Finite Group Databasebeta

← #11740219

Subgroups — #11740219

Order 896.

484 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 153 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×C22⋊Q8C211yesmaximal, characteristic
2C2×#6489C211yesmaximal, characteristic
3#90002C211yesmaximal, characteristic, complemented
4#90002C211yesmaximal, characteristic, complemented
5C4×C22⋊Dic7C211yesmaximal, characteristic
6#89750C214yesmaximal, complemented
7#89749C211yesmaximal, characteristic, complemented
8#89748C211yesmaximal, characteristic, complemented
9#89724C214yesmaximal
10C22×C7×Q8C2211yescharacteristic
11C23×C4×C7C2211yescharacteristic
12C7×C22⋊Q8—24no—
13C2×C7×C4⋊C4C2211yescharacteristic
14C2×C7×C4⋊C4C2212yes—
15C2×C7×C22⋊C4C2212yes—
16C2×C22⋊Dic7C2212yes—
17C2×C22⋊Dic7C2212yes—
18#6489—24no—
19C2×C4⋊Dic7C2211yescharacteristic
20C2×C4⋊Dic7C2211yescharacteristic
21C2×Dic7⋊C4C2214yes—
22C2×C4×Dic7C2212yes—
23C2×C4×Dic7C2211yescharacteristic
24C2×C4×Dic7C2211yescharacteristic
25C7⋊#95C2214yes—
26C7⋊#95C2214yes—
27C7⋊#95C2214yes—
28C7⋊#95C2212yes—
29#2499—71nomaximal, Sylow
30C24×C7C2311yescharacteristic
31C2×C7×Q8—22no—
32C2×C7×Q8—22no—
33C2×C7×Q8—22no—
34C22×C4×C7—22no—
35C22×C4×C7—22no—
36C22×C4×C7D414yes—
37C22×C4×C7C2312yes—
38C22×C4×C7C2312yes—
39C22×C4×C7C2311yescharacteristic
40C22×C4×C7C2311yescharacteristic
41C22×Dic7C2312yes—
42C22×Dic7C2312yes—
43C22×Dic7C2312yes—
44C22×Dic7C2312yes—
45C7×C4⋊C4—22no—
46C7×C4⋊C4—24no—
47C7×C22⋊C4—24no—
48C22⋊Dic7—24no—
49C22⋊Dic7—24no—
50C4⋊Dic7—22no—
51C4⋊Dic7—22no—
52Dic7⋊C4—28no—
53C4×Dic7—24no—
54C4×Dic7—22no—
55C4×Dic7—22no—
56C2×C22⋊Q8D711yescharacteristic
57C2×#117—71no—
58#397—74no—
59#391—71no—
60#391—71no—
61#387—74no—
62#384—71no—
63#382—71no—
64C4×C22⋊C4—71no—
65C23×C7—22no—
66C23×C7—21no—
67C23×C7—21no—
68C23×C7—22no—
69C23×C7C2×D412yes—
70C23×C7C2411yescharacteristic
71C7×Q8—44no—
72C2×C4×C7—21no—
73C2×C4×C7—21no—
74C2×C4×C7—28no—
75C2×C4×C7—22no—
76C2×C4×C7—22no—
77C2×C4×C7—24no—
78C2×C4×C7—22no—
79C2×C4×C7—22no—
80C2×C4×C7C2×D412yes—
81C2×C4×C7C2×D412yes—
82C2×C4×C7#4112yes—
83C2×C4×C7#4112yes—
84C2×Dic7—22no—
85C2×Dic7—22no—
86C2×Dic7—22no—
87C2×Dic7—22no—
88C2×Dic7—22no—
89C2×Dic7—24no—
90C2×Dic7—22no—
91C2×Dic7—24no—
92C2×Dic7—22no—
93C2×Dic7#4114yes—
94C22×Q8C2×D711yescharacteristic
95C23×C4C2×D711yescharacteristic
96C22⋊Q8—24no—
97#117—144no—
98C2×C4⋊C4—71no—
99C2×C4⋊C4—71no—
100C2×C4⋊C4—74no—
101C2×C4⋊C4C2×D711yescharacteristic
102C2×C4⋊C4C2×D712yes—
103C2×C22⋊C4—72no—
104C2×C22⋊C4—72no—
105C2×C22⋊C4C2×D712yes—
106C2×C42—72no—
107C2×C42—71no—
