Finite Group Databasebeta

← #11740097

Subgroups — #11740097

Order 896.

952 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 952 conjugacy classes of subgroups — too many to draw. Only its 169 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
1C22×#1157C211yesmaximal, characteristic, complemented
2C2×C7×#121C211yesmaximal, characteristic, complemented
3C2×#6503C211yesmaximal, characteristic, complemented
4C2×#6499C211yesmaximal, characteristic, complemented
5#89995C214yesmaximal, complemented
6C4×C22⋊Dic7C211yesmaximal, characteristic, complemented
7#89760C211yesmaximal, characteristic, complemented
8#89746C211yesmaximal, characteristic, complemented
9#89728C214yesmaximal, complemented
10C22×C7×D4C2211yescharacteristic
11C22×C7×D4C2212yes—
12C23×C4×C7C2211yescharacteristic
13C22×C22⋊D7C2212yes—
14C22×C22⋊D7C2214yes—
15C2×#1157—24no—
16C2×#1157—24no—
17C2×#1157—24no—
18C22×D28C2211yescharacteristic
19C22×C4×D7C2212yes—
20C22×Dic14C2211yescharacteristic
21C7×#121—24no—
22C2×C7×C4⋊C4C2211yescharacteristic
23C2×C7×C22⋊C4C2212yes—
24C2×C22⋊Dic7C2214yes—
25C2×C22⋊Dic7C2212yes—
26#6503—24no—
27#6499—24no—
28C2×(C2×D7)⋊C4C2214yes—
29C2×C4×Dic7C2211yescharacteristic
30C2×C4×Dic7C2211yescharacteristic
31C7⋊#95C2212yes—
32C7⋊#95C2214yes—
33#2431—71nomaximal, Sylow
34C24×C7C2312yes—
35C24×C7C2311yescharacteristic
36C23×D7C2312yes—
37C2×C7×D4—24no—
38C2×C7×D4—22no—
39C2×C7×D4—22no—
40C2×C7×D4—22no—
41C2×C7×D4—44no—
42C22×C4×C7—22no—
43C22×C4×C7—22no—
44C22×C4×C7D414yescomplemented
45C22×C4×C7C2312yes—
46C22×C4×C7C2311yescharacteristic
47C22×C4×C7C2311yescharacteristic
48C2×C22⋊D7—24no—
49C2×C22⋊D7—24no—
50C2×C22⋊D7—28no—
51C2×C22⋊D7—24no—
52C2×C22⋊D7—28no—
53C2×C22⋊D7—44no—
54C22×Dic7C2312yes—
55C22×Dic7C2312yes—
56C22×Dic7C2312yes—
57#1157—232no—
58C2×D28—22no—
59C2×D28—22no—
60C2×D28D414yescomplemented
61C2×C4×D7—24no—
62C2×C4×D7—24no—
63C2×C4×D7—24no—
64C2×Dic14—22no—
65C2×Dic14—22no—
66C2×Dic14D414yescomplemented
67C7×C4⋊C4—22no—
68C7×C22⋊C4—24no—
69C22⋊Dic7—28no—
70C22⋊Dic7—24no—
71(C2×D7)⋊C4—44no—
72C4×Dic7—22no—
73C4×Dic7—22no—
74C22×#41—71no—
75C2×#127—71no—
76C2×#124—71no—
77C2×#121D711yescharacteristic, complemented
78#392—74no—
79#391—71no—
80#390—71no—
81#386—74no—
82C4×C22⋊C4—71no—
83C23×C7—21no—
84C23×C7—21no—
85C23×C7—22no—
86C23×C7—22no—
87C23×C7—22no—
88C23×C7—22no—
89C23×C7—22no—
90C23×C7—24no—
91C23×C7—24no—
92C23×C7C2×D412yes—
93C23×C7C2411yescharacteristic
94C22×D7—22no—
95C22×D7—22no—
96C22×D7—24no—
97C22×D7—24no—
98C22×D7C2×D414yes—
99C7×D4—48no—
100C7×D4—44no—
101C2×C4×C7—21no—
102C2×C4×C7—21no—
103C2×C4×C7—28no—
104C2×C4×C7—22no—
105C2×C4×C7—24no—
106C2×C4×C7C2×D412yes—
107C2×C4×C7C2×D412yes—
108C2×C4×C7C2×D412yes—
109C2×C4×C7C2×D412yes—
110C22⋊D7—216no—
111C22⋊D7—416no—
112C2×Dic7—22no—
113C2×Dic7—22no—
114C2×Dic7—22no—
115C2×Dic7—22no—
116C2×Dic7—22no—
117C2×Dic7—22no—
118C2×Dic7—22no—
119C2×Dic7C2×D414yes—
120C2×Dic7#4114yescomplemented
121D28—28no—
122C4×D7—216no—
123Dic14—28no—
124C2×#41—144no—
125C2×#41—144no—
126C2×#41—144no—
127C22×Q8—71no—
128C22×D4—72no—
129C22×D4—74no—
130C22×D4—71no—
131C22×D4C2×D711yescharacteristic
132C22×D4C2×D712yes—
133C23×C4—72no—
134C23×C4C2×D711yescharacteristic
135#127—144no—
136#124—144no—
137#121—24no—
138C2×C4⋊C4C2×D711yescharacteristic
139C2×C22⋊C4—74no—
140C2×C22⋊C4—72no—
141C2×C22⋊C4—74no—
142C2×C22⋊C4C2×D712yes—
143C2×C42—71no—
144C2×C42—71no—
145#95—72no—
