C2×C8—71no—
237C2×C8—281no—
238C4⋊C4—21no—
239C4⋊C4—21no—
240C4⋊C4—21no—
241C4⋊C4—71no—
242C4⋊C4—71no—
243C4⋊C4—141no—
244C4⋊C4—141no—
245C4⋊C4—141no—
246C4⋊C4—141no—
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⋊C4—141no—
262C22⋊C4C22⋊D711yescharacteristic
263C22⋊C4C22⋊D711yescharacteristic
264C42—21no—
265C42—141no—
266C42—141no—
267C42C22×D711yescharacteristic
268C42C22×D711yescharacteristic
269C2×C7—41no—
270C2×C7—81no—
271C2×C7#53711yescharacteristic
272C2×C7#57711yescharacteristic
273C2×C7#57811yescharacteristic
274C23—21no—
275C23C2×C22⋊D711yescharacteristic
276Q8—21no—
277Q8—21no—
278Q8—81no—
279Q8—141no—
280Q8—141no—
281Q8—141no—
282Q8—141no—
283D4—21no—
284D4—21no—
285D4—81no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—81no—
292C2×C4—141no—
293C2×C4—141no—
294C2×C4—141no—
295C2×C4—141no—
296C2×C4C2×C22⋊D711yescharacteristic
297C2×C4C2×C22⋊D711yescharacteristic
298C2×C4C2×C22⋊D711yescharacteristic
299C2×C4C2×C22⋊D711yescharacteristic
300C2×C4C2×C22⋊D711yescharacteristic
301C2×C4C23×D711yescharacteristic
302C8—141no—
303C8—141no—
304C8—141no—
305C8—141no—
306C7#325811yescharacteristic, complemented, Sylow
307C22—21no—
308C22—21no—
309C22—21no—
310C22—41no—
311C22—41no—
312C22—41no—
313C22C22×C22⋊D711yescharacteristic
314C4—21no—
315C4—21no—
316C4—21no—
317C4—21no—
318C4—21no—
319C4—41no—
320C4—41no—
321C4—41no—
322C4—281no—
323C4—281no—
324C4—281no—
325C4—281no—
326C4#655311yescharacteristic
327C4#655411yescharacteristic
328C2—41no—
329C2—81no—
330C2#9052411yescharacteristic
331C2#9052711yescharacteristic
332C2#9052911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#542C2
socleC22×C7C22×D4
prime radicalC7×#121C22
hypercenter#542D7
hypocenterC7#3258
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. #90209C2semidirect
3. #89866C2semidirect
4. #89863C2semidirect
5. #89826C2semidirect
6. #89826C2semidirect
7. #89824C2semidirect
8. #89823C2semidirect
11. #89803C2semidirect
12. #89794C2semidirect
13. #89791C2semidirect
14. #89781C2semidirect
15. #89771C2semidirect
306. C7#3258semidirect

Maximal subgroups

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

subgroupindex
C7×#5422
#902092
#898662
#898632
#898262
#898242
#898232
#898142
#898132
#898032
#897942
#897912
#897812
#897712
#32587