Finite Group Databasebeta

← #11733580

Subgroups — #11733580

Order 896.

456 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 157 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×#6451C212yesmaximal
2C7×#389C211yesmaximal, characteristic
3#89756C212yesmaximal
4#89753C212yesmaximal
5#89751C212yesmaximal
6#89701C211yesmaximal, characteristic
7#89700C211yesmaximal, characteristic
8#89428C212yesmaximal
9#89426C212yesmaximal
10C22×Dic14C2212yes—
11C22×Dic14C2211yescharacteristic
12C2×C7×C4⋊C4C2212yes—
13C2×C7×C4⋊C4C2211yescharacteristic
14C2×C7×C4⋊C4C2211yescharacteristic
15C2×C42×C7C2211yescharacteristic
16C2×C4⋊Dic7C2212yes—
17C2×C4⋊Dic7C2211yescharacteristic
18C2×C4⋊Dic7C2211yescharacteristic
19C2×C4⋊Dic7C2211yescharacteristic
20C2×Dic7⋊C4C2212yes—
21C2×Dic7⋊C4C2212yes—
22C2×Dic7⋊C4C2212yes—
23C2×C4×Dic7C2212yes—
24#6451—28no—
25C7×#95C2212yes—
26C7⋊#95C2212yes—
27C7⋊#95C2212yes—
28C7⋊#95C2212yes—
29C7⋊#95C2212yes—
30C7⋊#95C2212yes—
31C7⋊#95C2212yes—
32#2734—71nomaximal, Sylow
33C22×C4×C7C2312yes—
34C22×C4×C7C2311yescharacteristic
35C22×C4×C7C2311yescharacteristic
36C22×C4×C7C2311yescharacteristic
37C22×C4×C7C2312yes—
38C22×Dic7C2312yes—
39C22×Dic7C2312yes—
40C22×Dic7C2312yes—
41C22×Dic7C2312yes—
42C2×Dic14—24no—
43C2×Dic14—41no—
44C2×Dic14—41no—
45C2×Dic14—42no—
46C2×Dic14—42no—
47C2×Dic14D414yes—
48C7×C4⋊C4—24no—
49C7×C4⋊C4—41no—
50C7×C4⋊C4Q814yescomplemented
51C42×C7—41no—
52C4⋊Dic7—24no—
53C4⋊Dic7—41no—
54C4⋊Dic7—41no—
55C4⋊Dic7Q814yescomplemented
56Dic7⋊C4—24no—
57Dic7⋊C4—24no—
58Dic7⋊C4—42no—
59C4×Dic7—24no—
60C2×C4⋊Q8—72no—
61#398—72no—
62#395—72no—
63#395—72no—
64#391—71no—
65#389—71no—
66#389D711yescharacteristic
67#387—72no—
68#387—72no—
69C23×C7C2411yescharacteristic
70C2×C4×C7—22no—
71C2×C4×C7—21no—
72C2×C4×C7—21no—
73C2×C4×C7—21no—
74C2×C4×C7—21no—
75C2×C4×C7—21no—
76C2×C4×C7—21no—
77C2×C4×C7—22no—
78C2×C4×C7—22no—
79C2×C4×C7—22no—
80C2×C4×C7—22no—
81C2×C4×C7C2×D412yes—
82C2×C4×C7C2×Q812yes—
83C2×C4×C7C2×Q814yes—
84C2×C4×C7C2×Q812yes—
85C2×Dic7—22no—
86C2×Dic7—22no—
87C2×Dic7—22no—
88C2×Dic7—22no—
89C2×Dic7—22no—
90C2×Dic7—22no—
91C2×Dic7—22no—
92C2×Dic7—22no—
93C2×Dic7—22no—
94C2×Dic7—22no—
95C2×Dic7C2×D414yes—
96C2×Dic7C2×Q814yes—
97Dic14—44no—
98Dic14—48no—
99C22×Q8—71no—
100C22×Q8—72no—
101C4⋊Q8—148no—
102C2×C4⋊C4—71no—
103C2×C4⋊C4—72no—
104C2×C4⋊C4—72no—
105C2×C4⋊C4—72no—
106C2×C4⋊C4—72no—
107C2×C4⋊C4—71no—
108C2×C4⋊C4—71no—
109C2×C4⋊C4C2×D712yes—
110C2×C4⋊C4C2×D711yescharacteristic
111C2×C4⋊C4C2×D711yescharacteristic
112C2×C42—72no—
113C2×C42C2×D711yescharacteristic
114#95—72no—
115#95—72no—
116#95—72no—
117#95—72no—
118#95—72no—
119#95—72no—
120#95C2×D712yes—
121C22×C7C22×D411yescharacteristic
