Finite Group Databasebeta

← #11738196

Subgroups — #11738196

Order 896.

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

normal subgroups only

This group has 346 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#90334C212yesmaximal, complemented
2C7×#467C211yesmaximal, characteristic, complemented
3#89957C212yesmaximal, complemented
4#89887C211yesmaximal, characteristic
5#89874C211yesmaximal, characteristic, complemented
6#89821C212yesmaximal, complemented
7#89662C212yesmaximal, complemented
8#89591C212yesmaximal, complemented
9#89566C212yesmaximal, complemented
10C2×C7×Q16C2211yescharacteristic
11C22×C7×C8C2211yescharacteristic
12C7×C22⋊Q8C2212yes—
13#6510C2212yes—
14Q8×Dic7C2212yes—
15C2×#1143C2212yes—
16#6493C2212yes—
17#6489C2211yescharacteristic
18C2×#1136C2211yescharacteristic
19C2×D56C2211yescharacteristic
20#6461—22no—
21#6460—22no—
22#6456C2212yes—
23#6451C2212yes—
24#6448—22no—
25#6446—22no—
26C7×#107C2211yescharacteristic
27C7×#103C2212yes—
28#6409C2212yes—
29#6395C2212yes—
30#6392C2211yescharacteristic
31Dic7⋊C8C2212yes—
32Dic7⋊C8C2212yes—
33#6383C2212yes—
34#6381C2212yes—
35#3262—71nomaximal, Sylow
36C2×C7×Q8C2312yes—
37C22×C4×C7C2311yescharacteristic
38C2×C22⋊D7—22no—
39C2×D28C2312yes—
40C2×C4×D7—22no—
41C2×Dic14—22no—
42C7×Q16—41no—
43C2×C7×C8—41no—
44C2×C7×C8C2311yescharacteristic
45C2×C7×C8C2311yescharacteristic
46C7×C4⋊C4—22no—
47C7×C4⋊C4C2312yes—
48C7×C22⋊C4—22no—
49C22⋊Dic7D412yes—
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⋊Dic7D411yescharacteristic
57C4⋊Dic7D411yescharacteristic
58C4⋊Dic7C2312yes—
59Dic7⋊C4—22no—
60C4×Dic7—22no—
61C4×Dic7C2312yes—
62#1136—41no—
63C2×C7⋊C8C2312yes—
64D56—41no—
65#542—72no—
66#477—72no—
67#474—72no—
68#468—72no—
69#467D711yescharacteristic, complemented
70#466—71no—
71#464—72no—
72#463—72no—
73#424—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—82no—
85C2×Dic7—22no—
86C2×Dic7C2×D412yes—
87C2×Dic7C2×D412yes—
88D28—22no—
89D28—22no—
90C4×D7—82no—
91Dic14—82no—
92C7×C8—21no—
93C7×C8—21no—
94C7⋊C8—22no—
95C2×Q16C2×D711yescharacteristic
96C2×SD16—72no—
97C2×D8—71no—
98C2×M16—71no—
99C22×C8C2×D711yescharacteristic
100C4⋊Q8—72no—
101#126—142no—
102#124—72no—
103#124—142no—
104#123—142no—
105C22⋊Q8—142no—
106C22⋊Q8C2×D712yes—
107#121—72no—
108C4×Q8—72no—
109C4×D4—72no—
110#117—71no—
111#107—71no—
112#107—72no—
113#107C2×D711yescharacteristic
114C4⋊C8—72no—
115C4⋊C8—72no—
116#103—72no—
117#103C2×D712yes—
118#102—72no—
119#102—72no—
120C22×C7—21no—
121C22×C7—21no—
122C22×C7—21no—
123C22×C7C22×D411yescharacteristic
124C2×D7—42no—
125C2×D7—42no—
126C2×D7—42no—
127C4×C7—21no—
128C4×C7—42no—
129C4×C7—42no—
130C4×C7H(5,𝔽2)11yescharacteristic
131C4×C72−1+411yescharacteristic
132Dic7—22no—
133Dic7—22no—
134Dic7—42no—
135C2×Q8—142no—
136C2×Q8C22×D712yes—
137C2×D4—72no—
138C2×D4—142no—
139C22×C4—142no—
140C22×C4C22×D711yescharacteristic
141Q16—41no—
142SD16—282no—
143D8—281no—
144M16—281no—
145C2×C8—41no—
146C2×C8—72no—
147C2×C8C22×D711yescharacteristic
148C2×C8C22×D711yescharacteristic
149C4⋊C4—22no—
150C4⋊C4—71no—
151C4⋊C4—71no—
152C4⋊C4—72no—
153C4⋊C4—142no—
154C4⋊C4—142no—
155C4⋊C4C22×D712yes—
156C22⋊C4—22no—
157C22⋊C4—72no—
158C22⋊C4—142no—
159C22⋊C4—142no—
160C22⋊C4—142no—
161C22⋊C4—142no—
162C42—72no—
163C42—142no—
164C2×C7—41no—
165C2×C7#53711yescharacteristic
166C2×C7#57611yescharacteristic
167C2×C7#57811yescharacteristic
168D7—82no—
169C23—142no—
170C23D4×D711yescharacteristic
171Q8—22no—
172Q8—22no—
173Q8—562no—
174D4—142no—
175D4—142no—
176D4—562no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—72no—
182C2×C4—72no—
183C2×C4—142no—
184C2×C4—562no—
185C2×C4D4×D711yescharacteristic
186C2×C4C23×D711yescharacteristic
187C8—21no—
188C8—21no—
189C8—142no—
190C7#326211yescharacteristic, complemented, Sylow
191C22—21no—
192C22—21no—
193C22—21no—
194C22—282no—
195C22—282no—
196C22—282no—
197C22C2×D4×D711yescharacteristic
198C4—21no—
199C4—42no—
200C4—42no—
201C4—142no—
202C4—142no—
203C4—282no—
204C4#654811yescharacteristic
205C4#655411yescharacteristic
206C2—41no—
207C2—562no—
208C2#9028811yescharacteristic
209C2#9045711yescharacteristic
210C2#9046011yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#467C2
socleC22×C7C22×D4
prime radical#89887C2
hypercenter#467D7
hypocenterC7#3262
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. #90334C2semidirect
2. C7×#467C2semidirect
3. #89957C2semidirect
5. #89874C2semidirect
6. #89821C2semidirect
7. #89662C2semidirect
8. #89591C2semidirect
9. #89566C2semidirect
69. #467D7semidirect
190. C7#3262semidirect

Maximal subgroups

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

subgroupindex
#903342
C7×#4672
#899572
#898872
#898742
#898212
#896622
#895912
#895662
#32627