Finite Group Databasebeta

← #11735492

Subgroups — #11735492

Order 896.

476 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 139 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×#6453C211yesmaximal, characteristic, complemented
2C2×#6440C211yesmaximal, characteristic
3C7×#399C211yesmaximal, characteristic
4#89754C211yesmaximal, characteristic, complemented
5#89753C212yesmaximal, complemented
6#89745C212yesmaximal, complemented
7#89725C211yesmaximal, characteristic
8#89722C212yesmaximal
9#89716C212yesmaximal
10#89433C211yesmaximal, characteristic, complemented
11#89420C211yesmaximal, characteristic, complemented
12C23×Dic7C2211yescharacteristic
13C2×C7×C4⋊C4C2211yescharacteristic
14C2×C7×C4⋊C4C2212yes—
15C2×C7×C22⋊C4C2211yescharacteristic
16C2×C7×C22⋊C4C2212yes—
17C2×C22⋊Dic7C2211yescharacteristic
18C2×C22⋊Dic7C2212yes—
19C2×C4⋊Dic7C2211yescharacteristic
20C2×C4⋊Dic7C2212yes—
21C2×C4⋊Dic7C2212yes—
22C2×Dic7⋊C4C2211yescharacteristic
23C2×Dic7⋊C4C2211yescharacteristic
24C2×Dic7⋊C4C2212yes—
25C2×C4×Dic7C2211yescharacteristic
26C2×C4×Dic7C2212yes—
27C2×C4×Dic7C2211yescharacteristic
28#6453—24no—
29#6440—24no—
30C7×#95C2211yescharacteristic
31C7⋊#95C2212yes—
32C7⋊#95C2211yescharacteristic
33C7⋊#95C2212yes—
34C7⋊#95C2212yes—
35C7⋊#95C2212yes—
36C7⋊#95C2212yes—
37#2611—71nomaximal, Sylow
38C24×C7C2311yescharacteristic
39C22×C4×C7C2311yescharacteristic
40C22×C4×C7C2312yes—
41C22×C4×C7C2311yescharacteristic
42C22×C4×C7C2312yes—
43C22×Dic7—22no—
44C22×Dic7—22no—
45C22×Dic7—22no—
46C22×Dic7C2312yes—
47C22×Dic7C2311yescharacteristic
48C22×Dic7C2312yes—
49C22×Dic7C2311yescharacteristic
50C22×Dic7C2311yescharacteristic
51C22×Dic7C2311yescharacteristic
52C7×C4⋊C4—22no—
53C7×C4⋊C4—24no—
54C7×C22⋊C4—22no—
55C7×C22⋊C4—24no—
56C22⋊Dic7—24no—
57C22⋊Dic7—22no—
58C4⋊Dic7—22no—
59C4⋊Dic7—24no—
60C4⋊Dic7—24no—
61Dic7⋊C4—22no—
62Dic7⋊C4—22no—
63Dic7⋊C4—24no—
64C4×Dic7—24no—
65C4×Dic7—22no—
66C4×Dic7D414yes—
67C2×#125—71no—
68C2×#117—71no—
69#400—71no—
70#399D711yescharacteristic
71#398—71no—
72#397—71no—
73#389—72no—
74#388—72no—
75#387—72no—
76#385—72no—
77#376—71no—
78C23×C7—21no—
79C23×C7—22no—
80C23×C7—21no—
81C23×C7—22no—
82C23×C7#4112yes—
83C23×C7C2411yescharacteristic
84C2×C4×C7—22no—
85C2×C4×C7—22no—
86C2×C4×C7—22no—
87C2×C4×C7—21no—
88C2×C4×C7—22no—
89C2×C4×C7—22no—
90C2×C4×C7—22no—
91C2×C4×C7—22no—
92C2×C4×C7C2×D412yes—
93C2×C4×C7#4114yes—
94C2×Dic7—28no—
95C2×Dic7—21no—
96C2×Dic7—22no—
97C2×Dic7—21no—
98C2×Dic7—21no—
99C2×Dic7—21no—
100C2×Dic7—21no—
101C2×Dic7—22no—
102C2×Dic7—22no—
103C2×Dic7—22no—
104C2×Dic7—22no—
105C2×Dic7—22no—
106C2×Dic7—22no—
107C2×Dic7—22no—
108C2×Dic7C2×D412yes—
109C2×Dic7C2×D412yes—
110C2×Dic7#4112yes—
111C23×C4—71no—
112#125—144no—
113#117—144no—
114C2×C4⋊C4—71no—
115C2×C4⋊C4—71no—
116C2×C4⋊C4—72no—
117C2×C4⋊C4—72no—
118C2×C4⋊C4—71no—
119C2×C4⋊C4—72no—
