Finite Group Databasebeta

← #11735772

Subgroups — #11735772

Order 896.

358 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#400C211yesmaximal, characteristic, complemented
2#89766C211yesmaximal, characteristic
3#89764C211yesmaximal, characteristic
4#89756C211yesmaximal, characteristic, complemented
5#89744C211yesmaximal, characteristic, complemented
6#89448C211yesmaximal, characteristic
7#89447C211yesmaximal, characteristic
8#89446C211yesmaximal, characteristic
9#89446C211yesmaximal, characteristic
10#89442C211yesmaximal, characteristic
11#89433C211yesmaximal, characteristic, complemented
12#89431C211yesmaximal, characteristic, complemented
13#89429C211yesmaximal, characteristic, complemented
14#89429C211yesmaximal, characteristic, complemented
15#89427C211yesmaximal, characteristic, complemented
16C2×C7×C4⋊C4C2211yescharacteristic
17C2×C7×C4⋊C4C2211yescharacteristic
18C2×(C2×D7)⋊C4C2211yescharacteristic
19C2×(C2×D7)⋊C4C2211yescharacteristic
20C2×(C2×D7)⋊C4C2211yescharacteristic
21C2×(C2×D7)⋊C4C2211yescharacteristic
22C2×(C2×D7)⋊C4C2211yescharacteristic
23C2×(C2×D7)⋊C4C2211yescharacteristic
24C2×(C2×D7)⋊C4C2211yescharacteristic
25C2×C4⋊Dic7C2211yescharacteristic
26C2×C4⋊Dic7C2211yescharacteristic
27C2×C4⋊Dic7C2211yescharacteristic
28C2×C4⋊Dic7C2211yescharacteristic
29C2×Dic7⋊C4C2211yescharacteristic
30C2×Dic7⋊C4C2211yescharacteristic
31C2×Dic7⋊C4C2211yescharacteristic
32C2×Dic7⋊C4C2211yescharacteristic
33C2×Dic7⋊C4C2211yescharacteristic
34C2×Dic7⋊C4C2211yescharacteristic
35C2×Dic7⋊C4C2211yescharacteristic
36C2×C4×Dic7C2211yescharacteristic
37C7×#95C2211yescharacteristic
38C7×#95C2211yescharacteristic
39C7×#95C2211yescharacteristic
40C7×#95C2211yescharacteristic
41C7×#95C2211yescharacteristic
42C7⋊#95C2211yescharacteristic
43C7⋊#95C2211yescharacteristic
44C7⋊#95C2211yescharacteristic
45C7⋊#95C2211yescharacteristic
46C7⋊#95C2211yescharacteristic
47C7⋊#95C2211yescharacteristic
48C7⋊#95C2211yescharacteristic
49C7⋊#95C2211yescharacteristic
50C7⋊#95C2211yescharacteristic
51#2836—71nomaximal, Sylow
52C23×D7C2311yescharacteristic
53C22×C4×C7C2311yescharacteristic
54C22×C4×C7C2311yescharacteristic
55C22×C4×C7C2311yescharacteristic
56C22×C4×C7C2311yescharacteristic
57C22×C4×C7C2311yescharacteristic
58C22×C4×C7C2311yescharacteristic
59C22×C4×C7C2311yescharacteristic
60C22×Dic7C2311yescharacteristic
61C22×Dic7C2311yescharacteristic
62C22×Dic7C2311yescharacteristic
63C22×Dic7C2311yescharacteristic
64C22×Dic7C2311yescharacteristic
65C22×Dic7C2311yescharacteristic
66C22×Dic7C2311yescharacteristic
67C7×C4⋊C4—41no—
68C7×C4⋊C4—41no—
69(C2×D7)⋊C4—22no—
70(C2×D7)⋊C4—41no—
71(C2×D7)⋊C4—41no—
72(C2×D7)⋊C4—41no—
73(C2×D7)⋊C4—41no—
74(C2×D7)⋊C4—41no—
