Finite Group Databasebeta

← #11735371

Subgroups — #11735371

Order 896.

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

normal subgroups only

This group has 380 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×#6453C211yesmaximal, characteristic
2C2×#6452C211yesmaximal, characteristic
3C7×#398C211yesmaximal, characteristic
4#89758C211yesmaximal, characteristic
5#89756C212yesmaximal
6#89752C211yesmaximal, characteristic
7#89752C212yesmaximal
8#89432C212yesmaximal
9#89431C211yesmaximal, characteristic
10#89429C212yesmaximal
11#89422C211yesmaximal, characteristic
12C2×C7×C4⋊C4C2211yescharacteristic
13C2×C7×C4⋊C4C2211yescharacteristic
14C2×C7×C4⋊C4C2212yes—
15C2×C4⋊Dic7C2211yescharacteristic
16C2×C4⋊Dic7C2211yescharacteristic
17C2×C4⋊Dic7C2212yes—
18C2×Dic7⋊C4C2211yescharacteristic
19C2×Dic7⋊C4C2211yescharacteristic
20C2×Dic7⋊C4C2211yescharacteristic
21C2×Dic7⋊C4C2211yescharacteristic
22C2×Dic7⋊C4C2212yes—
23C2×Dic7⋊C4C2212yes—
24C2×Dic7⋊C4C2212yes—
25C2×C4×Dic7C2211yescharacteristic
26#6453—24no—
27#6452—24no—
28C7×#95C2211yescharacteristic
29C7×#95C2212yes—
30C7⋊#95C2212yes—
31C7⋊#95C2211yescharacteristic
32C7⋊#95C2212yes—
33C7⋊#95C2212yes—
34C7⋊#95C2212yes—
35C7⋊#95C2212yes—
36C7⋊#95C2212yes—
37#2822—71nomaximal, Sylow
38C22×C4×C7C2312yes—
39C22×C4×C7C2311yescharacteristic
40C22×C4×C7C2311yescharacteristic
41C22×C4×C7C2312yes—
42C22×C4×C7C2311yescharacteristic
43C22×Dic7C2312yes—
44C22×Dic7C2311yescharacteristic
45C22×Dic7C2312yes—
46C22×Dic7C2311yescharacteristic
47C22×Dic7C2311yescharacteristic
48C22×Dic7C2311yescharacteristic
49C7×C4⋊C4—22no—
50C7×C4⋊C4—22no—
51C7×C4⋊C4—42no—
52C4⋊Dic7—22no—
53C4⋊Dic7—22no—
54C4⋊Dic7—24no—
55Dic7⋊C4—22no—
56Dic7⋊C4—22no—
57Dic7⋊C4—22no—
58Dic7⋊C4—22no—
59Dic7⋊C4—24no—
60Dic7⋊C4—42no—
61Dic7⋊C4—42no—
62C4×Dic7D414yes—
63C2×#125—71no—
64C2×#125—71no—
65#400—71no—
66#400—71no—
67#400—72no—
68#398—71no—
69#398—72no—
70#398—72no—
71#398—72no—
72#398D711yescharacteristic
73#382—71no—
74C23×C7C2411yescharacteristic
75C2×C4×C7—22no—
76C2×C4×C7—21no—
77C2×C4×C7—21no—
78C2×C4×C7—22no—
79C2×C4×C7—22no—
80C2×C4×C7—22no—
81C2×C4×C7—21no—
82C2×C4×C7—22no—
83C2×C4×C7—22no—
84C2×C4×C7—21no—
85C2×C4×C7—22no—
86C2×C4×C7—22no—
87C2×C4×C7C2×D412yes—
88C2×Dic7—21no—
89C2×Dic7—22no—
90C2×Dic7—21no—
91C2×Dic7—21no—
92C2×Dic7—21no—
93C2×Dic7—22no—
94C2×Dic7—21no—
95C2×Dic7—22no—
96C2×Dic7—24no—
97C2×Dic7—22no—
98C2×Dic7—22no—
99C2×Dic7—21no—
100C2×Dic7—22no—
101C2×Dic7C2×D412yes—
102C2×Dic7C2×D412yes—
103#125—144no—
104#125—144no—
105C2×C4⋊C4—71no—
106C2×C4⋊C4—71no—
107C2×C4⋊C4—72no—
