Finite Group Databasebeta

← #11734612

Subgroups — #11734612

Order 896.

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

normal subgroups only

This group has 508 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×#394C211yesmaximal, characteristic, complemented
2#89995C212yesmaximal, complemented
3#89760C211yesmaximal, characteristic, complemented
4#89758C211yesmaximal, characteristic, complemented
5#89733C212yesmaximal, complemented
6#89733C212yesmaximal, complemented
7#89732C212yesmaximal, complemented
8#89723C212yesmaximal, complemented
9#89446C212yesmaximal, complemented
10C22×C7×D4C2211yescharacteristic
11C22×C22⋊D7C2212yes—
12C22×C22⋊D7C2212yes—
13C22×D28C2211yescharacteristic
14C2×C7×C4⋊C4C2211yescharacteristic
15C2×C7×C22⋊C4C2212yes—
16C2×C7×C22⋊C4C2212yes—
17C2×C22⋊Dic7C2212yes—
18C2×C22⋊Dic7C2212yes—
19C2×(C2×D7)⋊C4C2212yes—
20C2×(C2×D7)⋊C4C2212yes—
21C2×(C2×D7)⋊C4C2212yes—
22C2×(C2×D7)⋊C4C2212yes—
23C2×C4⋊Dic7C2212yes—
24C2×C4⋊Dic7C2211yescharacteristic
25C2×C4×Dic7C2211yescharacteristic
26C7×#95C2211yescharacteristic
27C7⋊#95C2212yes—
28C7⋊#95C2212yes—
29C7⋊#95C2212yes—
30C7⋊#95C2211yescharacteristic
31#2828—71nomaximal, Sylow
32C24×C7C2312yes—
33C23×D7C2312yes—
34C2×C7×D4—42no—
35C2×C7×D4—41no—
36C22×C4×C7C2311yescharacteristic
37C22×C4×C7C2312yes—
38C22×C4×C7C2311yescharacteristic
39C22×C4×C7C2311yescharacteristic
40C2×C22⋊D7—42no—
41C2×C22⋊D7—42no—
42C2×C22⋊D7—42no—
43C2×C22⋊D7—42no—
44C2×C22⋊D7—42no—
45C2×C22⋊D7—42no—
46C22×Dic7C2312yes—
47C22×Dic7C2312yes—
48C22×Dic7C2312yes—
49C2×D28—22no—
50C2×D28—41no—
51C2×D28—41no—
52C7×C4⋊C4—41no—
53C7×C22⋊C4—24no—
54C7×C22⋊C4—42no—
55C22⋊Dic7—42no—
56C22⋊Dic7—42no—
57(C2×D7)⋊C4—42no—
58(C2×D7)⋊C4—42no—
59(C2×D7)⋊C4—42no—
60(C2×D7)⋊C4—42no—
61C4⋊Dic7—42no—
62C4⋊Dic7—41no—
63C4×Dic7—41no—
64#400—71no—
65#397—72no—
66#394—72no—
67#394—72no—
68#394D711yescharacteristic, complemented
69#392—72no—
70#392—72no—
71#390—71no—
72#388—72no—
73C23×C7—22no—
74C23×C7—22no—
75C23×C7—22no—
76C23×C7—22no—
77C23×C7—22no—
78C23×C7—22no—
79C23×C7—22no—
80C23×C7C2411yescharacteristic
81C22×D7—22no—
82C22×D7—22no—
83C22×D7—22no—
84C22×D7—22no—
85C22×D7—22no—
86C22×D7—22no—
87C22×D7—22no—
88C7×D4—44no—
89C2×C4×C7—21no—
90C2×C4×C7—21no—
91C2×C4×C7—22no—
92C2×C4×C7—22no—
93C2×C4×C7—21no—
94C2×C4×C7—22no—
95C2×C4×C7—22no—
96C2×C4×C7—22no—
97C2×C4×C7—21no—
98C2×C4×C7#4112yes—
99C22⋊D7—48no—
100C22⋊D7—48no—
101C2×Dic7—22no—
102C2×Dic7—22no—
103C2×Dic7—22no—
104C2×Dic7—22no—
105C2×Dic7—22no—
106C2×Dic7—22no—
107C2×Dic7—22no—
108C2×Dic7—22no—
109C2×Dic7—22no—
110D28—44no—
111C22×D4—72no—
112C22×D4—72no—
113C22×D4—71no—
114C22×D4C2×D711yescharacteristic
115C2×C4⋊C4—72no—
116C2×C4⋊C4—71no—
117C2×C4⋊C4C2×D711yescharacteristic
118C2×C22⋊C4—72no—
119C2×C22⋊C4—72no—
120C2×C22⋊C4—72no—
121C2×C22⋊C4—72no—
122C2×C22⋊C4—72no—
123C2×C22⋊C4—72no—
124C2×C22⋊C4C2×D712yes—
125C2×C22⋊C4C2×D712yes—
126C2×C42—71no—
127#95—72no—
128#95—72no—
