Finite Group Databasebeta

← #11733679

Subgroups — #11733679

Order 896.

412 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 137 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×#389C211yesmaximal, characteristic, complemented
2#89765C212yesmaximal
3#89759C211yesmaximal, characteristic
4#89759C211yesmaximal, characteristic
5#89750C212yesmaximal, complemented
6C4⋊C4×Dic7C211yesmaximal, characteristic, complemented
7#89745C211yesmaximal, characteristic, complemented
8C4×(C2×D7)⋊C4C211yesmaximal, characteristic
9#89708C211yesmaximal, characteristic, complemented
10#89441C212yesmaximal
11C7⋊#401C212yesmaximal, complemented
12C22×C4×D7C2211yescharacteristic
13C2×C7×C4⋊C4C2211yescharacteristic
14C2×C7×C4⋊C4C2211yescharacteristic
15C2×C7×C4⋊C4C2212yes—
16C2×C42×C7C2211yescharacteristic
17C2×(C2×D7)⋊C4C2211yescharacteristic
18C2×(C2×D7)⋊C4C2212yes—
19C2×(C2×D7)⋊C4C2211yescharacteristic
20C2×(C2×D7)⋊C4C2212yes—
21C2×C4⋊Dic7C2211yescharacteristic
22C2×C4⋊Dic7C2211yescharacteristic
23C2×Dic7⋊C4C2212yes—
24C2×C4×Dic7C2211yescharacteristic
25C2×C4×Dic7C2212yes—
26C7×#95C2212yes—
27C7⋊#95C2211yescharacteristic
28C7⋊#95C2212yes—
29C7⋊#95C2212yes—
30C7⋊#95C2212yes—
31C7⋊#95C2212yes—
32C7⋊#95C2212yes—
33C7⋊#95C2212yes—
34C7⋊#95C2211yescharacteristic
35#2494—71nomaximal, Sylow
36C23×D7C2311yescharacteristic
37C22×C4×C7C2311yescharacteristic
38C22×C4×C7C2311yescharacteristic
39C22×C4×C7C2312yes—
40C22×C4×C7C2312yes—
41C22×C4×C7C2311yescharacteristic
42C22×Dic7C2311yescharacteristic
43C22×Dic7C2311yescharacteristic
44C22×Dic7C2312yes—
45C22×Dic7C2312yes—
46C22×Dic7C2311yescharacteristic
47C2×C4×D7—22no—
48C2×C4×D7—22no—
49C2×C4×D7—22no—
50C7×C4⋊C4—22no—
51C7×C4⋊C4—22no—
52C7×C4⋊C4—42no—
53C42×C7—22no—
54(C2×D7)⋊C4—24no—
55(C2×D7)⋊C4—22no—
56(C2×D7)⋊C4—22no—
57(C2×D7)⋊C4—42no—
58C4⋊Dic7—22no—
59C4⋊Dic7—22no—
60Dic7⋊C4—24no—
61C4×Dic7—22no—
62C4×Dic7—42no—
63#401—72no—
64#397—72no—
65#389—71no—
66#389D711yescharacteristic, complemented
67#388—72no—
68#387—72no—
69#382—71no—
70#380—71no—
71#380—71no—
72C4×C4⋊C4—71no—
73C4×C22⋊C4—71no—
74C23×C7C2411yescharacteristic
75C22×D7—21no—
76C22×D7—21no—
77C22×D7—21no—
78C22×D7—21no—
79C22×D7—21no—
80C22×D7—21no—
81C22×D7#4112yes—
82C2×C4×C7—22no—
83C2×C4×C7—21no—
84C2×C4×C7—22no—
85C2×C4×C7—21no—
86C2×C4×C7—21no—
87C2×C4×C7—21no—
88C2×C4×C7—22no—
89C2×C4×C7—22no—
90C2×C4×C7—22no—
91C2×C4×C7—22no—
92C2×C4×C7—21no—
93C2×C4×C7—21no—
94C2×C4×C7#4112yes—
95C2×C4×C7#4112yes—
96C2×C4×C7#4112yes—
97C2×Dic7—21no—
98C2×Dic7—21no—
99C2×Dic7—21no—
100C2×Dic7—22no—
101C2×Dic7—22no—
102C2×Dic7—22no—
103C2×Dic7—22no—
104C2×Dic7—21no—
105C2×Dic7—21no—
106C2×Dic7—21no—
107C2×Dic7—21no—
108C2×Dic7—22no—
109C2×Dic7—22no—
110C2×Dic7—21no—
111C2×Dic7#4112yes—
112C4×D7—44no—
113C23×C4—71no—
114C2×C4⋊C4—72no—
115C2×C4⋊C4—71no—
116C2×C4⋊C4—71no—
117C2×C4⋊C4C2×D711yescharacteristic
118C2×C4⋊C4C2×D711yescharacteristic
119C2×C4⋊C4C2×D712yes—
120C2×C22⋊C4—71no—
121C2×C22⋊C4—71no—
122C2×C22⋊C4—72no—
123C2×C22⋊C4—72no—
124C2×C42—71no—
125C2×C42—72no—
126C2×C42C2×D711yescharacteristic
127#95—71no—
128#95—71no—
129#95—72no—
130#95—72no—
131#95—72no—
132#95—72no—
133#95—72no—
134#95—72no—
