Finite Group Databasebeta

← #11735241

Subgroups — #11735241

Order 896.

484 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 161 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×#6450C212yesmaximal
2C7×#398C211yesmaximal, characteristic
3#89756C212yesmaximal
4#89755C211yesmaximal, characteristic
5#89748C211yesmaximal, characteristic
6#89744C212yesmaximal
7#89419C212yesmaximal
8#89418C212yesmaximal
9#89418C212yesmaximal
10C22×Dic14C2212yes—
11C2×C7×C4⋊C4C2212yes—
12C2×C7×C4⋊C4C2211yescharacteristic
13C2×C7×C4⋊C4C2211yescharacteristic
14C2×C4⋊Dic7C2211yescharacteristic
15C2×C4⋊Dic7C2211yescharacteristic
16C2×Dic7⋊C4C2212yes—
17C2×Dic7⋊C4C2212yes—
18C2×Dic7⋊C4C2212yes—
19C2×Dic7⋊C4C2212yes—
20C2×Dic7⋊C4C2212yes—
21C2×C4×Dic7C2212yes—
22C2×C4×Dic7C2212yes—
23C2×C4×Dic7C2211yescharacteristic
24#6450—28no—
25C7×#95C2211yescharacteristic
26C7×#95C2212yes—
27C7⋊#95C2211yescharacteristic
28C7⋊#95C2212yes—
29C7⋊#95C2212yes—
30C7⋊#95C2212yes—
31C7⋊#95C2212yes—
32#2582—71nomaximal, Sylow
33C22×C4×C7C2311yescharacteristic
34C22×C4×C7C2311yescharacteristic
35C22×C4×C7C2312yes—
36C22×C4×C7C2311yescharacteristic
37C22×C4×C7C2312yes—
38C22×Dic7C2312yes—
39C22×Dic7C2312yes—
40C22×Dic7C2312yes—
41C22×Dic7C2312yes—
42C2×Dic14—24no—
43C2×Dic14—24no—
44C2×Dic14Q818yescomplemented
45C7×C4⋊C4—24no—
46C7×C4⋊C4—41no—
47C7×C4⋊C4—41no—
48C4⋊Dic7—41no—
49C4⋊Dic7—41no—
50Dic7⋊C4—24no—
51Dic7⋊C4—24no—
52Dic7⋊C4—24no—
53Dic7⋊C4—24no—
54Dic7⋊C4—42no—
55C4×Dic7—24no—
56C4×Dic7—24no—
57C4×Dic7D414yes—
58C2×C4×Q8—72no—
59#398—71no—
60#398—72no—
61#398D711yescharacteristic
62#391—72no—
63#391—72no—
64#389—72no—
65#389—72no—
66#384—71no—
67C23×C7C2411yescharacteristic
68C2×C4×C7—22no—
69C2×C4×C7—22no—
70C2×C4×C7—21no—
71C2×C4×C7—21no—
72C2×C4×C7—22no—
73C2×C4×C7—22no—
74C2×C4×C7—22no—
75C2×C4×C7—22no—
76C2×C4×C7—21no—
77C2×C4×C7—21no—
78C2×C4×C7—22no—
79C2×C4×C7C2×D412yes—
80C2×C4×C7C2×Q814yes—
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×D414yes—
90C2×Dic7C2×Q814yes—
91C2×Dic7C2×Q814yes—
92C2×Dic7#4114yes—
93Dic14—216no—
94C22×Q8—72no—
95C4×Q8—148no—
96C2×C4⋊C4—72no—
97C2×C4⋊C4—72no—
98C2×C4⋊C4—72no—
99C2×C4⋊C4—72no—
100C2×C4⋊C4—72no—
101C2×C4⋊C4—71no—
102C2×C4⋊C4—71no—
103C2×C4⋊C4C2×D712yes—
104C2×C4⋊C4C2×D711yescharacteristic
105C2×C4⋊C4C2×D711yescharacteristic
106C2×C42—71no—
107C2×C42—72no—
108C2×C42—72no—
109#95—72no—
110#95—72no—
111#95—72no—
112#95—71no—
113#95—72no—
114#95C2×D712yes—
115#95C2×D711yescharacteristic
116C22×C7C22×D411yescharacteristic
117C22×C7C22×Q812yes—
118C22×C7C2×#4112yes—
119C22×C7H(5,𝔽2)11yescharacteristic
120C22×C72−1+411yescharacteristic
121C4×C7—24no—
122C4×C7—22no—
123C4×C7—42no—
124C4×C7—41no—
125C4×C7—41no—
126Dic7—24no—
127Dic7—42no—
128Dic7—42no—
129Dic7C22⋊Q818yes—
130C2×Q8—78no—
131C2×Q8—144no—
132C2×Q8—144no—
133C22×C4—72no—
134C22×C4—72no—
135C22×C4—72no—
136C22×C4—72no—
137C22×C4C22×D711yescharacteristic
138C22×C4C22×D711yescharacteristic
139C22×C4C22×D712yes—
140C22×C4C22×D711yescharacteristic
141C22×C4C22×D712yes—
142C4⋊C4—24no—
143C4⋊C4—41no—
144C4⋊C4—41no—
145C4⋊C4—144no—
146C4⋊C4—144no—
147C4⋊C4—144no—
148C4⋊C4—144no—
149C4⋊C4—282no—
150C4⋊C4—281no—
151C4⋊C4—281no—
152C42—74no—
153C42—144no—
154C42—144no—
155C2×C7C2×C22⋊Q812yes—
156C2×C7#53711yescharacteristic
157C2×C7#55712yes—
158C2×C7Q8211yescharacteristic
159C2×C7#55911yescharacteristic
160C23C23×D711yescharacteristic
161Q8—1416no—
162C2×C4—22no—
163C2×C4—21no—
164C2×C4—22no—
165C2×C4—21no—
166C2×C4—22no—
167C2×C4—21no—
168C2×C4—22no—
169C2×C4—21no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—74no—
174C2×C4—74no—
175C2×C4—74no—
176C2×C4—74no—
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×C4D4×D712yes—
186C2×C4Q8×D714yes—
187C7#258211yescharacteristic, complemented, Sylow
188C22C2×D4×D711yescharacteristic
189C22#654811yescharacteristic
190C22C2×Q8×D712yes—
191C22D7×#4112yes—
192C22#655411yescharacteristic
193C4—22no—
194C4—24no—
195C4—42no—
196C4—41no—
197C4—41no—
198C4—78no—
199C4—144no—
200C4—282no—
201C4—282no—
202C2#9028811yescharacteristic
203C2D7×C22⋊Q812yes—
204C2#9035511yescharacteristic
205C2#9037612yes—
206C2#9041311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, prime radical, cosocleC23×C7C24
FittingC7×#398C2
hypercenter#398D7
hypocenterC7#2582
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
44. C2×Dic14Q8semidirect
187. C7#2582semidirect

Maximal subgroups

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

subgroupindex
C2×#64502
C7×#3982
#897562
#897552
#897482
#897442
#894192
#894182
#25827