Finite Group Databasebeta

← #11739944

Subgroups — #11739944

Order 896.

604 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 604 conjugacy classes of subgroups — too many to draw. Only its 169 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×C4×C7×Q8C211yesmaximal, characteristic, complemented
2C2×#6509C212yesmaximal, complemented
3C2×C4×D28C211yesmaximal, characteristic
4#89763C212yesmaximal
5#89759C212yesmaximal, complemented
6#89755C212yesmaximal, complemented
7#89712C214yesmaximal, complemented
8#89708C211yesmaximal, characteristic, complemented
9C22×C7×Q8C2211yescharacteristic
10C22×D28C2211yescharacteristic
11C22×C4×D7C2212yes—
12C4×C7×Q8—24no—
13C2×C7×C4⋊C4C2212yes—
14C2×C7×C4⋊C4C2211yescharacteristic
15C2×C42×C7C2212yes—
16C2×C42×C7C2211yescharacteristic
17#6509—28no—
18C2×(C2×D7)⋊C4C2212yes—
19C2×(C2×D7)⋊C4C2214yes—
20C2×(C2×D7)⋊C4C2214yes—
21C2×C4⋊Dic7C2211yescharacteristic
22C2×C4⋊Dic7C2212yes—
23C2×Dic7⋊C4C2214yes—
24C4×D28—24no—
25C7⋊#95C2212yes—
26C7⋊#95C2212yes—
27C7⋊#95C2214yes—
28#2536—71nomaximal, Sylow
29C23×D7C2312yes—
30C2×C7×Q8—22no—
31C2×C7×Q8—22no—
32C2×C7×Q8D414yescomplemented
33C22×C4×C7C2312yes—
34C22×C4×C7C2311yescharacteristic
35C22×C4×C7C2311yescharacteristic
36C22×C4×C7C2312yes—
37C22×C4×C7C2311yescharacteristic
38C22×Dic7C2314yes—
39C22×Dic7C2312yes—
40C2×D28—24no—
41C2×D28—22no—
42C2×C4×D7—24no—
43C2×C4×D7—24no—
44C2×C4×D7—24no—
45C7×C4⋊C4—24no—
46C7×C4⋊C4—22no—
47C42×C7—24no—
48C42×C7D414yes—
49(C2×D7)⋊C4—28no—
50(C2×D7)⋊C4—24no—
51(C2×D7)⋊C4—28no—
52C4⋊Dic7—22no—
53C4⋊Dic7—24no—
54Dic7⋊C4—28no—
55C2×C22⋊Q8—72no—
56C2×C4×Q8D711yescharacteristic, complemented
57C2×C4×D4—71no—
58#398—72no—
59#394—72no—
60#388—74no—
61#382—71no—
62#380—72no—
63C23×C7C2411yescharacteristic
64C22×D7—22no—
65C22×D7—22no—
66C22×D7—22no—
67C22×D7—22no—
68C22×D7—22no—
69C22×D7—22no—
70C22×D7#4114yes—
71C7×Q8—28no—
72C2×C4×C7—22no—
73C2×C4×C7—22no—
74C2×C4×C7—22no—
75C2×C4×C7—22no—
76C2×C4×C7—22no—
77C2×C4×C7—24no—
78C2×C4×C7C2×D414yes—
79C2×C4×C7C2×D412yes—
80C2×C4×C7C2×D412yes—
81C2×C4×C7C2×D412yes—
82C2×C4×C7C2×D412yes—
83C2×C4×C7#4112yes—
84C2×Dic7—22no—
85C2×Dic7—24no—
86C2×Dic7—22no—
87C2×Dic7—24no—
88C2×Dic7—22no—
89C2×Dic7—24no—
90D28—28no—
91C4×D7—48no—
92C22×Q8C2×D711yescharacteristic
93C22×D4—71no—
94C23×C4—72no—
95C22⋊Q8—148no—
96C4×Q8—24no—
97C4×D4—144no—
98C2×C4⋊C4—71no—
99C2×C4⋊C4—74no—
100C2×C4⋊C4—72no—
101C2×C4⋊C4C2×D711yescharacteristic
102C2×C4⋊C4C2×D712yes—
103C2×C22⋊C4—72no—
104C2×C22⋊C4—74no—
105C2×C22⋊C4—74no—
106C2×C42C2×D712yes—
107C2×C42C2×D711yescharacteristic
