Finite Group Databasebeta

← #11735486

Subgroups — #11735486

Order 896.

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

normal subgroups only

This group has 564 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×#6450C211yesmaximal, characteristic, complemented
2C2×(C22⋊D7)⋊C4C211yesmaximal, characteristic
3C7×#399C211yesmaximal, characteristic, complemented
4#89764C211yesmaximal, characteristic, complemented
5#89761C212yesmaximal, complemented
6#89744C212yesmaximal, complemented
7#89729C212yesmaximal
8#89719C211yesmaximal, characteristic, complemented
9#89718C212yesmaximal, complemented
10#89445C211yesmaximal, characteristic, complemented
11#89422C211yesmaximal, characteristic, complemented
12C22×C22⋊D7C2211yescharacteristic
13C23×Dic7C2211yescharacteristic
14C22×C4×D7C2211yescharacteristic
15C22×Dic14C2211yescharacteristic
16C2×C7×C4⋊C4C2211yescharacteristic
17C2×C7×C4⋊C4C2212yes—
18C2×C7×C22⋊C4C2211yescharacteristic
19C2×C7×C22⋊C4C2212yes—
20C2×C22⋊Dic7C2212yes—
21C2×(C2×D7)⋊C4C2211yescharacteristic
22C2×(C2×D7)⋊C4C2212yes—
23C2×(C2×D7)⋊C4C2212yes—
24C2×Dic7⋊C4C2211yescharacteristic
25C2×Dic7⋊C4C2212yes—
26C2×Dic7⋊C4C2212yes—
27C2×Dic7⋊C4C2212yes—
28C2×C4×Dic7C2212yes—
29C2×C4×Dic7C2211yescharacteristic
30#6450—24no—
31(C22⋊D7)⋊C4—24no—
32C7×#95C2211yescharacteristic
33C7⋊#95C2212yes—
34C7⋊#95C2211yescharacteristic
35C7⋊#95C2212yes—
36C7⋊#95C2212yes—
37#2580—71nomaximal, Sylow
38C24×C7C2311yescharacteristic
39C23×D7C2311yescharacteristic
40C22×C4×C7C2312yes—
41C22×C4×C7C2311yescharacteristic
42C22×C4×C7C2312yes—
43C22×C4×C7C2311yescharacteristic
44C2×C22⋊D7—22no—
45C2×C22⋊D7—22no—
46C2×C22⋊D7Q814yescomplemented
47C22×Dic7—22no—
48C22×Dic7—42no—
49C22×Dic7C2312yes—
50C22×Dic7C2311yescharacteristic
51C22×Dic7C2311yescharacteristic
52C22×Dic7C2312yes—
53C22×Dic7C2311yescharacteristic
54C2×C4×D7—22no—
55C2×C4×D7—42no—
56C2×Dic14—22no—
57C2×Dic14—24no—
58C7×C4⋊C4—22no—
59C7×C4⋊C4—42no—
60C7×C22⋊C4—22no—
61C7×C22⋊C4—42no—
62C22⋊Dic7—24no—
63(C2×D7)⋊C4—22no—
64(C2×D7)⋊C4—24no—
65(C2×D7)⋊C4—42no—
66Dic7⋊C4—22no—
67Dic7⋊C4—24no—
68Dic7⋊C4—24no—
69Dic7⋊C4—42no—
70C4×Dic7—24no—
71C4×Dic7D414yes—
72C2×C4×Q8—71no—
73C2×C4×D4—71no—
74#399—71no—
75#399D711yescharacteristic, complemented
76#393—71no—
77#393—71no—
78#390—72no—
79#389—72no—
80#385—72no—
81#385—72no—
82#382—71no—
83C23×C7—21no—
84C23×C7—22no—
85C23×C7—21no—
86C23×C7—22no—
87C23×C7C2×Q812yes—
88C23×C7C2411yescharacteristic
89C22×D7—21no—
90C22×D7—22no—
91C22×D7—21no—
92C22×D7—22no—
93C22×D7C2×Q812yes—
94C2×C4×C7—22no—
95C2×C4×C7—22no—
96C2×C4×C7—21no—
97C2×C4×C7—22no—
98C2×C4×C7—22no—
99C2×C4×C7—22no—
100C2×C4×C7—22no—
101C2×C4×C7—22no—
102C2×C4×C7—22no—
103C2×C4×C7C2×D412yes—
104C22⋊D7—28no—
105C2×Dic7—22no—
106C2×Dic7—22no—
107C2×Dic7—22no—
108C2×Dic7—21no—
109C2×Dic7—22no—
110C2×Dic7—22no—
111C2×Dic7—22no—
112C2×Dic7—22no—
113C2×Dic7—44no—
114C2×Dic7C2×D412yes—
115C2×Dic7C2×D412yes—
116C2×Dic7C2×Q812yes—
117C2×Dic7#4114yes—
118C2×Dic7#4112yes—
119C4×D7—44no—
120Dic14—28no—
121C22×Q8—71no—
122C22×D4—71no—
123C23×C4—71no—
124C23×C4—71no—
125C4×Q8—144no—
126C4×D4—144no—
127C2×C4⋊C4—71no—
128C2×C4⋊C4—72no—
129C2×C4⋊C4—72no—
130C2×C4⋊C4—72no—
131C2×C4⋊C4C2×D711yescharacteristic
132C2×C4⋊C4C2×D712yes—