122C22×C7C22×Q811yescharacteristic
123C22×C7C22×Q811yescharacteristic
124C22×C7H(5,𝔽2)11yescharacteristic
125C22×C7H(5,𝔽2)11yescharacteristic
126C22×C72−1+411yescharacteristic
127C22×C72−1+411yescharacteristic
128C4×C7—22no—
129C4×C7—22no—
130C4×C7—22no—
131C4×C7—24no—
132C4×C7—42no—
133Dic7—24no—
134Dic7—24no—
135Dic7—42no—
136Dic7—42no—
137C2×Q8—74no—
138C2×Q8—144no—
139C2×Q8—281no—
140C2×Q8—281no—
141C2×Q8—282no—
142C2×Q8—282no—
143C22×C4—72no—
144C22×C4—72no—
145C22×C4—72no—
146C22×C4—72no—
147C22×C4C22×D712yes—
148C22×C4C22×D711yescharacteristic
149C22×C4C22×D712yes—
150C22×C4C22×D711yescharacteristic
151C22×C4C22×D711yescharacteristic
152C4⋊C4—24no—
153C4⋊C4—41no—
154C4⋊C4—74no—
155C4⋊C4—144no—
156C4⋊C4—144no—
157C4⋊C4—144no—
158C4⋊C4—282no—
159C4⋊C4—281no—
160C4⋊C4—281no—
161C4⋊C4Dic1414yescomplemented
162C42—41no—
163C42—144no—
164C2×C7#53411yescharacteristic
165C2×C7#54411yescharacteristic
166C2×C7#54411yescharacteristic
167C2×C7D4×Q811yescharacteristic
168C2×C7D4×Q811yescharacteristic
169C2×C7Q8211yescharacteristic
170C2×C7#56311yescharacteristic
171C23C23×D711yescharacteristic
172Q8—284no—
173Q8—288no—
174C2×C4—22no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—22no—
181C2×C4—21no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—74no—
186C2×C4—74no—
187C2×C4—142no—
188C2×C4—142no—
189C2×C4—142no—
190C2×C4—142no—
191C2×C4—142no—
192C2×C4—142no—
193C2×C4—142no—
194C2×C4—142no—
195C2×C4—142no—
196C2×C4—142no—
197C2×C4C2×Dic1412yes—
198C2×C4C2×Dic1414yes—
199C2×C4D4×D712yes—
200C2×C4Q8×D712yes—
201C7#273411yescharacteristic, complemented, Sylow
202C22C22×Dic1411yescharacteristic
203C22C2×D4×D711yescharacteristic
204C22#654811yescharacteristic
205C22C2×Q8×D711yescharacteristic
206C22#655111yescharacteristic
207C22#655311yescharacteristic
208C22#655411yescharacteristic
209C4—22no—
210C4—24no—
211C4—22no—
212C4—22no—
213C4—42no—
214C4—144no—
215C4—144no—
216C4—282no—
217C4—282no—
218C2#9019111yescharacteristic
219C2D4×Dic1411yescharacteristic
220C2Q8×Dic1411yescharacteristic
221C2#9029611yescharacteristic
222C2#9031411yescharacteristic
223C2#9039011yescharacteristic
224C2#9041811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, prime radical, cosocleC23×C7C24
FittingC7×#389C2
hypercenter#389D7
hypocenterC7#2734
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #389

Complements

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

NH ≅ G/Ntype
50. C7×C4⋊C4Q8semidirect
55. C4⋊Dic7Q8semidirect
161. C4⋊C4Dic14semidirect
201. C7#2734semidirect

Maximal subgroups

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

subgroupindex
C2×#64512
C7×#3892
#897562
#897532
#897512
#897012
#897002
#894282
#894262
#27347