108#95—74no—
109#95—72no—
110#95—72no—
111C22×C7C22×D411yescharacteristic
112C22×C7C22×D411yescharacteristic
113C22×C7C2×#4111yescharacteristic
114C22×C7C2×#4112yes—
115C22×C72−1+412yes—
116C2×D7—24no—
117C2×D7—24no—
118C2×D7—24no—
119C2×D7—42no—
120C2×D7—42no—
121C2×D7—42no—
122C2×D7—42no—
123C4×C7—24no—
124C4×C7—22no—
125C4×C7—22no—
126C4×C7—42no—
127C4×C7#12114yes—
128Dic7—42no—
129Dic7—44no—
130C24—72no—
131C2×Q8—22no—
132C2×Q8—22no—
133C2×Q8D2814yescomplemented
134C2×D4—144no—
135C2×D4—142no—
136C22×C4—74no—
137C22×C4—72no—
138C22×C4—144no—
139C22×C4—144no—
140C22×C4—144no—
141C22×C4C22×D712yes—
142C22×C4C22×D712yes—
143C22×C4C22×D711yescharacteristic
144C22×C4C22×D711yescharacteristic
145C22×C4C22×D711yescharacteristic
146C4⋊C4—24no—
147C4⋊C4—22no—
148C4⋊C4—148no—
149C4⋊C4—144no—
150C4⋊C4—142no—
151C22⋊C4—148no—
152C22⋊C4—148no—
153C22⋊C4—144no—
154C42—24no—
155C42C22⋊D714yes—
156C2×C7C2×#12111yescharacteristic
157C2×C7#52511yescharacteristic
158C2×C7#53611yescharacteristic
159C2×C7#54812yes—
160C2×C7#55212yes—
161D7—44no—
162C23—74no—
163C23—142no—
164C23—142no—
165C23—142no—
166C23—142no—
167C23—142no—
168C23—142no—
169C23C23×D711yescharacteristic
170Q8—28no—
171D4—148no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—24no—
177C2×C4—22no—
178C2×C4—144no—
179C2×C4—142no—
180C2×C4—142no—
181C2×C4—144no—
182C2×C4—144no—
183C2×C4—142no—
184C2×C4—288no—
185C2×C4C2×D2814yes—
186C2×C4C2×D2812yes—
187C2×C4#115712yes—
188C2×C4C2×C22⋊D712yes—
189C2×C4C2×C22⋊D712yes—
190C2×C4C2×C22⋊D712yes—
191C7#253611yescharacteristic, complemented, Sylow
192C22—144no—
193C22—144no—
194C22—144no—
195C22—282no—
196C22—282no—
197C22—282no—
198C22—282no—
199C22C22×D2811yescharacteristic
200C22C2×#115711yescharacteristic
201C22#655112yes—
202C22D7×#4112yes—
203C22C22×C22⋊D711yescharacteristic
204C4—24no—
205C4—22no—
206C4—22no—
207C4—42no—
208C4—282no—
209C4—284no—
210C4#649314yes—
211C2—284no—
212C2#9026712yes—
213C2#9027212yes—
214C2C2×#649311yescharacteristic
215C2#9050711yescharacteristic
216C2#9051911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC2×C4×C7×Q8C2
prime radicalC22×D28C22
hypercenterC2×C4×Q8D7
hypocenterC7#2536
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×C4×C7×Q8C2semidirect
2. C2×#6509C2semidirect
5. #89759C2semidirect
6. #89755C2semidirect
7. #89712C2semidirect
8. #89708C2semidirect
32. C2×C7×Q8D4semidirect
56. C2×C4×Q8D7semidirect
133. C2×Q8D28semidirect
191. C7#2536semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C7×Q82
C2×#65092
C2×C4×D282
#897632
#897592
#897552
#897122
#897082
#25367