Finite Group Databasebeta

← #11734654

Subgroups — #11734654

Order 896.

534 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 534 conjugacy classes of subgroups — too many to draw. Only its 133 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×#6510C211yesmaximal, characteristic, complemented
2C2×#6461C211yesmaximal, characteristic, complemented
3C7×#395C211yesmaximal, characteristic, complemented
4#90003C211yesmaximal, characteristic, complemented
5#89766C212yesmaximal, complemented
6#89763C211yesmaximal, characteristic, complemented
7#89760C212yesmaximal, complemented
8#89444C212yesmaximal, complemented
9#89443C211yesmaximal, characteristic
10#89441C212yesmaximal, complemented
11#89421C211yesmaximal, characteristic, complemented
12C22×C7×Q8C2211yescharacteristic
13C22×D28C2212yes—
14C22×D28C2211yescharacteristic
15C2×C7×C4⋊C4C2211yescharacteristic
16C2×C7×C4⋊C4C2212yes—
17#6510—24no—
18C2×(C2×D7)⋊C4C2211yescharacteristic
19C2×(C2×D7)⋊C4C2211yescharacteristic
20C2×(C2×D7)⋊C4C2211yescharacteristic
21C2×(C2×D7)⋊C4C2212yes—
22C2×(C2×D7)⋊C4C2212yes—
23C2×(C2×D7)⋊C4C2212yes—
24C2×(C2×D7)⋊C4C2212yes—
25C2×(C2×D7)⋊C4C2212yes—
26C2×(C2×D7)⋊C4C2212yes—
27C2×C4⋊Dic7C2211yescharacteristic
28C2×Dic7⋊C4C2212yes—
29C2×Dic7⋊C4C2211yescharacteristic
30C2×C4×Dic7C2212yes—
31C2×C4×Dic7C2211yescharacteristic
32#6461—24no—
33C7×#95C2211yescharacteristic
34C7×#95C2212yes—
35C7⋊#95C2211yescharacteristic
36C7⋊#95C2212yes—
37#2662—71nomaximal, Sylow
38C23×D7C2311yescharacteristic
39C23×D7C2312yes—
40C2×C7×Q8—22no—
41C2×C7×Q8—42no—
42C22×C4×C7C2312yes—
43C22×C4×C7C2311yescharacteristic
44C22×C4×C7C2311yescharacteristic
45C22×C4×C7C2311yescharacteristic
46C22×C4×C7C2312yes—
47C22×Dic7C2312yes—
48C22×Dic7C2311yescharacteristic
49C22×Dic7C2311yescharacteristic
50C22×Dic7C2311yescharacteristic
51C2×D28—22no—
52C2×D28—24no—
53C2×D28—42no—
54C2×D28—42no—
55C2×D28—42no—
56C7×C4⋊C4—22no—
57C7×C4⋊C4—42no—
58(C2×D7)⋊C4—22no—
59(C2×D7)⋊C4—24no—
60(C2×D7)⋊C4—22no—
61(C2×D7)⋊C4—22no—
62(C2×D7)⋊C4—24no—
63(C2×D7)⋊C4—24no—
64(C2×D7)⋊C4—42no—
65(C2×D7)⋊C4—42no—
66(C2×D7)⋊C4—42no—
67C4⋊Dic7—22no—
68Dic7⋊C4—24no—
69Dic7⋊C4—22no—
70C4×Dic7—42no—
71C4×Dic7D414yescomplemented
72C2×#126—71no—
73C2×#124—71no—
74#396—72no—
75#395D711yescharacteristic, complemented
76#394—71no—
77#394—72no—
78#393—71no—
79#392—71no—
80#390—72no—
81#388—72no—
82C4×C4⋊C4—71no—
83C23×C7C2411yescharacteristic
84C22×D7—21no—
85C22×D7—22no—
86C22×D7—22no—
87C22×D7—22no—
88C22×D7—22no—
89C22×D7—22no—
90C22×D7—22no—
91C22×D7—22no—
92C22×D7—22no—
93C22×D7—21no—
94C22×D7—21no—
95C22×D7—22no—
96C7×Q8—44no—
97C2×C4×C7—22no—
98C2×C4×C7—22no—
99C2×C4×C7—22no—
100C2×C4×C7—21no—
101C2×C4×C7—21no—
102C2×C4×C7—22no—
103C2×C4×C7—21no—
104C2×C4×C7—22no—
105C2×C4×C7—22no—
106C2×C4×C7—22no—
107C2×C4×C7—21no—
108C2×C4×C7C2×D412yes—
109C2×C4×C7#4114yes—
110C2×Dic7—22no—
111C2×Dic7—21no—
112C2×Dic7—21no—
113C2×Dic7—21no—
114C2×Dic7—22no—
115C2×Dic7—22no—
116C2×Dic7—22no—
117C2×Dic7—22no—
118C2×Dic7C2×D412yes—
119C2×Dic7C2×D412yes—
120D28—44no—
121D28—48no—
122C22×Q8C2×D711yescharacteristic
123C22×D4—71no—
124C22×D4—72no—
125#126—144no—
126#124—144no—
127C2×C4⋊C4—71no—
128C2×C4⋊C4—71no—
129C2×C4⋊C4—72no—
130C2×C4⋊C4C2×D711yescharacteristic
131C2×C4⋊C4C2×D712yes—
