Finite Group Databasebeta

← #11732097

Subgroups — #11732097

Order 896.

464 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 117 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
1C7×#385C211yesmaximal, characteristic, complemented
2#89985C212yesmaximal, complemented
3#89766C212yesmaximal, complemented
4#89765C212yesmaximal, complemented
5#89732C211yesmaximal, characteristic
6#89731C212yesmaximal, complemented
7#89721C211yesmaximal, characteristic, complemented
8#89444C212yesmaximal, complemented
9#89430C212yesmaximal, complemented
10C23×C4×C7C2211yescharacteristic
11C22×C22⋊D7C2212yes—
12C22×D28C2211yescharacteristic
13C2×C7×C4⋊C4C2212yes—
14C2×C7×C22⋊C4C2211yescharacteristic
15C2×C7×C22⋊C4C2211yescharacteristic
16C2×C22⋊Dic7C2212yes—
17C2×(C2×D7)⋊C4C2212yes—
18C2×(C2×D7)⋊C4C2212yes—
19C2×(C2×D7)⋊C4C2212yes—
20C2×(C2×D7)⋊C4C2212yes—
21C2×(C2×D7)⋊C4C2212yes—
22C2×C4⋊Dic7C2211yescharacteristic
23C2×Dic7⋊C4C2212yes—
24C2×Dic7⋊C4C2212yes—
25C7×#95C2212yes—
26C7⋊#95C2211yescharacteristic
27C7⋊#95C2212yes—
28C7⋊#95C2212yes—
29C7⋊#95C2212yes—
30C7⋊#95C2211yescharacteristic
31#2826—71nomaximal, Sylow
32C24×C7C2311yescharacteristic
33C23×D7C2312yes—
34C22×C4×C7—41no—
35C22×C4×C7—42no—
36C22×C4×C7C2312yes—
37C22×C4×C7C2311yescharacteristic
38C22×C4×C7C2312yes—
39C22×C4×C7C2311yescharacteristic
40C2×C22⋊D7—24no—
41C2×C22⋊D7—42no—
42C2×C22⋊D7—42no—
43C22×Dic7C2312yes—
44C22×Dic7C2312yes—
45C22×Dic7C2312yes—
46C2×D28—42no—
47C2×D28—41no—
48C7×C4⋊C4—42no—
49C7×C22⋊C4—22no—
50C7×C22⋊C4—41no—
51C22⋊Dic7—42no—
52(C2×D7)⋊C4—42no—
53(C2×D7)⋊C4—42no—
54(C2×D7)⋊C4—42no—
55(C2×D7)⋊C4—42no—
56(C2×D7)⋊C4—42no—
57C4⋊Dic7—41no—
58Dic7⋊C4—42no—
59Dic7⋊C4—42no—
60#400—72no—
61#399—71no—
62#397—72no—
63#396—72no—
64#394—72no—
65#394—72no—
66#392—71no—
67#386—72no—
68#385D711yescharacteristic, complemented
69C23×C7—22no—
70C23×C7—21no—
71C23×C7—22no—
72C23×C7—21no—
73C23×C7#4112yes—
74C23×C7C2411yescharacteristic
75C22×D7—22no—
76C22×D7—22no—
77C22×D7—22no—
78C22×D7—22no—
79C22×D7—22no—
80C22×D7—22no—
81C22×D7—22no—
82C2×C4×C7—22no—
83C2×C4×C7—22no—
84C2×C4×C7—22no—
85C2×C4×C7—21no—
86C2×C4×C7—22no—
87C2×C4×C7—22no—
88C2×C4×C7—22no—
89C2×C4×C7—22no—
90C2×C4×C7—21no—
91C2×C4×C7—22no—
92C2×C4×C7—44no—
93C22⋊D7—48no—
94C2×Dic7—22no—
95C2×Dic7—22no—
96C2×Dic7—22no—
97C2×Dic7—22no—
98C2×Dic7—22no—
99C2×Dic7—22no—
100C2×Dic7—22no—
101C2×Dic7—22no—
102C2×Dic7—22no—
103D28—44no—
104C22×D4—72no—
105C22×D4—71no—
106C23×C4C2×D711yescharacteristic
107C2×C4⋊C4—72no—
108C2×C4⋊C4—72no—
109C2×C4⋊C4—71no—
110C2×C4⋊C4C2×D712yes—
111C2×C22⋊C4—72no—
112C2×C22⋊C4—72no—
113C2×C22⋊C4—72no—
114C2×C22⋊C4—72no—
115C2×C22⋊C4—72no—
116C2×C22⋊C4—72no—
117C2×C22⋊C4C2×D711yescharacteristic
118C2×C22⋊C4C2×D711yescharacteristic
119#95—72no—
120#95—72no—
