Finite Group Databasebeta

← #11734565

Subgroups — #11734565

Order 896.

724 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 724 conjugacy classes of subgroups — too many to draw. Only its 147 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#6501C211yesmaximal, characteristic, complemented
2C2×#6456C211yesmaximal, characteristic, complemented
3C2×#6447C212yesmaximal, complemented
4C7×#394C211yesmaximal, characteristic, complemented
5#89995C211yesmaximal, characteristic, complemented
6#89755C211yesmaximal, characteristic, complemented
7#89728C212yesmaximal, complemented
8#89728C212yesmaximal, complemented
9#89717C212yesmaximal, complemented
10#89446C211yesmaximal, characteristic, complemented
11D7×#95C211yesmaximal, characteristic, complemented
12C22×C7×D4C2211yescharacteristic
13C22×C22⋊D7C2212yes—
14C22×C22⋊D7C2212yes—
15C22×D28C2211yescharacteristic
16C22×C4×D7C2212yes—
17C22×C4×D7C2211yescharacteristic
18C22×C4×D7C2211yescharacteristic
19C2×C7×C4⋊C4C2211yescharacteristic
20C2×C7×C22⋊C4C2212yes—
21C2×C7×C22⋊C4C2212yes—
22C2×C22⋊Dic7C2212yes—
23C2×C22⋊Dic7C2212yes—
24#6501—24no—
25C2×(C2×D7)⋊C4C2212yes—
26C2×(C2×D7)⋊C4C2211yescharacteristic
27C2×(C2×D7)⋊C4C2211yescharacteristic
28C2×C4⋊Dic7C2211yescharacteristic
29C2×Dic7⋊C4C2212yes—
30C2×C4×Dic7C2211yescharacteristic
31#6456—24no—
32#6447—28no—
33C7×#95C2211yescharacteristic
34C7⋊#95C2212yes—
35C7⋊#95C2211yescharacteristic
36C7⋊#95C2212yes—
37C7⋊#95C2212yes—
38#2543—71nomaximal, Sylow
39C24×C7C2312yes—
40C23×D7C2311yescharacteristic
41C23×D7C2311yescharacteristic
42C2×C7×D4—22no—
43C2×C7×D4—42no—
44C22×C4×C7C2312yes—
45C22×C4×C7C2311yescharacteristic
46C22×C4×C7C2311yescharacteristic
47C22×C4×C7C2311yescharacteristic
48C2×C22⋊D7—24no—
49C2×C22⋊D7—24no—
50C2×C22⋊D7—24no—
51C2×C22⋊D7—24no—
52C2×C22⋊D7—24no—
53C2×C22⋊D7—42no—
54C22×Dic7C2311yescharacteristic
55C22×Dic7C2312yes—
56C22×Dic7C2311yescharacteristic
57C22×Dic7C2312yes—
58C2×D28—22no—
59C2×D28—22no—
60C2×D28D414yescomplemented
61C2×C4×D7—24no—
62C2×C4×D7—22no—
63C2×C4×D7—24no—
64C2×C4×D7—42no—
65C2×C4×D7—42no—
66C2×C4×D7—42no—
67C2×C4×D7D414yescomplemented
68C7×C4⋊C4—22no—
69C7×C22⋊C4—24no—
70C7×C22⋊C4—42no—
71C22⋊Dic7—24no—
72C22⋊Dic7—24no—
73(C2×D7)⋊C4—24no—
74(C2×D7)⋊C4—22no—
75(C2×D7)⋊C4—22no—
76C4⋊Dic7—22no—
77Dic7⋊C4—24no—
78C4×Dic7—22no—
79C2×#121—71no—
80C2×#121—72no—
81C2×C4×D4—71no—
82#398—71no—
83#397—71no—
84#394D711yescharacteristic, complemented
85#392—71no—
86#388—72no—
87#386—72no—
88#386—72no—
89C2×#95—71no—
90C23×C7—22no—
91C23×C7—22no—
92C23×C7—22no—
93C23×C7—22no—
94C23×C7—22no—
95C23×C7—22no—
96C23×C7—22no—
97C23×C7C2411yescharacteristic
98C22×D7—22no—
99C22×D7—22no—
100C22×D7—21no—
101C22×D7—21no—
102C22×D7—22no—
103C22×D7—22no—
104C22×D7C2×D412yes—
105C22×D7C2×D412yes—
106C22×D7C2×D412yes—
107C22×D7#4112yes—
108C7×D4—44no—
109C2×C4×C7—21no—
110C2×C4×C7—21no—
111C2×C4×C7—22no—
112C2×C4×C7—22no—
113C2×C4×C7—21no—
114C2×C4×C7—22no—
115C2×C4×C7—22no—
116C2×C4×C7—22no—
117C2×C4×C7C2×D412yes—
118C2×C4×C7C2×D412yes—
119C22⋊D7—48no—
120C22⋊D7—48no—
121C2×Dic7—22no—
122C2×Dic7—22no—
123C2×Dic7—22no—
124C2×Dic7—21no—
125C2×Dic7—22no—
126C2×Dic7—22no—
127C2×Dic7—22no—
128C2×Dic7—22no—
129C2×Dic7C2×D412yes—
130C2×Dic7#4114yes—
131D28—28no—
132C4×D7—44no—
133C4×D7—48no—
134C4×D7—44no—
135C22×D4—72no—
136C22×D4—72no—
137C22×D4—71no—
138C22×D4C2×D711yescharacteristic
139C23×C4—71no—
140C23×C4—72no—
141C23×C4—71no—
142#121—148no—
143#121—144no—
144C4×D4—144no—
145C2×C4⋊C4—72no—
146C2×C4⋊C4—71no—
147C2×C4⋊C4C2×D711yescharacteristic
148C2×C22⋊C4—72no—
149C2×C22⋊C4—72no—
150C2×C22⋊C4—71no—
