Finite Group Databasebeta

← #11734412

Subgroups — #11734412

Order 896.

582 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 137 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×D4×Dic7C211yesmaximal, characteristic, complemented
2C2×#6461C211yesmaximal, characteristic, complemented
3C7×#394C211yesmaximal, characteristic, complemented
4#89995C211yesmaximal, characteristic
5#89755C211yesmaximal, characteristic, complemented
6#89730C212yesmaximal, complemented
7#89729C212yesmaximal, complemented
8#89723C212yesmaximal, complemented
9#89718C212yesmaximal, complemented
10#89446C211yesmaximal, characteristic, complemented
11#89420C211yesmaximal, characteristic, complemented
12C22×C7×D4C2211yescharacteristic
13C22×C22⋊D7C2212yes—
14C23×Dic7C2212yes—
15C2×C7×C4⋊C4C2211yescharacteristic
16C2×C7×C22⋊C4C2212yes—
17C2×C7×C22⋊C4C2212yes—
18C2×C22⋊Dic7C2212yes—
19C2×C22⋊Dic7C2212yes—
20D4×Dic7—24no—
21C2×(C2×D7)⋊C4C2211yescharacteristic
22C2×(C2×D7)⋊C4C2211yescharacteristic
23C2×(C2×D7)⋊C4C2211yescharacteristic
24C2×(C2×D7)⋊C4C2212yes—
25C2×C4⋊Dic7C2211yescharacteristic
26C2×C4⋊Dic7C2211yescharacteristic
27C2×Dic7⋊C4C2211yescharacteristic
28C2×Dic7⋊C4C2212yes—
29C2×C4×Dic7C2212yes—
30C2×C4×Dic7C2211yescharacteristic
31#6461—24no—
32C7×#95C2211yescharacteristic
33C7⋊#95C2212yes—
34C7⋊#95C2211yescharacteristic
35C7⋊#95C2212yes—
36C7⋊#95C2212yes—
37#2603—71nomaximal, Sylow
38C24×C7C2312yes—
39C23×D7C2311yescharacteristic
40C2×C7×D4—24no—
41C2×C7×D4—22no—
42C22×C4×C7C2312yes—
43C22×C4×C7C2311yescharacteristic
44C22×C4×C7C2311yescharacteristic
45C22×C4×C7C2311yescharacteristic
46C2×C22⋊D7—24no—
47C2×C22⋊D7—24no—
48C2×C22⋊D7—42no—
49C22×Dic7—24no—
50C22×Dic7—42no—
51C22×Dic7—42no—
52C22×Dic7C2311yescharacteristic
53C22×Dic7C2312yes—
54C22×Dic7C2311yescharacteristic
55C22×Dic7C2311yescharacteristic
56C22×Dic7C2312yes—
57C7×C4⋊C4—22no—
58C7×C22⋊C4—24no—
59C7×C22⋊C4—42no—
60C22⋊Dic7—24no—
61C22⋊Dic7—24no—
62(C2×D7)⋊C4—22no—
63(C2×D7)⋊C4—22no—
64(C2×D7)⋊C4—22no—
65(C2×D7)⋊C4—42no—
66C4⋊Dic7—22no—
67C4⋊Dic7—22no—
68Dic7⋊C4—22no—
69Dic7⋊C4—42no—
70C4×Dic7—24no—
71C4×Dic7D414yescomplemented
72C2×#126—71no—
73C2×C4×D4—71no—
74#398—71no—
75#397—71no—
76#394D711yescharacteristic, complemented
77#392—71no—
78#390—72no—
79#388—72no—
80#386—72no—
81#385—72no—
82#376—71no—
83C23×C7—22no—
84C23×C7—22no—
85C23×C7—22no—
86C23×C7—22no—
87C23×C7—22no—
88C23×C7—22no—
89C23×C7#4114yes—
90C23×C7C2411yescharacteristic
91C22×D7—21no—
92C22×D7—22no—
93C22×D7—22no—
94C22×D7—21no—
95C22×D7—21no—
96C7×D4—28no—
97C2×C4×C7—22no—
98C2×C4×C7—21no—
99C2×C4×C7—21no—
100C2×C4×C7—22no—
101C2×C4×C7—22no—
102C2×C4×C7—22no—
103C2×C4×C7—21no—
104C2×C4×C7—22no—
105C2×C4×C7—21no—
106C2×C4×C7C2×D412yes—
107C22⋊D7—48no—
108C2×Dic7—22no—
109C2×Dic7—22no—
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×Dic7—48no—
119C2×Dic7C2×D412yes—
120C2×Dic7C2×D412yes—
121C2×Dic7#4114yes—
122C22×D4—72no—
123C22×D4C2×D711yescharacteristic
124C23×C4—72no—
125#126—144no—
126C4×D4—144no—
127C2×C4⋊C4—71no—
128C2×C4⋊C4—71no—
129C2×C4⋊C4—71no—
130C2×C4⋊C4—72no—
131C2×C4⋊C4C2×D711yescharacteristic
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⋊C4C2×D712yes—