120C2×C4⋊C4C2×D711yescharacteristic
121C2×C4⋊C4C2×D712yes—
122C2×C22⋊C4—71no—
123C2×C22⋊C4—72no—
124C2×C22⋊C4C2×D711yescharacteristic
125C2×C22⋊C4C2×D712yes—
126C2×C42—71no—
127C2×C42—71no—
128C2×C42—72no—
129#95—72no—
130#95—71no—
131#95—72no—
132#95—72no—
133#95—72no—
134#95—72no—
135#95C2×D711yescharacteristic
136C22×C7—22no—
137C22×C7—24no—
138C22×C7—41no—
139C22×C7—41no—
140C22×C7—42no—
141C22×C7C22×D411yescharacteristic
142C22×C7C2×#4112yes—
143C22×C7C2×#4111yescharacteristic
144C22×C7C2×#4111yescharacteristic
145C22×C72−1+412yes—
146C4×C7—22no—
147C4×C7—24no—
148C4×C7—41no—
149C4×C7—42no—
150Dic7—22no—
151Dic7—22no—
152Dic7—22no—
153Dic7—42no—
154Dic7—41no—
155Dic7—42no—
156C24C22×D711yescharacteristic
157C22×C4—71no—
158C22×C4—71no—
159C22×C4—71no—
160C22×C4—72no—
161C22×C4—71no—
162C22×C4—72no—
163C22×C4—142no—
164C22×C4—142no—
165C22×C4—142no—
166C22×C4C22×D711yescharacteristic
167C22×C4C22×D712yes—
168C22×C4C22×D711yescharacteristic
169C22×C4C22×D712yes—
170C4⋊C4—22no—
171C4⋊C4—24no—
172C4⋊C4—142no—
173C4⋊C4—144no—
174C4⋊C4—144no—
175C4⋊C4—144no—
176C4⋊C4—142no—
177C4⋊C4—142no—
178C22⋊C4—22no—
179C22⋊C4—24no—
180C22⋊C4—142no—
181C22⋊C4—144no—
182C42—74no—
183C42—142no—
184C42—144no—
185C2×C7—42no—
186C2×C7#52511yescharacteristic
187C2×C7#53211yescharacteristic
188C2×C7#53611yescharacteristic
189C2×C7#55212yes—
190C2×C7#55212yes—
191C23—21no—
192C23—22no—
193C23—21no—
194C23—22no—
195C23#116112yes—
196C23C23×D711yescharacteristic
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—72no—
206C2×C4—72no—
207C2×C4—72no—
208C2×C4—148no—
209C2×C4—141no—
210C2×C4—142no—
211C2×C4—141no—
212C2×C4—141no—
213C2×C4—141no—
214C2×C4—141no—
215C2×C4—142no—
216C2×C4—142no—
217C2×C4—142no—
218C2×C4—142no—
219C2×C4—142no—
220C2×C4—142no—
221C2×C4—142no—
222C2×C4D4×D712yes—
223C2×C4#115914yes—
224C7#261111yescharacteristic, complemented, Sylow
225C22—22no—
226C22—24no—
227C22—41no—
228C22—41no—
229C22—42no—
230C22C2×D4×D711yescharacteristic
231C22C2×#115912yes—
232C22C2×#116111yescharacteristic
233C22D7×#4111yescharacteristic
234C22#655412yes—
235C4—22no—
236C4—24no—
237C4—41no—
238C4—42no—
239C4—142no—
240C4—142no—
241C4—142no—
242C4—282no—
243C4—282no—
244C4—281no—
245C2—42no—
246C2#9031811yescharacteristic
247C2#9033312yes—
248C2#9033911yescharacteristic
249C2#9034112yes—
250C2#9040711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#399C2
prime radicalC24×C7C23
hypercenter#399D7
hypocenterC7#2611
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#6453C2semidirect
4. #89754C2semidirect
5. #89753C2semidirect
6. #89745C2semidirect
10. #89433C2semidirect
11. #89420C2semidirect
224. C7#2611semidirect

Maximal subgroups

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

subgroupindex
C2×#64532
C2×#64402
C7×#3992
#897542
#897532
#897452
#897252
#897222
#897162
#894332
#894202
#26117