151C2×C22⋊C4—71no—
152C2×C22⋊C4—72no—
153C2×C22⋊C4C2×D712yes—
154C2×C22⋊C4C2×D712yes—
155C2×C42—71no—
156#95—72no—
157#95—71no—
158#95—72no—
159#95—72no—
160#95C2×D711yescharacteristic
161C22×C7—42no—
162C22×C7—42no—
163C22×C7—42no—
164C22×C7—42no—
165C22×C7—42no—
166C22×C7—42no—
167C22×C7—42no—
168C22×C7C22×D411yescharacteristic
169C22×C7C22×D411yescharacteristic
170C22×C7C2×#4111yescharacteristic
171C22×C7C2×#4112yes—
172C22×C7H(5,𝔽2)12yes—
173C2×D7—22no—
174C2×D7—22no—
175C2×D7—24no—
176C2×D7—24no—
177C2×D7—22no—
178C2×D7—24no—
179C2×D7—41no—
180C2×D7—41no—
181C2×D7—42no—
182C2×D7#12114yes—
183C4×C7—22no—
184C4×C7—22no—
185C4×C7—41no—
186C4×C7—42no—
187Dic7—22no—
188Dic7—24no—
189Dic7—42no—
190Dic7—41no—
191C24—71no—
192C24—71no—
193C24C22×D712yes—
194C2×D4—22no—
195C2×D4—42no—
196C2×D4—74no—
197C2×D4—142no—
198C2×D4—142no—
199C2×D4—144no—
200C2×D4—144no—
201C2×D4—144no—
202C2×D4—144no—
203C2×D4—144no—
204C2×D4—282no—
205C22×C4—72no—
206C22×C4—71no—
207C22×C4—74no—
208C22×C4—71no—
209C22×C4—72no—
210C22×C4—142no—
211C22×C4—144no—
212C22×C4—144no—
213C22×C4—282no—
214C22×C4—282no—
215C22×C4—282no—
216C22×C4C22×D712yes—
217C22×C4C22×D711yescharacteristic
218C22×C4C22×D711yescharacteristic
219C22×C4C22×D711yescharacteristic
220C4⋊C4—22no—
221C4⋊C4—144no—
222C4⋊C4—142no—
223C22⋊C4—24no—
224C22⋊C4—42no—
225C22⋊C4—142no—
226C22⋊C4—144no—
227C22⋊C4—144no—
228C22⋊C4—144no—
229C22⋊C4—142no—
230C42—142no—
231C2×C7—82no—
232C2×C7C2×#12111yescharacteristic
233C2×C7#53211yescharacteristic
234C2×C7#53411yescharacteristic
235C2×C7#54612yes—
236C2×C7#55312yes—
237D7—24no—
238D7—42no—
239C23—22no—
240C23—22no—
241C23—22no—
242C23—22no—
243C23—22no—
244C23—22no—
245C23—22no—
246C23—72no—
247C23—72no—
248C23—72no—
249C23—72no—
250C23—142no—
251C23—142no—
252C23—141no—
253C23—142no—
254C23—141no—
255C23—142no—
256C23C23×D711yescharacteristic
257D4—44no—
258D4—148no—
259D4—288no—
260D4—288no—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—21no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—72no—
270C2×C4—74no—
271C2×C4—142no—
272C2×C4—142no—
273C2×C4—142no—
274C2×C4—141no—
275C2×C4—142no—
276C2×C4—142no—
277C2×C4—142no—
278C2×C4—142no—
279C2×C4—284no—
280C2×C4—288no—
281C2×C4—284no—
282C2×C4D4×D712yes—
283C2×C4D4×D712yes—
284C7#254311yescharacteristic, complemented, Sylow
285C22—42no—
286C22—42no—
287C22—42no—
288C22—42no—
289C22—42no—
290C22—42no—
291C22—42no—
292C22—74no—
293C22—142no—
294C22—144no—
295C22—144no—
296C22—142no—
297C22—142no—
298C22—144no—
299C22—281no—
300C22—282no—
301C22—281no—
302C22C2×D4×D711yescharacteristic
303C22C2×D4×D711yescharacteristic
304C22#654812yes—
305C22D7×#4111yescharacteristic
306C22D7×#4112yes—
307C4—22no—
308C4—22no—
309C4—41no—
310C4—42no—
311C4—142no—
312C4—144no—
313C4—282no—
314C4—281no—
315C2—82no—
316C2—144no—
317C2—282no—
318C2#9028011yescharacteristic
319C2D7×#12111yescharacteristic
320C2#9030312yes—
321C2#9035212yes—
322C2#9035911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#394C2
hypercenter#394D7
hypocenterC7#2543
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×#6501C2semidirect
2. C2×#6456C2semidirect
3. C2×#6447C2semidirect
4. C7×#394C2semidirect
5. #89995C2semidirect
6. #89755C2semidirect
7. #89728C2semidirect
8. #89728C2semidirect
9. #89717C2semidirect
10. #89446C2semidirect
11. D7×#95C2semidirect
60. C2×D28D4semidirect
67. C2×C4×D7D4semidirect
84. #394D7semidirect
284. C7#2543semidirect

Maximal subgroups

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

subgroupindex
C2×#65012
C2×#64562
C2×#64472
C7×#3942
#899952
#897552
#897282
#897172
#894462
D7×#952
#25437