Finite Group Databasebeta

← #11734119

Subgroups — #11734119

Order 896.

660 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 660 conjugacy classes of subgroups — too many to draw. Only its 149 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×#6498C212yesmaximal, complemented
2C2×D4×Dic7C212yesmaximal, complemented
3C7×#392C211yesmaximal, characteristic
4#89994C212yesmaximal, complemented
5#89722C212yesmaximal, complemented
6#89721C211yesmaximal, characteristic, complemented
7#89718C211yesmaximal, characteristic, complemented
8#89717C212yesmaximal, complemented
9#89419C212yesmaximal, complemented
10C22×C7×D4C2212yes—
11C22×C7×D4C2211yescharacteristic
12C23×Dic7C2211yescharacteristic
13C23×Dic7C2212yes—
14C2×C7×C22⋊C4C2212yes—
15C2×C7×C22⋊C4C2211yescharacteristic
16C2×C22⋊Dic7C2212yes—
17C2×C22⋊Dic7C2211yescharacteristic
18C2×C22⋊Dic7C2212yes—
19C2×C22⋊Dic7C2212yes—
20C2×C22⋊Dic7C2212yes—
21#6498—28no—
22D4×Dic7—28no—
23C2×C4⋊Dic7C2212yes—
24C2×C4⋊Dic7C2211yescharacteristic
25C2×Dic7⋊C4C2212yes—
26C2×Dic7⋊C4C2212yes—
27C2×Dic7⋊C4C2212yes—
28C2×C4×Dic7C2212yes—
29C7×#95C2211yescharacteristic
30C7⋊#95C2212yes—
31C7⋊#95C2211yescharacteristic
32C7⋊#95C2212yes—
33#2455—71nomaximal, Sylow
34C24×C7C2312yes—
35C24×C7C2311yescharacteristic
36C2×C7×D4—24no—
37C2×C7×D4—24no—
38C2×C7×D4—24no—
39C2×C7×D4—24no—
40C2×C7×D4—41no—
41C22×C4×C7C2311yescharacteristic
42C22×C4×C7C2311yescharacteristic
43C22×C4×C7C2312yes—
44C22×Dic7—24no—
45C22×Dic7—24no—
46C22×Dic7—24no—
47C22×Dic7—41no—
48C22×Dic7—42no—
49C22×Dic7C2312yes—
50C22×Dic7C2312yes—
51C22×Dic7C2312yes—
52C22×Dic7C2312yes—
53C7×C22⋊C4—24no—
54C7×C22⋊C4—41no—
55C22⋊Dic7—24no—
56C22⋊Dic7—22no—
57C22⋊Dic7—24no—
58C22⋊Dic7—24no—
59C22⋊Dic7D418yescomplemented
60C4⋊Dic7—24no—
61C4⋊Dic7—41no—
62Dic7⋊C4—24no—
63Dic7⋊C4—24no—
64Dic7⋊C4—24no—
65C4×Dic7—24no—
66C2×#123—72no—
67C2×C4×D4—72no—
68#399—71no—
69#394—72no—
70#392D711yescharacteristic
71#389—72no—
72#388—72no—
73#385—71no—
74#385—72no—
75C23×C7—22no—
76C23×C7—22no—
77C23×C7—22no—
78C23×C7—22no—
79C23×C7—22no—
80C23×C7—22no—
81C23×C7—21no—
82C23×C7—22no—
83C23×C7—21no—
84C23×C7—22no—
85C23×C7C2×D414yes—
86C23×C7#4112yes—
87C23×C7C2411yescharacteristic
88C7×D4—216no—
89C7×D4—44no—
90C2×C4×C7—22no—
91C2×C4×C7—22no—
92C2×C4×C7—22no—
93C2×C4×C7—22no—
94C2×C4×C7—21no—
95C2×C4×C7—21no—
96C2×C4×C7#4114yes—
97C2×Dic7—22no—
98C2×Dic7—22no—
99C2×Dic7—22no—
100C2×Dic7—22no—
101C2×Dic7—22no—
102C2×Dic7—22no—
103C2×Dic7—22no—
104C2×Dic7—22no—
105C2×Dic7—22no—
106C2×Dic7—22no—
107C2×Dic7—44no—
108C2×Dic7—48no—
109C2×Dic7C2×D414yes—
110C2×Dic7C2×D414yes—
111C22×D4C2×D712yes—
112C22×D4C2×D711yescharacteristic
113C23×C4—71no—
114C23×C4—72no—
115#123—148no—
116C4×D4—148no—
117C2×C4⋊C4—72no—
118C2×C4⋊C4—72no—
119C2×C4⋊C4—72no—
120C2×C4⋊C4—72no—
121C2×C4⋊C4—71no—
122C2×C22⋊C4—72no—
123C2×C22⋊C4—72no—
124C2×C22⋊C4—72no—
125C2×C22⋊C4—72no—
126C2×C22⋊C4—71no—
127C2×C22⋊C4C2×D712yes—
128C2×C22⋊C4C2×D711yescharacteristic