133C2×C22⋊C4—72no—
134C2×C22⋊C4—71no—
135C2×C22⋊C4—72no—
136C2×C22⋊C4—72no—
137C2×C22⋊C4C2×D711yescharacteristic
138C2×C22⋊C4C2×D712yes—
139C2×C42—71no—
140C2×C42—72no—
141#95—72no—
142#95—71no—
143#95—72no—
144#95—72no—
145#95C2×D711yescharacteristic
146C22×C7—22no—
147C22×C7—24no—
148C22×C7—41no—
149C22×C7—41no—
150C22×C7—42no—
151C22×C7C22×D411yescharacteristic
152C22×C7C22×Q811yescharacteristic
153C22×C7C2×#4111yescharacteristic
154C22×C7C2×#4112yes—
155C22×C7H(5,𝔽2)12yes—
156C2×D7—22no—
157C2×D7—24no—
158C2×D7—41no—
159C2×D7—41no—
160C2×D7—42no—
161C4×C7—22no—
162C4×C7—41no—
163C4×C7—42no—
164C4×C7—42no—
165Dic7—24no—
166Dic7—22no—
167Dic7—41no—
168Dic7—42no—
169Dic7C22⋊Q814yes—
170C24—71no—
171C24C22×D711yescharacteristic
172C2×Q8—144no—
173C2×Q8—142no—
174C2×D4—74no—
175C2×D4—142no—
176C2×D4—142no—
177C22×C4—71no—
178C22×C4—72no—
179C22×C4—71no—
180C22×C4—72no—
181C22×C4—71no—
182C22×C4—142no—
183C22×C4—142no—
184C22×C4—282no—
185C22×C4—282no—
186C22×C4C22×D712yes—
187C22×C4C22×D711yescharacteristic
188C22×C4C22×D712yes—
189C22×C4C22×D711yescharacteristic
190C4⋊C4—22no—
191C4⋊C4—42no—
192C4⋊C4—144no—
193C4⋊C4—142no—
194C4⋊C4—144no—
195C4⋊C4—282no—
196C22⋊C4—22no—
197C22⋊C4—42no—
198C22⋊C4—144no—
199C22⋊C4—144no—
200C22⋊C4—142no—
201C22⋊C4—282no—
202C42—74no—
203C42—144no—
204C2×C7—42no—
205C2×C7C2×C22⋊Q811yescharacteristic
206C2×C7#52511yescharacteristic
207C2×C7#53511yescharacteristic
208C2×C7#55412yes—
209C2×C7#55912yes—
210D7—42no—
211C23—21no—
212C23—22no—
213C23—22no—
214C23—21no—
215C23—72no—
216C23—141no—
217C23—142no—
218C23—142no—
219C23—141no—
220C23Q8×D712yes—
221C23C23×D711yescharacteristic
222Q8—148no—
223D4—148no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—21no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—72no—
234C2×C4—72no—
235C2×C4—72no—
236C2×C4—72no—
237C2×C4—74no—
238C2×C4—142no—
239C2×C4—142no—
240C2×C4—142no—
241C2×C4—141no—
242C2×C4—142no—
243C2×C4—142no—
244C2×C4—142no—
245C2×C4—142no—
246C2×C4—284no—
247C2×C4—284no—
248C2×C4D4×D712yes—
249C7#258011yescharacteristic, complemented, Sylow
250C22—22no—
251C22—24no—
252C22—41no—
253C22—41no—
254C22—42no—
255C22—142no—
256C22—144no—
257C22—281no—
258C22—281no—
259C22—282no—
260C22C2×D4×D711yescharacteristic
261C22#654812yes—
262C22C2×Q8×D711yescharacteristic
263C22D7×#4111yescharacteristic
264C22D7×#4112yes—
265C4—22no—
266C4—41no—
267C4—42no—
268C4—42no—
269C4—74no—
270C4—142no—
271C4—144no—
272C4—281no—
273C4—282no—
274C2—42no—
275C2—282no—
276C2D7×C22⋊Q811yescharacteristic
277C2#9032712yes—
278C2#9034711yescharacteristic
279C2#9035512yes—
280C2#9040611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#399C2
prime radicalC22×C22⋊D7C22
hypercenter#399D7
hypocenterC7#2580
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#6450C2semidirect
3. C7×#399C2semidirect
4. #89764C2semidirect
5. #89761C2semidirect
6. #89744C2semidirect
8. #89719C2semidirect
9. #89718C2semidirect
10. #89445C2semidirect
11. #89422C2semidirect
46. C2×C22⋊D7Q8semidirect
75. #399D7semidirect
249. C7#2580semidirect

Maximal subgroups

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

subgroupindex
C2×#64502
C2×(C22⋊D7)⋊C42
C7×#3992
#897642
#897612
#897442
#897292
#897192
#897182
#894452
#894222
#25807