Finite Group Databasebeta

← #11735317

Subgroups — #11735317

Order 896.

456 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 149 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×#6451C212yesmaximal
2C7×#398C211yesmaximal, characteristic
3#89755C211yesmaximal, characteristic
4#89751C212yesmaximal
5#89746C211yesmaximal, characteristic
6#89744C212yesmaximal
7#89428C212yesmaximal
8#89427C212yesmaximal
9#89424C212yesmaximal
10C22×Dic14C2212yes—
11C22×Dic14C2211yescharacteristic
12C2×C7×C4⋊C4C2212yes—
13C2×C7×C4⋊C4C2211yescharacteristic
14C2×C7×C4⋊C4C2211yescharacteristic
15C2×C4⋊Dic7C2212yes—
16C2×C4⋊Dic7C2211yescharacteristic
17C2×Dic7⋊C4C2212yes—
18C2×Dic7⋊C4C2212yes—
19C2×Dic7⋊C4C2212yes—
20C2×Dic7⋊C4C2212yes—
21C2×C4×Dic7C2212yes—
22C2×C4×Dic7C2211yescharacteristic
23#6451—28no—
24C7×#95C2212yes—
25C7×#95C2211yescharacteristic
26C7⋊#95C2211yescharacteristic
27C7⋊#95C2212yes—
28C7⋊#95C2212yes—
29C7⋊#95C2212yes—
30C7⋊#95C2212yes—
31C7⋊#95C2212yes—
32#2734—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—41no—
44C2×Dic14—41no—
45C2×Dic14—42no—
46C2×Dic14—42no—
47C2×Dic14D414yes—
48C7×C4⋊C4—24no—
49C7×C4⋊C4—41no—
50C7×C4⋊C4—41no—
51C4⋊Dic7—24no—
52C4⋊Dic7—41no—
53Dic7⋊C4—24no—
54Dic7⋊C4—24no—
55Dic7⋊C4—42no—
56Dic7⋊C4Q818yescomplemented
57C4×Dic7—24no—
58C4×Dic7—41no—
59C2×C4⋊Q8—72no—
60#398—71no—
61#398D711yescharacteristic
62#395—72no—
63#395—72no—
64#391—71no—
65#389—72no—
66#387—72no—
67#387—72no—
68C23×C7C2411yescharacteristic
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—21no—
76C2×C4×C7—21no—
77C2×C4×C7—22no—
78C2×C4×C7—22no—
79C2×C4×C7—22no—
80C2×C4×C7C2×D412yes—
81C2×C4×C7C2×Q814yes—
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×Dic7C2×D414yes—
92C2×Dic7C2×Q814yes—
93C2×Dic7C2×Q814yes—
94Dic14—44no—
95Dic14—48no—
96C22×Q8—71no—
97C22×Q8—72no—
98C4⋊Q8—148no—
99C2×C4⋊C4—72no—
100C2×C4⋊C4—72no—
101C2×C4⋊C4—72no—
102C2×C4⋊C4—72no—
103C2×C4⋊C4—72no—
104C2×C4⋊C4—71no—
105C2×C4⋊C4C2×D712yes—
106C2×C4⋊C4C2×D711yescharacteristic
107C2×C4⋊C4C2×D711yescharacteristic
108C2×C42—72no—
109C2×C42—71no—
110#95—71no—
111#95—72no—
112#95—72no—
113#95—72no—
114#95—72no—
115#95—72no—
116#95C2×D711yescharacteristic
117#95C2×D712yes—
118C22×C7C22×D411yescharacteristic
119C22×C7C22×Q812yes—
120C22×C7H(5,𝔽2)11yescharacteristic
121C22×C7H(5,𝔽2)11yescharacteristic
122C22×C72−1+412yes—
123C4×C7—22no—
124C4×C7—24no—
125C4×C7—42no—
126C4×C7—41no—
127C4×C7—41no—
128Dic7—24no—
129Dic7—24no—
130Dic7—24no—
131Dic7—42no—
132C2×Q8—74no—
133C2×Q8—144no—
134C2×Q8—281no—
135C2×Q8—281no—
136C2×Q8—282no—
137C2×Q8—282no—
138C22×C4—72no—
139C22×C4—72no—
140C22×C4—72no—
141C22×C4—72no—
142C22×C4C22×D711yescharacteristic
143C22×C4C22×D711yescharacteristic
144C22×C4C22×D712yes—
145C22×C4C22×D711yescharacteristic
146C22×C4C22×D712yes—
147C4⋊C4—24no—
148C4⋊C4—41no—
149C4⋊C4—41no—
150C4⋊C4—78no—
151C4⋊C4—144no—
152C4⋊C4—144no—
153C4⋊C4—144no—
154C4⋊C4—282no—
155C4⋊C4—281no—
156C42—144no—
157C42—281no—
158C2×C7#53411yescharacteristic
159C2×C7#54412yes—
160C2×C7D4×Q812yes—
161C2×C7Q8211yescharacteristic
162C2×C7#56311yescharacteristic
163C23C23×D711yescharacteristic
164Q8—284no—
165Q8—288no—
166C2×C4—22no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—74no—
178C2×C4—74no—
179C2×C4—74no—
180C2×C4—142no—
181C2×C4—142no—
182C2×C4—142no—
183C2×C4—142no—
184C2×C4—142no—
185C2×C4—142no—
186C2×C4—142no—
187C2×C4—142no—
188C2×C4—142no—
189C2×C4D4×D712yes—
190C2×C4Q8×D714yes—
191C7#273411yescharacteristic, complemented, Sylow
192C22C2×D4×D711yescharacteristic
193C22#654811yescharacteristic
194C22C2×Q8×D712yes—
195C22#655311yescharacteristic
196C22#655412yes—
197C4—22no—
198C4—24no—
199C4—42no—
200C4—41no—
201C4—41no—
202C4—144no—
203C4—144no—
204C4—144no—
205C4—282no—
206C2#9029611yescharacteristic
207C2#9032312yes—
208C2#9034211yescharacteristic
209C2#9038012yes—
210C2#9041311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, prime radical, cosocleC23×C7C24
FittingC7×#398C2
hypercenter#398D7
hypocenterC7#2734
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
56. Dic7⋊C4Q8semidirect
191. C7#2734semidirect

Maximal subgroups

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

subgroupindex
C2×#64512
C7×#3982
#897552
#897512
#897462
#897442
#894282
#894272
#894242
#27347