Finite Group Databasebeta

← #11740197

Subgroups — #11740197

Order 896.

648 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×C22⋊Q8C211yesmaximal, characteristic, complemented
2C2×#6509C211yesmaximal, characteristic, complemented
3C2×C4×C22⋊D7C211yesmaximal, characteristic
4C2×D7×C4⋊C4C211yesmaximal, characteristic, complemented
5#90002C211yesmaximal, characteristic, complemented
6#89984C211yesmaximal, characteristic, complemented
7#89759C212yesmaximal, complemented
8#89757C211yesmaximal, characteristic, complemented
9#89752C212yesmaximal, complemented
10#89728C212yesmaximal
11#89720C212yesmaximal, complemented
12C22×C7×Q8C2211yescharacteristic
13C23×C4×C7C2211yescharacteristic
14C22×C22⋊D7C2211yescharacteristic
15C22×C4×D7C2212yes—
16C22×C4×D7C2211yescharacteristic
17C7×C22⋊Q8—24no—
18C2×C7×C4⋊C4C2212yes—
19C2×C7×C4⋊C4C2211yescharacteristic
20C2×C7×C22⋊C4C2212yes—
21C2×C22⋊Dic7C2211yescharacteristic
22C2×C22⋊Dic7C2212yes—
23#6509—24no—
24C4×C22⋊D7—24no—
25C2×(C2×D7)⋊C4C2211yescharacteristic
26C2×(C2×D7)⋊C4C2212yes—
27C2×C4⋊Dic7C2211yescharacteristic
28C2×C4⋊Dic7C2211yescharacteristic
29C2×C4⋊Dic7C2212yes—
30C2×Dic7⋊C4C2212yes—
31C2×Dic7⋊C4C2211yescharacteristic
32C2×Dic7⋊C4C2212yes—
33C2×C4×Dic7C2211yescharacteristic
34D7×C4⋊C4—24no—
35C7⋊#95C2212yes—
36C7⋊#95C2212yes—
37C7⋊#95C2212yes—
38C7⋊#95C2212yes—
39#2417—71nomaximal, Sylow
40C24×C7C2311yescharacteristic
41C23×D7C2311yescharacteristic
42C2×C7×Q8—22no—
43C2×C7×Q8—22no—
44C2×C7×Q8—22no—
45C22×C4×C7—22no—
46C22×C4×C7—22no—
47C22×C4×C7—22no—
48C22×C4×C7C2312yes—
49C22×C4×C7C2311yescharacteristic
50C22×C4×C7C2312yes—
51C22×C4×C7C2311yescharacteristic
52C2×C22⋊D7—22no—
53C2×C22⋊D7—22no—
54C2×C22⋊D7Q814yescomplemented
55C22×Dic7C2312yes—
56C22×Dic7C2311yescharacteristic
57C22×Dic7C2312yes—
58C22×Dic7C2311yescharacteristic
59C22×Dic7C2311yescharacteristic
60C2×C4×D7—22no—
61C2×C4×D7—22no—
62C2×C4×D7—24no—
63C2×C4×D7—24no—
64C2×C4×D7—42no—
65C2×C4×D7D414yes—
66C7×C4⋊C4—24no—
67C7×C4⋊C4D414yescomplemented
68C7×C22⋊C4—24no—
69C22⋊Dic7—22no—
70C22⋊Dic7—42no—
71(C2×D7)⋊C4—22no—
72(C2×D7)⋊C4—24no—
73C4⋊Dic7—22no—
74C4⋊Dic7—22no—
75C4⋊Dic7—42no—
76Dic7⋊C4—24no—
77Dic7⋊C4—22no—
78Dic7⋊C4—24no—
79C4×Dic7—22no—
80C2×C22⋊Q8—71no—
81C2×C22⋊Q8D711yescharacteristic, complemented
82C2×C4×D4—71no—
83C22×C4⋊C4—71no—
84#398—72no—
85#396—72no—
86#391—71no—
87#389—71no—
88#386—72no—
89#380—71no—
90#380—72no—
91C23×C7—21no—
92C23×C7—21no—
93C23×C7—21no—
94C23×C7—21no—
95C23×C7—21no—
96C23×C7—21no—
97C23×C7C2×Q812yes—
98C23×C7C2411yescharacteristic
99C22×D7—21no—
