Finite Group Databasebeta

← #11731981

Subgroups — #11731981

Order 896.

448 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 141 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×#6442C212yesmaximal
2C7×#385C211yesmaximal, characteristic
3#89984C212yesmaximal
4#89757C212yesmaximal, complemented
5#89755C212yesmaximal, complemented
6#89725C211yesmaximal, characteristic
7#89725C211yesmaximal, characteristic
8#89433C212yesmaximal, complemented
9#89425C212yesmaximal, complemented
10C23×C4×C7C2211yescharacteristic
11C2×C7×C4⋊C4C2212yes—
12C2×C7×C22⋊C4C2211yescharacteristic
13C2×C7×C22⋊C4C2211yescharacteristic
14C2×C22⋊Dic7C2212yes—
15C2×C22⋊Dic7C2212yes—
16C2×C4⋊Dic7C2212yes—
17C2×C4⋊Dic7C2212yes—
18C2×C4⋊Dic7C2211yescharacteristic
19C2×C4⋊Dic7C2212yes—
20C2×C4⋊Dic7C2211yescharacteristic
21C2×Dic7⋊C4C2212yes—
22C2×Dic7⋊C4C2212yes—
23C2×C4×Dic7C2212yes—
24#6442—28no—
25C7×#95C2212yes—
26C7⋊#95C2211yescharacteristic
27C7⋊#95C2212yes—
28C7⋊#95C2212yes—
29C7⋊#95C2212yes—
30C7⋊#95C2211yescharacteristic
31C7⋊#95C2212yes—
32#2722—71nomaximal, Sylow
33C24×C7C2311yescharacteristic
34C22×C4×C7—24no—
35C22×C4×C7—41no—
36C22×C4×C7C2311yescharacteristic
37C22×C4×C7C2311yescharacteristic
38C22×C4×C7C2312yes—
39C22×C4×C7C2312yes—
40C22×Dic7C2312yes—
41C22×Dic7C2312yes—
42C22×Dic7C2312yes—
43C22×Dic7C2312yes—
44C7×C4⋊C4—24no—
45C7×C22⋊C4—41no—
46C7×C22⋊C4D414yes—
47C22⋊Dic7—24no—
48C22⋊Dic7—24no—
49C4⋊Dic7—24no—
50C4⋊Dic7—24no—
51C4⋊Dic7—42no—
52C4⋊Dic7—41no—
53C4⋊Dic7—41no—
54Dic7⋊C4—24no—
55Dic7⋊C4—24no—
56C4×Dic7—24no—
57C2×#126—72no—
58#398—72no—
59#398—72no—
60#397—71no—
61#397—71no—
62#389—72no—
63#387—72no—
64#385D711yescharacteristic
65#380—72no—
66C23×C7—22no—
67C23×C7—21no—
68C23×C7—21no—
69C23×C7—22no—
70C23×C7C2×D412yes—
71C23×C7C2411yescharacteristic
72C2×C4×C7—22no—
73C2×C4×C7—22no—
74C2×C4×C7—22no—
75C2×C4×C7—22no—
76C2×C4×C7—22no—
77C2×C4×C7—21no—
78C2×C4×C7—21no—
79C2×C4×C7—22no—
80C2×C4×C7—44no—
81C2×C4×C7C2×D414yes—
82C2×C4×C7#4114yes—
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—22no—
93C2×Dic7—22no—
94C2×Dic7—22no—
95C23×C4C2×D711yescharacteristic
96#126—148no—
97C2×C4⋊C4—72no—
98C2×C4⋊C4—72no—
99C2×C4⋊C4—72no—
100C2×C4⋊C4—72no—
101C2×C4⋊C4—72no—
102C2×C4⋊C4—71no—
103C2×C4⋊C4—71no—
104C2×C4⋊C4C2×D712yes—
105C2×C22⋊C4—72no—
106C2×C22⋊C4—72no—
107C2×C22⋊C4C2×D711yescharacteristic
108C2×C22⋊C4C2×D711yescharacteristic
109C2×C42—72no—
110#95—72no—
111#95—72no—
112#95—71no—
113#95—72no—
114#95—72no—
115#95—71no—
116#95C2×D712yes—
117C22×C7—22no—
118C22×C7—24no—
119C22×C7—42no—
120C22×C7—41no—
121C22×C7—41no—
122C22×C7C22×D411yescharacteristic
123C22×C7C2×#4112yes—
124C22×C7H(5,𝔽2)11yescharacteristic
125C22×C72−1+411yescharacteristic
126C22×C72−1+412yes—
127C4×C7—24no—
128C4×C7—24no—
129C4×C7—41no—
130C4×C7—41no—
131Dic7—42no—
132Dic7—42no—
133Dic7—42no—
134Dic7—42no—
135C24C22×D711yescharacteristic
136C22×C4—24no—
137C22×C4—41no—
138C22×C4—72no—
139C22×C4—72no—
140C22×C4—72no—
141C22×C4—72no—
142C22×C4C22×D711yescharacteristic
143C22×C4C22×D711yescharacteristic
144C22×C4C22×D712yes—
145C22×C4C22×D712yes—
146C4⋊C4—24no—
147C4⋊C4—144no—
148C4⋊C4—144no—
149C4⋊C4—144no—
150C4⋊C4—144no—
151C4⋊C4—282no—
152C4⋊C4—281no—
153C4⋊C4—281no—
154C22⋊C4—41no—
155C22⋊C4—144no—
156C22⋊C4—144no—
157C22⋊C4D2814yes—
158C42—144no—
159C2×C7—42no—
160C2×C7#53711yescharacteristic
161C2×C7#54112yes—
162C2×C7#54712yes—
163C2×C7#55111yescharacteristic
164C2×C7#56311yescharacteristic
165C23—22no—
166C23—21no—
167C23—21no—
168C23—22no—
169C23C2×D2812yes—
170C23C23×D711yescharacteristic
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—22no—
179C2×C4—44no—
180C2×C4—142no—
181C2×C4—142no—
182C2×C4—142no—
183C2×C4—142no—
184C2×C4—142no—
185C2×C4—142no—
186C2×C4—142no—
187C2×C4—142no—
188C2×C4—142no—
189C2×C4—142no—
190C2×C4—142no—
191C2×C4—142no—
192C2×C4C2×D2814yes—
193C2×C4#115914yes—
194C7#272211yescharacteristic, complemented, Sylow
195C22—22no—
196C22—24no—
197C22—41no—
198C22—42no—
199C22—41no—
200C22C22×D2811yescharacteristic
201C22C2×#115912yes—
202C22#654811yescharacteristic
203C22#655111yescharacteristic
204C22#655412yes—
205C4—24no—
206C4—24no—
207C4—41no—
208C4—41no—
209C4—282no—
210C4—282no—
211C4—282no—
212C4—282no—
213C2—42no—
214C2#9020111yescharacteristic
215C2#9024612yes—
216C2#9027811yescharacteristic
217C2#9031612yes—
218C2#9034211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#385C2
prime radicalC24×C7C23
hypercenter#385D7
hypocenterC7#2722
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
4. #89757C2semidirect
5. #89755C2semidirect
8. #89433C2semidirect
9. #89425C2semidirect
194. C7#2722semidirect

Maximal subgroups

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

subgroupindex
C2×#64422
C7×#3852
#899842
#897572
#897552
#897252
#894332
#894252
#27227