Finite Group Databasebeta

← #11734193

Subgroups — #11734193

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×#392C211yesmaximal, characteristic
2#89994C212yesmaximal, complemented
3#89994C212yesmaximal, complemented
4#89992C212yesmaximal, complemented
5#89724C212yesmaximal, complemented
6#89721C211yesmaximal, characteristic, complemented
7#89720C212yesmaximal, complemented
8#89718C211yesmaximal, characteristic, complemented
9#89430C212yesmaximal, complemented
10C22×C7×D4C2212yes—
11C22×C7×D4C2211yescharacteristic
12C23×Dic7C2211yescharacteristic
13C2×C7×C22⋊C4C2211yescharacteristic
14C2×C7×C22⋊C4C2212yes—
15C2×C22⋊Dic7C2211yescharacteristic
16C2×C22⋊Dic7C2212yes—
17C2×C22⋊Dic7C2212yes—
18C2×C22⋊Dic7C2212yes—
19C2×C22⋊Dic7C2212yes—
20C2×C22⋊Dic7C2212yes—
21C2×C4⋊Dic7C2211yescharacteristic
22C2×Dic7⋊C4C2212yes—
23C2×Dic7⋊C4C2212yes—
24C2×Dic7⋊C4C2212yes—
25C7×#95C2211yescharacteristic
26C7⋊#95C2212yes—
27C7⋊#95C2212yes—
28C7⋊#95C2212yes—
29C7⋊#95C2212yes—
30C7⋊#95C2211yescharacteristic
31#2826—71nomaximal, Sylow
32C24×C7C2312yes—
33C24×C7C2311yescharacteristic
34C2×C7×D4—24no—
35C2×C7×D4—42no—
36C2×C7×D4—42no—
37C2×C7×D4—42no—
38C2×C7×D4—41no—
39C22×C4×C7C2312yes—
40C22×C4×C7C2311yescharacteristic
41C22×C4×C7C2311yescharacteristic
42C22×Dic7—42no—
43C22×Dic7—41no—
44C22×Dic7C2312yes—
45C22×Dic7C2312yes—
46C22×Dic7C2312yes—
47C22×Dic7C2312yes—
48C7×C22⋊C4—42no—
49C7×C22⋊C4—41no—
50C22⋊Dic7—22no—
51C22⋊Dic7—42no—
52C22⋊Dic7—42no—
53C22⋊Dic7—42no—
54C22⋊Dic7—42no—
55C22⋊Dic7—42no—
56C4⋊Dic7—41no—
57Dic7⋊C4—42no—
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#392D711yescharacteristic
67#386—72no—
68#385—71no—
69C23×C7—22no—
70C23×C7—21no—
71C23×C7—22no—
72C23×C7—21no—
73C23×C7—22no—
74C23×C7—22no—
75C23×C7—22no—
76C23×C7—22no—
77C23×C7—22no—
78C23×C7—22no—
79C23×C7—22no—
80C23×C7#4112yes—
81C23×C7C2411yescharacteristic
82C7×D4—48no—
83C7×D4—44no—
84C2×C4×C7—22no—
85C2×C4×C7—22no—
86C2×C4×C7—22no—
87C2×C4×C7—22no—
88C2×C4×C7—21no—
89C2×C4×C7—22no—
90C2×C4×C7—21no—
91C2×Dic7—22no—
92C2×Dic7—22no—
93C2×Dic7—22no—
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—
103C2×Dic7—44no—
104C22×D4C2×D712yes—
105C22×D4C2×D711yescharacteristic
106C23×C4—71no—
107C2×C4⋊C4—72no—
108C2×C4⋊C4—72no—
109C2×C4⋊C4—72no—
110C2×C4⋊C4—71no—
111C2×C22⋊C4—71no—
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×D712yes—
119#95—72no—
120#95—72no—
121#95—72no—
122#95—72no—
123#95—71no—
124#95C2×D711yescharacteristic
125C22×C7—24no—
126C22×C7—22no—
127C22×C7—41no—
128C22×C7—41no—
129C22×C7—42no—
130C22×C7—42no—
131C22×C7—42no—
132C22×C7—42no—
133C22×C7—42no—
134C22×C7—42no—
135C22×C7—42no—
136C22×C7—42no—
137C22×C7C2×#4111yescharacteristic
138C22×C7H(5,𝔽2)12yes—
139C22×C7H(5,𝔽2)12yes—
140C22×C7H(5,𝔽2)11yescharacteristic
141C22×C72−1+411yescharacteristic
142C4×C7—42no—
143C4×C7—41no—
144C4×C7—41no—
145Dic7—42no—
146Dic7—42no—
147Dic7—42no—
148Dic7—42no—
149C24C22×D712yes—
150C24C22×D711yescharacteristic
151C2×D4—24no—
152C2×D4—42no—
153C2×D4—41no—
154C2×D4—42no—
155C2×D4—42no—
156C22×C4—72no—
157C22×C4—72no—
158C22×C4—72no—
159C22×C4—72no—
160C22×C4—281no—
161C22×C4—282no—
162C22×C4C22×D712yes—
163C22×C4C22×D711yescharacteristic
164C22×C4C22×D711yescharacteristic
165C4⋊C4—282no—
166C4⋊C4—282no—
167C4⋊C4—282no—
168C4⋊C4—281no—
169C22⋊C4—41no—
170C22⋊C4—42no—
171C22⋊C4—142no—
172C22⋊C4—282no—
173C22⋊C4—282no—
174C22⋊C4—282no—
175C22⋊C4—282no—
176C22⋊C4—282no—
177C2×C7—42no—
178C2×C7—82no—
179C2×C7#53812yes—
180C2×C7#53911yescharacteristic
181C2×C7#56011yescharacteristic
182C2×C7#56011yescharacteristic
183C2×C7#56212yes—
184C23—21no—
185C23—22no—
186C23—22no—
187C23—21no—
188C23—22no—
189C23—22no—
190C23—22no—
191C23—22no—
192C23—22no—
193C23—22no—
194C23—22no—
195C23#115912yes—
196C23C23×D711yescharacteristic
197D4—48no—
198D4—44no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—21no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—142no—
207C2×C4—142no—
208C2×C4—142no—
209C2×C4—142no—
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—284no—
219C7#282611yescharacteristic, complemented, Sylow
220C22—24no—
221C22—22no—
222C22—41no—
223C22—41no—
224C22—42no—
225C22—42no—
226C22—42no—
227C22—42no—
228C22—42no—
229C22—42no—
230C22—42no—
231C22—42no—
232C22C2×#115911yescharacteristic
233C22#654812yes—
234C22#654811yescharacteristic
235C22#654812yes—
236C22#655411yescharacteristic
237C4—41no—
238C4—42no—
239C4—41no—
240C4—282no—
241C4—282no—
242C4—282no—
243C4—282no—
244C2—42no—
245C2—82no—
246C2#9028412yes—
247C2#9028611yescharacteristic
248C2#9029311yescharacteristic
249C2#9031212yes—
250C2#9041111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
Fitting, prime radicalC7×#392C2
hypercenter#392D7
hypocenterC7#2826
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
2. #89994C2semidirect
3. #89994C2semidirect
4. #89992C2semidirect
5. #89724C2semidirect
6. #89721C2semidirect
7. #89720C2semidirect
8. #89718C2semidirect
9. #89430C2semidirect
219. C7#2826semidirect

Maximal subgroups

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

subgroupindex
C7×#3922
#899942
#899922
#897242
#897212
#897202
#897182
#894302
#28267