Finite Group Databasebeta

← #11735752

Subgroups — #11735752

Order 896.

376 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 129 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×#400C211yesmaximal, characteristic
2#89751C211yesmaximal, characteristic
3#89750C212yesmaximal
4#89746C211yesmaximal, characteristic
5#89431C212yesmaximal
6#89429C212yesmaximal
7#89427C212yesmaximal
8#89418C212yesmaximal
9#89418C212yesmaximal
10C22×Dic14C2211yescharacteristic
11C22×Dic14C2211yescharacteristic
12C2×C7×C4⋊C4C2211yescharacteristic
13C2×C7×C4⋊C4C2211yescharacteristic
14C2×C4⋊Dic7C2212yes—
15C2×Dic7⋊C4C2212yes—
16C2×Dic7⋊C4C2212yes—
17C2×Dic7⋊C4C2212yes—
18C2×C4×Dic7C2212yes—
19C2×C4×Dic7C2211yescharacteristic
20C7×#95C2212yes—
21C7×#95C2212yes—
22C7×#95C2211yescharacteristic
23C7⋊#95C2212yes—
24C7⋊#95C2212yes—
25C7⋊#95C2212yes—
26C7⋊#95C2212yes—
27C7⋊#95C2212yes—
28C7⋊#95C2212yes—
29C7⋊#95C2212yes—
30C7⋊#95C2211yescharacteristic
31#2799—71nomaximal, Sylow
32C22×C4×C7C2312yes—
33C22×C4×C7C2311yescharacteristic
34C22×C4×C7C2311yescharacteristic
35C22×C4×C7C2312yes—
36C22×C4×C7C2311yescharacteristic
37C22×Dic7C2312yes—
38C22×Dic7C2312yes—
39C22×Dic7C2312yes—
40C22×Dic7C2312yes—
41C2×Dic14—24no—
42C2×Dic14—41no—
43C2×Dic14—41no—
44C2×Dic14—41no—
45C2×Dic14Q814yescomplemented
46C7×C4⋊C4—41no—
47C7×C4⋊C4—41no—
48C4⋊Dic7—42no—
49Dic7⋊C4—24no—
50Dic7⋊C4—24no—
51Dic7⋊C4—42no—
52C4×Dic7—42no—
53C4×Dic7—41no—
54#400—72no—
55#400D711yescharacteristic
56#398—72no—
57#395—71no—
58#391—71no—
59#391—72no—
60#391—72no—
61#387—72no—
62#387—72no—
63C23×C7C2411yescharacteristic
64C2×C4×C7—21no—
65C2×C4×C7—21no—
66C2×C4×C7—22no—
67C2×C4×C7—22no—
68C2×C4×C7—22no—
69C2×C4×C7—22no—
70C2×C4×C7—22no—
71C2×C4×C7—21no—
72C2×C4×C7—21no—
73C2×C4×C7—22no—
74C2×C4×C7—24no—
75C2×C4×C7C2×Q812yes—
76C2×Dic7—22no—
77C2×Dic7—22no—
78C2×Dic7—22no—
79C2×Dic7—22no—
80C2×Dic7—22no—
81C2×Dic7—22no—
82C2×Dic7—22no—
83C2×Dic7—22no—
84C2×Dic7—22no—
85C2×Dic7—22no—
86C2×Dic7C2×Q814yes—
87C2×Dic7#4114yes—
88Dic14—44no—
89Dic14—44no—
90C22×Q8—71no—
91C22×Q8—71no—
92C2×C4⋊C4—72no—
93C2×C4⋊C4—72no—
94C2×C4⋊C4—72no—
95C2×C4⋊C4—72no—
96C2×C4⋊C4C2×D711yescharacteristic
97C2×C4⋊C4C2×D711yescharacteristic
98C2×C42—71no—
99C2×C42—72no—
100#95—72no—
101#95—72no—
102#95—72no—
103#95—72no—
104#95—72no—
105#95—72no—
106#95—72no—
107#95—71no—
108#95C2×D712yes—
109#95C2×D712yes—
110#95C2×D711yescharacteristic
111C22×C7C22×Q811yescharacteristic
112C22×C7C2×#4112yes—
113C22×C7H(5,𝔽2)11yescharacteristic
114C22×C7H(5,𝔽2)11yescharacteristic
115C22×C72−1+412yes—
116C4×C7—22no—
117C4×C7—42no—
118C4×C7—41no—
119C4×C7—41no—
120C4×C7—42no—
121Dic7—24no—
122Dic7—24no—
123Dic7—42no—
124Dic7—42no—
125C2×Q8—74no—
126C2×Q8—144no—
127C2×Q8—281no—
128C2×Q8—281no—
129C2×Q8—281no—
130C22×C4—72no—
131C22×C4—72no—
132C22×C4—72no—
133C22×C4—72no—
134C22×C4C22×D712yes—
135C22×C4C22×D711yescharacteristic
136C22×C4C22×D711yescharacteristic
137C22×C4C22×D712yes—
138C22×C4C22×D711yescharacteristic
139C4⋊C4—41no—
140C4⋊C4—41no—
141C4⋊C4—144no—
142C4⋊C4—144no—
143C4⋊C4—282no—
144C4⋊C4—282no—
145C42—281no—
146C42—282no—
147C2×C7#54212yes—
148C2×C7#54311yescharacteristic
149C2×C7#55111yescharacteristic
150C2×C7#55712yes—
151C2×C7#56311yescharacteristic
152C23C23×D711yescharacteristic
153Q8—284no—
154Q8—284no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—22no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—24no—
166C2×C4—74no—
167C2×C4—74no—
168C2×C4—142no—
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×C4Q8×D712yes—
179C7#279911yescharacteristic, complemented, Sylow
180C22#654811yescharacteristic
181C22C2×Q8×D711yescharacteristic
182C22D7×#4112yes—
183C22#655311yescharacteristic
184C22#655412yes—
185C4—22no—
186C4—42no—
187C4—41no—
188C4—41no—
189C4—42no—
190C4—144no—
191C4—144no—
192C4—282no—
193C4—282no—
194C2#9032511yescharacteristic
195C2#9035011yescharacteristic
196C2#9036311yescharacteristic
197C2#9037612yes—
198C2#9040112yes—
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, prime radical, cosocleC23×C7C24
FittingC7×#400C2
hypercenter#400D7
hypocenterC7#2799
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
45. C2×Dic14Q8semidirect
179. C7#2799semidirect

Maximal subgroups

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

subgroupindex
C7×#4002
#897512
#897502
#897462
#894312
#894292
#894272
#894182
#27997