Finite Group Databasebeta

← #11733625

Subgroups — #11733625

Order 896.

368 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 141 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×#389C211yesmaximal, characteristic
2#89758C212yesmaximal
3#89758C212yesmaximal
4#89757C212yesmaximal
5#89753C212yesmaximal
6#89701C211yesmaximal, characteristic
7#89700C211yesmaximal, characteristic
8#89428C212yesmaximal
9#89427C212yesmaximal
10C22×Dic14C2211yescharacteristic
11C2×C7×C4⋊C4C2212yes—
12C2×C7×C4⋊C4C2211yescharacteristic
13C2×C7×C4⋊C4C2211yescharacteristic
14C2×C42×C7C2211yescharacteristic
15C2×C4⋊Dic7C2211yescharacteristic
16C2×C4⋊Dic7C2211yescharacteristic
17C2×C4⋊Dic7C2212yes—
18C2×C4⋊Dic7C2212yes—
19C2×C4⋊Dic7C2211yescharacteristic
20C2×Dic7⋊C4C2212yes—
21C2×C4×Dic7C2212yes—
22C7×#95C2212yes—
23C7⋊#95C2212yes—
24C7⋊#95C2212yes—
25C7⋊#95C2212yes—
26C7⋊#95C2212yes—
27C7⋊#95C2212yes—
28C7⋊#95C2212yes—
29C7⋊#95C2212yes—
30C7⋊#95C2212yes—
31#2805—71nomaximal, Sylow
32C22×C4×C7C2312yes—
33C22×C4×C7C2311yescharacteristic
34C22×C4×C7C2312yes—
35C22×C4×C7C2311yescharacteristic
36C22×C4×C7C2311yescharacteristic
37C22×Dic7C2312yes—
38C22×Dic7C2312yes—
39C22×Dic7C2312yes—
40C22×Dic7C2312yes—
41C2×Dic14—22no—
42C2×Dic14—41no—
43C2×Dic14—41no—
44C7×C4⋊C4—24no—
45C7×C4⋊C4—41no—
46C7×C4⋊C4Q814yescomplemented
47C42×C7—41no—
48C4⋊Dic7—22no—
49C4⋊Dic7—41no—
50C4⋊Dic7—42no—
51C4⋊Dic7—41no—
52C4⋊Dic7—42no—
53Dic7⋊C4—24no—
54C4×Dic7—42no—
55#400—72no—
56#400—72no—
57#395—72no—
58#391—71no—
59#389—72no—
60#389—71no—
61#389D711yescharacteristic
62#387—72no—
63#387—72no—
64C23×C7C2411yescharacteristic
65C2×C4×C7—22no—
66C2×C4×C7—22no—
67C2×C4×C7—21no—
68C2×C4×C7—21no—
69C2×C4×C7—21no—
70C2×C4×C7—21no—
71C2×C4×C7—21no—
72C2×C4×C7—21no—
73C2×C4×C7—22no—
74C2×C4×C7—22no—
75C2×C4×C7—22no—
76C2×C4×C7C2×Q812yes—
77C2×C4×C7C2×Q814yes—
78C2×C4×C7#4112yes—
79C2×C4×C7#4112yes—
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—22no—
92Dic14—44no—
93C22×Q8—71no—
94C2×C4⋊C4—72no—
95C2×C4⋊C4—71no—
96C2×C4⋊C4—71no—
97C2×C4⋊C4—72no—
98C2×C4⋊C4—71no—
99C2×C4⋊C4—72no—
100C2×C4⋊C4C2×D712yes—
101C2×C4⋊C4C2×D711yescharacteristic
102C2×C4⋊C4C2×D711yescharacteristic
103C2×C42—72no—
104C2×C42C2×D711yescharacteristic
105#95—72no—
106#95—72no—
107#95—72no—
108#95—72no—
109#95—72no—
110#95—72no—
111#95—72no—
112#95—72no—
113#95C2×D712yes—
114C22×C7C22×Q811yescharacteristic
115C22×C7C2×#4111yescharacteristic
116C22×C7C2×#4111yescharacteristic
117C22×C7H(5,𝔽2)11yescharacteristic
118C22×C7H(5,𝔽2)11yescharacteristic
119C22×C72−1+411yescharacteristic
120C22×C72−1+411yescharacteristic
121C4×C7—22no—
122C4×C7—22no—
123C4×C7—22no—
124C4×C7—24no—
125C4×C7—42no—
126Dic7—42no—
127Dic7—42no—
128Dic7—42no—
129Dic7—42no—
130C2×Q8—142no—
131C2×Q8—281no—
132C2×Q8—281no—
133C22×C4—72no—
134C22×C4—72no—
135C22×C4—72no—
136C22×C4—72no—
137C22×C4C22×D712yes—
138C22×C4C22×D711yescharacteristic
139C22×C4C22×D712yes—
140C22×C4C22×D711yescharacteristic
141C22×C4C22×D711yescharacteristic
142C4⋊C4—24no—
143C4⋊C4—41no—
144C4⋊C4—142no—
145C4⋊C4—144no—
146C4⋊C4—281no—
147C4⋊C4—282no—
148C4⋊C4—281no—
149C4⋊C4—282no—
150C4⋊C4Dic1414yescomplemented
151C42—41no—
152C42—282no—
153C2×C7#53811yescharacteristic
154C2×C7#53911yescharacteristic
155C2×C7#54411yescharacteristic
156C2×C7#55411yescharacteristic
157C2×C7#55611yescharacteristic
158C2×C7#55711yescharacteristic
159C2×C7#56311yescharacteristic
160C23C23×D711yescharacteristic
161Q8—284no—
162C2×C4—22no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—22no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—142no—
174C2×C4—142no—
175C2×C4—142no—
176C2×C4—142no—
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×C4C2×Dic1412yes—
186C2×C4C2×Dic1414yes—
187C2×C4#115912yes—
188C2×C4#116112yes—
189C7#280511yescharacteristic, complemented, Sylow
190C22C22×Dic1411yescharacteristic
191C22C2×#115911yescharacteristic
192C22#654811yescharacteristic
193C22C2×#116111yescharacteristic
194C22#655111yescharacteristic
195C22#655311yescharacteristic
196C22#655411yescharacteristic
197C4—22no—
198C4—24no—
199C4—22no—
200C4—22no—
201C4—42no—
202C4—282no—
203C4—282no—
204C4—282no—
205C4—282no—
206C2#9019111yescharacteristic
207C2#9023411yescharacteristic
208C2#9026111yescharacteristic
209C2#9029311yescharacteristic
210C2#9033511yescharacteristic
211C2#9039011yescharacteristic
212C2#9042311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, prime radical, cosocleC23×C7C24
FittingC7×#389C2
hypercenter#389D7
hypocenterC7#2805
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #389

Complements

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

NH ≅ G/Ntype
46. C7×C4⋊C4Q8semidirect
150. C4⋊C4Dic14semidirect
189. C7#2805semidirect

Maximal subgroups

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

subgroupindex
C7×#3892
#897582
#897572
#897532
#897012
#897002
#894282
#894272
#28057