129#95—72no—
130#95—71no—
131#95C2×D711yescharacteristic
132C22×C7—42no—
133C22×C7—42no—
134C22×C7—42no—
135C22×C7—42no—
136C22×C7—42no—
137C22×C7—42no—
138C22×C7—42no—
139C22×C7C2×#4111yescharacteristic
140C22×C7H(5,𝔽2)11yescharacteristic
141C22×C7H(5,𝔽2)12yes—
142C22×C7H(5,𝔽2)11yescharacteristic
143C22×C7H(5,𝔽2)12yes—
144C2×D7—42no—
145C2×D7—42no—
146C2×D7—42no—
147C2×D7—42no—
148C2×D7—42no—
149C2×D7—42no—
150C2×D7—42no—
151C4×C7—22no—
152C4×C7—42no—
153C4×C7—41no—
154C4×C7—41no—
155Dic7—42no—
156Dic7—42no—
157Dic7—42no—
158C24—72no—
159C24C22×D712yes—
160C2×D4—42no—
161C2×D4—41no—
162C2×D4—142no—
163C2×D4—281no—
164C2×D4—281no—
165C2×D4—282no—
166C2×D4—282no—
167C2×D4—282no—
168C2×D4—282no—
169C2×D4—282no—
170C2×D4—282no—
171C22×C4—72no—
172C22×C4—72no—
173C22×C4—72no—
174C22×C4C22×D711yescharacteristic
175C22×C4C22×D712yes—
176C22×C4C22×D711yescharacteristic
177C22×C4C22×D711yescharacteristic
178C4⋊C4—41no—
179C4⋊C4—281no—
180C4⋊C4—282no—
181C22⋊C4—24no—
182C22⋊C4—42no—
183C22⋊C4—282no—
184C22⋊C4—282no—
185C22⋊C4—282no—
186C22⋊C4—282no—
187C22⋊C4—282no—
188C22⋊C4—282no—
189C42—281no—
190C2×C7—82no—
191C2×C7#53811yescharacteristic
192C2×C7#54012yes—
193C2×C7#56011yescharacteristic
194C2×C7#56012yes—
195C2×C7H(3,𝔽4)11yescharacteristic
196D7—82no—
197C23—22no—
198C23—22no—
199C23—22no—
200C23—22no—
201C23—22no—
202C23—22no—
203C23—22no—
204C23—142no—
205C23—142no—
206C23—142no—
207C23—142no—
208C23—142no—
209C23—142no—
210C23—142no—
211C23C23×D711yescharacteristic
212D4—44no—
213D4—284no—
214D4—288no—
215D4—288no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—22no—
225C2×C4—142no—
226C2×C4—142no—
227C2×C4—142no—
228C2×C4—142no—
229C2×C4—142no—
230C2×C4—142no—
231C2×C4—142no—
232C2×C4—142no—
233C2×C4—142no—
234C2×C4#115912yes—
235C7#282811yescharacteristic, complemented, Sylow
236C22—42no—
237C22—42no—
238C22—42no—
239C22—42no—
240C22—42no—
241C22—42no—
242C22—42no—
243C22—282no—
244C22—282no—
245C22—282no—
246C22—282no—
247C22—282no—
248C22—282no—
249C22—282no—
250C22C2×#115911yescharacteristic
251C22#654811yescharacteristic
252C22#654812yes—
253C22#655312yes—
254C22#655311yescharacteristic
255C4—22no—
256C4—41no—
257C4—41no—
258C4—42no—
259C4—282no—
260C4—282no—
261C4—282no—
262C2—82no—
263C2—562no—
264C2#9028611yescharacteristic
265C2#9030811yescharacteristic
266C2#9031012yes—
267C2#9035712yes—
268C2#9037411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#394C2
hypercenter#394D7
hypocenterC7#2828
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C7×#394C2semidirect
2. #89995C2semidirect
3. #89760C2semidirect
4. #89758C2semidirect
5. #89733C2semidirect
6. #89733C2semidirect
7. #89732C2semidirect
8. #89723C2semidirect
9. #89446C2semidirect
68. #394D7semidirect
235. C7#2828semidirect

Maximal subgroups

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

subgroupindex
C7×#3942
#899952
#897602
#897582
#897332
#897322
#897232
#894462
#28287