Finite Group Databasebeta

← #11736831

Subgroups — #11736831

Order 896.

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

normal subgroups only

This group has 418 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#90226C211yesmaximal, characteristic, complemented
2#90212C211yesmaximal, characteristic, complemented
3C7×#433C211yesmaximal, characteristic, complemented
4#89619C212yesmaximal, complemented
5#89615C212yesmaximal, complemented
6#89509C212yesmaximal, complemented
7#89504C212yesmaximal, complemented
8#89483C212yesmaximal, complemented
9C4×D56C212yesmaximal, complemented
10C22×D28C2211yescharacteristic
11C7×#117C2211yescharacteristic
12#6493C2212yes—
13#6489C2211yescharacteristic
14C2×D56C2212yes—
15C2×D56C2212yes—
16#6457—22no—
17#6456—22no—
18#6447—22no—
19D7×C22⋊C4—22no—
20#6437C2211yescharacteristic, complemented
21C4×D28C2212yes—
22C4×D28C2212yes—
23C4×D28C2212yescomplemented
24#6433C2211yescharacteristic, complemented
25C7×C4⋊C8C2212yescomplemented
26C7×C22⋊C8C2212yes—
27C7×#97C2211yescharacteristic, complemented
28C4×C7×C8C2211yescharacteristic, complemented
29#6395C2212yes—
30#6395C2212yes—
31#6395C2212yescomplemented
32#6395C2212yescomplemented
33#6392C2212yescomplemented
34#6391C2212yescomplemented
35#3122—71nomaximal, Sylow
36C23×D7—21no—
37C22×C4×C7C2311yescharacteristic
38C2×C22⋊D7—22no—
39C2×D28—21no—
40C2×D28—24no—
41C2×D28—22no—
42C2×D28C2312yes—
43C2×D28C2312yes—
44C2×C4×D7—22no—
45C2×C4×D7—22no—
46C2×C4×D7—22no—
47C2×C7×C8C2312yes—
48C2×C7×C8C2312yes—
49C7×C4⋊C4D412yes—
50C7×C22⋊C4D412yes—
51C42×C7C2311yescharacteristic
52C42×C7C2311yescharacteristic
53C22⋊Dic7—21no—
54(C2×D7)⋊C4—22no—
55(C2×D7)⋊C4—22no—
56(C2×D7)⋊C4—22no—
57(C2×D7)⋊C4—22no—
58C4⋊Dic7C2312yes—
59C4⋊Dic7C2312yes—
60Dic7⋊C4—22no—
61C4×Dic7—21no—
62D56—24no—
63D56—24no—
64#540—71no—
65#518—71no—
66#482—72no—
67#478—72no—
68#459—72no—
69#447—72no—
70#442—72no—
71C4×D8—72no—
72#433D711yescharacteristic, complemented
73C23×C7C2×D411yescharacteristic
74C22×D7—22no—
75C22×D7—24no—
76C22×D7—22no—
77C22×D7—82no—
78C2×C4×C7—22no—
79C2×C4×C7C2×D411yescharacteristic
80C2×C4×C7C2×D412yes—
81C2×C4×C7C2×D412yes—
82C2×C4×C7C2411yescharacteristic
83C22⋊D7—82no—
84C2×Dic7—22no—
85C2×Dic7—22no—
86D28—24no—
87D28—22no—
88D28—22no—
89D28—22no—
90D28—82no—
91D28#4114yes—
92C4×D7—82no—
93C4×D7—82no—
94C4×D7—82no—
95C7×C8—22no—
96C7×C8—22no—
97C22×D4—71no—
98C2×D8—72no—
99C2×D8—72no—
100#127—71no—
101#125—71no—
102#123—142no—
103#121—72no—
104#121—142no—
105C4×D4—72no—
106C4×D4—72no—
107C4×D4—72no—
108C4×D4—142no—
109#117—71no—
110#117C2×D711yescharacteristic
111C2×C22⋊C4—142no—
112#107—72no—
113#106—72no—
114C4⋊C8C2×D712yescomplemented
115#102—72no—
116#102—72no—
117#102—72no—
118#102—72no—
119C22⋊C8C2×D712yes—
