Finite Group Databasebeta

← #11740682

Subgroups — #11740682

Order 896.

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

normal subgroups only

This group has 576 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×C7×C4⋊Q8C211yesmaximal, characteristic, complemented
2C2×#6510C211yesmaximal, characteristic
3C2×#6510C211yesmaximal, characteristic
4C2×Q8×Dic7C211yesmaximal, characteristic, complemented
5#90002C211yesmaximal, characteristic, complemented
6#89766C214yesmaximal, complemented
7#89762C214yesmaximal, complemented
8#89711C211yesmaximal, characteristic
9C4×Dic7⋊C4C211yesmaximal, characteristic, complemented
10C22×C7×Q8C2211yescharacteristic
11C22×C7×Q8C2211yescharacteristic
12C22×D28C2211yescharacteristic
13C7×C4⋊Q8—24no—
14C2×C7×C4⋊C4C2214yes—
15C2×C42×C7C2211yescharacteristic
16#6510—24no—
17#6510—24no—
18Q8×Dic7—24no—
19C2×(C2×D7)⋊C4C2214yes—
20C2×(C2×D7)⋊C4C2212yes—
21C2×(C2×D7)⋊C4C2214yes—
22C2×(C2×D7)⋊C4C2212yes—
23C2×C4⋊Dic7C2211yescharacteristic
24C2×C4⋊Dic7C2212yes—
25C2×Dic7⋊C4C2212yes—
26C2×C4×Dic7C2211yescharacteristic
27C2×C4×Dic7C2212yes—
28C2×C4×Dic7C2211yescharacteristic
29C7⋊#95C2212yes—
30C7⋊#95C2214yes—
31#2545—71nomaximal, Sylow
32C23×D7C2312yes—
33C2×C7×Q8—22no—
34C2×C7×Q8—22no—
35C2×C7×Q8—22no—
36C2×C7×Q8—22no—
37C2×C7×Q8—22no—
38C2×C7×Q8D414yes—
39C22×C4×C7C2312yes—
40C22×C4×C7C2312yes—
41C22×C4×C7C2312yes—
42C22×C4×C7C2311yescharacteristic
43C22×Dic7C2312yes—
44C22×Dic7C2312yes—
45C22×Dic7C2311yescharacteristic
46C22×Dic7C2311yescharacteristic
47C2×D28—22no—
48C2×D28—22no—
49C2×D28—22no—
50C7×C4⋊C4—28no—
51C42×C7—22no—
52(C2×D7)⋊C4—28no—
53(C2×D7)⋊C4—28no—
54(C2×D7)⋊C4—24no—
55(C2×D7)⋊C4—24no—
56C4⋊Dic7—22no—
57C4⋊Dic7—24no—
58Dic7⋊C4—24no—
59C4×Dic7—22no—
60C4×Dic7—24no—
61C4×Dic7D414yes—
62C2×C4⋊Q8D711yescharacteristic, complemented
63C2×#124—71no—
64C2×#124—71no—
65C2×C4×Q8—71no—
66#396—74no—
67#391—71no—
68#390—71no—
69#388—74no—
70C4×C4⋊C4—71no—
71C23×C7C2411yescharacteristic
72C22×D7—22no—
73C22×D7—22no—
74C22×D7—24no—
75C22×D7—22no—
76C22×D7—24no—
77C7×Q8—28no—
78C7×Q8—44no—
79C2×C4×C7—22no—
80C2×C4×C7—22no—
81C2×C4×C7—22no—
82C2×C4×C7—22no—
83C2×C4×C7—22no—
84C2×C4×C7—22no—
85C2×C4×C7—22no—
86C2×C4×C7C2×D412yes—
87C2×C4×C7C2×D414yes—
88C2×C4×C7C2×D412yes—
89C2×C4×C7#4112yes—
90C2×C4×C7#4114yes—
91C2×Dic7—22no—
92C2×Dic7—22no—
93C2×Dic7—22no—
94C2×Dic7—22no—
95C2×Dic7—22no—
96C2×Dic7—24no—
97C2×Dic7—22no—
98C2×Dic7C2×D412yes—
99C2×Dic7C2×D412yes—
100D28—44no—
101C22×Q8C2×D711yescharacteristic
102C22×Q8C2×D711yescharacteristic
103C22×D4—71no—
104C4⋊Q8—24no—
105#124—144no—
106#124—144no—
107C4×Q8—144no—
108C2×C4⋊C4—71no—
109C2×C4⋊C4—72no—
110C2×C4⋊C4—72no—
111C2×C4⋊C4C2×D714yes—
