Finite Group Databasebeta

← #11734514

Subgroups — #11734514

Order 896.

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

normal subgroups only

This group has 448 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
1C7×#394C211yesmaximal, characteristic
2#89992C212yesmaximal
3#89758C211yesmaximal, characteristic, complemented
4#89749C211yesmaximal, characteristic, complemented
5#89724C212yesmaximal, complemented
6#89722C212yesmaximal, complemented
7#89722C212yesmaximal, complemented
8#89716C212yesmaximal, complemented
9#89431C212yesmaximal, complemented
10C22×C7×D4C2211yescharacteristic
11C23×Dic7C2212yes—
12C2×C7×C4⋊C4C2211yescharacteristic
13C2×C7×C22⋊C4C2212yes—
14C2×C7×C22⋊C4C2212yes—
15C2×C22⋊Dic7C2212yes—
16C2×C22⋊Dic7C2212yes—
17C2×C22⋊Dic7C2212yes—
18C2×C4⋊Dic7C2212yes—
19C2×C4⋊Dic7C2211yescharacteristic
20C2×C4⋊Dic7C2211yescharacteristic
21C2×Dic7⋊C4C2212yes—
22C2×C4×Dic7C2211yescharacteristic
23C7×#95C2211yescharacteristic
24C7⋊#95C2212yes—
25C7⋊#95C2212yes—
26C7⋊#95C2212yes—
27C7⋊#95C2212yes—
28C7⋊#95C2212yes—
29C7⋊#95C2212yes—
30C7⋊#95C2211yescharacteristic
31#2782—71nomaximal, Sylow
32C24×C7C2312yes—
33C2×C7×D4—24no—
34C2×C7×D4—41no—
35C22×C4×C7C2311yescharacteristic
36C22×C4×C7C2311yescharacteristic
37C22×C4×C7C2312yes—
38C22×C4×C7C2311yescharacteristic
39C22×Dic7—42no—
40C22×Dic7—42no—
41C22×Dic7—42no—
42C22×Dic7C2312yes—
43C22×Dic7C2312yes—
44C22×Dic7C2312yes—
45C22×Dic7C2312yes—
46C7×C4⋊C4—41no—
47C7×C22⋊C4—24no—
48C7×C22⋊C4—24no—
49C22⋊Dic7—24no—
50C22⋊Dic7—42no—
51C22⋊Dic7—42no—
52C4⋊Dic7—41no—
53C4⋊Dic7—42no—
54C4⋊Dic7—41no—
55Dic7⋊C4—42no—
56C4×Dic7—22no—
57#400—71no—
58#400—72no—
59#397—72no—
60#394D711yescharacteristic
61#388—72no—
62#386—72no—
63#385—72no—
64#385—72no—
65#382—71no—
66C23×C7—22no—
67C23×C7—22no—
68C23×C7—22no—
69C23×C7—22no—
70C23×C7—22no—
71C23×C7—22no—
72C23×C7#4114yes—
73C23×C7C2411yescharacteristic
74C7×D4—44no—
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×C7—22no—
82C2×C4×C7—21no—
83C2×C4×C7—21no—
84C2×C4×C7#4112yes—
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×Dic7—22no—
96C2×Dic7—22no—
97C2×Dic7—48no—
98C22×D4C2×D711yescharacteristic
99C23×C4—72no—
100C2×C4⋊C4—72no—
101C2×C4⋊C4—71no—
102C2×C4⋊C4—71no—
103C2×C4⋊C4—72no—
104C2×C4⋊C4C2×D711yescharacteristic
105C2×C22⋊C4—72no—
106C2×C22⋊C4—72no—
107C2×C22⋊C4—72no—
108C2×C22⋊C4C2×D712yes—
109C2×C22⋊C4C2×D712yes—
110C2×C42—71no—
111#95—72no—
112#95—72no—
113#95—72no—
114#95—72no—
115#95—72no—
116#95—72no—
117#95—71no—
118#95C2×D711yescharacteristic
119C22×C7—24no—
120C22×C7—24no—
121C22×C7—24no—
122C22×C7—42no—
123C22×C7—42no—
124C22×C7—42no—
125C22×C7—42no—
126C22×C7C2×#4112yes—
127C22×C7C2×#4111yescharacteristic
128C22×C7H(5,𝔽2)11yescharacteristic
129C22×C7H(5,𝔽2)12yes—
130C22×C72−1+411yescharacteristic
131C4×C7—22no—
132C4×C7—41no—
133C4×C7—41no—
134C4×C7—42no—
135Dic7—42no—
136Dic7—42no—
137Dic7—42no—
138Dic7—42no—
139C24C22×D712yes—
140C2×D4—24no—
141C2×D4—41no—
142C22×C4—72no—
143C22×C4—72no—
144C22×C4—72no—
145C22×C4—72no—
146C22×C4—282no—
147C22×C4—282no—
148C22×C4—282no—
149C22×C4C22×D711yescharacteristic
150C22×C4C22×D711yescharacteristic
151C22×C4C22×D712yes—
152C22×C4C22×D711yescharacteristic
153C4⋊C4—41no—
154C4⋊C4—282no—
155C4⋊C4—281no—
156C4⋊C4—281no—
157C4⋊C4—282no—
158C22⋊C4—24no—
159C22⋊C4—24no—
160C22⋊C4—144no—
161C22⋊C4—282no—
162C22⋊C4—282no—
163C42—142no—
164C2×C7—44no—
165C2×C7#53911yescharacteristic
166C2×C7#53912yes—
167C2×C7#55111yescharacteristic
168C2×C7#55312yes—
169C2×C7H(3,𝔽4)11yescharacteristic
170C23—22no—
171C23—22no—
172C23—22no—
173C23—22no—
174C23—22no—
175C23—22no—
176C23#115914yes—
177C23C23×D711yescharacteristic
178D4—44no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—22no—
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×C4—142no—
198C2×C4—142no—
199C2×C4—142no—
200C2×C4—288no—
201C2×C4#115912yes—
202C7#278211yescharacteristic, complemented, Sylow
203C22—24no—
204C22—24no—
205C22—24no—
206C22—42no—
207C22—42no—
208C22—42no—
209C22—42no—
210C22C2×#115912yes—
211C22C2×#115911yescharacteristic
212C22#654811yescharacteristic
213C22#655312yes—
214C22#655411yescharacteristic
215C4—22no—
216C4—41no—
217C4—41no—
218C4—42no—
219C4—282no—
220C4—282no—
221C4—282no—
222C4—282no—
223C2—44no—
224C2#9027811yescharacteristic
225C2#9029311yescharacteristic
226C2#9030912yes—
227C2#9035612yes—
228C2#9037411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#394C2
prime radicalC22×C7×D4C22
hypercenter#394D7
hypocenterC7#2782
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
3. #89758C2semidirect
4. #89749C2semidirect
5. #89724C2semidirect
6. #89722C2semidirect
7. #89722C2semidirect
8. #89716C2semidirect
9. #89431C2semidirect
202. C7#2782semidirect

Maximal subgroups

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

subgroupindex
C7×#3942
#899922
#897582
#897492
#897242
#897222
#897162
#894312
#27827