Finite Group Databasebeta

← #11733989

Subgroups — #11733989

Order 896.

468 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 165 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×Q8×Dic7C211yesmaximal, characteristic
2C2×#6451C211yesmaximal, characteristic
3C7×#391C211yesmaximal, characteristic
4#90002C211yesmaximal, characteristic
5#89749C211yesmaximal, characteristic
6#89708C211yesmaximal, characteristic
7C4×Dic7⋊C4C211yesmaximal, characteristic
8#89432C214yesmaximal
9#89418C214yesmaximal
10C22×C7×Q8C2211yescharacteristic
11C22×Dic14C2212yes—
12C2×C7×C4⋊C4C2211yescharacteristic
13C2×C42×C7C2211yescharacteristic
14Q8×Dic7—24no—
15C2×C4⋊Dic7C2211yescharacteristic
16C2×C4⋊Dic7C2212yes—
17C2×C4⋊Dic7C2211yescharacteristic
18C2×Dic7⋊C4C2212yes—
19C2×Dic7⋊C4C2212yes—
20C2×C4×Dic7C2212yes—
21C2×C4×Dic7C2211yescharacteristic
22C2×C4×Dic7C2211yescharacteristic
23#6451—24no—
24C7×#95C2214yes—
25C7⋊#95C2214yes—
26C7⋊#95C2214yes—
27C7⋊#95C2214yes—
28C7⋊#95C2212yes—
29#2568—71nomaximal, Sylow
30C2×C7×Q8—22no—
31C2×C7×Q8—22no—
32C2×C7×Q8Q814yescomplemented
33C22×C4×C7C2312yes—
34C22×C4×C7C2312yes—
35C22×C4×C7C2312yes—
36C22×C4×C7C2311yescharacteristic
37C22×Dic7C2312yes—
38C22×Dic7C2312yes—
39C22×Dic7C2312yes—
40C22×Dic7C2311yescharacteristic
41C22×Dic7C2311yescharacteristic
42C2×Dic14—24no—
43C2×Dic14—28no—
44C7×C4⋊C4—22no—
45C42×C7—22no—
46C4⋊Dic7—22no—
47C4⋊Dic7—22no—
48C4⋊Dic7—24no—
49Dic7⋊C4—24no—
50Dic7⋊C4—24no—
51C4×Dic7—22no—
52C4×Dic7—24no—
53C4×Dic7D414yes—
54C2×C4⋊Q8—71no—
55C2×C4×Q8—71no—
56#400—74no—
57#391—71no—
58#391—74no—
59#391D711yescharacteristic
60#382—71no—
61#382—71no—
62C4×C4⋊C4—71no—
63C23×C7C2411yescharacteristic
64C7×Q8—28no—
65C2×C4×C7—22no—
66C2×C4×C7—22no—
67C2×C4×C7—22no—
68C2×C4×C7—22no—
69C2×C4×C7—22no—
70C2×C4×C7—24no—
71C2×C4×C7—22no—
72C2×C4×C7C2×D412yes—
73C2×C4×C7C2×Q814yes—
74C2×C4×C7C2×Q812yes—
75C2×C4×C7#4112yes—
76C2×Dic7—22no—
77C2×Dic7—22no—
78C2×Dic7—22no—
79C2×Dic7—22no—
80C2×Dic7—22no—
81C2×Dic7—24no—
82C2×Dic7—24no—
83C2×Dic7—22no—
84C2×Dic7C2×D412yes—
85C2×Dic7C2×D412yes—
86C2×Dic7#4114yes—
87Dic14—48no—
88C22×Q8—72no—
89C22×Q8C2×D711yescharacteristic
90C4⋊Q8—144no—
91C4×Q8—144no—
92C2×C4⋊C4—71no—
93C2×C4⋊C4—72no—
94C2×C4⋊C4—72no—
95C2×C4⋊C4—71no—
96C2×C4⋊C4—72no—
97C2×C4⋊C4C2×D711yescharacteristic
98C2×C42—71no—
99C2×C42—72no—
100C2×C42—71no—
101C2×C42C2×D711yescharacteristic
102#95—72no—
103#95—74no—
104#95—74no—
105#95—74no—
106#95C2×D714yes—
107C22×C7C22×D411yescharacteristic
108C22×C7C22×Q811yescharacteristic
109C22×C7C2×#4111yescharacteristic
110C22×C7C2×#4112yes—
111C22×C72−1+412yes—
112C4×C7—24no—
113C4×C7—42no—
114C4×C7—42no—
115C4×C7C22⋊Q814yes—
116Dic7—22no—
117Dic7—42no—
118Dic7—42no—
119Dic7—42no—
120Dic7#12414yes—
121C2×Q8—22no—
122C2×Q8—22no—
123C2×Q8—144no—
124C2×Q8—148no—
125C2×Q8Dic1414yescomplemented
126C22×C4—71no—
127C22×C4—71no—
128C22×C4—72no—
129C22×C4—72no—
130C22×C4—72no—
131C22×C4C22×D712yes—
132C22×C4C22×D712yes—
133C22×C4C22×D712yes—
134C22×C4C22×D711yescharacteristic
135C4⋊C4—22no—
136C4⋊C4—144no—
137C4⋊C4—144no—
138C4⋊C4—144no—
139C4⋊C4—142no—
140C4⋊C4—142no—
141C42—22no—
142C42—74no—
143C42—144no—
144C42—142no—
145C2×C7C2×C22⋊Q811yescharacteristic
146C2×C7C2×#12411yescharacteristic
147C2×C7#53611yescharacteristic
148C2×C7#55612yes—
149C2×C7#55712yes—
150C23C23×D711yescharacteristic
151Q8—28no—
152Q8—288no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—24no—
160C2×C4—72no—
161C2×C4—74no—
162C2×C4—72no—
163C2×C4—142no—
164C2×C4—142no—
165C2×C4—142no—
166C2×C4—142no—
167C2×C4—144no—
168C2×C4—142no—
169C2×C4—142no—
170C2×C4—144no—
171C2×C4C2×Dic1414yes—
172C2×C4C2×Dic1412yes—
173C2×C4D4×D712yes—
174C2×C4#115912yes—
175C7#256811yescharacteristic, complemented, Sylow
176C22C22×Dic1411yescharacteristic
177C22C2×D4×D711yescharacteristic
178C22C2×#115911yescharacteristic
179C22#655112yes—
180C22D7×#4112yes—
181C4—24no—
182C4—42no—
183C4—42no—
184C4—74no—
185C4—142no—
186C4—282no—
187C4—282no—
188C4—282no—
189C4#644114yes—
190C2C2×#644111yescharacteristic
191C2#9026012yes—
192C2#9033212yes—
193C2D7×#12411yescharacteristic
194C2#9041511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, prime radical, cosocleC23×C7C24
FittingC7×#391C2
hypercenter#391D7
hypocenterC7#2568
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
32. C2×C7×Q8Q8semidirect
125. C2×Q8Dic14semidirect
175. C7#2568semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×Dic72
C2×#64512
C7×#3912
#900022
#897492
#897082
C4×Dic7⋊C42
#894322
#894182
#25687