Finite Group Databasebeta

← #11733965

Subgroups — #11733965

Order 896.

512 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 149 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×D7×C4⋊C4C211yesmaximal, characteristic
3C7×#391C211yesmaximal, characteristic, complemented
4#90002C211yesmaximal, characteristic, complemented
5#89749C211yesmaximal, characteristic, complemented
6C4×(C2×D7)⋊C4C211yesmaximal, characteristic
7#89708C211yesmaximal, characteristic, complemented
8#89436C214yesmaximal
9#89430C214yesmaximal, complemented
10C22×C7×Q8C2211yescharacteristic
11C22×C4×D7C2212yes—
12C22×C4×D7C2211yescharacteristic
13C2×C7×C4⋊C4C2211yescharacteristic
14C2×C42×C7C2211yescharacteristic
15#6509—24no—
16C2×(C2×D7)⋊C4C2212yes—
17C2×(C2×D7)⋊C4C2212yes—
18C2×C4⋊Dic7C2211yescharacteristic
19C2×C4⋊Dic7C2211yescharacteristic
20C2×Dic7⋊C4C2212yes—
21C2×Dic7⋊C4C2212yes—
22C2×C4×Dic7C2211yescharacteristic
23D7×C4⋊C4—24no—
24C7×#95C2214yes—
25C7⋊#95C2212yes—
26C7⋊#95C2214yes—
27C7⋊#95C2214yes—
28C7⋊#95C2214yes—
29#2503—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×Dic7C2312yes—
41C22×Dic7C2311yescharacteristic
42C2×C4×D7—22no—
43C2×C4×D7—22no—
44C2×C4×D7—24no—
45C2×C4×D7—28no—
46C2×C4×D7D414yes—
47C7×C4⋊C4—22no—
48C42×C7—22no—
49(C2×D7)⋊C4—24no—
50(C2×D7)⋊C4—24no—
51C4⋊Dic7—22no—
52C4⋊Dic7—22no—
53Dic7⋊C4—24no—
54Dic7⋊C4—24no—
55C4×Dic7—22no—
56C2×C22⋊Q8—71no—
57C22×C4⋊C4—71no—
58#400—74no—
59#391—71no—
60#391D711yescharacteristic, complemented
61#382—71no—
62#382—71no—
63#381—74no—
64C4×C22⋊C4—71no—
65C23×C7C2411yescharacteristic
66C22×D7—21no—
67C22×D7—21no—
68C22×D7—22no—
69C22×D7C2×D412yes—
70C22×D7#4114yes—
71C7×Q8—44no—
72C2×C4×C7—22no—
73C2×C4×C7—22no—
74C2×C4×C7—22no—
75C2×C4×C7—22no—
76C2×C4×C7—24no—
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—21no—
84C2×Dic7—21no—
85C2×Dic7—22no—
86C2×Dic7—22no—
87C2×Dic7—22no—
88C2×Dic7—22no—
89C2×Dic7—24no—
90C2×Dic7—24no—
91C2×Dic7—22no—
92C2×Dic7C2×D412yes—
93C4×D7—28no—
94C4×D7—48no—
95C22×Q8C2×D711yescharacteristic
96C23×C4—71no—
97C23×C4—72no—
98C22⋊Q8—144no—
99C2×C4⋊C4—72no—
100C2×C4⋊C4—71no—
101C2×C4⋊C4—72no—
102C2×C4⋊C4—71no—
103C2×C4⋊C4—144no—
104C2×C4⋊C4C2×D711yescharacteristic
105C2×C22⋊C4—72no—
106C2×C22⋊C4—72no—
107C2×C42—71no—
108C2×C42C2×D711yescharacteristic
109#95—72no—
110#95—74no—
111#95—74no—
112#95—74no—
113#95C2×D714yes—
114C22×C7C22×D411yescharacteristic
115C22×C7C2×#4111yescharacteristic
116C22×C7C2×#4111yescharacteristic
117C22×C7C2×#4112yes—
118C22×C72−1+412yes—
119C2×D7—22no—
120C2×D7—22no—
121C2×D7—24no—
122C2×D7—24no—
123C2×D7#12414yes—
124C4×C7—42no—
125C4×C7—42no—
126C4×C7—42no—
127C4×C7#12314yes—
128Dic7—22no—
129Dic7—42no—
130Dic7—42no—
131Dic7—42no—
132C24—71no—
133C2×Q8—22no—
134C2×Q8—22no—
135C2×Q8—22no—
136C22×C4—71no—
137C22×C4—74no—
138C22×C4—72no—
139C22×C4—72no—
140C22×C4—72no—
141C22×C4—144no—
142C22×C4—148no—
143C22×C4—142no—
144C22×C4—142no—
145C22×C4C22×D712yes—
146C22×C4C22×D712yes—
147C22×C4C22×D712yes—
148C22×C4C22×D711yescharacteristic
149C4⋊C4—22no—
150C4⋊C4—144no—
151C4⋊C4—144no—
152C4⋊C4—142no—
153C4⋊C4—142no—
154C22⋊C4—144no—
155C22⋊C4—144no—
156C42—22no—
157C42—142no—
158C2×C7C2×#12311yescharacteristic
159C2×C7C2×#12411yescharacteristic
160C2×C7#53611yescharacteristic
161C2×C7#55212yes—
162C2×C7#55212yes—
163D7—24no—
164C23—72no—
165C23—74no—
166C23—141no—
167C23—141no—
168C23—142no—
169C23C23×D711yescharacteristic
170Q8—44no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—24no—
179C2×C4—72no—
180C2×C4—141no—
181C2×C4—141no—
182C2×C4—148no—
183C2×C4—142no—
184C2×C4—142no—
185C2×C4—142no—
186C2×C4—144no—
187C2×C4—142no—
188C2×C4—142no—
189C2×C4—144no—
190C2×C4—288no—
191C2×C4#115712yes—
192C2×C4D4×D712yes—
193C2×C4#115912yes—
194C7#250311yescharacteristic, complemented, Sylow
195C22—74no—
196C22—142no—
197C22—142no—
198C22—144no—
199C22—144no—
200C22C2×#115711yescharacteristic
201C22C2×D4×D711yescharacteristic
202C22C2×#115911yescharacteristic
203C22#655112yes—
204C22D7×#4112yes—
205C4—42no—
206C4—42no—
207C4—42no—
208C4—142no—
209C4—282no—
210C4—282no—
211C4—282no—
212C4#644614yes—
213C2—144no—
214C2C2×#644611yescharacteristic
215C2#9027212yes—
216C2#9033012yes—
217C2D7×#12411yescharacteristic
218C2#9041511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#391C2
prime radicalC23×D7C23
hypercenter#391D7
hypocenterC7#2503
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. #89749C2semidirect
7. #89708C2semidirect
9. #89430C2semidirect
60. #391D7semidirect
194. C7#2503semidirect

Maximal subgroups

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

subgroupindex
C2×#65092
C2×D7×C4⋊C42
C7×#3912
#900022
#897492
C4×(C2×D7)⋊C42
#897082
#894362
#894302
#25037