Finite Group Databasebeta

← #11733963

Subgroups — #11733963

Order 896.

484 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 145 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×#6509C211yesmaximal, characteristic
2C2×#6455C211yesmaximal, characteristic
3C7×#391C211yesmaximal, characteristic, complemented
4#90002C211yesmaximal, characteristic, complemented
5#89748C211yesmaximal, characteristic, complemented
6C4×(C2×D7)⋊C4C211yesmaximal, characteristic
7#89708C211yesmaximal, characteristic, complemented
8#89447C214yesmaximal
9#89424C214yesmaximal, complemented
10C22×C7×Q8C2211yescharacteristic
11C22×C4×D7C2211yescharacteristic
12C2×C7×C4⋊C4C2211yescharacteristic
13C2×C42×C7C2211yescharacteristic
14#6509—24no—
15C2×(C2×D7)⋊C4C2212yes—
16C2×(C2×D7)⋊C4C2212yes—
17C2×(C2×D7)⋊C4C2212yes—
18C2×C4⋊Dic7C2211yescharacteristic
19C2×C4⋊Dic7C2211yescharacteristic
20C2×Dic7⋊C4C2212yes—
21C2×Dic7⋊C4C2214yes—
22C2×C4×Dic7C2212yes—
23C2×C4×Dic7C2211yescharacteristic
24#6455—24no—
25C7×#95C2214yes—
26C7⋊#95C2212yes—
27C7⋊#95C2214yes—
28C7⋊#95C2214yes—
29#2499—71nomaximal, Sylow
30C23×D7C2311yescharacteristic
31C2×C7×Q8—22no—
32C2×C7×Q8—22no—
33C2×C7×Q8—22no—
34C22×C4×C7C2312yes—
35C22×C4×C7C2312yes—
36C22×C4×C7C2312yes—
37C22×C4×C7C2311yescharacteristic
38C22×Dic7C2312yes—
39C22×Dic7C2312yes—
40C22×Dic7C2311yescharacteristic
41C22×Dic7C2312yes—
42C2×C4×D7—22no—
43C2×C4×D7—22no—
44C2×C4×D7D414yes—
45C7×C4⋊C4—22no—
46C42×C7—22no—
47(C2×D7)⋊C4—24no—
48(C2×D7)⋊C4—24no—
49(C2×D7)⋊C4—24no—
50C4⋊Dic7—22no—
51C4⋊Dic7—22no—
52Dic7⋊C4—24no—
53Dic7⋊C4—28no—
54C4×Dic7—22no—
55C4×Dic7—24no—
56C2×C22⋊Q8—71no—
57C2×#117—71no—
58#397—74no—
59#391—71no—
60#391D711yescharacteristic, complemented
61#387—74no—
62#384—71no—
63#382—71no—
64C4×C22⋊C4—71no—
65C23×C7C2411yescharacteristic
66C22×D7—22no—
67C22×D7—21no—
68C22×D7—21no—
69C22×D7—22no—
70C22×D7C2×D412yes—
71C7×Q8—44no—
72C2×C4×C7—22no—
73C2×C4×C7—22no—
74C2×C4×C7—24no—
75C2×C4×C7—22no—
76C2×C4×C7—22no—
77C2×C4×C7—22no—
78C2×C4×C7—22no—
79C2×C4×C7—22no—
80C2×C4×C7C2×D412yes—
81C2×C4×C7#4112yes—
82C2×C4×C7#4112yes—
83C2×Dic7—22no—
84C2×Dic7—21no—
85C2×Dic7—21no—
86C2×Dic7—22no—
87C2×Dic7—22no—
88C2×Dic7—22no—
89C2×Dic7—24no—
90C2×Dic7—24no—
91C2×Dic7C2×D412yes—
92C2×Dic7#4114yes—
93C4×D7—28no—
94C22×Q8C2×D711yescharacteristic
95C23×C4—71no—
96C22⋊Q8—144no—
97#117—144no—
98C2×C4⋊C4—71no—
99C2×C4⋊C4—72no—
100C2×C4⋊C4—71no—
101C2×C4⋊C4—74no—
102C2×C4⋊C4C2×D711yescharacteristic
103C2×C22⋊C4—72no—
104C2×C22⋊C4—72no—