146#95—74no—
147C22×C7—22no—
148C22×C7—22no—
149C22×C7—22no—
150C22×C7—42no—
151C22×C7—42no—
152C22×C7—42no—
153C22×C7—42no—
154C22×C7—42no—
155C22×C7—44no—
156C22×C7—44no—
157C22×C7C22×D411yescharacteristic
158C22×C7C22×D411yescharacteristic
159C22×C7C22×D411yescharacteristic
160C22×C7C2×#4112yes—
161C22×C7H(5,𝔽2)12yes—
162C2×D7—24no—
163C2×D7—28no—
164C2×D7—42no—
165C2×D7—42no—
166C2×D7—44no—
167C4×C7—22no—
168C4×C7—42no—
169C4×C7#12014yes—
170Dic7—24no—
171Dic7—24no—
172Dic7—42no—
173C24—72no—
174C24C22×D712yes—
175C24C22×D711yescharacteristic
176#41—1432no—
177C2×Q8—74no—
178C2×Q8—142no—
179C2×Q8—142no—
180C2×D4—24no—
181C2×D4—22no—
182C2×D4—22no—
183C2×D4—22no—
184C2×D4—44no—
185C2×D4—74no—
186C2×D4—144no—
187C2×D4—148no—
188C2×D4—144no—
189C2×D4—144no—
190C2×D4—148no—
191C2×D4—142no—
192C2×D4—142no—
193C2×D4—284no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—72no—
197C22×C4—72no—
198C22×C4—72no—
199C22×C4—144no—
200C22×C4—144no—
201C22×C4—144no—
202C22×C4C22⋊D714yescomplemented
203C22×C4C22×D712yes—
204C22×C4C22×D711yescharacteristic
205C22×C4C22×D711yescharacteristic
206C4⋊C4—22no—
207C22⋊C4—24no—
208C22⋊C4—144no—
209C22⋊C4—148no—
210C22⋊C4—284no—
211C42—142no—
212C42—142no—
213C2×C7—42no—
214C2×C7—82no—
215C2×C7C2×#12011yescharacteristic
216C2×C7#53211yescharacteristic
217C2×C7#53511yescharacteristic
218C2×C7#54612yes—
219C2×C7#55012yes—
220D7—44no—
221C23—21no—
222C23—21no—
223C23—22no—
224C23—22no—
225C23—22no—
226C23—22no—
227C23—24no—
228C23—24no—
229C23—22no—
230C23—74no—
231C23—142no—
232C23—144no—
233C23—144no—
234C23—142no—
235C23C2×C22⋊D712yes—
236C23C23×D711yescharacteristic
237Q8—148no—
238D4—48no—
239D4—44no—
240D4—1416no—
241D4—148no—
242D4—2816no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—28no—
246C2×C4—24no—
247C2×C4—22no—
248C2×C4—74no—
249C2×C4—74no—
250C2×C4—1416no—
251C2×C4—142no—
252C2×C4—142no—
253C2×C4—142no—
254C2×C4—142no—
255C2×C4—142no—
256C2×C4—142no—
257C2×C4—142no—
258C2×C4D4×D712yes—
259C2×C4D4×D712yes—
260C2×C4C2×C22⋊D712yes—
261C2×C4C2×C22⋊D712yes—
262C7#243111yescharacteristic, complemented, Sylow
263C22—22no—
264C22—22no—
265C22—22no—
266C22—42no—
267C22—42no—
268C22—42no—
269C22—42no—
270C22—42no—
271C22—44no—
272C22—44no—
273C22—144no—
274C22—148no—
275C22—282no—
276C22—282no—
277C22—284no—
278C22C2×D4×D711yescharacteristic
279C22C2×D4×D711yescharacteristic
280C22#654812yes—
281C22D7×#4112yes—
282C22C22×C22⋊D711yescharacteristic
283C4—22no—
284C4—42no—
285C4—144no—
286C4—144no—
287C4—282no—
288C4#650014yes—
289C2—42no—
290C2—82no—
291C2—284no—
292C2#9029012yes—
293C2#9030312yes—
294C2C2×#650011yescharacteristic
295C2#9049611yescharacteristic
296C2#9052311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC2×C7×#121C2
hypercenterC2×#121D7
hypocenterC7#2431
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C22×#1157C2semidirect
2. C2×C7×#121C2semidirect
3. C2×#6503C2semidirect
4. C2×#6499C2semidirect
5. #89995C2semidirect
6. C4×C22⋊Dic7C2semidirect
7. #89760C2semidirect
8. #89746C2semidirect
9. #89728C2semidirect
44. C22×C4×C7D4semidirect
60. C2×D28D4semidirect
66. C2×Dic14D4semidirect
77. C2×#121D7semidirect
120. C2×Dic7#41semidirect
202. C22×C4C22⋊D7semidirect
262. C7#2431semidirect

Maximal subgroups

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

subgroupindex
C22×#11572
C2×C7×#1212
C2×#65032
C2×#64992
#899952
C4×C22⋊Dic72
#897602
#897462
#897282
#24317