Finite Group Databasebeta

← #11734534

Subgroups — #11734534

Order 896.

456 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 125 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×#6442C212yesmaximal, complemented
2C7×#394C211yesmaximal, characteristic
3#89994C212yesmaximal
4#89755C211yesmaximal, characteristic, complemented
5#89748C211yesmaximal, characteristic, complemented
6#89720C212yesmaximal, complemented
7#89720C212yesmaximal, complemented
8#89720C212yesmaximal, complemented
9#89429C212yesmaximal, complemented
10C22×C7×D4C2211yescharacteristic
11C2×C7×C4⋊C4C2211yescharacteristic
12C2×C7×C22⋊C4C2212yes—
13C2×C7×C22⋊C4C2212yes—
14C2×C22⋊Dic7C2212yes—
15C2×C22⋊Dic7C2212yes—
16C2×C22⋊Dic7C2212yes—
17C2×C22⋊Dic7C2212yes—
18C2×C4⋊Dic7C2212yes—
19C2×C4⋊Dic7C2211yescharacteristic
20C2×C4⋊Dic7C2211yescharacteristic
21C2×Dic7⋊C4C2212yes—
22C2×Dic7⋊C4C2212yes—
23C2×Dic7⋊C4C2212yes—
24C2×Dic7⋊C4C2212yes—
25C2×Dic7⋊C4C2212yes—
26C2×C4×Dic7C2211yescharacteristic
27#6442—28no—
28C7×#95C2211yescharacteristic
29C7⋊#95C2212yes—
30C7⋊#95C2212yes—
31C7⋊#95C2211yescharacteristic
32#2823—71nomaximal, Sylow
33C24×C7C2312yes—
34C2×C7×D4—42no—
35C2×C7×D4—41no—
36C22×C4×C7C2311yescharacteristic
37C22×C4×C7C2312yes—
38C22×C4×C7C2311yescharacteristic
39C22×C4×C7C2311yescharacteristic
40C22×Dic7C2312yes—
41C22×Dic7C2312yes—
42C22×Dic7C2312yes—
43C22×Dic7C2312yes—
44C7×C4⋊C4—41no—
45C7×C22⋊C4—24no—
46C7×C22⋊C4—42no—
47C22⋊Dic7—24no—
48C22⋊Dic7—24no—
49C22⋊Dic7—42no—
50C22⋊Dic7—42no—
51C4⋊Dic7—24no—
52C4⋊Dic7—41no—
53C4⋊Dic7—41no—
54Dic7⋊C4—24no—
55Dic7⋊C4—24no—
56Dic7⋊C4—42no—
57Dic7⋊C4—42no—
58Dic7⋊C4—42no—
59C4×Dic7D414yescomplemented
60C2×#126—72no—
61#398—71no—
62#398—72no—
63#396—72no—
64#396—72no—
65#396—72no—
66#394—72no—
67#394D711yescharacteristic
68#384—71no—
69C23×C7—22no—
70C23×C7—22no—
71C23×C7—22no—
72C23×C7—22no—
73C23×C7—22no—
74C23×C7—22no—
75C23×C7—22no—
76C23×C7C2411yescharacteristic
77C7×D4—44no—
78C2×C4×C7—21no—
79C2×C4×C7—21no—
80C2×C4×C7—22no—
81C2×C4×C7—22no—
82C2×C4×C7—22no—
83C2×C4×C7—22no—
84C2×C4×C7—21no—
85C2×C4×C7—21no—
86C2×C4×C7—22no—
87C2×C4×C7C2×D412yes—
88C2×Dic7—22no—
89C2×Dic7—22no—
90C2×Dic7—22no—
91C2×Dic7—22no—
92C2×Dic7—22no—
93C2×Dic7—22no—
94C2×Dic7—22no—
95C2×Dic7—22no—
96C2×Dic7—22no—
97C2×Dic7—22no—
98C2×Dic7—22no—
99C2×Dic7C2×D414yes—
100C22×D4C2×D711yescharacteristic
101#126—148no—
102C2×C4⋊C4—72no—
103C2×C4⋊C4—72no—
104C2×C4⋊C4—72no—
105C2×C4⋊C4—72no—
106C2×C4⋊C4—72no—
107C2×C4⋊C4—72no—
108C2×C4⋊C4—71no—
109C2×C4⋊C4—71no—
110C2×C4⋊C4C2×D711yescharacteristic
111C2×C22⋊C4—72no—
112C2×C22⋊C4—72no—
113C2×C22⋊C4—72no—