105C2×C22⋊C4—72no—
106C2×C42—72no—
107C2×C42—71no—
108C2×C42C2×D711yescharacteristic
109#95—72no—
110#95—74no—
111#95—74no—
112#95C2×D714yes—
113C22×C7C22×D411yescharacteristic
114C22×C7C2×#4111yescharacteristic
115C22×C7C2×#4111yescharacteristic
116C22×C7C2×#4112yes—
117C22×C72−1+412yes—
118C2×D7—22no—
119C2×D7—22no—
120C2×D7—22no—
121C2×D7—42no—
122C2×D7—42no—
123C4×C7—42no—
124C4×C7—42no—
125C4×C7—42no—
126C4×C7#12314yes—
127Dic7—22no—
128Dic7—24no—
129Dic7—42no—
130Dic7—42no—
131C24—71no—
132C2×Q8—22no—
133C2×Q8—22no—
134C2×Q8—22no—
135C22×C4—72no—
136C22×C4—71no—
137C22×C4—74no—
138C22×C4—72no—
139C22×C4—72no—
140C22×C4—142no—
141C22×C4—142no—
142C22×C4C22×D712yes—
143C22×C4C22×D712yes—
144C22×C4C22×D712yes—
145C22×C4C22×D711yescharacteristic
146C4⋊C4—22no—
147C4⋊C4—148no—
148C4⋊C4—144no—
149C4⋊C4—142no—
150C4⋊C4—142no—
151C22⋊C4—144no—
152C22⋊C4—144no—
153C22⋊C4—144no—
154C42—22no—
155C42—144no—
156C42—142no—
157C2×C7C2×#12311yescharacteristic
158C2×C7#53211yescharacteristic
159C2×C7#53611yescharacteristic
160C2×C7#55212yes—
161C2×C7#55612yes—
162D7—42no—
163C23—72no—
164C23—141no—
165C23—141no—
166C23—142no—
167C23—142no—
168C23C23×D711yescharacteristic
169Q8—44no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—24no—
178C2×C4—72no—
179C2×C4—74no—
180C2×C4—142no—
181C2×C4—141no—
182C2×C4—141no—
183C2×C4—142no—
184C2×C4—148no—
185C2×C4—142no—
186C2×C4—142no—
187C2×C4—144no—
188C2×C4—144no—
189C2×C4#115712yes—
190C2×C4D4×D712yes—
191C2×C4#115912yes—
192C7#249911yescharacteristic, complemented, Sylow
193C22—142no—
194C22—142no—
195C22—142no—
196C22—282no—
197C22—282no—
198C22C2×#115711yescharacteristic
199C22C2×D4×D711yescharacteristic
200C22C2×#115911yescharacteristic
201C22#655112yes—
202C22D7×#4112yes—
203C4—42no—
204C4—42no—
205C4—42no—
206C4—142no—
207C4—144no—
208C4—282no—
209C4—282no—
210C4#644614yes—
211C2—282no—
212C2C2×#644611yescharacteristic
213C2#9027212yes—
214C2#9033212yes—
215C2#9035911yescharacteristic
216C2#9041511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#391C2
prime radicalC23×D7C23
hypercenter#391D7
hypocenterC7#2499
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
3. C7×#391C2semidirect
4. #90002C2semidirect
5. #89748C2semidirect
7. #89708C2semidirect
9. #89424C2semidirect
60. #391D7semidirect
192. C7#2499semidirect

Maximal subgroups

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

subgroupindex
C2×#65092
C2×#64552
C7×#3912
#900022
#897482
C4×(C2×D7)⋊C42
#897082
#894472
#894242
#24997