75(C2×D7)⋊C4—41no—
76C4⋊Dic7—41no—
77C4⋊Dic7—41no—
78C4⋊Dic7—41no—
79C4⋊Dic7—41no—
80Dic7⋊C4—22no—
81Dic7⋊C4—22no—
82Dic7⋊C4—41no—
83Dic7⋊C4—41no—
84Dic7⋊C4—41no—
85Dic7⋊C4—41no—
86Dic7⋊C4—41no—
87C4×Dic7—41no—
88#400—71no—
89#400D711yescharacteristic, complemented
90#399—71no—
91#398—71no—
92#398—71no—
93#398—71no—
94#398—71no—
95#397—71no—
96#397—71no—
97#397—71no—
98#396—71no—
99#396—71no—
100#389—71no—
101#388—71no—
102#387—71no—
103C23×C7C2411yescharacteristic
104C22×D7—21no—
105C22×D7—21no—
106C22×D7—21no—
107C22×D7—21no—
108C22×D7—21no—
109C22×D7—21no—
110C22×D7—21no—
111C2×C4×C7—21no—
112C2×C4×C7—21no—
113C2×C4×C7—21no—
114C2×C4×C7—21no—
115C2×C4×C7—21no—
116C2×C4×C7—21no—
117C2×C4×C7—21no—
118C2×C4×C7—21no—
119C2×C4×C7—21no—
120C2×C4×C7—21no—
121C2×C4×C7—21no—
122C2×C4×C7—21no—
123C2×C4×C7—21no—
124C2×C4×C7—21no—
125C2×C4×C7—21no—
126C2×C4×C7—21no—
127C2×C4×C7—21no—
128C2×C4×C7—21no—
129C2×C4×C7—21no—
130C2×C4×C7—21no—
131C2×C4×C7—21no—
132C2×Dic7—21no—
133C2×Dic7—21no—
134C2×Dic7—21no—
135C2×Dic7—21no—
136C2×Dic7—21no—
137C2×Dic7—21no—
138C2×Dic7—21no—
139C2×Dic7—21no—
140C2×Dic7—21no—
141C2×Dic7—21no—
142C2×Dic7—21no—
143C2×Dic7—21no—
144C2×Dic7—21no—
145C2×Dic7—21no—
146C2×Dic7—21no—
147C2×Dic7—21no—
148C2×Dic7—21no—
149C2×Dic7—21no—
150C2×Dic7—21no—
151C2×Dic7—21no—
152C2×Dic7#4112yes—
153C2×C4⋊C4—71no—
154C2×C4⋊C4—71no—
155C2×C4⋊C4—71no—
156C2×C4⋊C4—71no—
157C2×C4⋊C4—71no—
158C2×C4⋊C4—71no—
159C2×C4⋊C4—71no—
160C2×C4⋊C4—71no—
161C2×C4⋊C4—71no—
162C2×C4⋊C4—71no—
163C2×C4⋊C4—71no—
164C2×C4⋊C4C2×D711yescharacteristic
165C2×C4⋊C4C2×D711yescharacteristic
166C2×C22⋊C4—71no—
167C2×C22⋊C4—71no—
168C2×C22⋊C4—71no—
169C2×C22⋊C4—71no—
170C2×C22⋊C4—71no—
171C2×C22⋊C4—71no—
172C2×C22⋊C4—71no—
173C2×C42—71no—
174#95—71no—
175#95—71no—
176#95—71no—
177#95—71no—
178#95—71no—
179#95—71no—
180#95—71no—
181#95—71no—
182#95—71no—
183#95C2×D711yescharacteristic
184#95C2×D711yescharacteristic
185#95C2×D711yescharacteristic
186#95C2×D711yescharacteristic
187#95C2×D711yescharacteristic
188C22×C7C2×#4111yescharacteristic
189C22×C7H(5,𝔽2)11yescharacteristic
190C22×C7H(5,𝔽2)11yescharacteristic
191C22×C7H(5,𝔽2)11yescharacteristic
192C22×C72−1+411yescharacteristic
193C22×C72−1+411yescharacteristic
194C22×C72−1+411yescharacteristic
195C2×D7—41no—
196C2×D7—41no—
197C2×D7—41no—
198C2×D7—41no—
199C2×D7—41no—
200C2×D7—41no—
201C2×D7—41no—
202C4×C7—41no—
203C4×C7—41no—
204C4×C7—41no—
205C4×C7—41no—
206C4×C7—41no—
207C4×C7—41no—
208C4×C7—41no—
209Dic7—22no—
210Dic7—41no—
211Dic7—41no—
212Dic7—41no—
213Dic7—41no—
214Dic7—41no—
215Dic7—41no—
216C24—71no—
217C22×C4—71no—
218C22×C4—71no—
219C22×C4—71no—
220C22×C4—71no—
221C22×C4—71no—
222C22×C4—71no—
223C22×C4—71no—
