Finite Group Databasebeta

← #11733794

Subgroups — #11733794

Order 896.

676 conjugacy classes of subgroups; only the 185 normal subgroups are shown.

normal subgroups only

This group has 676 conjugacy classes of subgroups — too many to draw. Only its 185 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×D4×Dic7C211yesmaximal, characteristic
2C2×#6451C211yesmaximal, characteristic, complemented
3C2×#6441C214yesmaximal, complemented
4C7×#390C211yesmaximal, characteristic
5#89993C211yesmaximal, characteristic
6#89748C211yesmaximal, characteristic, complemented
7#89721C214yesmaximal, complemented
8#89707C211yesmaximal, characteristic, complemented
9C4×Dic7⋊C4C211yesmaximal, characteristic, complemented
10C22×C7×D4C2211yescharacteristic
11C23×Dic7C2212yes—
12C22×Dic14C2212yes—
13C2×C7×C4⋊C4C2211yescharacteristic
14C2×C7×C22⋊C4C2214yes—
15C2×C42×C7C2211yescharacteristic
16C2×C22⋊Dic7C2214yes—
17C2×C22⋊Dic7C2212yes—
18D4×Dic7—24no—
19C2×C4⋊Dic7C2214yes—
20C2×C4⋊Dic7C2211yescharacteristic
21C2×C4⋊Dic7C2211yescharacteristic
22C2×Dic7⋊C4C2214yes—
23C2×Dic7⋊C4C2212yes—
24C2×Dic7⋊C4C2212yes—
25C2×C4×Dic7C2211yescharacteristic
26C2×C4×Dic7C2211yescharacteristic
27#6451—24no—
28#6441—216no—
29C7⋊#95C2212yes—
30#2542—71nomaximal, Sylow
31C24×C7C2312yes—
32C2×C7×D4—22no—
33C2×C7×D4—22no—
34C2×C7×D4Q814yescomplemented
35C22×C4×C7C2312yes—
36C22×C4×C7C2312yes—
37C22×C4×C7C2311yescharacteristic
38C22×Dic7—28no—
39C22×Dic7—24no—
40C22×Dic7C2312yes—
41C22×Dic7C2312yes—
42C22×Dic7C2311yescharacteristic
43C22×Dic7C2312yes—
44C22×Dic7C2311yescharacteristic
45C2×Dic14—28no—
46C2×Dic14—24no—
47C7×C4⋊C4—22no—
48C7×C22⋊C4—28no—
49C42×C7—22no—
50C22⋊Dic7—28no—
51C22⋊Dic7—24no—
52C4⋊Dic7—28no—
53C4⋊Dic7—22no—
54C4⋊Dic7—22no—
55Dic7⋊C4—28no—
56Dic7⋊C4—24no—
57Dic7⋊C4D418yescomplemented
58C4×Dic7—22no—
59C4×Dic7D414yes—
60C2×C4⋊Q8—71no—
61C2×C22⋊Q8—74no—
62C2×C4×D4—71no—
63#399—74no—
64#390—71no—
65#390D711yescharacteristic
66#384—71no—
67#384—71no—
68C4×C4⋊C4—71no—
69C23×C7—24no—
70C23×C7—22no—
71C23×C7—22no—
72C23×C7—24no—
73C23×C7C2×Q814yes—
74C23×C7C2411yescharacteristic
75C7×D4—28no—
76C2×C4×C7—22no—
77C2×C4×C7—22no—
78C2×C4×C7—24no—
79C2×C4×C7—22no—
80C2×C4×C7C2×D414yes—
81C2×C4×C7C2×D412yes—
82C2×C4×C7C2×Q812yes—
83C2×C4×C7#4112yes—
84C2×Dic7—22no—
85C2×Dic7—22no—
86C2×Dic7—22no—
87C2×Dic7—22no—
88C2×Dic7—24no—
89C2×Dic7—24no—
90C2×Dic7—22no—
91C2×Dic7—48no—
92C2×Dic7C2×D412yes—
93C2×Dic7C2×D414yes—
94C2×Dic7C2×D412yes—
95C2×Dic7C2×D414yes—
96Dic14—48no—
97C22×Q8—72no—
98C22×D4C2×D711yescharacteristic
99C23×C4—72no—
100C4⋊Q8—144no—
101C22⋊Q8—1416no—
102C4×D4—144no—
103C2×C4⋊C4—72no—
104C2×C4⋊C4—71no—
105C2×C4⋊C4—74no—
106C2×C4⋊C4—74no—
107C2×C4⋊C4—72no—
108C2×C4⋊C4—71no—
109C2×C4⋊C4C2×D711yescharacteristic
