Finite Group Databasebeta

← #11729604

Subgroups — #11729604

Order 896.

382 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 71 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
1C7×#534C211yesmaximal, characteristic
2#89992C212yesmaximal, complemented
3C2×#6408C212yesmaximal, complemented
4#89718C211yesmaximal, characteristic, complemented
5#89321C211yesmaximal, characteristic, complemented
6C22×C7×D4C2212yes—
7C23×Dic7C2211yescharacteristic
8C7×#123—22no—
9C7×#121—21no—
10C7×#121—21no—
11C7×#120—22no—
12C2×C7×C22⋊C4C2211yescharacteristic
13C2×C22⋊Dic7—22no—
14C2×C22⋊Dic7C2212yes—
15C2×C22⋊Dic7C2211yescharacteristic
16C2×Dic7⋊C4—21no—
17#6408—24no—
18C7⋊#95—21no—
19C7⋊#95—22no—
20#2044—71nomaximal, Sylow
21C24×C7D412yes—
22C24×C7C2311yescharacteristic
23C2×C7×D4—22no—
24C2×C7×D4—22no—
25C2×C7×D4—22no—
26C2×C7×D4—22no—
27C2×C7×D4—22no—
28C2×C7×D4—22no—
29C2×C7×D4—21no—
30C2×C7×D4—21no—
31C22×C4×C7D412yes—
32C22×C4×C7D411yescharacteristic
33C22×C4×C7D411yescharacteristic
34C22×Dic7—21no—
35C22×Dic7—21no—
36C22×Dic7—22no—
37C22×Dic7—22no—
38C22×Dic7—42no—
39C7×C4⋊C4—21no—
40C7×C4⋊C4—21no—
41C7×C22⋊C4—22no—
42C7×C22⋊C4—22no—
43C7×C22⋊C4—22no—
44C22⋊Dic7—21no—
45C22⋊Dic7—22no—
46C22⋊Dic7—22no—
47C22⋊Dic7—44no—
48C22⋊Dic7D412yescomplemented
49Dic7⋊C4—42no—
50#534D711yescharacteristic
51C2×#99—72no—
52#386—72no—
53#385—71no—
54#342—71no—
55C23×C7—22no—
56C23×C7—22no—
57C23×C7—22no—
58C23×C7—41no—
59C23×C7—41no—
60C23×C7—42no—
61C23×C7—42no—
62C23×C7C2×D411yescharacteristic
63C23×C7C2×D412yes—
64C23×C7C2×D411yescharacteristic
65C23×C7#4111yescharacteristic
66C23×C7#4112yes—
67C7×D4—42no—
68C7×D4—42no—
69C7×D4—42no—
70C7×D4—42no—
71C7×D4—81no—
72C7×D4—81no—
73C2×C4×C7—21no—
74C2×C4×C7—21no—
75C2×C4×C7—22no—
76C2×C4×C7—22no—
77C2×C4×C7—22no—
78C2×C4×C7—41no—
79C2×C4×C7—41no—
80C2×Dic7—22no—
81C2×Dic7—24no—
82C2×Dic7—22no—
83C2×Dic7—41no—
84C2×Dic7—41no—
85C2×Dic7—41no—
86C2×Dic7—42no—
87C2×Dic7—41no—
88C2×Dic7—42no—
89C2×Dic7—42no—
90C2×Dic7—42no—
91C22×D4C2×D712yes—
92C23×C4—71no—
93#123—22no—
94#121—21no—
95#121—21no—
96#120—22no—
97C2×C4⋊C4—141no—
98C2×C22⋊C4—71no—
99C2×C22⋊C4—72no—
100C2×C22⋊C4—142no—
101C2×C22⋊C4C2×D711yescharacteristic
102#99—144no—
103#95—141no—
104#95—142no—
105C22×C7—21no—
106C22×C7—22no—
107C22×C7—21no—
108C22×C7—22no—
109C22×C7—41no—
110C22×C7—41no—
111C22×C7—42no—
112C22×C7—44no—
113C22×C7—42no—
114C22×C7—42no—
115C22×C7—42no—
116C22×C7—82no—
117C22×C7#12011yescharacteristic
118C22×C7#12111yescharacteristic
119C22×C7#12112yes—
