Finite Group Databasebeta

← #11734687

Subgroups — #11734687

Order 896.

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

normal subgroups only

This group has 416 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×#395C211yesmaximal, characteristic, complemented
2#90003C212yesmaximal, complemented
3#89763C212yesmaximal
4#89763C211yesmaximal, characteristic
5#89754C212yesmaximal, complemented
6#89746C211yesmaximal, characteristic, complemented
7#89447C212yesmaximal, complemented
8#89445C212yesmaximal, complemented
9#89441C212yesmaximal, complemented
10C22×C7×Q8C2211yescharacteristic
11C22×D28C2211yescharacteristic
12C22×Dic14C2211yescharacteristic
13C2×C7×C4⋊C4C2211yescharacteristic
14C2×C7×C4⋊C4C2212yes—
15C2×(C2×D7)⋊C4C2212yes—
16C2×(C2×D7)⋊C4C2212yes—
17C2×(C2×D7)⋊C4C2212yes—
18C2×(C2×D7)⋊C4C2212yes—
19C2×(C2×D7)⋊C4C2212yes—
20C2×(C2×D7)⋊C4C2212yes—
21C2×Dic7⋊C4C2212yes—
22C2×C4×Dic7C2211yescharacteristic
23C7×#95C2212yes—
24C7×#95C2211yescharacteristic
25C7⋊#95C2212yes—
26C7⋊#95C2212yes—
27C7⋊#95C2212yes—
28C7⋊#95C2212yes—
29C7⋊#95C2212yes—
30C7⋊#95C2211yescharacteristic
31#2832—71nomaximal, Sylow
32C23×D7C2312yes—
33C2×C7×Q8—41no—
34C2×C7×Q8—42no—
35C22×C4×C7C2311yescharacteristic
36C22×C4×C7C2312yes—
37C22×C4×C7C2311yescharacteristic
38C22×C4×C7C2312yes—
39C22×C4×C7C2311yescharacteristic
40C22×Dic7C2312yes—
41C22×Dic7C2312yes—
42C22×Dic7C2312yes—
43C2×D28—41no—
44C2×D28—42no—
45C2×Dic14—22no—
46C2×Dic14—41no—
47C2×Dic14—41no—
48C7×C4⋊C4—41no—
49C7×C4⋊C4—42no—
50(C2×D7)⋊C4—24no—
51(C2×D7)⋊C4—42no—
52(C2×D7)⋊C4—42no—
53(C2×D7)⋊C4—42no—
54(C2×D7)⋊C4—42no—
55(C2×D7)⋊C4—42no—
56Dic7⋊C4—42no—
57C4×Dic7—41no—
58#400—72no—
59#397—72no—
60#395D711yescharacteristic, complemented
61#394—72no—
62#394—71no—
63#393—72no—
64#393—72no—
65#391—71no—
66#388—72no—
67C23×C7C2411yescharacteristic
68C22×D7—22no—
69C22×D7—22no—
70C22×D7—22no—
71C22×D7—22no—
72C22×D7—22no—
73C22×D7—22no—
74C22×D7—22no—
75C7×Q8—44no—
76C2×C4×C7—21no—
77C2×C4×C7—21no—
78C2×C4×C7—22no—
79C2×C4×C7—22no—
80C2×C4×C7—22no—
81C2×C4×C7—22no—
82C2×C4×C7—22no—
83C2×C4×C7—21no—
84C2×C4×C7—22no—
85C2×C4×C7—21no—
86C2×C4×C7—22no—
87C2×C4×C7—22no—
88C2×C4×C7#4112yes—
89C2×Dic7—22no—
90C2×Dic7—22no—
91C2×Dic7—22no—
92C2×Dic7—22no—
93C2×Dic7—22no—
94C2×Dic7—22no—
95C2×Dic7—22no—
96C2×Dic7—22no—
97C2×Dic7—22no—
98D28—44no—
99Dic14—44no—
100C22×Q8—71no—
101C22×Q8C2×D711yescharacteristic
102C22×D4—71no—
103C2×C4⋊C4—72no—
104C2×C4⋊C4C2×D711yescharacteristic
105C2×C4⋊C4C2×D712yes—
106C2×C22⋊C4—72no—
107C2×C22⋊C4—72no—
108C2×C22⋊C4—72no—
109C2×C22⋊C4—72no—
110C2×C22⋊C4—72no—
111C2×C22⋊C4—72no—
112C2×C42—71no—
113#95—72no—
