Finite Group Databasebeta

← #11736764

Subgroups — #11736764

Order 896.

366 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 121 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
1#90168C211yesmaximal, characteristic
2#90161C211yesmaximal, characteristic, complemented
3C7×#432C211yesmaximal, characteristic, complemented
4#89618C212yesmaximal, complemented
5#89614C212yesmaximal, complemented
6#89509C214yesmaximal, complemented
7#89485C212yesmaximal, complemented
8#89474C212yesmaximal, complemented
9C2×#1157—21no—
10C2×C42×C7C2211yescharacteristic
11#6493—21no—
12#6493C2212yes—
13#6489C2212yes—
14#6487—21no—
15#6487—21no—
16#6438—21no—
17#6437C2211yescharacteristic
18C4×D28—21no—
19C4×D28C2212yes—
20#6433C2212yes—
21#6432C2211yescharacteristic
22#6432C2211yescharacteristic
23#6432C2211yescharacteristic
24C4×Dic14—21no—
25C4×Dic14—21no—
26C7×C4⋊C8C2212yes—
27C7×C22⋊C8C2212yes—
28C7×#97C2212yes—
29#6395C2214yes—
30#6395C2214yes—
31#6392C2214yes—
32#6391C2214yes—
33#3235—71nomaximal, Sylow
34C22×C4×C7D411yescharacteristic
35C22×C4×C7D411yescharacteristic
36C22×C4×C7C2311yescharacteristic
37C2×C22⋊D7—22no—
38#1157—44no—
39C2×D28—21no—
40C2×D28—21no—
41C2×D28C2312yes—
42C2×C4×D7—22no—
43C2×Dic14—21no—
44C2×Dic14—21no—
45C2×Dic14—21no—
46C2×C7×C8C2314yes—
47C42×C7D411yescharacteristic
48C42×C7D411yescharacteristic
49C42×C7C2311yescharacteristic
50C42×C7C2311yescharacteristic
51C22⋊Dic7—22no—
52(C2×D7)⋊C4—22no—
53(C2×D7)⋊C4—22no—
54C4⋊Dic7—21no—
55C4⋊Dic7—21no—
56C4⋊Dic7—21no—
57C4⋊Dic7C2312yes—
58C4⋊Dic7C2312yes—
59C4⋊Dic7C2312yes—
60Dic7⋊C4—22no—
61Dic7⋊C4—22no—
62C4×Dic7—22no—
63#533—71no—
64#532—71no—
65#501—72no—
66#490—72no—
67#482—74no—
68#476—72no—
69#474—72no—
70#432D711yescharacteristic, complemented
71C23×C7C2×D411yescharacteristic
72C22×D7—22no—
73C2×C4×C7—22no—
74C2×C4×C7—22no—
75C2×C4×C7—41no—
76C2×C4×C7C2×D411yescharacteristic
77C2×C4×C7C2×D411yescharacteristic
78C2×C4×C7C2×D411yescharacteristic
79C2×C4×C7C2×D411yescharacteristic
80C2×C4×C7C2×D411yescharacteristic
81C2×C4×C7C2411yescharacteristic
82C22⋊D7—44no—
83C2×Dic7—22no—
84C2×Dic7—22no—
85C2×Dic7—22no—
86D28—24no—
87D28—42no—
88D28—42no—
89C4×D7—44no—
90Dic14—42no—
91Dic14—42no—
92Dic14—42no—
93C7×C8—24no—
94C2×#41—141no—
95C4⋊Q8—71no—
96C4⋊Q8—71no—
97C4⋊Q8—71no—
98#127—71no—
99#125—72no—
100#124—141no—
101C22⋊Q8—141no—
102C22⋊Q8—141no—
103#121—72no—
104#121—141no—
105C4×Q8—141no—
106C4×Q8—141no—
107C4×D4—72no—
108C4×D4—141no—
109#117—72no—
110C2×C42C2×D711yescharacteristic
111#107—74no—
112#106—74no—
113C4⋊C8C2×D712yes—
114#102—74no—
115#102—74no—
116C22⋊C8C2×D712yes—
117#97C2×D712yes—
118C22×C7—21no—
119C22×C7—21no—
120C22×C7—21no—
121C22×C7C22×D411yescharacteristic
122C2×D7—42no—
123C2×D7—42no—
124C2×D7—42no—
125C4×C7—21no—
126C4×C7—21no—
127C4×C7—21no—
128C4×C7—21no—
129C4×C7AGL(1,ℤ/8)12yes—
130C4×C72−1+412yes—
131Dic7—42no—
132Dic7—42no—
133Dic7—42no—
134#41—284no—
135C2×Q8—141no—
136C2×Q8—141no—
137C2×Q8—141no—
138C2×D4—72no—
139C2×D4—142no—
140C2×D4—141no—
141C2×D4—141no—
142C22×C4—142no—
143C22×C4D2811yescharacteristic
144C22×C4D2811yescharacteristic
145C22×C4C22×D711yescharacteristic
146C2×C8C22×D714yes—
147C4⋊C4—72no—
148C4⋊C4—72no—
149C4⋊C4—72no—
150C4⋊C4—142no—
151C4⋊C4—142no—
152C4⋊C4—141no—
153C4⋊C4—141no—
154C4⋊C4—141no—
155C22⋊C4—142no—
156C22⋊C4—142no—
157C22⋊C4—142no—
158C42—142no—
159C42D2811yescharacteristic
160C42D2811yescharacteristic
161C42C22×D711yescharacteristic
162C42C22×D711yescharacteristic
163C2×C7—41no—
164C2×C7#53611yescharacteristic
165C2×C7C2×AGL(1,ℤ/8)11yescharacteristic
166C2×C7#57511yescharacteristic
167D7—82no—
168C23—142no—
169C23C2×D2811yescharacteristic
170Q8—282no—
171Q8—282no—
172Q8—282no—
173D4—144no—
174D4—282no—
175D4—282no—
176D4—284no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—41no—
180C2×C4—142no—
181C2×C4—142no—
182C2×C4—142no—
183C2×C4—284no—
184C2×C4C2×D2811yescharacteristic
185C2×C4C2×D2811yescharacteristic
186C2×C4C2×D2811yescharacteristic
187C2×C4C2×D2811yescharacteristic
188C2×C4C2×D2811yescharacteristic
189C2×C4C23×D711yescharacteristic
190C8—24no—
191C7#323511yescharacteristic, complemented, Sylow
192C22—21no—
193C22—21no—
194C22—21no—
195C22—282no—
196C22—282no—
197C22—282no—
198C22C22×D2811yescharacteristic
199C4—21no—
200C4—21no—
201C4—21no—
202C4—21no—
203C4—282no—
204C4—282no—
205C4—282no—
206C4#647112yes—
207C4#655412yes—
208C2—41no—
209C2—562no—
210C2#9021711yescharacteristic
211C2C2×#647111yescharacteristic
212C2#9043911yescharacteristic
subgroupG/H
centerC22C22×D28
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#432C2
socleC22×C7C22×D4
prime radical#90168C2
hypercenter#432D7
hypocenterC7#3235
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #432

Complements

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

NH ≅ G/Ntype
2. #90161C2semidirect
3. C7×#432C2semidirect
4. #89618C2semidirect
5. #89614C2semidirect
6. #89509C2semidirect
7. #89485C2semidirect
8. #89474C2semidirect
70. #432D7semidirect
191. C7#3235semidirect

Maximal subgroups

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

subgroupindex
#901682
#901612
C7×#4322
#896182
#896142
#895092
#894852
#894742
#32357