Finite Group Databasebeta

← #11734005

Subgroups — #11734005

Order 896.

352 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 352 conjugacy classes of subgroups — too many to draw. Only its 133 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×#391C211yesmaximal, characteristic
2#90002C211yesmaximal, characteristic
3#90002C211yesmaximal, characteristic
4#89757C211yesmaximal, characteristic
5#89745C211yesmaximal, characteristic
6#89704C211yesmaximal, characteristic
7#89700C211yesmaximal, characteristic
8C7⋊#401C212yesmaximal
9#89432C212yesmaximal
10#89431C212yesmaximal
11#89430C212yesmaximal
12C22×C7×Q8C2211yescharacteristic
13C2×C7×C4⋊C4C2211yescharacteristic
14C2×C42×C7C2211yescharacteristic
15C2×C4⋊Dic7C2211yescharacteristic
16C2×C4⋊Dic7C2211yescharacteristic
17C2×C4⋊Dic7C2212yes—
18C2×Dic7⋊C4C2212yes—
19C2×Dic7⋊C4C2212yes—
20C2×C4×Dic7C2211yescharacteristic
21C2×C4×Dic7C2211yescharacteristic
22C7×#95C2212yes—
23C7×#95C2212yes—
24C7⋊#95C2211yescharacteristic
25C7⋊#95C2212yes—
26C7⋊#95C2212yes—
27C7⋊#95C2212yes—
28C7⋊#95C2211yescharacteristic
29C7⋊#95C2212yes—
30C7⋊#95C2212yes—
31C7⋊#95C2211yescharacteristic
32C7⋊#95C2212yes—
33C7⋊#95C2212yes—
34C7⋊#95C2211yescharacteristic
35#2645—71nomaximal, Sylow
36C2×C7×Q8—22no—
37C2×C7×Q8—22no—
38C2×C7×Q8—22no—
39C22×C4×C7C2311yescharacteristic
40C22×C4×C7C2312yes—
41C22×C4×C7C2311yescharacteristic
42C22×C4×C7C2312yes—
43C22×C4×C7C2311yescharacteristic
44C22×Dic7C2311yescharacteristic
45C22×Dic7C2311yescharacteristic
46C22×Dic7C2311yescharacteristic
47C22×Dic7C2311yescharacteristic
48C22×Dic7C2312yes—
49C22×Dic7C2312yes—
50C7×C4⋊C4—22no—
51C42×C7—22no—
52C4⋊Dic7—22no—
53C4⋊Dic7—22no—
54C4⋊Dic7—42no—
55Dic7⋊C4—42no—
56Dic7⋊C4—42no—
57C4×Dic7—22no—
58C4×Dic7—22no—
59#401—72no—
60#400—72no—
61#400—72no—
62#400—72no—
63#391—71no—
64#391—71no—
65#391D711yescharacteristic
66#389—71no—
67#389—71no—
68#389—71no—
69#376—71no—
70C23×C7C2411yescharacteristic
71C7×Q8—44no—
72C2×C4×C7—21no—
73C2×C4×C7—21no—
74C2×C4×C7—22no—
75C2×C4×C7—22no—
76C2×C4×C7—22no—
77C2×C4×C7—22no—
78C2×C4×C7—21no—
79C2×C4×C7—22no—
80C2×C4×C7—21no—
81C2×C4×C7—21no—
82C2×C4×C7—21no—
83C2×C4×C7—22no—
84C2×C4×C7#4112yes—
85C2×C4×C7#4112yes—
86C2×C4×C7#4112yes—
87C2×Dic7—21no—
88C2×Dic7—21no—
89C2×Dic7—21no—
90C2×Dic7—21no—
91C2×Dic7—22no—
92C2×Dic7—22no—
93C2×Dic7—21no—
94C2×Dic7—22no—
95C2×Dic7—22no—
96C2×Dic7—21no—
97C2×Dic7—21no—
98C2×Dic7—21no—
99C2×Dic7—21no—
100C2×Dic7—21no—
101C2×Dic7—22no—
102C2×Dic7—22no—
103C2×Dic7—21no—
104C2×Dic7—21no—
105C22×Q8C2×D711yescharacteristic
106C2×C4⋊C4—71no—
107C2×C4⋊C4—71no—
108C2×C4⋊C4—72no—
109C2×C4⋊C4—72no—
110C2×C4⋊C4—72no—
111C2×C4⋊C4C2×D711yescharacteristic
112C2×C42—71no—
113C2×C42—71no—
114C2×C42C2×D711yescharacteristic
115#95—71no—