100C22×D7—21no—
101C22×D7—21no—
102C22×D7—21no—
103C22×D7C2×D412yes—
104C22×D7C2×Q812yes—
105C22×D7#4112yes—
106C7×Q8—44no—
107C2×C4×C7—22no—
108C2×C4×C7—22no—
109C2×C4×C7—21no—
110C2×C4×C7—22no—
111C2×C4×C7—22no—
112C2×C4×C7—21no—
113C2×C4×C7—21no—
114C2×C4×C7—22no—
115C2×C4×C7—44no—
116C2×C4×C7C2×D412yes—
117C2×C4×C7C2×D414yes—
118C2×C4×C7C2×D412yes—
119C2×C4×C7#4112yes—
120C22⋊D7—28no—
121C2×Dic7—21no—
122C2×Dic7—21no—
123C2×Dic7—21no—
124C2×Dic7—21no—
125C2×Dic7—21no—
126C2×Dic7—22no—
127C2×Dic7—22no—
128C2×Dic7—22no—
129C2×Dic7—21no—
130C2×Dic7—22no—
131C2×Dic7—22no—
132C2×Dic7—21no—
133C2×Dic7—22no—
134C2×Dic7C2×D412yes—
135C2×Dic7C2×Q812yes—
136C4×D7—28no—
137C4×D7—48no—
138C22×Q8C2×D711yescharacteristic
139C22×D4—71no—
140C23×C4—71no—
141C23×C4—72no—
142C23×C4C2×D711yescharacteristic
143C22⋊Q8—24no—
144C22⋊Q8—144no—
145C4×D4—144no—
146C2×C4⋊C4—72no—
147C2×C4⋊C4—71no—
148C2×C4⋊C4—71no—
149C2×C4⋊C4—72no—
150C2×C4⋊C4—71no—
151C2×C4⋊C4—72no—
152C2×C4⋊C4—144no—
153C2×C4⋊C4C2×D712yes—
154C2×C4⋊C4C2×D711yescharacteristic
155C2×C22⋊C4—71no—
156C2×C22⋊C4—71no—
157C2×C22⋊C4—72no—
158C2×C22⋊C4—72no—
159C2×C22⋊C4C2×D712yes—
160C2×C42—71no—
161#95—72no—
162#95—72no—
163#95—72no—
164#95—72no—
165C22×C7—22no—
166C22×C7—22no—
167C22×C7—22no—
168C22×C7—41no—
169C22×C7—41no—
170C22×C7—41no—
171C22×C7—41no—
172C22×C7C22×D411yescharacteristic
173C22×C7C22×D411yescharacteristic
174C22×C7C22×Q811yescharacteristic
175C22×C7C2×#4111yescharacteristic
176C22×C7C2×#4111yescharacteristic
177C22×C7H(5,𝔽2)11yescharacteristic
178C22×C72−1+411yescharacteristic
179C2×D7—22no—
180C2×D7—22no—
181C2×D7—22no—
182C2×D7—22no—
183C2×D7—22no—
184C2×D7—22no—
185C2×D7C22⋊Q814yescomplemented
186C4×C7—24no—
187C4×C7—41no—
188C4×C7—42no—
189C4×C7#12114yes—
190Dic7—22no—
191Dic7—22no—
192Dic7—41no—
193Dic7—42no—
194Dic7—42no—
195C24—71no—
196C24C22×D711yescharacteristic
197C2×Q8—22no—
198C2×Q8—22no—
199C2×Q8—22no—
200C2×D4—74no—
201C2×D4—142no—
202C2×D4—142no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—71no—
207C22×C4—71no—
208C22×C4—74no—
209C22×C4—72no—
210C22×C4—71no—
211C22×C4—72no—
212C22×C4—144no—
213C22×C4—142no—
214C22×C4—142no—
215C22×C4—144no—
216C22×C4—282no—
217C22×C4C22×D712yes—
218C22×C4C22×D711yescharacteristic
219C22×C4C22×D712yes—
220C22×C4C22×D711yescharacteristic
221C4⋊C4—24no—
222C4⋊C4—142no—
223C4⋊C4—144no—
224C4⋊C4—142no—
225C4⋊C4—144no—
226C4⋊C4—142no—
227C4⋊C4—282no—
228C4⋊C4C22⋊D714yescomplemented
229C22⋊C4—24no—
230C22⋊C4—142no—
231C22⋊C4—144no—
232C22⋊C4—142no—
233C22⋊C4—282no—
234C42—142no—