132C2×C22⋊C4—72no—
133C2×C22⋊C4—72no—
134C2×C22⋊C4—71no—
135C2×C22⋊C4—71no—
136C2×C22⋊C4—71no—
137C2×C22⋊C4—72no—
138C2×C22⋊C4—72no—
139C2×C22⋊C4—72no—
140C2×C22⋊C4—72no—
141C2×C42—71no—
142C2×C42—72no—
143#95—71no—
144#95—72no—
145#95C2×D711yescharacteristic
146#95C2×D712yes—
147C22×C7C22×D411yescharacteristic
148C22×C7C2×#4112yes—
149C22×C7H(5,𝔽2)12yes—
150C22×C7H(5,𝔽2)11yescharacteristic
151C22×C72−1+411yescharacteristic
152C2×D7—41no—
153C2×D7—42no—
154C2×D7—42no—
155C2×D7—41no—
156C2×D7—41no—
157C2×D7—42no—
158C2×D7—42no—
159C2×D7—42no—
160C2×D7—42no—
161C2×D7—42no—
162C2×D7—42no—
163C2×D7—42no—
164C4×C7—22no—
165C4×C7—24no—
166C4×C7—41no—
167C4×C7—42no—
168C4×C7—41no—
169Dic7—22no—
170Dic7—22no—
171Dic7—41no—
172Dic7—42no—
173C24—71no—
174C24—72no—
175C2×Q8—22no—
176C2×Q8—42no—
177C2×D4—142no—
178C2×D4—144no—
179C2×D4—282no—
180C2×D4—282no—
181C2×D4—282no—
182C22×C4—71no—
183C22×C4—71no—
184C22×C4—72no—
185C22×C4—71no—
186C22×C4C22×D712yes—
187C22×C4C22×D711yescharacteristic
188C22×C4C22×D711yescharacteristic
189C22×C4C22×D711yescharacteristic
190C22×C4C22×D712yes—
191C4⋊C4—22no—
192C4⋊C4—42no—
193C4⋊C4—142no—
194C4⋊C4—142no—
195C4⋊C4—144no—
196C22⋊C4—144no—
197C22⋊C4—144no—
198C22⋊C4—142no—
199C22⋊C4—142no—
200C22⋊C4—142no—
201C22⋊C4—144no—
202C22⋊C4—282no—
203C22⋊C4—282no—
204C22⋊C4—282no—
205C42—74no—
206C42—282no—
207C2×C7#53211yescharacteristic
208C2×C7#53511yescharacteristic
209C2×C7#53711yescharacteristic
210C2×C7#53812yes—
211C2×C7#54212yes—
212D7—82no—
213D7—81no—
214C23—141no—
215C23—142no—
216C23—142no—
217C23—142no—
218C23—142no—
219C23—142no—
220C23—142no—
221C23—142no—
222C23—141no—
223C23—142no—
224C23—142no—
225C23—141no—
226C23C23×D711yescharacteristic
227Q8—44no—
228D4—284no—
229D4—288no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—72no—
242C2×C4—72no—
243C2×C4—142no—
244C2×C4—141no—
245C2×C4—141no—
246C2×C4—141no—
247C2×C4—142no—
248C2×C4—142no—
249C2×C4—142no—
250C2×C4—142no—
251C2×C4D4×D712yes—
252C2×C4#116114yes—
253C7#266211yescharacteristic, complemented, Sylow
254C22—281no—
255C22—282no—
256C22—282no—
257C22—282no—
258C22—282no—
259C22—282no—
260C22—281no—
261C22—281no—
262C22—282no—
263C22—282no—
264C22—282no—
265C22—282no—
266C22C2×D4×D711yescharacteristic
267C22#654812yes—
268C22C2×#116112yes—
269C22#655111yescharacteristic
270C22#655311yescharacteristic
271C4—22no—
272C4—24no—
273C4—42no—
274C4—41no—
275C4—41no—
276C4—142no—
277C4—142no—
278C4—281no—
279C4—282no—
280C2—562no—
281C2—561no—
282C2#9028311yescharacteristic
283C2#9032011yescharacteristic
284C2#9033512yes—
285C2#9041611yescharacteristic
286C2#9042412yes—
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#395C2
prime radical#89443C2
hypercenter#395D7
hypocenterC7#2662
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
1. C2×#6510C2semidirect
2. C2×#6461C2semidirect
3. C7×#395C2semidirect
4. #90003C2semidirect
5. #89766C2semidirect
6. #89763C2semidirect
7. #89760C2semidirect
8. #89444C2semidirect
10. #89441C2semidirect
11. #89421C2semidirect
71. C4×Dic7D4semidirect
75. #395D7semidirect
253. C7#2662semidirect

Maximal subgroups

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

subgroupindex
C2×#65102
C2×#64612
C7×#3952
#900032
#897662
#897632
#897602
#894442
#894432
#894412
#894212
#26627