110C2×C22⋊C4—72no—
111C2×C22⋊C4—74no—
112C2×C22⋊C4C2×D714yes—
113C2×C42—71no—
114C2×C42—71no—
115C2×C42C2×D711yescharacteristic
116#95—72no—
117C22×C7—24no—
118C22×C7—28no—
119C22×C7—42no—
120C22×C7—42no—
121C22×C7—44no—
122C22×C7C22×D411yescharacteristic
123C22×C7C22×D412yes—
124C22×C7C22×Q811yescharacteristic
125C22×C7C2×#4111yescharacteristic
126C22×C72−1+412yes—
127C4×C7—24no—
128C4×C7—42no—
129C4×C7C22⋊Q814yescomplemented
130Dic7—22no—
131Dic7—24no—
132Dic7—42no—
133Dic7—42no—
134Dic7#12714yes—
135C24C22×D712yes—
136C2×Q8—148no—
137C2×Q8—144no—
138C2×D4—22no—
139C2×D4—22no—
140C2×D4Dic1414yescomplemented
141C22×C4—71no—
142C22×C4—71no—
143C22×C4—72no—
144C22×C4—72no—
145C22×C4—72no—
146C22×C4—144no—
147C22×C4—148no—
148C22×C4C22×D712yes—
149C22×C4C22×D712yes—
150C22×C4C22×D711yescharacteristic
151C4⋊C4—22no—
152C4⋊C4—78no—
153C4⋊C4—148no—
154C4⋊C4—144no—
155C4⋊C4—142no—
156C4⋊C4—148no—
157C4⋊C4—142no—
158C22⋊C4—28no—
159C22⋊C4—144no—
160C22⋊C4—148no—
161C42—22no—
162C42—74no—
163C42—142no—
164C2×C7—44no—
165C2×C7C2×C22⋊Q811yescharacteristic
166C2×C7C2×#12711yescharacteristic
167C2×C7#53611yescharacteristic
168C2×C7#54712yes—
169C2×C7D4×Q812yes—
170C23—24no—
171C23—22no—
172C23—22no—
173C23—24no—
174C23C2×Dic1414yes—
175C23C23×D711yescharacteristic
176Q8—288no—
177D4—28no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—24no—
181C2×C4—22no—
182C2×C4—72no—
183C2×C4—74no—
184C2×C4—72no—
185C2×C4—74no—
186C2×C4—142no—
187C2×C4—142no—
188C2×C4—142no—
189C2×C4—142no—
190C2×C4—144no—
191C2×C4—144no—
192C2×C4—142no—
193C2×C4—288no—
194C2×C4C2×Dic1412yes—
195C2×C4D4×D714yes—
196C2×C4D4×D712yes—
197C2×C4#115912yes—
198C7#254211yescharacteristic, complemented, Sylow
199C22—24no—
200C22—28no—
201C22—42no—
202C22—44no—
203C22—42no—
204C22C22×Dic1411yescharacteristic
205C22C2×D4×D711yescharacteristic
206C22C2×D4×D712yes—
207C22C2×#115911yescharacteristic
208C22#655412yes—
209C4—24no—
210C4—42no—
211C4—74no—
212C4—144no—
213C4—142no—
214C4—282no—
215C4—282no—
216C4#644114yescomplemented
217C2—44no—
218C2C2×#644111yescharacteristic
219C2D4×Dic1412yes—
220C2#9030212yes—
221C2#9036611yescharacteristic
222C2D7×#12711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#390C2
prime radicalC22×C7×D4C22
hypercenter#390D7
hypocenterC7#2542
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
2. C2×#6451C2semidirect
3. C2×#6441C2semidirect
6. #89748C2semidirect
7. #89721C2semidirect
8. #89707C2semidirect
9. C4×Dic7⋊C4C2semidirect
34. C2×C7×D4Q8semidirect
57. Dic7⋊C4D4semidirect
129. C4×C7C22⋊Q8semidirect
140. C2×D4Dic14semidirect
198. C7#2542semidirect
216. C4#6441semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Dic72
C2×#64512
C2×#64412
C7×#3902
#899932
#897482
#897212
#897072
C4×Dic7⋊C42
#25427