139C2×C22⋊C4C2×D712yes—
140C2×C42—71no—
141C2×C42—72no—
142#95—72no—
143#95—71no—
144#95—72no—
145#95—72no—
146#95C2×D711yescharacteristic
147C22×C7—24no—
148C22×C7—24no—
149C22×C7—24no—
150C22×C7—42no—
151C22×C7—42no—
152C22×C7—42no—
153C22×C7—42no—
154C22×C7C22×D411yescharacteristic
155C22×C7C2×#4112yes—
156C22×C7C2×#4112yes—
157C22×C7H(5,𝔽2)11yescharacteristic
158C22×C7H(5,𝔽2)11yescharacteristic
159C2×D7—41no—
160C2×D7—41no—
161C2×D7—41no—
162C2×D7—42no—
163C2×D7—42no—
164C4×C7—22no—
165C4×C7—41no—
166C4×C7—41no—
167C4×C7—42no—
168Dic7—22no—
169Dic7—22no—
170Dic7—24no—
171Dic7—41no—
172Dic7—42no—
173C24—71no—
174C24C22×D712yes—
175C2×D4—24no—
176C2×D4—22no—
177C2×D4—144no—
178C2×D4—144no—
179C2×D4—282no—
180C22×C4—72no—
181C22×C4—71no—
182C22×C4—71no—
183C22×C4—71no—
184C22×C4—72no—
185C22×C4—144no—
186C22×C4—282no—
187C22×C4—282no—
188C22×C4C22×D712yes—
189C22×C4C22×D711yescharacteristic
190C22×C4C22×D711yescharacteristic
191C22×C4C22×D711yescharacteristic
192C4⋊C4—22no—
193C4⋊C4—142no—
194C4⋊C4—142no—
195C4⋊C4—142no—
196C4⋊C4—282no—
197C22⋊C4—24no—
198C22⋊C4—42no—
199C22⋊C4—142no—
200C22⋊C4—142no—
201C22⋊C4—144no—
202C22⋊C4—144no—
203C22⋊C4—142no—
204C22⋊C4—282no—
205C42—74no—
206C42—144no—
207C2×C7—44no—
208C2×C7#52511yescharacteristic
209C2×C7#53211yescharacteristic
210C2×C7#53511yescharacteristic
211C2×C7#55112yes—
212C2×C7#55312yes—
213D7—81no—
214C23—22no—
215C23—22no—
216C23—22no—
217C23—22no—
218C23—22no—
219C23—22no—
220C23—141no—
221C23—142no—
222C23—141no—
223C23—142no—
224C23—141no—
225C23#115914yes—
226C23C23×D711yescharacteristic
227D4—28no—
228D4—288no—
229C2×C4—22no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—21no—
238C2×C4—72no—
239C2×C4—72no—
240C2×C4—74no—
241C2×C4—142no—
242C2×C4—141no—
243C2×C4—142no—
244C2×C4—142no—
245C2×C4—141no—
246C2×C4—141no—
247C2×C4—142no—
248C2×C4—142no—
249C2×C4—142no—
250C2×C4—142no—
251C2×C4—288no—
252C2×C4D4×D712yes—
253C7#260311yescharacteristic, complemented, Sylow
254C22—24no—
255C22—24no—
256C22—24no—
257C22—42no—
258C22—42no—
259C22—42no—
260C22—42no—
261C22—281no—
262C22—282no—
263C22—281no—
264C22—282no—
265C22—281no—
266C22C2×D4×D711yescharacteristic
267C22C2×#115912yes—
268C22#654811yescharacteristic
269C22D7×#4112yes—
270C22#655311yescharacteristic
271C4—22no—
272C4—41no—
273C4—42no—
274C4—41no—
275C4—142no—
276C4—144no—
277C4—142no—
278C4—282no—
279C4—281no—
280C2—44no—
281C2—561no—
282C2#9027711yescharacteristic
283C2#9029212yes—
284C2#9029811yescharacteristic
285C2#9034912yes—
286C2#9035911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#394C2
prime radical#89995C2
hypercenter#394D7
hypocenterC7#2603
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×D4×Dic7C2semidirect
2. C2×#6461C2semidirect
3. C7×#394C2semidirect
5. #89755C2semidirect
6. #89730C2semidirect
7. #89729C2semidirect
8. #89723C2semidirect
9. #89718C2semidirect
10. #89446C2semidirect
11. #89420C2semidirect
71. C4×Dic7D4semidirect
76. #394D7semidirect
253. C7#2603semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Dic72
C2×#64612
C7×#3942
#899952
#897552
#897302
#897292
#897232
#897182
#894462
#894202
#26037