Finite Group Databasebeta

← #11738341

Subgroups — #11738341

Order 896.

354 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 109 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
1#90315C212yesmaximal, complemented
2C7×#470C211yesmaximal, characteristic, complemented
3#89957C212yesmaximal, complemented
4#89907C211yesmaximal, characteristic
5#89890C211yesmaximal, characteristic, complemented
6#89821C212yesmaximal, complemented
7#89643C212yesmaximal, complemented
8#89591C212yesmaximal, complemented
9#89571C212yesmaximal, complemented
10C2×C7×Q16C2211yescharacteristic
11C2×C7×M16C2211yescharacteristic
12C7×C22⋊Q8C2212yes—
13#6510C2212yes—
14Q8×Dic7C2212yes—
15C2×#1143C2212yes—
16#6493C2212yes—
17C2×C4×Dic7C2211yescharacteristic
18C2×#1136C2211yescharacteristic
19C2×D56C2211yescharacteristic
20#6461—22no—
21#6456C2212yes—
22#6455—22no—
23#6453C2212yes—
24#6449—22no—
25(C22⋊D7)⋊C4—22no—
26C7×#106C2211yescharacteristic
27C7×#103C2212yes—
28#6409C2212yes—
29#6395C2212yes—
30#6391C2211yescharacteristic
31Dic7⋊C8C2212yes—
32Dic7⋊C8C2212yes—
33#6383C2212yes—
34#6381C2212yes—
35#3256—71nomaximal, Sylow
36C2×C7×Q8C2312yes—
37C22×C4×C7C2311yescharacteristic
38C2×C22⋊D7—22no—
39C22×Dic7D412yes—
40C2×D28C2312yes—
41C2×C4×D7—22no—
42C7×Q16—41no—
43C7×M16—41no—
44C2×C7×C8C2311yescharacteristic
45C2×C7×C8C2311yescharacteristic
46C7×C4⋊C4—22no—
47C7×C4⋊C4C2312yes—
48C7×C22⋊C4—22no—
49#1143—42no—
50(C2×D7)⋊C4—22no—
51(C2×D7)⋊C4—22no—
52(C2×D7)⋊C4—22no—
53(C2×D7)⋊C4—22no—
54C4⋊Dic7—22no—
55C4⋊Dic7—22no—
56C4⋊Dic7C2312yes—
57Dic7⋊C4—22no—
58C4×Dic7—22no—
59C4×Dic7D411yescharacteristic
60C4×Dic7D411yescharacteristic
61C4×Dic7C2312yes—
62#1136—41no—
63C2×C7⋊C8C2312yes—
64D56—41no—
65#529—72no—
66#479—72no—
67#478—72no—
68#470D711yescharacteristic, complemented
69#469—71no—
70#468—72no—
71#464—72no—
72#463—72no—
73#423—71no—
74C23×C7C2×D411yescharacteristic
75C22×D7—22no—
76C7×Q8—22no—
77C7×Q8—22no—
78C2×C4×C7—21no—
79C2×C4×C7—21no—
80C2×C4×C7—22no—
81C2×C4×C7—22no—
82C2×C4×C7C2×D411yescharacteristic
83C2×C4×C7C2411yescharacteristic
84C22⋊D7—44no—
85C2×Dic7—22no—
86C2×Dic7—42no—
87C2×Dic7C2×D412yes—
88C2×Dic7C2×D412yes—
89D28—22no—
90D28—22no—
91C4×D7—44no—
92C7×C8—21no—
93C7×C8—21no—
94C7⋊C8—22no—
95C2×Q16C2×D711yescharacteristic
96C2×SD16—72no—
97C2×D8—71no—
98C2×M16—71no—
99C2×M16C2×D711yescharacteristic
100#126—142no—
101#125—72no—
102#124—72no—
103#123—142no—
104C22⋊Q8C2×D712yes—
105#121—72no—
106C4×Q8—72no—
107C4×D4—72no—
108C4×D4—142no—
109#117—142no—
110C2×C42—71no—
111#107—72no—
112#106—71no—
113#106C2×D711yescharacteristic
114C4⋊C8—72no—
115C4⋊C8—72no—
116#103—72no—
117#103C2×D712yes—
118#102—72no—
119#102—72no—
120C22×C7—21no—
121C22×C7—22no—
122C22×C7C22×D411yescharacteristic
123C2×D7—42no—
124C2×D7—42no—
125C2×D7—42no—
126C4×C7—21no—
127C4×C7—42no—
128C4×C7—42no—
129C4×C7H(5,𝔽2)11yescharacteristic
130C4×C72−1+411yescharacteristic
131Dic7—22no—
132Dic7—22no—
133Dic7—42no—
134C2×Q8C22×D712yes—
135C2×D4—72no—
136C2×D4—142no—
137C22×C4—72no—
138C22×C4—142no—
139C22×C4C22×D711yescharacteristic
140Q16—41no—
141SD16—282no—
142D8—281no—
143M16—41no—
144M16—281no—
145C2×C8—72no—
146C2×C8C22×D711yescharacteristic
147C2×C8C22×D711yescharacteristic
148C4⋊C4—22no—
149C4⋊C4—72no—
150C4⋊C4—142no—
151C4⋊C4—142no—
152C4⋊C4—142no—
153C4⋊C4C22×D712yes—
154C22⋊C4—22no—
155C22⋊C4—142no—
156C22⋊C4—142no—
157C22⋊C4—142no—
158C22⋊C4—142no—
159C42—71no—
160C42—71no—
161C42—72no—
162C42—142no—
163C2×C7—41no—
164C2×C7#53711yescharacteristic
165C2×C7#57512yes—
166D7—82no—
167C23—142no—
168C23D4×D711yescharacteristic
169Q8—22no—
170Q8—22no—
171D4—142no—
172D4—142no—
173D4—284no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—72no—
179C2×C4—72no—
180C2×C4—142no—
181C2×C4—284no—
182C2×C4—282no—
183C2×C4D4×D711yescharacteristic
184C2×C4C23×D711yescharacteristic
185C8—21no—
186C8—21no—
187C8—142no—
188C7#325611yescharacteristic, complemented, Sylow
189C22—22no—
190C22—21no—
191C22—282no—
192C22—282no—
193C22—282no—
194C22C2×D4×D711yescharacteristic
195C4—21no—
196C4—42no—
197C4—42no—
198C4—142no—
199C4—142no—
200C4—282no—
201C4#654811yescharacteristic
202C4#655411yescharacteristic
203C2—41no—
204C2—562no—
205C2#9028811yescharacteristic
206C2#9046812yes—
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#470C2
socleC22×C7C22×D4
prime radical#89907C2
hypercenter#470D7
hypocenterC7#3256
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
1. #90315C2semidirect
2. C7×#470C2semidirect
3. #89957C2semidirect
5. #89890C2semidirect
6. #89821C2semidirect
7. #89643C2semidirect
8. #89591C2semidirect
9. #89571C2semidirect
68. #470D7semidirect
188. C7#3256semidirect

Maximal subgroups

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

subgroupindex
#903152
C7×#4702
#899572
#899072
#898902
#898212
#896432
#895912
#895712
#32567