120#97C2×D711yescharacteristic, complemented
121C4×C8C2×D711yescharacteristic, complemented
122C22×C7—21no—
123C22×C7—22no—
124C22×C7C22×D411yescharacteristic
125C2×D7—24no—
126C2×D7—42no—
127C2×D7—42no—
128C2×D7—42no—
129C2×D7—42no—
130C2×D7—42no—
131C2×D7—84no—
132C4×C7—21no—
133C4×C7—22no—
134C4×C7—22no—
135C4×C7C2×#4112yes—
136Dic7—42no—
137Dic7—42no—
138C24—141no—
139C2×D4—72no—
140C2×D4—72no—
141C2×D4—144no—
142C2×D4—142no—
143C2×D4—141no—
144C2×D4—142no—
145C22×C4—142no—
146C22×C4—142no—
147C22×C4—142no—
148C22×C4C22×D711yescharacteristic
149D8—144no—
150D8—144no—
151C2×C8C22×D712yes—
152C2×C8C22×D712yes—
153C4⋊C4—72no—
154C4⋊C4—72no—
155C4⋊C4—142no—
156C4⋊C4D2812yes—
157C22⋊C4—141no—
158C22⋊C4—142no—
159C22⋊C4—142no—
160C22⋊C4—142no—
161C22⋊C4—142no—
162C22⋊C4D2812yes—
163C42—141no—
164C42C22×D711yescharacteristic
165C42C22×D711yescharacteristic
166C2×C7—41no—
167C2×C7#52511yescharacteristic
168C2×C7#57612yes—
169D7—44no—
170D7—82no—
171C23—144no—
172C23—142no—
173C23—142no—
174C23—562no—
175C23C2×D2811yescharacteristic
176D4—74no—
177D4—144no—
178D4—142no—
179D4—142no—
180D4—142no—
181D4—562no—
182D4—562no—
183C2×C4—22no—
184C2×C4—142no—
185C2×C4—142no—
186C2×C4—562no—
187C2×C4—562no—
188C2×C4—562no—
189C2×C4C2×D2811yescharacteristic
190C2×C4C2×D2812yes—
191C2×C4C2×D2812yes—
192C2×C4C23×D711yescharacteristic
193C8—22no—
194C8—22no—
195C7#312211yescharacteristic, complemented, Sylow
196C22—22no—
197C22—21no—
198C22—144no—
199C22—282no—
200C22—282no—
201C22—282no—
202C22—282no—
203C22—282no—
204C22—564no—
205C22C22×D2811yescharacteristic
206C4—22no—
207C4—22no—
208C4—21no—
209C4—282no—
210C4—282no—
211C4D7×#4112yes—
212C2—41no—
213C2—284no—
214C2—562no—
215C2#9021511yescharacteristic
216C2#9044312yes—
subgroupG/H
centerC22C22×D28
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#433C2
socleC22×C7C22×D4
hypercenter#433D7
hypocenterC7#3122
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ #117 ◁ #433

Complements

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

NH ≅ G/Ntype
1. #90226C2semidirect
2. #90212C2semidirect
3. C7×#433C2semidirect
4. #89619C2semidirect
5. #89615C2semidirect
6. #89509C2semidirect
7. #89504C2semidirect
8. #89483C2semidirect
9. C4×D56C2semidirect
20. #6437C22semidirect
23. C4×D28C22semidirect
24. #6433C22semidirect
25. C7×C4⋊C8C22semidirect
27. C7×#97C22semidirect
28. C4×C7×C8C22semidirect
31. #6395C22semidirect
32. #6395C22semidirect
33. #6392C22semidirect
34. #6391C22semidirect
72. #433D7semidirect
114. C4⋊C8C2×D7semidirect
120. #97C2×D7semidirect
121. C4×C8C2×D7semidirect
195. C7#3122semidirect

Maximal subgroups

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

subgroupindex
#902262
#902122
C7×#4332
#896192
#896152
#895092
#895042
#894832
C4×D562
#31227