108C2×C4⋊C4—71no—
109C2×C4⋊C4—71no—
110C2×C4⋊C4—71no—
111C2×C4⋊C4—72no—
112C2×C4⋊C4—71no—
113C2×C4⋊C4—72no—
114C2×C4⋊C4—72no—
115C2×C4⋊C4C2×D711yescharacteristic
116C2×C4⋊C4C2×D711yescharacteristic
117C2×C4⋊C4C2×D712yes—
118C2×C42—71no—
119#95—72no—
120#95—71no—
121#95—72no—
122#95—72no—
123#95—72no—
124#95—72no—
125#95—72no—
126#95C2×D711yescharacteristic
127#95C2×D712yes—
128C22×C7C22×D411yescharacteristic
129C22×C7H(5,𝔽2)11yescharacteristic
130C22×C72−1+412yes—
131C22×C72−1+412yes—
132C22×C72−1+411yescharacteristic
133C4×C7—22no—
134C4×C7—41no—
135C4×C7—41no—
136C4×C7—42no—
137C4×C7—42no—
138Dic7—22no—
139Dic7—22no—
140Dic7—42no—
141Dic7—41no—
142Dic7—41no—
143Dic7—42no—
144C22×C4—71no—
145C22×C4—71no—
146C22×C4—72no—
147C22×C4—72no—
148C22×C4—71no—
149C22×C4—71no—
150C22×C4C22×D712yes—
151C22×C4C22×D711yescharacteristic
152C22×C4C22×D711yescharacteristic
153C22×C4C22×D712yes—
154C22×C4C22×D711yescharacteristic
155C4⋊C4—22no—
156C4⋊C4—22no—
157C4⋊C4—42no—
158C4⋊C4—142no—
159C4⋊C4—142no—
160C4⋊C4—142no—
161C4⋊C4—144no—
162C4⋊C4—142no—
163C4⋊C4—144no—
164C4⋊C4—142no—
165C4⋊C4—142no—
166C4⋊C4—282no—
167C4⋊C4—282no—
168C42—74no—
169C2×C7#53611yescharacteristic
170C2×C7#53611yescharacteristic
171C2×C7#53711yescharacteristic
172C2×C7#56312yes—
173C2×C7#56412yes—
174C23C23×D711yescharacteristic
175C2×C4—22no—
176C2×C4—21no—
177C2×C4—22no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—21no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—72no—
188C2×C4—72no—
189C2×C4—141no—
190C2×C4—142no—
191C2×C4—141no—
192C2×C4—142no—
193C2×C4—141no—
194C2×C4—141no—
195C2×C4—141no—
196C2×C4—142no—
197C2×C4—144no—
198C2×C4—142no—
199C2×C4—141no—
200C2×C4—142no—
201C2×C4—142no—
202C2×C4D4×D712yes—
203C7#282211yescharacteristic, complemented, Sylow
204C22C2×D4×D711yescharacteristic
205C22#654811yescharacteristic
206C22#655112yes—
207C22#655411yescharacteristic
208C22#655412yes—
209C4—22no—
210C4—41no—
211C4—42no—
212C4—42no—
213C4—41no—
214C4—142no—
215C4—142no—
216C4—282no—
217C4—281no—
218C4—282no—
219C4—281no—
220C2#9029911yescharacteristic
221C2#9033712yes—
222C2#9033911yescharacteristic
223C2C7⋊#56412yes—
224C2#9041511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, prime radical, cosocleC23×C7C24
FittingC7×#398C2
hypercenter#398D7
hypocenterC7#2822
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
203. C7#2822semidirect

Maximal subgroups

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

subgroupindex
C2×#64532
C2×#64522
C7×#3982
#897582
#897562
#897522
#894322
#894312
#894292
#894222
#28227