Finite Group Databasebeta

← #11738193

Subgroups — #11738193

Order 896.

360 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 113 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×#467C211yesmaximal, characteristic, complemented
3#89957C212yesmaximal, complemented
4#89887C211yesmaximal, characteristic
5C2×Dic7⋊C8C211yesmaximal, characteristic, complemented
6#89820C212yesmaximal, complemented
7#89663C212yesmaximal, complemented
8#89591C212yesmaximal, complemented
9#89568C212yesmaximal, complemented
10C2×C7×Q16C2211yescharacteristic
11C22×C7×C8C2211yescharacteristic
12C7×C22⋊Q8C2212yes—
13#6510C2212yes—
14Q8×Dic7C2212yes—
15C2×#1143C2212yes—
16#6493C2212yes—
17C2×C4×Dic7C2211yescharacteristic
18C22×C7⋊C8C2211yescharacteristic
19C2×D56C2211yescharacteristic
20#6461—22no—
21#6456C2212yes—
22#6455—22no—
23#6453C2212yes—
24#6449—22no—
25(C22⋊D7)⋊C4—22no—
26C7×#107C2211yescharacteristic
27C7×#103C2212yes—
28#6409C2212yes—
29#6395C2212yes—
30#6392C2211yescharacteristic
31Dic7⋊C8C2212yes—
32Dic7⋊C8C2212yes—
33#6383C2212yes—
34#6382C2212yes—
35#3250—71nomaximal, Sylow
36C2×C7×Q8C2312yes—
37C22×C4×C7C2311yescharacteristic
38C2×C22⋊D7—22no—
39C22×Dic7D411yescharacteristic
40C22×Dic7D411yescharacteristic
41C2×D28C2312yes—
42C2×C4×D7—22no—
43C7×Q16—41no—
44C2×C7×C8—41no—
45C2×C7×C8C2311yescharacteristic
46C2×C7×C8C2311yescharacteristic
47C7×C4⋊C4—22no—
48C7×C4⋊C4C2312yes—
49C7×C22⋊C4—22no—
50#1143—42no—
51(C2×D7)⋊C4—22no—
52(C2×D7)⋊C4—22no—
53(C2×D7)⋊C4—22no—
54(C2×D7)⋊C4—22no—
55C4⋊Dic7—22no—
56C4⋊Dic7—22no—
57C4⋊Dic7C2312yes—
58Dic7⋊C4—22no—
59C4×Dic7—22no—
60C4×Dic7D412yes—
61C4×Dic7C2312yes—
62C2×C7⋊C8—41no—
63C2×C7⋊C8C2312yes—
64D56—41no—
65#529—72no—
66#479—72no—
67#478—72no—
68#467D711yescharacteristic, complemented
69#466—71no—
70#465—72no—
71#464—72no—
72#463—72no—
73C2×C4⋊C8—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—22no—
87C2×Dic7—41no—
88C2×Dic7C2×D412yes—
89C2×Dic7C2×D412yes—
90D28—22no—
91D28—22no—
92C4×D7—44no—
93C7×C8—21no—
94C7×C8—21no—
95C7⋊C8—22no—
96C2×Q16C2×D711yescharacteristic
97C2×SD16—72no—
98C2×D8—71no—
99C22×C8—71no—
100C22×C8C2×D711yescharacteristic
101#126—142no—
102#125—72no—
103#124—72no—
104#123—142no—
105C22⋊Q8C2×D712yes—
106#121—72no—
107C4×Q8—72no—
108C4×D4—72no—
109C4×D4—142no—
110#117—142no—
111C2×C42—71no—
112#107—71no—
113#107C2×D711yescharacteristic
114#106—72no—
115C4⋊C8—72no—
116C4⋊C8—72no—
117#103—72no—
118#103C2×D712yes—
119#102—72no—
120#102—72no—
121C22×C7—21no—
122C22×C7—21no—
123C22×C7#13512yes—
124C22×C7C22×D411yescharacteristic
125C2×D7—42no—
126C2×D7—42no—
127C2×D7—42no—
128C4×C7—21no—
129C4×C7—42no—
130C4×C7—42no—
131C4×C7H(5,𝔽2)11yescharacteristic
132C4×C72−1+411yescharacteristic
133Dic7—22no—
134Dic7—22no—
135Dic7—42no—
136C2×Q8C22×D712yes—
137C2×D4—72no—
138C2×D4—142no—
139C22×C4—71no—
140C22×C4—71no—
141C22×C4—142no—
142C22×C4C22×D711yescharacteristic
143Q16—41no—
144SD16—282no—
145D8—281no—
146C2×C8—41no—
147C2×C8—72no—
148C2×C8—281no—
149C2×C8C22×D711yescharacteristic
150C2×C8C22×D711yescharacteristic
151C4⋊C4—22no—
152C4⋊C4—72no—
153C4⋊C4—142no—
154C4⋊C4—142no—
155C4⋊C4—142no—
156C4⋊C4C22×D712yes—
157C22⋊C4—22no—
158C22⋊C4—142no—
159C22⋊C4—142no—
160C22⋊C4—142no—
161C22⋊C4—142no—
162C42—72no—
163C42—72no—
164C42—142no—
165C2×C7—22no—
166C2×C7#53711yescharacteristic
167C2×C7C2×#13511yescharacteristic
168C2×C7#57511yescharacteristic
169D7—82no—
170C23—142no—
171C23D4×D711yescharacteristic
172Q8—22no—
173Q8—22no—
174D4—142no—
175D4—142no—
176D4—284no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—72no—
182C2×C4—72no—
183C2×C4—142no—
184C2×C4—142no—
185C2×C4—284no—
186C2×C4—281no—
187C2×C4D4×D711yescharacteristic
188C2×C4C23×D711yescharacteristic
189C8—21no—
190C8—21no—
191C8—142no—
192C7#325011yescharacteristic, complemented, Sylow
193C22—21no—
194C22—21no—
195C22—282no—
196C22—282no—
197C22—282no—
198C22#648212yes—
199C22C2×D4×D711yescharacteristic
200C4—21no—
201C4—42no—
202C4—42no—
203C4—142no—
204C4—142no—
205C4—282no—
206C4#654811yescharacteristic
207C4#655411yescharacteristic
208C2—22no—
209C2—562no—
210C2#9028811yescharacteristic
211C2C2×#648211yescharacteristic
212C2#9045911yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#467C2
socleC22×C7C22×D4
prime radical#89887C2
hypercenter#467D7
hypocenterC7#3250
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

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

NH ≅ G/Ntype
1. #90315C2semidirect
2. C7×#467C2semidirect
3. #89957C2semidirect
5. C2×Dic7⋊C8C2semidirect
6. #89820C2semidirect
7. #89663C2semidirect
8. #89591C2semidirect
9. #89568C2semidirect
68. #467D7semidirect
192. C7#3250semidirect

Maximal subgroups

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

subgroupindex
#903152
C7×#4672
#899572
#898872
C2×Dic7⋊C82
#898202
#896632
#895912
#895682
#32507