Finite Group Databasebeta

← #11734647

Subgroups — #11734647

Order 896.

488 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 143 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×#6507C211yesmaximal, characteristic, complemented
2C2×#6455C211yesmaximal, characteristic
3C7×#395C211yesmaximal, characteristic, complemented
4#90003C211yesmaximal, characteristic
5#89764C212yesmaximal
6#89751C211yesmaximal, characteristic, complemented
7#89744C212yesmaximal, complemented
8#89445C211yesmaximal, characteristic
9#89439C212yesmaximal
10#89432C212yesmaximal, complemented
11#89422C211yesmaximal, characteristic, complemented
12C22×C7×Q8C2211yescharacteristic
13C22×C4×D7C2211yescharacteristic
14C22×Dic14C2211yescharacteristic
15C2×C7×C4⋊C4C2211yescharacteristic
16C2×C7×C4⋊C4C2212yes—
17#6507—24no—
18C2×(C2×D7)⋊C4C2211yescharacteristic
19C2×(C2×D7)⋊C4C2211yescharacteristic
20C2×(C2×D7)⋊C4C2212yes—
21C2×(C2×D7)⋊C4C2212yes—
22C2×C4⋊Dic7C2211yescharacteristic
23C2×Dic7⋊C4C2212yes—
24C2×Dic7⋊C4C2212yes—
25C2×Dic7⋊C4C2212yes—
26C2×Dic7⋊C4C2212yes—
27C2×C4×Dic7C2211yescharacteristic
28C2×C4×Dic7C2211yescharacteristic
29#6455—24no—
30C7×#95C2211yescharacteristic
31C7×#95C2212yes—
32C7⋊#95C2212yes—
33C7⋊#95C2211yescharacteristic
34C7⋊#95C2212yes—
35C7⋊#95C2212yes—
36C7⋊#95C2212yes—
37#2628—71nomaximal, Sylow
38C23×D7C2311yescharacteristic
39C2×C7×Q8—22no—
40C2×C7×Q8—42no—
41C22×C4×C7C2312yes—
42C22×C4×C7C2312yes—
43C22×C4×C7C2311yescharacteristic
44C22×C4×C7C2311yescharacteristic
45C22×C4×C7C2311yescharacteristic
46C22×Dic7C2311yescharacteristic
47C22×Dic7C2312yes—
48C22×Dic7C2312yes—
49C22×Dic7C2311yescharacteristic
50C22×Dic7C2311yescharacteristic
51C2×C4×D7—22no—
52C2×C4×D7—22no—
53C2×C4×D7Q814yescomplemented
54C2×Dic14—22no—
55C2×Dic14—42no—
56C7×C4⋊C4—22no—
57C7×C4⋊C4—42no—
58(C2×D7)⋊C4—22no—
59(C2×D7)⋊C4—22no—
60(C2×D7)⋊C4—42no—
61(C2×D7)⋊C4—42no—
62C4⋊Dic7—22no—
63Dic7⋊C4—24no—
64Dic7⋊C4—24no—
65Dic7⋊C4—24no—
66Dic7⋊C4—42no—
67C4×Dic7—22no—
68C4×Dic7D414yes—
69C2×C4⋊Q8—71no—
70C2×#117—71no—
71#400—72no—
72#399—72no—
73#395—71no—
74#395D711yescharacteristic, complemented
75#393—71no—
76#393—71no—
77#389—72no—
78#382—71no—
79#380—72no—
80C23×C7C2411yescharacteristic
81C22×D7—21no—
82C22×D7—21no—
83C22×D7—22no—
84C22×D7—22no—
85C22×D7C2×Q812yes—
86C7×Q8—44no—
87C2×C4×C7—21no—
88C2×C4×C7—21no—
89C2×C4×C7—22no—
90C2×C4×C7—21no—
91C2×C4×C7—22no—
92C2×C4×C7—22no—
93C2×C4×C7—22no—
94C2×C4×C7—22no—
95C2×C4×C7—22no—
96C2×C4×C7—22no—
97C2×C4×C7—22no—
98C2×C4×C7C2×D412yes—
99C2×C4×C7C2×Q812yes—
100C2×Dic7—22no—
101C2×Dic7—21no—
102C2×Dic7—22no—
103C2×Dic7—22no—
104C2×Dic7—22no—
105C2×Dic7—22no—
106C2×Dic7—22no—
107C2×Dic7—22no—
108C2×Dic7—22no—
109C2×Dic7C2×D412yes—
110C2×Dic7C2×D412yes—
111C2×Dic7C2×Q812yes—
112C2×Dic7#4112yes—
113C4×D7—28no—
114Dic14—44no—
115C22×Q8—71no—
116C22×Q8C2×D711yescharacteristic
117C23×C4—71no—
118C4⋊Q8—144no—
119#117—144no—
120C2×C4⋊C4—71no—
121C2×C4⋊C4—72no—
122C2×C4⋊C4—72no—
