Finite Group Databasebeta

← #11733988

Subgroups — #11733988

Order 896.

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

normal subgroups only

This group has 500 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×Q8×Dic7C211yesmaximal, characteristic
2C2×#6451C211yesmaximal, characteristic
3C7×#391C211yesmaximal, characteristic
4#90002C211yesmaximal, characteristic
5#89748C211yesmaximal, characteristic
6#89707C211yesmaximal, characteristic
7C4×Dic7⋊C4C211yesmaximal, characteristic
8#89428C214yesmaximal
9#89419C214yesmaximal
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×C4⋊Dic7C2214yes—
19C2×Dic7⋊C4C2212yes—
20C2×Dic7⋊C4C2214yes—
21C2×Dic7⋊C4C2212yes—
22C2×C4×Dic7C2212yes—
23C2×C4×Dic7C2211yescharacteristic
24C2×C4×Dic7C2211yescharacteristic
25#6451—24no—
26C7×#95C2214yes—
27C7⋊#95C2214yes—
28C7⋊#95C2212yes—
29#2567—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×Dic7C2311yescharacteristic
40C22×Dic7C2312yes—
41C22×Dic7C2311yescharacteristic
42C2×Dic14—24no—
43C2×Dic14—44no—
44C7×C4⋊C4—22no—
45C42×C7—22no—
46C4⋊Dic7—22no—
47C4⋊Dic7—22no—
48C4⋊Dic7—24no—
49C4⋊Dic7—44no—
50Dic7⋊C4—24no—
51Dic7⋊C4—44no—
52Dic7⋊C4Q818yescomplemented
53C4×Dic7—22no—
54C4×Dic7—24no—
55C4×Dic7D414yes—
56C2×C4⋊Q8—71no—
57C2×C4×Q8—71no—
58#395—74no—
59#391—71no—
60#391D711yescharacteristic
61#389—74no—
62#384—71no—
63#384—71no—
64C4×C4⋊C4—71no—
65C23×C7C2411yescharacteristic
66C7×Q8—28no—
67C2×C4×C7—22no—
68C2×C4×C7—22no—
69C2×C4×C7—22no—
70C2×C4×C7—22no—
71C2×C4×C7—22no—
72C2×C4×C7—24no—
73C2×C4×C7C2×D412yes—
74C2×C4×C7C2×Q814yes—
75C2×C4×C7C2×Q812yes—
76C2×C4×C7C2×Q814yes—
77C2×C4×C7#4112yes—
78C2×Dic7—22no—
79C2×Dic7—22no—
80C2×Dic7—22no—
81C2×Dic7—22no—
82C2×Dic7—22no—
83C2×Dic7—24no—
84C2×Dic7—24no—
85C2×Dic7C2×D412yes—
86C2×Dic7C2×D412yes—
87C2×Dic7C2×Q814yes—
88C2×Dic7C2×Q814yes—
89Dic14—48no—
90C22×Q8—72no—
91C22×Q8C2×D711yescharacteristic
92C4⋊Q8—144no—
93C4×Q8—144no—
94C2×C4⋊C4—72no—
95C2×C4⋊C4—71no—
96C2×C4⋊C4—72no—
97C2×C4⋊C4—72no—
98C2×C4⋊C4—71no—
99C2×C4⋊C4—74no—
100C2×C4⋊C4—74no—
101C2×C4⋊C4C2×D711yescharacteristic
102C2×C42—71no—
103C2×C42—72no—
104C2×C42—71no—
105C2×C42C2×D711yescharacteristic
106#95—72no—
107#95—74no—
108#95C2×D714yes—
109C22×C7C22×D411yescharacteristic
110C22×C7C22×Q811yescharacteristic
111C22×C7C22×Q812yes—
112C22×C7C2×#4111yescharacteristic
113C22×C7H(5,𝔽2)12yes—
114C4×C7—24no—
115C4×C7—24no—
116C4×C7—42no—
117C4×C7C22⋊Q814yes—
118Dic7—22no—
119Dic7—24no—
120Dic7—42no—
121Dic7—42no—
122Dic7C4⋊Q814yes—
123C2×Q8—22no—
124C2×Q8—22no—
125C2×Q8—144no—
126C2×Q8—284no—
127C2×Q8Dic1414yescomplemented
128C22×C4—71no—
129C22×C4—72no—
130C22×C4—71no—
131C22×C4—72no—
132C22×C4—72no—
133C22×C4C22×D712yes—
134C22×C4C22×D712yes—
135C22×C4C22×D712yes—
136C22×C4C22×D711yescharacteristic
137C4⋊C4—22no—
138C4⋊C4—78no—
139C4⋊C4—144no—
140C4⋊C4—144no—
141C4⋊C4—142no—
142C4⋊C4—142no—
143C4⋊C4—284no—
144C4⋊C4—284no—
145C42—22no—
146C42—74no—
147C42—144no—
148C42—142no—
149C2×C7C2×C22⋊Q811yescharacteristic
150C2×C7C2×C4⋊Q811yescharacteristic
151C2×C7#53511yescharacteristic
152C2×C7#55412yes—
153C2×C7Q8212yes—
154C23C23×D711yescharacteristic
155Q8—28no—
156Q8—288no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—24no—
163C2×C4—72no—
164C2×C4—74no—
165C2×C4—72no—
166C2×C4—74no—
167C2×C4—142no—
168C2×C4—142no—
169C2×C4—142no—
170C2×C4—142no—
171C2×C4—142no—
172C2×C4—144no—
173C2×C4—144no—
174C2×C4C2×Dic1414yes—
175C2×C4C2×Dic1412yes—
176C2×C4D4×D712yes—
177C2×C4#115912yes—
178C2×C4Q8×D714yes—
179C7#256711yescharacteristic, complemented, Sylow
180C22C22×Dic1411yescharacteristic
181C22C2×D4×D711yescharacteristic
182C22C2×#115911yescharacteristic
183C22C2×Q8×D712yes—
184C22#655312yes—
185C4—24no—
186C4—24no—
187C4—42no—
188C4—74no—
189C4—144no—
190C4—142no—
191C4—282no—
192C4—282no—
193C4#644114yes—
194C2C2×#644111yescharacteristic
195C2Q8×Dic1412yes—
196C2#9032612yes—
197C2#9036511yescharacteristic
198C2D7×C4⋊Q811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, prime radical, cosocleC23×C7C24
FittingC7×#391C2
hypercenter#391D7
hypocenterC7#2567
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
52. Dic7⋊C4Q8semidirect
127. C2×Q8Dic14semidirect
179. C7#2567semidirect

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
#897482
#897072
C4×Dic7⋊C42
#894282
#894192
#25677