235C2×C7—42no—
236C2×C7C2×#12111yescharacteristic
237C2×C7C2×C22⋊Q811yescharacteristic
238C2×C7#53711yescharacteristic
239C2×C7#54611yescharacteristic
240C2×C7D4×Q811yescharacteristic
241C2×C7#55211yescharacteristic
242C2×C7#55411yescharacteristic
243D7—24no—
244C23—21no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—72no—
251C23—72no—
252C23—72no—
253C23—141no—
254C23—141no—
255C23—141no—
256C23—141no—
257C23Q8×D712yes—
258C23C23×D711yescharacteristic
259Q8—44no—
260D4—148no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—21no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—21no—
269C2×C4—44no—
270C2×C4—72no—
271C2×C4—72no—
272C2×C4—141no—
273C2×C4—141no—
274C2×C4—141no—
275C2×C4—141no—
276C2×C4—148no—
277C2×C4—141no—
278C2×C4—142no—
279C2×C4—141no—
280C2×C4—142no—
281C2×C4—142no—
282C2×C4—142no—
283C2×C4—142no—
284C2×C4—141no—
285C2×C4—142no—
286C2×C4—288no—
287C2×C4D4×D712yes—
288C2×C4#115912yes—
289C2×C4C2×C22⋊D714yes—
290C2×C4C2×C22⋊D712yes—
291C7#241711yescharacteristic, complemented, Sylow
292C22—22no—
293C22—22no—
294C22—22no—
295C22—41no—
296C22—41no—
297C22—41no—
298C22—41no—
299C22—74no—
300C22—142no—
301C22—142no—
302C22—142no—
303C22—142no—
304C22—142no—
305C22—142no—
306C22C2×D4×D711yescharacteristic
307C22C2×#115911yescharacteristic
308C22C2×Q8×D711yescharacteristic
309C22#655111yescharacteristic
310C22D7×#4111yescharacteristic
311C22#655311yescharacteristic
312C22C22×C22⋊D711yescharacteristic
313C4—24no—
314C4—41no—
315C4—42no—
316C4—142no—
317C4—142no—
318C4—282no—
319C4—281no—
320C4—282no—
321C4#650114yes—
322C2—42no—
323C2—144no—
324C2D7×C22⋊Q811yescharacteristic
325C2#9032011yescharacteristic
326C2#9032611yescharacteristic
327C2#9033011yescharacteristic
328C2C2×#650111yescharacteristic
329C2Q8×C22⋊D711yescharacteristic
330C2#9052011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC2×C7×C22⋊Q8C2
prime radicalC22×C22⋊D7C22
hypercenterC2×C22⋊Q8D7
hypocenterC7#2417
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×C7×C22⋊Q8C2semidirect
2. C2×#6509C2semidirect
4. C2×D7×C4⋊C4C2semidirect
5. #90002C2semidirect
6. #89984C2semidirect
7. #89759C2semidirect
8. #89757C2semidirect
9. #89752C2semidirect
11. #89720C2semidirect
54. C2×C22⋊D7Q8semidirect
67. C7×C4⋊C4D4semidirect
81. C2×C22⋊Q8D7semidirect
185. C2×D7C22⋊Q8semidirect
228. C4⋊C4C22⋊D7semidirect
291. C7#2417semidirect

Maximal subgroups

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

subgroupindex
C2×C7×C22⋊Q82
C2×#65092
C2×C4×C22⋊D72
C2×D7×C4⋊C42
#900022
#899842
#897592
#897572
#897522
#897282
#897202
#24177