Finite Group Databasebeta

← #11730992

Subgroups — #11730992

Order 896.

544 conjugacy classes of subgroups; only the 185 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 185 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×#6487C211yesmaximal, characteristic
2C22×Dic7⋊C4C211yesmaximal, characteristic
3C4×C7×C22⋊C4C211yesmaximal, characteristic
4#89722C212yesmaximal
5#89722C212yesmaximal
6#89700C214yesmaximal, complemented
7#89433C212yesmaximal, complemented
8#89428C212yesmaximal, complemented
9C23×C4×C7C2211yescharacteristic
10C23×Dic7C2211yescharacteristic
11C23×Dic7C2211yescharacteristic
12C22×Dic14C2211yescharacteristic
13C2×C7×C22⋊C4C2212yes—
14C2×C42×C7C2212yes—
15C2×C22⋊Dic7C2212yes—
16C2×C4⋊Dic7C2211yescharacteristic
17C2×C4⋊Dic7C2214yes—
18C2×C4⋊Dic7C2214yes—
19#6487—24no—
20C2×Dic7⋊C4—24no—
21C2×Dic7⋊C4C2212yes—
22C2×Dic7⋊C4C2212yes—
23C2×Dic7⋊C4C2212yes—
24C7×#95C2212yes—
25C7⋊#95C2214yes—
26C7⋊#95C2214yes—
27#2583—71nomaximal, Sylow
28C24×C7C2311yescharacteristic
29C22×C4×C7—42no—
30C22×C4×C7D414yes—
31C22×C4×C7C2311yescharacteristic
32C22×C4×C7C2311yescharacteristic
33C22×C4×C7C2314yes—
34C22×Dic7—22no—
35C22×Dic7—24no—
36C22×Dic7—22no—
37C22×Dic7—42no—
38C22×Dic7C2312yes—
39C22×Dic7C2312yes—
40C22×Dic7C2312yes—
41C22×Dic7C2312yes—
42C2×Dic14—22no—
43C2×Dic14—42no—
44C7×C22⋊C4Q818yescomplemented
45C42×C7—42no—
46C22⋊Dic7—24no—
47C4⋊Dic7—22no—
48C4⋊Dic7—28no—
49C4⋊Dic7—44no—
50Dic7⋊C4—24no—
51Dic7⋊C4—24no—
52Dic7⋊C4—24no—
53C2×C22⋊Q8—71no—
54C22×C4⋊C4—71no—
55#398—72no—
56#395—72no—
57#389—74no—
58#385—72no—
59#385—72no—
60C4×C22⋊C4D711yescharacteristic
61C23×C7—22no—
62C23×C7—22no—
63C23×C7C2×D412yes—
64C23×C7C2×Q814yes—
65C23×C7C2411yescharacteristic
66C2×C4×C7—24no—
67C2×C4×C7—24no—
68C2×C4×C7—22no—
69C2×C4×C7—22no—
70C2×C4×C7—44no—
71C2×C4×C7C2×D412yes—
72C2×C4×C7C2×D412yes—
73C2×C4×C7C2×Q818yes—
74C2×Dic7—28no—
75C2×Dic7—22no—
76C2×Dic7—22no—
77C2×Dic7—22no—
78C2×Dic7—22no—
79C2×Dic7—22no—
80C2×Dic7—24no—
81C2×Dic7—24no—
82C2×Dic7—24no—
83C2×Dic7—44no—
84C2×Dic7#4114yes—
85Dic14—44no—
86C22×Q8—71no—
87C23×C4—71no—
88C23×C4—71no—
89C23×C4C2×D711yescharacteristic
90C22⋊Q8—144no—
91C2×C4⋊C4—72no—
92C2×C4⋊C4—72no—
93C2×C4⋊C4—72no—
94C2×C4⋊C4—71no—
95C2×C4⋊C4—74no—
96C2×C4⋊C4—74no—
97C2×C4⋊C4—144no—
98C2×C22⋊C4—72no—
99C2×C22⋊C4C2×D712yes—
100C2×C42C2×D712yes—
101#95—74no—
102#95—74no—
103#95C2×D712yes—
104C22×C7—22no—
105C22×C7—22no—
106C22×C7—28no—
107C22×C7C4⋊Q814yes—
108C22×C7C22×D411yescharacteristic
109C22×C7C22×Q812yes—
110C22×C7C2×#4112yes—
111C22×C7H(5,𝔽2)12yes—
112C4×C7—22no—
113C4×C7—22no—
114C4×C7—28no—
115Dic7—24no—
116Dic7—42no—
117Dic7—42no—
118Dic7—42no—
119C24C22×D711yescharacteristic
120C2×Q8—142no—
121C2×Q8—282no—
122C22×C4—42no—
123C22×C4—72no—
124C22×C4—72no—
125C22×C4—72no—
126C22×C4—72no—
127C22×C4—144no—
128C22×C4—142no—
129C22×C4—142no—
130C22×C4—282no—
131C22×C4D2814yes—
132C22×C4C22×D714yes—
133C22×C4C22×D711yescharacteristic
134C22×C4C22×D711yescharacteristic
135C4⋊C4—148no—
136C4⋊C4—144no—
137C4⋊C4—144no—
138C4⋊C4—144no—
139C4⋊C4—142no—
140C4⋊C4—284no—
141C22⋊C4—144no—
142C22⋊C4Dic1418yescomplemented
143C42—42no—
144C2×C7—24no—
145C2×C7#52511yescharacteristic
146C2×C7C2×C4⋊Q811yescharacteristic
147C2×C7#53411yescharacteristic
148C2×C7#55414yes—
149C23—22no—
150C23—22no—
151C23C2×Dic1414yes—
152C23C2×D2812yes—
153C23C23×D711yescharacteristic
154Q8—284no—
155C2×C4—24no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—24no—
159C2×C4—44no—
160C2×C4—74no—
161C2×C4—148no—
162C2×C4—142no—
163C2×C4—142no—
164C2×C4—142no—
165C2×C4—142no—
166C2×C4—142no—
167C2×C4—144no—
168C2×C4—144no—
169C2×C4—144no—
170C2×C4—284no—
171C2×C4C2×Dic1418yes—
172C2×C4C2×D2812yes—
173C2×C4C2×D2812yes—
174C7#258311yescharacteristic, complemented, Sylow
175C22—22no—
176C22—22no—
177C22—28no—
178C22#643214yes—
179C22C22×Dic1412yes—
180C22C22×D2811yescharacteristic
181C22#654812yes—
182C22D7×#4112yes—
183C4—22no—
184C4—22no—
185C4—28no—
186C4—144no—
187C4—282no—
188C4—282no—
189C4—282no—
190C2—24no—
191C2C2×#643211yescharacteristic
192C2#9018411yescharacteristic
193C2#9021511yescharacteristic
194C2#9023014yes—
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC4×C7×C22⋊C4C2
prime radicalC24×C7C23
hypercenterC4×C22⋊C4D7
hypocenterC7#2583
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
6. #89700C2semidirect
7. #89433C2semidirect
8. #89428C2semidirect
44. C7×C22⋊C4Q8semidirect
142. C22⋊C4Dic14semidirect
174. C7#2583semidirect

Maximal subgroups

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

subgroupindex
C2×#64872
C22×Dic7⋊C42
C4×C7×C22⋊C42
#897222
#897002
#894332
#894282
#25837