123C2×C4⋊C4—72no—
124C2×C4⋊C4—72no—
125C2×C4⋊C4C2×D711yescharacteristic
126C2×C4⋊C4C2×D712yes—
127C2×C22⋊C4—71no—
128C2×C22⋊C4—71no—
129C2×C22⋊C4—72no—
130C2×C22⋊C4—72no—
131C2×C42—71no—
132C2×C42—71no—
133#95—72no—
134#95—71no—
135#95—72no—
136#95—72no—
137#95—72no—
138#95C2×D712yes—
139#95C2×D711yescharacteristic
140C22×C7C22×D411yescharacteristic
141C22×C7C22×Q811yescharacteristic
142C22×C7C2×#4111yescharacteristic
143C22×C7H(5,𝔽2)12yes—
144C22×C72−1+412yes—
145C2×D7—22no—
146C2×D7—24no—
147C2×D7—41no—
148C2×D7—41no—
149C2×D7—42no—
150C4×C7—22no—
151C4×C7—22no—
152C4×C7—41no—
153C4×C7—42no—
154C4×C7—42no—
155Dic7—22no—
156Dic7—42no—
157Dic7—41no—
158Dic7—42no—
159Dic7C22⋊Q814yes—
160C24—71no—
161C2×Q8—22no—
162C2×Q8—42no—
163C2×Q8—142no—
164C2×Q8—282no—
165C22×C4—71no—
166C22×C4—71no—
167C22×C4—74no—
168C22×C4—71no—
169C22×C4—72no—
170C22×C4—72no—
171C22×C4—142no—
172C22×C4—142no—
173C22×C4C22×D712yes—
174C22×C4C22×D712yes—
175C22×C4C22×D711yescharacteristic
176C22×C4C22×D711yescharacteristic
177C22×C4C22×D711yescharacteristic
178C4⋊C4—22no—
179C4⋊C4—42no—
180C4⋊C4—144no—
181C4⋊C4—144no—
182C4⋊C4—144no—
183C4⋊C4—142no—
184C4⋊C4—282no—
185C22⋊C4—142no—
186C22⋊C4—142no—
187C22⋊C4—282no—
188C22⋊C4—282no—
189C42—74no—
190C42—142no—
191C2×C7C2×C22⋊Q811yescharacteristic
192C2×C7#53511yescharacteristic
193C2×C7#53611yescharacteristic
194C2×C7#54212yes—
195C2×C7#54412yes—
196D7—42no—
197C23—72no—
198C23—141no—
199C23—142no—
200C23—142no—
201C23—141no—
202C23C23×D711yescharacteristic
203Q8—44no—
204Q8—284no—
205C2×C4—21no—
206C2×C4—22no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—72no—
217C2×C4—72no—
218C2×C4—72no—
219C2×C4—72no—
220C2×C4—148no—
221C2×C4—142no—
222C2×C4—141no—
223C2×C4—142no—
224C2×C4—142no—
225C2×C4—142no—
226C2×C4—142no—
227C2×C4—142no—
228C2×C4—142no—
229C2×C4—142no—
230C2×C4D4×D712yes—
231C2×C4Q8×D712yes—
232C7#262811yescharacteristic, complemented, Sylow
233C22—144no—
234C22—142no—
235C22—281no—
236C22—281no—
237C22—282no—
238C22C2×D4×D711yescharacteristic
239C22#654812yes—
240C22C2×Q8×D711yescharacteristic
241C22#655112yes—
242C22D7×#4111yescharacteristic
243C4—22no—
244C4—22no—
245C4—42no—
246C4—41no—
247C4—42no—
248C4—74no—
249C4—142no—
250C4—281no—
251C4—282no—
252C4—282no—
253C2—282no—
254C2#9028311yescharacteristic
255C2D7×C22⋊Q811yescharacteristic
256C2#9033112yes—
257C2#9041511yescharacteristic
258C2#9042012yes—
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#395C2
prime radicalC23×D7C23
hypercenter#395D7
hypocenterC7#2628
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
1. C2×#6507C2semidirect
3. C7×#395C2semidirect
6. #89751C2semidirect
7. #89744C2semidirect
10. #89432C2semidirect
11. #89422C2semidirect
53. C2×C4×D7Q8semidirect
74. #395D7semidirect
232. C7#2628semidirect

Maximal subgroups

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

subgroupindex
C2×#65072
C2×#64552
C7×#3952
#900032
#897642
#897512
#897442
#894452
#894392
#894322
#894222
#26287