112C2×C22⋊C4—72no—
113C2×C22⋊C4—74no—
114C2×C22⋊C4—72no—
115C2×C22⋊C4—74no—
116C2×C42—71no—
117C2×C42—72no—
118C2×C42—71no—
119C2×C42C2×D711yescharacteristic
120#95—72no—
121#95—74no—
122C22×C7C22×D411yescharacteristic
123C22×C7C22×D411yescharacteristic
124C22×C7C2×#4111yescharacteristic
125C22×C7C2×#4112yes—
126C22×C72−1+412yes—
127C2×D7—42no—
128C2×D7—42no—
129C2×D7—42no—
130C2×D7—44no—
131C2×D7—44no—
132C4×C7—24no—
133C4×C7—24no—
134C4×C7—42no—
135C4×C7#12114yes—
136Dic7—22no—
137Dic7—22no—
138Dic7—42no—
139Dic7—42no—
140C24—72no—
141C2×Q8—22no—
142C2×Q8—22no—
143C2×Q8—22no—
144C2×Q8—22no—
145C2×Q8—22no—
146C2×Q8C22⋊D714yes—
147C2×D4—142no—
148C2×D4—142no—
149C2×D4—142no—
150C22×C4—71no—
151C22×C4—71no—
152C22×C4—72no—
153C22×C4—72no—
154C22×C4C22×D712yes—
155C22×C4C22×D712yes—
156C22×C4C22×D712yes—
157C22×C4C22×D711yescharacteristic
158C4⋊C4—28no—
159C4⋊C4—144no—
160C4⋊C4—142no—
161C4⋊C4—144no—
162C22⋊C4—144no—
163C22⋊C4—144no—
164C22⋊C4—148no—
165C22⋊C4—148no—
166C42—22no—
167C42—74no—
168C42—144no—
169C42—142no—
170C2×C7C2×#12111yescharacteristic
171C2×C7#53211yescharacteristic
172C2×C7#53611yescharacteristic
173C2×C7#54812yes—
174C2×C7#55612yes—
175D7—82no—
176C23—142no—
177C23—142no—
178C23—144no—
179C23—144no—
180C23—142no—
181C23C23×D711yescharacteristic
182Q8—28no—
183Q8—44no—
184D4—284no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—72no—
193C2×C4—72no—
194C2×C4—142no—
195C2×C4—142no—
196C2×C4—142no—
197C2×C4—144no—
198C2×C4—142no—
199C2×C4—142no—
200C2×C4—142no—
201C2×C4D4×D712yes—
202C2×C4#115912yes—
203C2×C4#116114yes—
204C2×C4C2×C22⋊D714yes—
205C2×C4C2×C22⋊D712yes—
206C7#254511yescharacteristic, complemented, Sylow
207C22—282no—
208C22—282no—
209C22—284no—
210C22—282no—
211C22—284no—
212C22C2×D4×D711yescharacteristic
213C22C2×#115911yescharacteristic
214C22C2×#116112yes—
215C22#655112yes—
216C22C22×C22⋊D711yescharacteristic
217C4—24no—
218C4—24no—
219C4—42no—
220C4—142no—
221C4—142no—
222C4—282no—
223C4—282no—
224C4#650214yes—
225C2—562no—
226C2#9041511yescharacteristic
227C2#9041611yescharacteristic
228C2#9042312yes—
229C2C2×#650211yescharacteristic
230C2#9050812yes—
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC2×C7×C4⋊Q8C2
prime radicalC22×D28C22
hypercenterC2×C4⋊Q8D7
hypocenterC7#2545
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×C7×C4⋊Q8C2semidirect
4. C2×Q8×Dic7C2semidirect
5. #90002C2semidirect
6. #89766C2semidirect
7. #89762C2semidirect
9. C4×Dic7⋊C4C2semidirect
62. C2×C4⋊Q8D7semidirect
206. C7#2545semidirect

Maximal subgroups

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

subgroupindex
C2×C7×C4⋊Q82
C2×#65102
C2×Q8×Dic72
#900022
#897662
#897622
#897112
C4×Dic7⋊C42
#25457