Finite Group Databasebeta

← #11733964

Subgroups — #11733964

Order 896.

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

normal subgroups only

This group has 544 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×#6509C211yesmaximal, characteristic
2C2×D7×C4⋊C4C211yesmaximal, characteristic
3C7×#391C211yesmaximal, characteristic, complemented
4#90002C211yesmaximal, characteristic, complemented
5#89748C211yesmaximal, characteristic, complemented
6C4×(C2×D7)⋊C4C211yesmaximal, characteristic
7#89707C211yesmaximal, characteristic, complemented
8#89440C214yesmaximal
9#89429C214yesmaximal, complemented
10C22×C7×Q8C2211yescharacteristic
11C22×C4×D7C2211yescharacteristic
12C22×C4×D7C2212yes—
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×C4⋊Dic7C2214yes—
21C2×Dic7⋊C4C2212yes—
22C2×Dic7⋊C4C2212yes—
23C2×Dic7⋊C4C2214yes—
24C2×C4×Dic7C2211yescharacteristic
25D7×C4⋊C4—24no—
26C7×#95C2214yes—
27C7⋊#95C2212yes—
28C7⋊#95C2214yes—
29#2502—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×D7—24no—
45C2×C4×D7—44no—
46C2×C4×D7D414yes—
47C7×C4⋊C4—22no—
48C42×C7—22no—
49(C2×D7)⋊C4—24no—
50(C2×D7)⋊C4Q818yescomplemented
51C4⋊Dic7—22no—
52C4⋊Dic7—22no—
53C4⋊Dic7—44no—
54Dic7⋊C4—24no—
55Dic7⋊C4—24no—
56Dic7⋊C4—44no—
57C4×Dic7—22no—
58C2×C22⋊Q8—71no—
59C22×C4⋊C4—71no—
60#398—74no—
61#391—71no—
62#391D711yescharacteristic, complemented
63#385—74no—
64#384—71no—
65#384—71no—
66C4×C22⋊C4—71no—
67C23×C7C2411yescharacteristic
68C22×D7—21no—
69C22×D7—21no—
70C22×D7—22no—
71C22×D7C2×D412yes—
72C22×D7C2×Q814yes—
73C7×Q8—44no—
74C2×C4×C7—22no—
75C2×C4×C7—22no—
76C2×C4×C7—22no—
77C2×C4×C7—22no—
78C2×C4×C7—22no—
79C2×C4×C7—22no—
80C2×C4×C7—24no—
81C2×C4×C7C2×D412yes—
82C2×C4×C7C2×Q814yes—
83C2×C4×C7#4112yes—
84C2×C4×C7#4112yes—
85C2×Dic7—22no—
86C2×Dic7—21no—
87C2×Dic7—21no—
88C2×Dic7—22no—
89C2×Dic7—22no—
90C2×Dic7—22no—
91C2×Dic7—24no—
92C2×Dic7—24no—
93C2×Dic7C2×D412yes—
94C2×Dic7C2×Q814yes—
95C4×D7—28no—
96C4×D7—48no—
97C22×Q8C2×D711yescharacteristic
98C23×C4—71no—
99C23×C4—72no—
100C22⋊Q8—144no—
101C2×C4⋊C4—72no—
102C2×C4⋊C4—71no—
103C2×C4⋊C4—72no—
104C2×C4⋊C4—71no—
105C2×C4⋊C4—74no—
106C2×C4⋊C4—74no—
107C2×C4⋊C4—144no—
108C2×C4⋊C4C2×D711yescharacteristic
109C2×C22⋊C4—72no—
110C2×C22⋊C4—72no—
111C2×C42—71no—
112C2×C42C2×D711yescharacteristic
113#95—72no—
114#95—74no—
115#95C2×D714yes—
116C22×C7C22×D411yescharacteristic
117C22×C7C22×Q812yes—
118C22×C7C2×#4111yescharacteristic
119C22×C7C2×#4111yescharacteristic
120C22×C7H(5,𝔽2)12yes—
121C2×D7—22no—
122C2×D7—22no—
123C2×D7—24no—
124C2×D7—24no—
125C2×D7C4⋊Q814yes—
126C4×C7—24no—
127C4×C7—42no—
128C4×C7—42no—
129C4×C7#12314yes—
130Dic7—22no—
131Dic7—24no—
132Dic7—42no—
133Dic7—42no—
134C24—71no—
135C2×Q8—22no—
136C2×Q8—22no—
137C2×Q8—22no—
138C22×C4—72no—
139C22×C4—71no—
140C22×C4—74no—
141C22×C4—72no—
142C22×C4—72no—
143C22×C4—144no—
144C22×C4—142no—
145C22×C4—142no—
146C22×C4—284no—
147C22×C4C22×D712yes—
148C22×C4C22×D712yes—
149C22×C4C22×D712yes—
150C22×C4C22×D711yescharacteristic
151C4⋊C4—22no—
152C4⋊C4—144no—
153C4⋊C4—144no—
154C4⋊C4—142no—
155C4⋊C4—142no—
156C4⋊C4—284no—
157C4⋊C4—284no—
158C22⋊C4—78no—
159C22⋊C4—144no—
160C42—22no—
161C42—142no—
162C2×C7C2×#12311yescharacteristic
163C2×C7C2×C4⋊Q811yescharacteristic
164C2×C7#53511yescharacteristic
165C2×C7#55412yes—
166C2×C7#55412yes—
167D7—24no—
168C23—72no—
169C23—74no—
170C23—141no—
171C23—141no—
172C23—142no—
173C23C23×D711yescharacteristic
174Q8—44no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—24no—
182C2×C4—72no—
183C2×C4—74no—
184C2×C4—142no—
185C2×C4—141no—
186C2×C4—141no—
187C2×C4—142no—
188C2×C4—148no—
189C2×C4—142no—
190C2×C4—142no—
191C2×C4—144no—
192C2×C4—144no—
193C2×C4—288no—
194C2×C4#115712yes—
195C2×C4D4×D712yes—
196C2×C4#115912yes—
197C2×C4Q8×D714yes—
198C7#250211yescharacteristic, complemented, Sylow
199C22—74no—
200C22—142no—
201C22—144no—
202C22—144no—
203C22—142no—
204C22C2×#115711yescharacteristic
205C22C2×D4×D711yescharacteristic
206C22C2×#115911yescharacteristic
207C22C2×Q8×D712yes—
208C22#655312yes—
209C4—24no—
210C4—42no—
211C4—42no—
212C4—142no—
213C4—144no—
214C4—282no—
215C4—282no—
216C4#644614yes—
217C2—144no—
218C2C2×#644611yescharacteristic
219C2#9027012yes—
220C2#9032612yes—
221C2#9036511yescharacteristic
222C2D7×C4⋊Q811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#391C2
prime radicalC23×D7C23
hypercenter#391D7
hypocenterC7#2502
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. #89707C2semidirect
9. #89429C2semidirect
50. (C2×D7)⋊C4Q8semidirect
62. #391D7semidirect
198. C7#2502semidirect

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
#897482
C4×(C2×D7)⋊C42
#897072
#894402
#894292
#25027