Finite Group Databasebeta

← #11734289

Subgroups — #11734289

Order 896.

424 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 129 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×#393C211yesmaximal, characteristic
2#90001C212yesmaximal, complemented
3#89719C211yesmaximal, characteristic
4#89718C212yesmaximal
5#89718C211yesmaximal, characteristic
6#89717C212yesmaximal
7#89430C212yesmaximal, complemented
8#89422C212yesmaximal, complemented
9#89418C212yesmaximal, complemented
10C22×C7×Q8C2211yescharacteristic
11C23×Dic7C2211yescharacteristic
12C22×Dic14C2211yescharacteristic
13C2×C7×C22⋊C4C2212yes—
14C2×C7×C22⋊C4C2211yescharacteristic
15C2×C22⋊Dic7C2212yes—
16C2×C22⋊Dic7C2211yescharacteristic
17C2×Dic7⋊C4C2212yes—
18C2×Dic7⋊C4C2212yes—
19C2×Dic7⋊C4C2212yes—
20C2×Dic7⋊C4C2212yes—
21C2×C4×Dic7C2212yes—
22C7×#95C2212yes—
23C7×#95C2211yescharacteristic
24C7⋊#95C2212yes—
25C7⋊#95C2212yes—
26C7⋊#95C2212yes—
27C7⋊#95C2212yes—
28C7⋊#95C2212yes—
29C7⋊#95C2211yescharacteristic
30C7⋊#95C2212yes—
31#2763—71nomaximal, Sylow
32C24×C7C2311yescharacteristic
33C2×C7×Q8—42no—
34C2×C7×Q8—41no—
35C22×C4×C7C2311yescharacteristic
36C22×C4×C7C2312yes—
37C22×C4×C7C2312yes—
38C22×C4×C7C2311yescharacteristic
39C22×Dic7—42no—
40C22×Dic7—41no—
41C22×Dic7C2312yes—
42C22×Dic7C2312yes—
43C22×Dic7C2312yes—
44C22×Dic7C2312yes—
45C2×Dic14—24no—
46C2×Dic14—41no—
47C7×C22⋊C4—42no—
48C7×C22⋊C4—41no—
49C22⋊Dic7—24no—
50C22⋊Dic7Q814yescomplemented
51Dic7⋊C4—42no—
52Dic7⋊C4—42no—
53Dic7⋊C4—42no—
54Dic7⋊C4—42no—
55C4×Dic7—24no—
56#400—72no—
57#395—72no—
58#393—71no—
59#393D711yescharacteristic
60#391—72no—
61#388—72no—
62#385—72no—
63#385—71no—
64#382—72no—
65C23×C7—22no—
66C23×C7—21no—
67C23×C7—22no—
68C23×C7—21no—
69C23×C7C2×Q812yes—
70C23×C7C2411yescharacteristic
71C7×Q8—44no—
72C2×C4×C7—22no—
73C2×C4×C7—22no—
74C2×C4×C7—22no—
75C2×C4×C7—22no—
76C2×C4×C7—22no—
77C2×C4×C7—22no—
78C2×C4×C7—21no—
79C2×C4×C7—22no—
80C2×C4×C7—21no—
81C2×C4×C7—22no—
82C2×Dic7—22no—
83C2×Dic7—22no—
84C2×Dic7—22no—
85C2×Dic7—22no—
86C2×Dic7—22no—
87C2×Dic7—22no—
88C2×Dic7—22no—
89C2×Dic7—22no—
90C2×Dic7—22no—
91C2×Dic7—22no—
92C2×Dic7—44no—
93C2×Dic7C2×Q814yes—
94C2×Dic7#4114yes—
95Dic14—44no—
96C22×Q8—71no—
97C22×Q8C2×D711yescharacteristic
98C23×C4—71no—
99C2×C4⋊C4—72no—
100C2×C4⋊C4—72no—
101C2×C4⋊C4—72no—
102C2×C4⋊C4—72no—
103C2×C22⋊C4—71no—
104C2×C22⋊C4—72no—
105C2×C22⋊C4C2×D712yes—
106C2×C22⋊C4C2×D711yescharacteristic
107C2×C42—72no—
108#95—72no—
