Finite Group Databasebeta

← #11735104

Subgroups — #11735104

Order 896.

380 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 117 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×#397C211yesmaximal, characteristic
2#89758C212yesmaximal, complemented
3#89724C212yesmaximal
4#89722C212yesmaximal
5#89719C211yesmaximal, characteristic
6#89718C211yesmaximal, characteristic
7#89431C212yesmaximal, complemented
8#89430C212yesmaximal, complemented
9#89428C212yesmaximal, complemented
10C23×Dic7C2211yescharacteristic
11C22×Dic14C2211yescharacteristic
12C2×C7×C4⋊C4C2211yescharacteristic
13C2×C7×C22⋊C4C2212yes—
14C2×C7×C22⋊C4C2211yescharacteristic
15C2×C22⋊Dic7C2211yescharacteristic
16C2×C22⋊Dic7C2212yes—
17C2×C4⋊Dic7C2212yes—
18C2×C4⋊Dic7C2212yes—
19C2×Dic7⋊C4C2212yes—
20C2×Dic7⋊C4C2212yes—
21C7×#95C2212yes—
22C7×#95C2211yescharacteristic
23C7⋊#95C2212yes—
24C7⋊#95C2212yes—
25C7⋊#95C2211yescharacteristic
26C7⋊#95C2212yes—
27C7⋊#95C2212yes—
28C7⋊#95C2212yes—
29C7⋊#95C2212yes—
30C7⋊#95C2212yes—
31#2835—71nomaximal, Sylow
32C24×C7C2311yescharacteristic
33C22×C4×C7C2311yescharacteristic
34C22×C4×C7C2312yes—
35C22×C4×C7C2312yes—
36C22×C4×C7C2311yescharacteristic
37C22×Dic7—42no—
38C22×Dic7—41no—
39C22×Dic7C2312yes—
40C22×Dic7C2312yes—
41C22×Dic7C2312yes—
42C22×Dic7C2312yes—
43C2×Dic14—41no—
44C2×Dic14—42no—
45C7×C4⋊C4—41no—
46C7×C22⋊C4—24no—
47C7×C22⋊C4—41no—
48C22⋊Dic7—22no—
49C22⋊Dic7—42no—
50C4⋊Dic7—42no—
51C4⋊Dic7—42no—
52Dic7⋊C4—42no—
53Dic7⋊C4—42no—
54#400—72no—
55#400—72no—
56#400—72no—
57#397—72no—
58#397D711yescharacteristic
59#395—72no—
60#393—71no—
61#385—71no—
62#385—72no—
63C23×C7—22no—
64C23×C7—21no—
65C23×C7—21no—
66C23×C7—22no—
67C23×C7#4112yes—
68C23×C7C2411yescharacteristic
69C2×C4×C7—22no—
70C2×C4×C7—22no—
71C2×C4×C7—22no—
72C2×C4×C7—22no—
73C2×C4×C7—22no—
74C2×C4×C7—21no—
75C2×C4×C7—21no—
76C2×C4×C7—22no—
77C2×C4×C7—22no—
78C2×C4×C7—22no—
79C2×Dic7—22no—
80C2×Dic7—22no—
81C2×Dic7—22no—
82C2×Dic7—22no—
83C2×Dic7—22no—
84C2×Dic7—22no—
85C2×Dic7—22no—
86C2×Dic7—22no—
87C2×Dic7—22no—
88C2×Dic7—22no—
89C2×Dic7—22no—
90C2×Dic7—22no—
91C2×Dic7—44no—
92Dic14—44no—
93C22×Q8—71no—
94C23×C4—71no—
95C2×C4⋊C4—72no—
96C2×C4⋊C4—72no—
97C2×C4⋊C4—72no—
98C2×C4⋊C4—72no—
99C2×C4⋊C4C2×D711yescharacteristic
100C2×C22⋊C4—71no—
101C2×C22⋊C4—72no—
102C2×C22⋊C4C2×D712yes—
103C2×C22⋊C4C2×D711yescharacteristic
104#95—72no—
105#95—72no—
106#95—72no—
107#95—71no—
108#95—72no—
109#95—72no—
110#95—72no—
111#95—72no—
112#95C2×D712yes—
113#95C2×D711yescharacteristic
114C22×C7—22no—
115C22×C7—24no—
116C22×C7—41no—
117C22×C7—42no—
118C22×C7—41no—
119C22×C7C2×#4111yescharacteristic
120C22×C7H(5,𝔽2)11yescharacteristic
121C22×C7H(5,𝔽2)12yes—
122C22×C72−1+411yescharacteristic
123C22×C72−1+412yes—
124C4×C7—42no—
125C4×C7—42no—
126C4×C7—41no—
127C4×C7—41no—
128Dic7—42no—
129Dic7—42no—
130Dic7—42no—
131Dic7—42no—
132C24C22×D711yescharacteristic
133C2×Q8—281no—
134C2×Q8—282no—
135C22×C4—72no—
136C22×C4—72no—
137C22×C4—72no—
138C22×C4—72no—
139C22×C4—281no—
140C22×C4—282no—
141C22×C4C22×D711yescharacteristic
142C22×C4C22×D712yes—
143C22×C4C22×D712yes—
144C22×C4C22×D711yescharacteristic
145C4⋊C4—41no—
146C4⋊C4—282no—
147C4⋊C4—282no—
148C4⋊C4—282no—
149C4⋊C4—282no—
150C22⋊C4—24no—
151C22⋊C4—41no—
152C22⋊C4—142no—
153C22⋊C4—282no—
154C2×C7—42no—
155C2×C7#53911yescharacteristic
156C2×C7#53912yes—
157C2×C7H(3,𝔽4)11yescharacteristic
158C2×C7#56312yes—
159C2×C7#56311yescharacteristic
160C23—21no—
161C23—22no—
162C23—21no—
163C23—22no—
164C23#115912yes—
165C23C23×D711yescharacteristic
166Q8—284no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—142no—
178C2×C4—142no—
179C2×C4—142no—
180C2×C4—142no—
181C2×C4—142no—
182C2×C4—142no—
183C2×C4—142no—
184C2×C4—142no—
185C2×C4—142no—
186C2×C4—142no—
187C2×C4—142no—
188C2×C4—142no—
189C2×C4—284no—
190C7#283511yescharacteristic, complemented, Sylow
191C22—22no—
192C22—24no—
193C22—41no—
194C22—42no—
195C22—41no—
196C22C2×#115911yescharacteristic
197C22#654811yescharacteristic
198C22#655312yes—
199C22#655411yescharacteristic
200C22#655412yes—
201C4—41no—
202C4—42no—
203C4—41no—
204C4—42no—
205C4—282no—
206C4—282no—
207C4—282no—
208C4—282no—
209C2—42no—
210C2#9029311yescharacteristic
211C2#9034211yescharacteristic
212C2#9035612yes—
213C2#9037411yescharacteristic
214C2#9040312yes—
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#397C2
prime radicalC24×C7C23
hypercenter#397D7
hypocenterC7#2835
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
2. #89758C2semidirect
7. #89431C2semidirect
8. #89430C2semidirect
9. #89428C2semidirect
190. C7#2835semidirect

Maximal subgroups

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

subgroupindex
C7×#3972
#897582
#897242
#897222
#897192
#897182
#894312
#894302
#894282
#28357