224C22×C4C22×D711yescharacteristic
225C22×C4C22×D711yescharacteristic
226C22×C4C22×D711yescharacteristic
227C22×C4C22×D711yescharacteristic
228C22×C4C22×D711yescharacteristic
229C22×C4C22×D711yescharacteristic
230C22×C4C22×D711yescharacteristic
231C4⋊C4—41no—
232C4⋊C4—41no—
233C4⋊C4—142no—
234C4⋊C4—142no—
235C4⋊C4—281no—
236C4⋊C4—281no—
237C4⋊C4—281no—
238C4⋊C4—281no—
239C4⋊C4—281no—
240C4⋊C4—281no—
241C4⋊C4—281no—
242C4⋊C4—281no—
243C4⋊C4—281no—
244C22⋊C4—142no—
245C22⋊C4—281no—
246C22⋊C4—281no—
247C22⋊C4—281no—
248C22⋊C4—281no—
249C22⋊C4—281no—
250C22⋊C4—281no—
251C42—281no—
252C2×C7#53911yescharacteristic
253C2×C7#54011yescharacteristic
254C2×C7#54111yescharacteristic
255C2×C7#56211yescharacteristic
256C2×C7#56211yescharacteristic
257C2×C7#56311yescharacteristic
258C2×C7#56311yescharacteristic
259D7—81no—
260C23—141no—
261C23—141no—
262C23—141no—
263C23—141no—
264C23—141no—
265C23—141no—
266C23—141no—
267C23C23×D711yescharacteristic
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—72no—
290C2×C4—141no—
291C2×C4—141no—
292C2×C4—141no—
293C2×C4—141no—
294C2×C4—141no—
295C2×C4—141no—
296C2×C4—141no—
297C2×C4—141no—
298C2×C4—141no—
299C2×C4—141no—
300C2×C4—141no—
301C2×C4—141no—
302C2×C4—141no—
303C2×C4—141no—
304C2×C4—141no—
305C2×C4—141no—
306C2×C4—141no—
307C2×C4—141no—
308C2×C4—141no—
309C2×C4—141no—
310C7#283611yescharacteristic, complemented, Sylow
311C22—281no—
312C22—281no—
313C22—281no—
314C22—281no—
315C22—281no—
316C22—281no—
317C22—281no—
318C22#654811yescharacteristic
319C22#655111yescharacteristic
320C22D7×#4111yescharacteristic
321C22#655311yescharacteristic
322C22#655311yescharacteristic
323C22#655411yescharacteristic
324C22#655411yescharacteristic
325C4—41no—
326C4—41no—
327C4—41no—
328C4—41no—
329C4—41no—
330C4—41no—
331C4—41no—
332C4—142no—
333C4—281no—
334C4—281no—
335C4—281no—
336C4—281no—
337C4—281no—
338C4—281no—
339C2—561no—
340C2#9033711yescharacteristic
341C2#9035411yescharacteristic
342C2#9037511yescharacteristic
343C2#9038711yescharacteristic
344C2#9039111yescharacteristic
345C2#9039811yescharacteristic
346C2#9040311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#400C2
prime radicalC23×D7C23
hypercenter#400D7
hypocenterC7#2836
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
1. C7×#400C2semidirect
4. #89756C2semidirect
5. #89744C2semidirect
11. #89433C2semidirect
12. #89431C2semidirect
13. #89429C2semidirect
14. #89429C2semidirect
15. #89427C2semidirect
89. #400D7semidirect
310. C7#2836semidirect

Maximal subgroups

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

subgroupindex
C7×#4002
#897662
#897642
#897562
#897442
#894482
#894472
#894462
#894422
#894332
#894312
#894292
#894272
#28367