129C2×C42—72no—
130#95—72no—
131#95—72no—
132#95—71no—
133#95C2×D711yescharacteristic
134C22×C7—24no—
135C22×C7—24no—
136C22×C7—22no—
137C22×C7—24no—
138C22×C7—24no—
139C22×C7—42no—
140C22×C7—42no—
141C22×C7—41no—
142C22×C7—42no—
143C22×C7—42no—
144C22×C7—42no—
145C22×C7—41no—
146C22×C7C22×D412yes—
147C22×C7C2×#4111yescharacteristic
148C22×C7C2×#4112yes—
149C22×C7H(5,𝔽2)11yescharacteristic
150C22×C72−1+411yescharacteristic
151C4×C7—24no—
152C4×C7—41no—
153C4×C7—41no—
154Dic7—24no—
155Dic7—24no—
156Dic7—42no—
157Dic7—42no—
158C24C22×D712yes—
159C24C22×D711yescharacteristic
160C2×D4—24no—
161C2×D4—24no—
162C2×D4—24no—
163C2×D4—24no—
164C2×D4—41no—
165C22×C4—72no—
166C22×C4—72no—
167C22×C4—72no—
168C22×C4—72no—
169C22×C4—144no—
170C22×C4—144no—
171C22×C4—144no—
172C22×C4—281no—
173C22×C4—282no—
174C22×C4C22×D711yescharacteristic
175C22×C4C22×D711yescharacteristic
176C22×C4C22×D712yes—
177C4⋊C4—144no—
178C4⋊C4—144no—
179C4⋊C4—144no—
180C4⋊C4—144no—
181C4⋊C4—281no—
182C22⋊C4—24no—
183C22⋊C4—41no—
184C22⋊C4—78no—
185C22⋊C4—144no—
186C22⋊C4—144no—
187C22⋊C4—144no—
188C22⋊C4—142no—
189C42—144no—
190C2×C7—44no—
191C2×C7—42no—
192C2×C7#52512yes—
193C2×C7#53911yescharacteristic
194C2×C7D4211yescharacteristic
195C2×C7#54712yes—
196C2×C7#55911yescharacteristic
197C23—22no—
198C23—22no—
199C23—22no—
200C23—22no—
201C23—22no—
202C23—22no—
203C23—21no—
204C23—22no—
205C23—21no—
206C23—22no—
207C23D4×D714yes—
208C23#115912yes—
209C23C23×D711yescharacteristic
210D4—216no—
211D4—44no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—74no—
219C2×C4—74no—
220C2×C4—142no—
221C2×C4—142no—
222C2×C4—142no—
223C2×C4—142no—
224C2×C4—142no—
225C2×C4—142no—
226C2×C4—142no—
227C2×C4—142no—
228C2×C4—142no—
229C2×C4—142no—
230C2×C4—284no—
231C2×C4—288no—
232C2×C4#115914yes—
233C7#245511yescharacteristic, complemented, Sylow
234C22—24no—
235C22—24no—
236C22—24no—
237C22—22no—
238C22—24no—
239C22—42no—
240C22—41no—
241C22—42no—
242C22—41no—
243C22—42no—
244C22—42no—
245C22—42no—
246C22C2×D4×D712yes—
247C22C2×#115912yes—
248C22C2×#115911yescharacteristic
249C22#654811yescharacteristic
250C22#655411yescharacteristic
251C4—24no—
252C4—41no—
253C4—41no—
254C4—144no—
255C4—144no—
256C4—282no—
257C4—282no—
258C2—44no—
259C2—42no—
260C2#9027712yes—
261C2#9028111yescharacteristic
262C2#9029311yescharacteristic
263C2#9030212yes—
264C2#9040411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
Fitting, prime radicalC7×#392C2
hypercenter#392D7
hypocenterC7#2455
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C2×#6498C2semidirect
2. C2×D4×Dic7C2semidirect
4. #89994C2semidirect
5. #89722C2semidirect
6. #89721C2semidirect
7. #89718C2semidirect
8. #89717C2semidirect
9. #89419C2semidirect
59. C22⋊Dic7D4semidirect
233. C7#2455semidirect

Maximal subgroups

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

subgroupindex
C2×#64982
C2×D4×Dic72
C7×#3922
#899942
#897222
#897212
#897182
#897172
#894192
#24557