114#95—72no—
115#95—72no—
116#95—72no—
117#95—72no—
118#95—71no—
119#95C2×D712yes—
120#95C2×D711yescharacteristic
121C22×C7C2×#4111yescharacteristic
122C22×C7H(5,𝔽2)11yescharacteristic
123C22×C7H(5,𝔽2)12yes—
124C22×C7H(5,𝔽2)11yescharacteristic
125C22×C72−1+412yes—
126C2×D7—42no—
127C2×D7—42no—
128C2×D7—42no—
129C2×D7—42no—
130C2×D7—42no—
131C2×D7—42no—
132C2×D7—42no—
133C4×C7—22no—
134C4×C7—41no—
135C4×C7—42no—
136C4×C7—41no—
137C4×C7—42no—
138Dic7—42no—
139Dic7—42no—
140Dic7—42no—
141C24—72no—
142C2×Q8—42no—
143C2×Q8—41no—
144C2×Q8—142no—
145C2×Q8—281no—
146C2×Q8—281no—
147C2×D4—282no—
148C2×D4—281no—
149C22×C4—72no—
150C22×C4—72no—
151C22×C4—72no—
152C22×C4C22×D711yescharacteristic
153C22×C4C22×D712yes—
154C22×C4C22×D711yescharacteristic
155C22×C4C22×D712yes—
156C22×C4C22×D711yescharacteristic
157C4⋊C4—41no—
158C4⋊C4—42no—
159C4⋊C4—282no—
160C22⋊C4—144no—
161C22⋊C4—282no—
162C22⋊C4—282no—
163C22⋊C4—282no—
164C22⋊C4—282no—
165C22⋊C4—282no—
166C42—281no—
167C2×C7#53811yescharacteristic
168C2×C7#54212yes—
169C2×C7H(3,𝔽4)11yescharacteristic
170C2×C7#56212yes—
171C2×C7#56311yescharacteristic
172D7—82no—
173C23—142no—
174C23—142no—
175C23—142no—
176C23—142no—
177C23—142no—
178C23—142no—
179C23—142no—
180C23C23×D711yescharacteristic
181Q8—44no—
182Q8—284no—
183D4—284no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—21no—
192C2×C4—22no—
193C2×C4—21no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—142no—
197C2×C4—142no—
198C2×C4—142no—
199C2×C4—142no—
200C2×C4—142no—
201C2×C4—142no—
202C2×C4—142no—
203C2×C4—142no—
204C2×C4—142no—
205C2×C4#116112yes—
206C7#283211yescharacteristic, complemented, Sylow
207C22—282no—
208C22—282no—
209C22—282no—
210C22—282no—
211C22—282no—
212C22—282no—
213C22—282no—
214C22#654812yes—
215C22#654811yescharacteristic
216C22C2×#116111yescharacteristic
217C22#655112yes—
218C22#655311yescharacteristic
219C4—22no—
220C4—41no—
221C4—41no—
222C4—42no—
223C4—42no—
224C4—282no—
225C4—282no—
226C4—282no—
227C2—562no—
228C2#9028511yescharacteristic
229C2#9033511yescharacteristic
230C2#9033812yes—
231C2#9042412yes—
232C2#9042611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#395C2
prime radicalC22×D28C22
hypercenter#395D7
hypocenterC7#2832
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
1. C7×#395C2semidirect
2. #90003C2semidirect
5. #89754C2semidirect
6. #89746C2semidirect
7. #89447C2semidirect
8. #89445C2semidirect
9. #89441C2semidirect
60. #395D7semidirect
206. C7#2832semidirect

Maximal subgroups

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

subgroupindex
C7×#3952
#900032
#897632
#897542
#897462
#894472
#894452
#894412
#28327