114C2×C22⋊C4—72no—
115C2×C22⋊C4C2×D712yes—
116C2×C22⋊C4C2×D712yes—
117C2×C42—71no—
118#95—72no—
119#95—71no—
120#95—72no—
121#95C2×D711yescharacteristic
122C22×C7—42no—
123C22×C7—42no—
124C22×C7—42no—
125C22×C7—42no—
126C22×C7—42no—
127C22×C7—42no—
128C22×C7—42no—
129C22×C7C22×D411yescharacteristic
130C22×C7H(5,𝔽2)11yescharacteristic
131C22×C7H(5,𝔽2)12yes—
132C22×C72−1+411yescharacteristic
133C22×C72−1+412yes—
134C4×C7—22no—
135C4×C7—42no—
136C4×C7—41no—
137C4×C7—41no—
138Dic7—24no—
139Dic7—42no—
140Dic7—42no—
141Dic7—42no—
142C24C22×D712yes—
143C2×D4—41no—
144C2×D4—42no—
145C22×C4—72no—
146C22×C4—72no—
147C22×C4—72no—
148C22×C4—72no—
149C22×C4C22×D711yescharacteristic
150C22×C4C22×D712yes—
151C22×C4C22×D711yescharacteristic
152C22×C4C22×D711yescharacteristic
153C4⋊C4—41no—
154C4⋊C4—144no—
155C4⋊C4—144no—
156C4⋊C4—144no—
157C4⋊C4—282no—
158C4⋊C4—282no—
159C4⋊C4—282no—
160C4⋊C4—281no—
161C4⋊C4—281no—
162C22⋊C4—24no—
163C22⋊C4—42no—
164C22⋊C4—144no—
165C22⋊C4—144no—
166C22⋊C4—282no—
167C22⋊C4—282no—
168C42—74no—
169C2×C7—82no—
170C2×C7#53711yescharacteristic
171C2×C7#53712yes—
172C2×C7#56011yescharacteristic
173C2×C7#56312yes—
174C2×C7#56311yescharacteristic
175C23—22no—
176C23—22no—
177C23—22no—
178C23—22no—
179C23—22no—
180C23—22no—
181C23—22no—
182C23C23×D711yescharacteristic
183D4—44no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—74no—
194C2×C4—142no—
195C2×C4—142no—
196C2×C4—142no—
197C2×C4—142no—
198C2×C4—142no—
199C2×C4—142no—
200C2×C4—142no—
201C2×C4—142no—
202C2×C4—142no—
203C2×C4—142no—
204C2×C4—142no—
205C2×C4D4×D712yes—
206C7#282311yescharacteristic, complemented, Sylow
207C22—42no—
208C22—42no—
209C22—42no—
210C22—42no—
211C22—42no—
212C22—42no—
213C22—42no—
214C22C2×D4×D711yescharacteristic
215C22#654811yescharacteristic
216C22#654812yes—
217C22#655411yescharacteristic
218C22#655412yes—
219C4—22no—
220C4—41no—
221C4—41no—
222C4—42no—
223C4—144no—
224C4—282no—
225C4—282no—
226C4—282no—
227C2—82no—
228C2#9028611yescharacteristic
229C2#9028811yescharacteristic
230C2#9029912yes—
231C2#9034212yes—
232C2#9036311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#394C2
prime radicalC22×C7×D4C22
hypercenter#394D7
hypocenterC7#2823
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×#6442C2semidirect
4. #89755C2semidirect
5. #89748C2semidirect
6. #89720C2semidirect
7. #89720C2semidirect
8. #89720C2semidirect
9. #89429C2semidirect
59. C4×Dic7D4semidirect
206. C7#2823semidirect

Maximal subgroups

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

subgroupindex
C2×#64422
C7×#3942
#899942
#897552
#897482
#897202
#894292
#28237