116#95—71no—
117#95—71no—
118#95—71no—
119#95—72no—
120#95—72no—
121#95—72no—
122#95—72no—
123#95—72no—
124#95—72no—
125#95—72no—
126#95C2×D712yes—
127#95C2×D712yes—
128C22×C7C2×#4111yescharacteristic
129C22×C7C2×#4111yescharacteristic
130C22×C7C2×#4111yescharacteristic
131C22×C7H(5,𝔽2)11yescharacteristic
132C22×C72−1+411yescharacteristic
133C22×C72−1+411yescharacteristic
134C22×C72−1+411yescharacteristic
135C4×C7—42no—
136C4×C7—41no—
137C4×C7—42no—
138C4×C7—41no—
139C4×C7#12614yes—
140Dic7—41no—
141Dic7—41no—
142Dic7—41no—
143Dic7—41no—
144Dic7—42no—
145Dic7—42no—
146C2×Q8—22no—
147C2×Q8—22no—
148C2×Q8—22no—
149C22×C4—71no—
150C22×C4—71no—
151C22×C4—71no—
152C22×C4—71no—
153C22×C4—72no—
154C22×C4—72no—
155C22×C4C22×D711yescharacteristic
156C22×C4C22×D712yes—
157C22×C4C22×D711yescharacteristic
158C22×C4C22×D712yes—
159C22×C4C22×D711yescharacteristic
160C4⋊C4—22no—
161C4⋊C4—142no—
162C4⋊C4—142no—
163C4⋊C4—282no—
164C4⋊C4—282no—
165C4⋊C4—282no—
166C42—22no—
167C42—142no—
168C42—142no—
169C2×C7C2×#12611yescharacteristic
170C2×C7#54111yescharacteristic
171C2×C7#54111yescharacteristic
172C2×C7#54111yescharacteristic
173C2×C7#54211yescharacteristic
174C2×C7#54211yescharacteristic
175C2×C7#54211yescharacteristic
176C23C23×D711yescharacteristic
177Q8—44no—
178C2×C4—22no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—22no—
187C2×C4—21no—
188C2×C4—22no—
189C2×C4—21no—
190C2×C4—141no—
191C2×C4—141no—
192C2×C4—142no—
193C2×C4—141no—
194C2×C4—141no—
195C2×C4—142no—
196C2×C4—141no—
197C2×C4—141no—
198C2×C4—141no—
199C2×C4—142no—
200C2×C4—141no—
201C2×C4—142no—
202C2×C4—141no—
203C2×C4—142no—
204C2×C4—141no—
205C2×C4—142no—
206C2×C4—141no—
207C2×C4—141no—
208C2×C4#115712yes—
209C2×C4#115912yes—
210C2×C4#115912yes—
211C7#264511yescharacteristic, complemented, Sylow
212C22C2×#115711yescharacteristic
213C22C2×#115911yescharacteristic
214C22C2×#115911yescharacteristic
215C22#655111yescharacteristic
216C22#655111yescharacteristic
217C22#655311yescharacteristic
218C22#655411yescharacteristic
219C4—42no—
220C4—42no—
221C4—41no—
222C4—41no—
223C4—281no—
224C4—281no—
225C4—281no—
226C4—282no—
227C4—281no—
228C4—282no—
229C4#644214yes—
230C2C2×#644211yescharacteristic
231C2#9027311yescharacteristic
232C2#9027411yescharacteristic
233C2#9031611yescharacteristic
234C2#9033411yescharacteristic
235C2#9037311yescharacteristic
236C2#9042211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, prime radical, cosocleC23×C7C24
FittingC7×#391C2
hypercenter#391D7
hypocenterC7#2645
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
211. C7#2645semidirect

Maximal subgroups

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

subgroupindex
C7×#3912
#900022
#897572
#897452
#897042
#897002
C7⋊#4012
#894322
#894312
#894302
#26457