Finite Group Databasebeta

← #11726122

Subgroups — #11726122

Order 896.

200 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 57 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#90382C211yesmaximal, characteristic
2#89477C211yesmaximal, characteristic, complemented
3#89477C211yesmaximal, characteristic, complemented
4C7×#330C211yesmaximal, characteristic, complemented
5#89338C211yesmaximal, characteristic, complemented
6#89259C211yesmaximal, characteristic, complemented
7#89259C211yesmaximal, characteristic, complemented
8C7×#125C2211yescharacteristic
9#6461—21no—
10#6460C411yescharacteristic
11#6458C411yescharacteristic
12#6457—21no—
13#6456—21no—
14#6456C411yescharacteristic
15#6455—21no—
16#6450C411yescharacteristic
17#6438C2211yescharacteristic
18C42×D7C2211yescharacteristic
19C7×#97C2211yescharacteristic
20C7×#97C2211yescharacteristic
21(C2×D7)⋊C8—21no—
22(C2×D7)⋊C8—21no—
23Dic7⋊C8—21no—
24Dic7⋊C8—21no—
25#6377C2211yescharacteristic
26#6377C2211yescharacteristic
27#1535—71nomaximal, Sylow
28C2×D28—21no—
29C2×D28C2×C411yescharacteristic
30C2×C4×D7—21no—
31C2×C4×D7C2×C411yescharacteristic
32C2×C4×D7D411yescharacteristic
33C2×C4×D7D411yescharacteristic
34C2×Dic14C2×C411yescharacteristic
35C2×C7×C8—21no—
36C2×C7×C8—21no—
37C7×C4⋊C4—21no—
38C7×C4⋊C4—21no—
39C7×C4⋊C4C2×C411yescharacteristic
40C7×C4⋊C4C2×C411yescharacteristic
41C42×C7C2311yescharacteristic
42(C2×D7)⋊C4—21no—
43(C2×D7)⋊C4—21no—
44(C2×D7)⋊C4—21no—
45(C2×D7)⋊C4—21no—
46(C2×D7)⋊C4—21no—
47C4⋊Dic7—21no—
48Dic7⋊C4—21no—
49C4×Dic7—21no—
50C4×Dic7C2×C411yescharacteristic
51C4×Dic7D411yescharacteristic
52C4×Dic7D411yescharacteristic
53C2×C7⋊C8—21no—
54C2×C7⋊C8—21no—
55#529—71no—
56#432—71no—
57#432—71no—
58#330—71no—
59#330D711yescharacteristic, complemented
60#329—71no—
61#329—71no—
62C22×D7—21no—
63C22×D7C22⋊C411yescharacteristic
64C2×C4×C7—21no—
65C2×C4×C7—21no—
66C2×C4×C7C22×C411yescharacteristic
67C2×C4×C7C2×D411yescharacteristic
68C2×C4×C7C2×D411yescharacteristic
69C2×Dic7—21no—
70C2×Dic7C22⋊C411yescharacteristic
71C2×Dic7C22⋊C411yescharacteristic
72C2×Dic7C22⋊C411yescharacteristic
73D28—41no—
74D28—42no—
75C4×D7—41no—
76C4×D7—41no—
77C4×D7—41no—
78C4×D7—42no—
79Dic14—41no—
80C7×C8—41no—
81C7×C8—41no—
82C7⋊C8—41no—
83C7⋊C8—41no—
84#126—141no—
85#125C2×D711yescharacteristic
86#124—71no—
87#123—141no—
88C22⋊Q8—71no—
89#121—71no—
90C4×Q8—71no—
91C4×D4—71no—
92C4×D4—141no—
93#117—141no—
94C2×C42—71no—
95C4⋊C8—141no—
96C4⋊C8—141no—
97C22⋊C8—141no—
98C22⋊C8—141no—
99#97—71no—
100#97—71no—
101#97C2×D711yescharacteristic
102#97C2×D711yescharacteristic
103C22×C7C2×C22⋊C411yescharacteristic
104C2×D7—21no—
105C2×D7—21no—
106C2×D7—21no—
107C2×D7—41no—
108C2×D7—41no—
109C2×D7—41no—
110C4×C7—21no—
111C4×C7—21no—
112C4×C7—21no—
113C4×C7—41no—
114C4×C7—41no—
115Dic7—21no—
116Dic7—21no—
117Dic7—21no—
118Dic7—41no—
119C2×Q8—71no—
120C2×D4—71no—
121C2×D4—141no—
122C22×C4—71no—
123C22×C4—71no—
124C22×C4—71no—
125C22×C4—141no—
126C2×C8—21no—
127C2×C8—21no—
128C2×C8—141no—
129C2×C8—141no—
130C4⋊C4—21no—
131C4⋊C4—21no—
132C4⋊C4—141no—
133C4⋊C4—141no—
134C4⋊C4C4×D711yescharacteristic
135C4⋊C4C4×D711yescharacteristic
136C22⋊C4—141no—
137C22⋊C4—141no—
138C22⋊C4—141no—
139C22⋊C4—141no—
140C22⋊C4—141no—
141C42—71no—
142C42—71no—
143C42—71no—
144C42—141no—
145C42C22×D711yescharacteristic
146C2×C7#41311yescharacteristic
147C2×C7#42111yescharacteristic
148C2×C7#42111yescharacteristic
149D7—41no—
150D7—81no—
151C23—71no—
152C23—141no—
153Q8—281no—
154D4—281no—
155D4—282no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—71no—
159C2×C4—71no—
160C2×C4—71no—
161C2×C4—141no—
162C2×C4—281no—
163C2×C4—281no—
164C2×C4—281no—
165C2×C4—282no—
166C2×C4C2×C4×D711yescharacteristic
167C2×C4D4×D711yescharacteristic
168C2×C4D4×D711yescharacteristic
169C8—41no—
170C8—41no—
171C8—281no—
172C8—281no—
173C7#153511yescharacteristic, complemented, Sylow
174C22—141no—
175C22—141no—
176C22—141no—
177C22—281no—
178C22—281no—
179C22—281no—
180C22D7×C22⋊C411yescharacteristic
181C4—21no—
182C4—21no—
183C4—21no—
184C4—41no—
185C4—41no—
186C4—141no—
187C4—141no—
188C4—141no—
189C4—281no—
190C2—281no—
191C2—561no—
192C2#8952311yescharacteristic
193C2#8959311yescharacteristic
194C2#8959311yescharacteristic
subgroupG/H
centerC22D7×C22⋊C4
commutatorC2×C4×C7C22×C4
FrattiniC42C22×D7
FittingC7×#330C2
socleC22×C7C2×C22⋊C4
prime radical#6458C4
cosocleC42×C7C23
hypercenter#330D7
hypocenterC7#1535
LCSG ▷ C2×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #330

Complements

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

NH ≅ G/Ntype
2. #89477C2semidirect
3. #89477C2semidirect
4. C7×#330C2semidirect
5. #89338C2semidirect
6. #89259C2semidirect
7. #89259C2semidirect
59. #330D7semidirect
173. C7#1535semidirect

Maximal subgroups

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

subgroupindex
#903822
#894772
C7×#3302
#893382
#892592
#15357