120C22×C7#12312yes—
121C22×C7#12411yescharacteristic
122C4×C7—42no—
123C4×C7—41no—
124C4×C7—41no—
125Dic7—42no—
126Dic7—42no—
127Dic7—82no—
128C24C22⋊D712yes—
129C24C22×D711yescharacteristic
130C2×D4—22no—
131C2×D4—22no—
132C2×D4—22no—
133C2×D4—22no—
134C2×D4—22no—
135C2×D4—22no—
136C2×D4—21no—
137C2×D4—21no—
138C22×C4—141no—
139C22×C4—142no—
140C22×C4—141no—
141C22×C4—142no—
142C22×C4—282no—
143C22×C4C22⋊D712yes—
144C22×C4C22⋊D711yescharacteristic
145C22×C4C22⋊D711yescharacteristic
146C4⋊C4—21no—
147C4⋊C4—21no—
148C4⋊C4—282no—
149C22⋊C4—22no—
150C22⋊C4—22no—
151C22⋊C4—22no—
152C22⋊C4—72no—
153C22⋊C4—141no—
154C22⋊C4—142no—
155C22⋊C4—142no—
156C22⋊C4—284no—
157C2×C7—21no—
158C2×C7—22no—
159C2×C7—22no—
160C2×C7—21no—
161C2×C7—82no—
162C2×C7#39411yescharacteristic
163C2×C7B(4,𝔽2)11yescharacteristic
164C2×C7#45811yescharacteristic
165C23—22no—
166C23—22no—
167C23—22no—
168C23—41no—
169C23—41no—
170C23—42no—
171C23—42no—
172C23D4×D711yescharacteristic
173C23#115911yescharacteristic
174C23#115912yes—
175C23C2×C22⋊D712yes—
176C23C2×C22⋊D711yescharacteristic
177D4—42no—
178D4—42no—
179D4—42no—
180D4—42no—
181D4—81no—
182D4—81no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—144no—
191C2×C4—142no—
192C2×C4—142no—
193C2×C4—281no—
194C2×C4—281no—
195C2×C4—281no—
196C2×C4—282no—
197C2×C4—281no—
198C2×C4—282no—
199C2×C4—282no—
200C2×C4—282no—
201C7#204411yescharacteristic, complemented, Sylow
202C22—21no—
203C22—22no—
204C22—21no—
205C22—22no—
206C22—41no—
207C22—41no—
208C22—42no—
209C22—44no—
210C22—42no—
211C22—42no—
212C22—42no—
213C22—82no—
214C22#649812yes—
215C22#649911yescharacteristic
216C22#650111yescharacteristic
217C22#650212yes—
218C22#651611yescharacteristic
219C4—42no—
220C4—41no—
221C4—41no—
222C4—282no—
223C4—282no—
224C4—562no—
225C2—21no—
226C2—22no—
227C2—22no—
228C2—21no—
229C2—82no—
230C2#8999411yescharacteristic
231C2#9001511yescharacteristic
232C2#9001611yescharacteristic
subgroupG/H
centerC22#6516
commutator, cosocleC24×C7C23
FrattiniC24C22×D7
Fitting, prime radicalC7×#534C2
socleC22×C7#120
hypercenter#534D7
hypocenterC7#2044
LCSG ▷ C24×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C24 ◁ #534

Complements

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

NH ≅ G/Ntype
2. #89992C2semidirect
3. C2×#6408C2semidirect
4. #89718C2semidirect
5. #89321C2semidirect
48. C22⋊Dic7D4semidirect
201. C7#2044semidirect

Maximal subgroups

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

subgroupindex
C7×#5342
#899922
C2×#64082
#897182
#893212
#20447