135#95C2×D712yes—
136C22×C7C2×#4111yescharacteristic
137C22×C7C2×#4111yescharacteristic
138C22×C7C2×#4111yescharacteristic
139C22×C7C2×#4111yescharacteristic
140C22×C7C2×#4111yescharacteristic
141C22×C7H(5,𝔽2)11yescharacteristic
142C22×C72−1+411yescharacteristic
143C2×D7—22no—
144C2×D7—22no—
145C2×D7—22no—
146C2×D7—41no—
147C2×D7—41no—
148C2×D7—41no—
149C2×D7—41no—
150C4×C7—42no—
151C4×C7—41no—
152C4×C7—42no—
153C4×C7—41no—
154C4×C7#12614yes—
155Dic7—22no—
156Dic7—41no—
157Dic7—42no—
158Dic7—42no—
159Dic7—41no—
160C24—71no—
161C22×C4—71no—
162C22×C4—71no—
163C22×C4—71no—
164C22×C4—72no—
165C22×C4—72no—
166C22×C4—142no—
167C22×C4—142no—
168C22×C4—142no—
169C22×C4C22×D711yescharacteristic
170C22×C4C22×D711yescharacteristic
171C22×C4C22×D712yes—
172C22×C4C22×D712yes—
173C22×C4C22×D711yescharacteristic
174C4⋊C4—22no—
175C4⋊C4—22no—
176C4⋊C4—42no—
177C4⋊C4—144no—
178C4⋊C4—142no—
179C4⋊C4—142no—
180C22⋊C4—142no—
181C22⋊C4—142no—
182C22⋊C4—144no—
183C22⋊C4—282no—
184C42—22no—
185C42—142no—
186C42—282no—
187C2×C7C2×#12611yescharacteristic
188C2×C7#52911yescharacteristic
189C2×C7#54211yescharacteristic
190C2×C7#55211yescharacteristic
191C2×C7#55311yescharacteristic
192C2×C7#55311yescharacteristic
193C2×C7#55611yescharacteristic
194D7—42no—
195C23—72no—
196C23—141no—
197C23—141no—
198C23—141no—
199C23—141no—
200C23—141no—
201C23—141no—
202C23C23×D711yescharacteristic
203C2×C4—22no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—22no—
213C2×C4—21no—
214C2×C4—22no—
215C2×C4—72no—
216C2×C4—141no—
217C2×C4—141no—
218C2×C4—141no—
219C2×C4—141no—
220C2×C4—141no—
221C2×C4—142no—
222C2×C4—142no—
223C2×C4—142no—
224C2×C4—141no—
225C2×C4—142no—
226C2×C4—141no—
227C2×C4—142no—
228C2×C4—141no—
229C2×C4—142no—
230C2×C4—284no—
231C2×C4#115712yes—
232C2×C4#115912yes—
233C2×C4#116112yes—
234C7#249411yescharacteristic, complemented, Sylow
235C22—142no—
236C22—142no—
237C22—142no—
238C22—281no—
239C22—281no—
240C22—281no—
241C22—281no—
242C22C2×#115711yescharacteristic
243C22C2×#115911yescharacteristic
244C22#654811yescharacteristic
245C22C2×#116111yescharacteristic
246C22#655111yescharacteristic
247C22D7×#4111yescharacteristic
248C22D7×#4111yescharacteristic
249C4—42no—
250C4—41no—
251C4—42no—
252C4—41no—
253C4—142no—
254C4—282no—
255C4—282no—
256C4—281no—
257C4—281no—
258C4#646114yes—
259C2—282no—
260C2C2×#646111yescharacteristic
261C2#9024911yescharacteristic
262C2#9027211yescharacteristic
263C2#9030511yescharacteristic
264C2#9033211yescharacteristic
265C2#9038211yescharacteristic
266C2#9042411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#389C2
prime radicalC23×D7C23
hypercenter#389D7
hypocenterC7#2494
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #389

Complements

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

NH ≅ G/Ntype
1. C7×#389C2semidirect
5. #89750C2semidirect
6. C4⋊C4×Dic7C2semidirect
7. #89745C2semidirect
9. #89708C2semidirect
11. C7⋊#401C2semidirect
66. #389D7semidirect
234. C7#2494semidirect

Maximal subgroups

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

subgroupindex
C7×#3892
#897652
#897592
#897502
C4⋊C4×Dic72
#897452
C4×(C2×D7)⋊C42
#897082
#894412
C7⋊#4012
#24947