121#95—71no—
122#95—72no—
123#95—71no—
124#95C2×D712yes—
125C22×C7—24no—
126C22×C7—22no—
127C22×C7—41no—
128C22×C7—41no—
129C22×C7—42no—
130C22×C7C2×#4111yescharacteristic
131C22×C7H(5,𝔽2)12yes—
132C22×C7H(5,𝔽2)11yescharacteristic
133C22×C7H(5,𝔽2)12yes—
134C22×C72−1+411yescharacteristic
135C2×D7—42no—
136C2×D7—42no—
137C2×D7—42no—
138C2×D7—42no—
139C2×D7—42no—
140C2×D7—42no—
141C2×D7—42no—
142C4×C7—42no—
143C4×C7—41no—
144C4×C7—41no—
145C4×C7—42no—
146Dic7—42no—
147Dic7—42no—
148Dic7—42no—
149C24—72no—
150C24C22×D711yescharacteristic
151C2×D4—144no—
152C2×D4—281no—
153C2×D4—282no—
154C2×D4—282no—
155C2×D4—282no—
156C22×C4—41no—
157C22×C4—42no—
158C22×C4—72no—
159C22×C4—72no—
160C22×C4—72no—
161C22×C4C22×D712yes—
162C22×C4C22×D711yescharacteristic
163C22×C4C22×D712yes—
164C22×C4C22×D711yescharacteristic
165C4⋊C4—42no—
166C4⋊C4—282no—
167C4⋊C4—282no—
168C4⋊C4—281no—
169C22⋊C4—22no—
170C22⋊C4—41no—
171C22⋊C4—282no—
172C22⋊C4—282no—
173C22⋊C4—282no—
174C22⋊C4—282no—
175C22⋊C4—282no—
176C22⋊C4—282no—
177C2×C7—42no—
178C2×C7#53812yes—
179C2×C7#53911yescharacteristic
180C2×C7#56011yescharacteristic
181C2×C7#56011yescharacteristic
182C2×C7#56212yes—
183D7—82no—
184C23—21no—
185C23—22no—
186C23—21no—
187C23—22no—
188C23—142no—
189C23—142no—
190C23—142no—
191C23—142no—
192C23—142no—
193C23—142no—
194C23—142no—
195C23#115712yes—
196C23C23×D711yescharacteristic
197D4—284no—
198D4—288no—
199C2×C4—22no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—44no—
210C2×C4—142no—
211C2×C4—142no—
212C2×C4—142no—
213C2×C4—142no—
214C2×C4—142no—
215C2×C4—142no—
216C2×C4—142no—
217C2×C4—142no—
218C2×C4—142no—
219C7#282611yescharacteristic, complemented, Sylow
220C22—22no—
221C22—24no—
222C22—41no—
223C22—42no—
224C22—41no—
225C22—282no—
226C22—282no—
227C22—282no—
228C22—282no—
229C22—282no—
230C22—282no—
231C22—282no—
232C22C2×#115711yescharacteristic
233C22#654811yescharacteristic
234C22#654812yes—
235C22#655111yescharacteristic
236C22#655312yes—
237C4—42no—
238C4—42no—
239C4—41no—
240C4—41no—
241C4—282no—
242C4—282no—
243C4—282no—
244C2—42no—
245C2—562no—
246C2#9020611yescharacteristic
247C2#9025112yes—
248C2#9028611yescharacteristic
249C2#9033812yes—
250C2#9035711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#385C2
prime radical#89732C2
hypercenter#385D7
hypocenterC7#2826
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
1. C7×#385C2semidirect
2. #89985C2semidirect
3. #89766C2semidirect
4. #89765C2semidirect
6. #89731C2semidirect
7. #89721C2semidirect
8. #89444C2semidirect
9. #89430C2semidirect
68. #385D7semidirect
219. C7#2826semidirect

Maximal subgroups

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

subgroupindex
C7×#3852
#899852
#897662
#897652
#897322
#897312
#897212
#894442
#894302
#28267