109#95—72no—
110#95—72no—
111#95—72no—
112#95—72no—
113#95—72no—
114#95—71no—
115#95C2×D711yescharacteristic
116#95C2×D712yes—
117C22×C7—22no—
118C22×C7—24no—
119C22×C7—42no—
120C22×C7—41no—
121C22×C7—41no—
122C22×C7C22×Q811yescharacteristic
123C22×C7C2×#4112yes—
124C22×C7H(5,𝔽2)12yes—
125C22×C7H(5,𝔽2)11yescharacteristic
126C22×C72−1+411yescharacteristic
127C4×C7—42no—
128C4×C7—41no—
129C4×C7—41no—
130C4×C7—42no—
131Dic7—24no—
132Dic7—24no—
133Dic7—42no—
134Dic7—42no—
135C24C22×D711yescharacteristic
136C2×Q8—41no—
137C2×Q8—42no—
138C2×Q8—144no—
139C2×Q8—281no—
140C22×C4—72no—
141C22×C4—72no—
142C22×C4—72no—
143C22×C4—72no—
144C22×C4—281no—
145C22×C4—282no—
146C22×C4C22×D711yescharacteristic
147C22×C4C22×D712yes—
148C22×C4C22×D711yescharacteristic
149C22×C4C22×D712yes—
150C4⋊C4—282no—
151C4⋊C4—282no—
152C4⋊C4—282no—
153C4⋊C4—282no—
154C22⋊C4—42no—
155C22⋊C4—41no—
156C22⋊C4—74no—
157C22⋊C4—144no—
158C42—144no—
159C2×C7—42no—
160C2×C7#54212yes—
161C2×C7#54411yescharacteristic
162C2×C7#55111yescharacteristic
163C2×C7#55412yes—
164C2×C7H(3,𝔽4)11yescharacteristic
165C23—21no—
166C23—22no—
167C23—22no—
168C23—21no—
169C23Q8×D712yes—
170C23C23×D711yescharacteristic
171Q8—44no—
172Q8—284no—
173C2×C4—22no—
174C2×C4—21no—
175C2×C4—22no—
176C2×C4—21no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—74no—
184C2×C4—74no—
185C2×C4—142no—
186C2×C4—142no—
187C2×C4—142no—
188C2×C4—142no—
189C2×C4—142no—
190C2×C4—142no—
191C2×C4—142no—
192C2×C4—142no—
193C2×C4—142no—
194C2×C4—142no—
195C2×C4—284no—
196C7#276311yescharacteristic, complemented, Sylow
197C22—22no—
198C22—24no—
199C22—41no—
200C22—41no—
201C22—42no—
202C22#654811yescharacteristic
203C22#654812yes—
204C22C2×Q8×D711yescharacteristic
205C22D7×#4112yes—
206C22#655411yescharacteristic
207C4—41no—
208C4—41no—
209C4—42no—
210C4—42no—
211C4—144no—
212C4—144no—
213C4—282no—
214C4—282no—
215C2—42no—
216C2#9028511yescharacteristic
217C2#9031411yescharacteristic
218C2#9032712yes—
219C2#9036012yes—
220C2#9037011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#393C2
prime radicalC24×C7C23
hypercenter#393D7
hypocenterC7#2763
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
2. #90001C2semidirect
7. #89430C2semidirect
8. #89422C2semidirect
9. #89418C2semidirect
50. C22⋊Dic7Q8semidirect
196. C7#2763semidirect

Maximal subgroups

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

subgroupindex
C7×#3932
#900012
#897192
#897182
#897172
#894302
#894222
#894182
#27637