108C2×C42—71no—
109#95—72no—
110#95—74no—
111#95—74no—
112#95—74no—
113C22×C7—22no—
114C22×C7—22no—
115C22×C7—22no—
116C22×C7—42no—
117C22×C7—42no—
118C22×C7C22×D411yescharacteristic
119C22×C7C2×#4111yescharacteristic
120C22×C7C2×#4111yescharacteristic
121C22×C7C2×#4112yes—
122C22×C72−1+412yes—
123C4×C7—22no—
124C4×C7—42no—
125C4×C7—42no—
126C4×C7#12314yes—
127Dic7—24no—
128Dic7—42no—
129Dic7—42no—
130Dic7—42no—
131C24C22×D711yescharacteristic
132C2×Q8—22no—
133C2×Q8—22no—
134C2×Q8—22no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—72no—
138C22×C4—72no—
139C22×C4—72no—
140C22×C4—72no—
141C22×C4C22⋊D714yes—
142C22×C4C22×D712yes—
143C22×C4C22×D712yes—
144C22×C4C22×D711yescharacteristic
145C22×C4C22×D711yescharacteristic
146C4⋊C4—22no—
147C4⋊C4—24no—
148C4⋊C4—148no—
149C4⋊C4—142no—
150C4⋊C4—142no—
151C22⋊C4—24no—
152C22⋊C4—144no—
153C22⋊C4—144no—
154C42—142no—
155C42—142no—
156C42—144no—
157C2×C7—42no—
158C2×C7C2×#12311yescharacteristic
159C2×C7#53211yescharacteristic
160C2×C7#53611yescharacteristic
161C2×C7#55212yes—
162C2×C7#55612yes—
163C23—21no—
164C23—21no—
165C23—22no—
166C23—22no—
167C23C2×C22⋊D712yes—
168C23C23×D711yescharacteristic
169Q8—44no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—28no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—24no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—74no—
179C2×C4—142no—
180C2×C4—142no—
181C2×C4—142no—
182C2×C4—144no—
183C2×C4—142no—
184C2×C4—142no—
185C2×C4—144no—
186C2×C4—142no—
187C2×C4—142no—
188C2×C4#115912yes—
189C2×C4#115912yes—
190C2×C4C2×C22⋊D712yes—
191C2×C4C2×C22⋊D712yes—
192C7#249911yescharacteristic, complemented, Sylow
193C22—22no—
194C22—22no—
195C22—22no—
196C22—42no—
197C22—42no—
198C22C2×#115911yescharacteristic
199C22C2×#115911yescharacteristic
200C22#655112yes—
201C22D7×#4112yes—
202C22C22×C22⋊D711yescharacteristic
203C4—22no—
204C4—42no—
205C4—42no—
206C4—144no—
207C4—282no—
208C4—282no—
209C4—282no—
210C4#649814yes—
211C2—42no—
212C2#9033012yes—
213C2#9033212yes—
214C2C2×#649811yescharacteristic
215C2#9050711yescharacteristic
216C2#9052311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC2×C7×C22⋊Q8C2
prime radicalC24×C7C23
hypercenterC2×C22⋊Q8D7
hypocenterC7#2499
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #90002C2semidirect
4. #90002C2semidirect
6. #89750C2semidirect
7. #89749C2semidirect
8. #89748C2semidirect
192. C7#2499semidirect

Maximal subgroups

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

subgroupindex
C2×C7×C22⋊Q82
C2×#64892
#900022
C4×C22⋊Dic72
#897502
#897492
#897482
#897242
#24997