Finite Group Databasebeta

← #11734680

Subgroups — #11734680

Order 896.

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

normal subgroups only

This group has 384 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×#395C211yesmaximal, characteristic
2#90001C212yesmaximal
3#89756C212yesmaximal
4#89755C211yesmaximal, characteristic
5#89748C211yesmaximal, characteristic
6#89745C212yesmaximal
7#89426C212yesmaximal
8#89425C212yesmaximal
9#89419C212yesmaximal
10C22×C7×Q8C2211yescharacteristic
11C2×C7×C4⋊C4C2212yes—
12C2×C7×C4⋊C4C2211yescharacteristic
13C2×C4⋊Dic7C2211yescharacteristic
14C2×C4⋊Dic7C2211yescharacteristic
15C2×C4⋊Dic7C2212yes—
16C2×Dic7⋊C4C2212yes—
17C2×Dic7⋊C4C2212yes—
18C2×Dic7⋊C4C2212yes—
19C2×Dic7⋊C4C2212yes—
20C2×Dic7⋊C4C2212yes—
21C2×C4×Dic7C2211yescharacteristic
22C2×C4×Dic7C2212yes—
23C7×#95C2212yes—
24C7×#95C2211yescharacteristic
25C7⋊#95C2212yes—
26C7⋊#95C2212yes—
27C7⋊#95C2212yes—
28C7⋊#95C2212yes—
29C7⋊#95C2212yes—
30C7⋊#95C2211yescharacteristic
31#2792—71nomaximal, Sylow
32C2×C7×Q8—41no—
33C2×C7×Q8—42no—
34C22×C4×C7C2312yes—
35C22×C4×C7C2311yescharacteristic
36C22×C4×C7C2311yescharacteristic
37C22×C4×C7C2312yes—
38C22×C4×C7C2311yescharacteristic
39C22×Dic7C2312yes—
40C22×Dic7C2312yes—
41C22×Dic7C2312yes—
42C22×Dic7C2312yes—
43C7×C4⋊C4—42no—
44C7×C4⋊C4—41no—
45C4⋊Dic7—41no—
46C4⋊Dic7—41no—
47C4⋊Dic7Q818yescomplemented
48Dic7⋊C4—24no—
49Dic7⋊C4—42no—
50Dic7⋊C4—42no—
51Dic7⋊C4—42no—
52Dic7⋊C4—42no—
53C4×Dic7—22no—
54C4×Dic7—42no—
55#398—71no—
56#398—72no—
57#395—72no—
58#395D711yescharacteristic
59#389—72no—
60#389—72no—
61#387—72no—
62#387—72no—
63#384—71no—
64C23×C7C2411yescharacteristic
65C7×Q8—44no—
66C2×C4×C7—22no—
67C2×C4×C7—21no—
68C2×C4×C7—21no—
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×C7C2×Q814yes—
78C2×C4×C7#4112yes—
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×Dic7C2×Q814yes—
90C2×Dic7C2×Q814yes—
91C22×Q8C2×D711yescharacteristic
92C2×C4⋊C4—72no—
93C2×C4⋊C4—72no—
94C2×C4⋊C4—72no—
95C2×C4⋊C4—72no—
96C2×C4⋊C4—72no—
97C2×C4⋊C4—72no—
98C2×C4⋊C4—71no—
99C2×C4⋊C4—71no—
100C2×C4⋊C4C2×D711yescharacteristic
101C2×C4⋊C4C2×D712yes—
102C2×C42—71no—
103C2×C42—72no—
104#95—72no—
105#95—72no—
106#95—72no—
107#95—72no—
108#95—72no—
109#95—71no—
110#95C2×D711yescharacteristic
111#95C2×D712yes—
112C22×C7C22×Q812yes—
113C22×C7C2×#4111yescharacteristic
114C22×C7H(5,𝔽2)11yescharacteristic
115C22×C7H(5,𝔽2)12yes—
116C22×C72−1+411yescharacteristic
117C4×C7—22no—
118C4×C7—24no—
119C4×C7—41no—
120C4×C7—42no—
121C4×C7—41no—
122Dic7—24no—
123Dic7—24no—
124Dic7—42no—
125Dic7—42no—
126C2×Q8—41no—
127C2×Q8—42no—
128C22×C4—72no—
129C22×C4—72no—
130C22×C4—72no—
131C22×C4—72no—
132C22×C4C22×D712yes—
133C22×C4C22×D711yescharacteristic
134C22×C4C22×D711yescharacteristic
135C22×C4C22×D712yes—
136C22×C4C22×D711yescharacteristic
137C4⋊C4—42no—
138C4⋊C4—41no—
139C4⋊C4—78no—
140C4⋊C4—144no—
141C4⋊C4—282no—
142C4⋊C4—282no—
143C4⋊C4—282no—
144C4⋊C4—282no—
145C4⋊C4—281no—
146C4⋊C4—281no—
147C42—142no—
148C42—282no—
149C2×C7#53911yescharacteristic
150C2×C7#54412yes—
151C2×C7#55412yes—
152C2×C7Q8211yescharacteristic
153C2×C7#56011yescharacteristic
154C23C23×D711yescharacteristic
155Q8—44no—
156C2×C4—22no—
157C2×C4—21no—
158C2×C4—22no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—22no—
162C2×C4—21no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—74no—
168C2×C4—74no—
169C2×C4—142no—
170C2×C4—142no—
171C2×C4—142no—
172C2×C4—142no—
173C2×C4—142no—
174C2×C4—142no—
175C2×C4—142no—
176C2×C4—142no—
177C2×C4—142no—
178C2×C4—142no—
179C2×C4Q8×D714yes—
180C2×C4#116112yes—
181C7#279211yescharacteristic, complemented, Sylow
182C22#654811yescharacteristic
183C22#654812yes—
184C22C2×Q8×D712yes—
185C22C2×#116111yescharacteristic
186C22#655411yescharacteristic
187C4—24no—
188C4—22no—
189C4—41no—
190C4—42no—
191C4—41no—
192C4—144no—
193C4—144no—
194C4—282no—
195C4—282no—
196C2#9028611yescharacteristic
197C2#9031412yes—
198C2#9033611yescharacteristic
199C2#9041311yescharacteristic
200C2#9042112yes—
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, prime radical, cosocleC23×C7C24
FittingC7×#395C2
hypercenter#395D7
hypocenterC7#2792
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
47. C4⋊Dic7Q8semidirect
181. C7#2792semidirect

Maximal subgroups

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

subgroupindex
C7×#3952
#900012
#897562
#897552
#897482
#897452
#894262
#894252
#894192
#27927