Finite Group Databasebeta

← #13274148

Subgroups — #13274148

Order 1792.

326 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 59 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#11739161C211yesmaximal, characteristic
2#11726905C212yesmaximal
3C7×#1412C211yesmaximal, characteristic, complemented
4#11725590C211yesmaximal, characteristic, complemented
5#11725345C212yesmaximal, complemented
6#90407C2211yescharacteristic
7D7×#125C411yescharacteristic, complemented
8C7×#490C2211yescharacteristic
9#89843C2211yescharacteristic
10#89841C411yescharacteristic, complemented
11#89543—22no—
12#89540—22no—
13#89512—22no—
14#89479C411yescharacteristic, complemented
15#89475C411yescharacteristic, complemented
16C7×#353C2212yes—
17#89340C2212yes—
18#89268—22no—
19#13596—71nomaximal, Sylow
20C2×#1159D412yes—
21C7×#127C2311yescharacteristic
22C7×#125C2×C411yescharacteristic
23#6501—41no—
24#6499D412yes—
25D4×Dic7—41no—
26C2×#1131—21no—
27D7×C4⋊C4—21no—
28D7×C4⋊C4—41no—
29#6453—21no—
30#6452—41no—
31#6440—42no—
32C42×D7C2×C411yescharacteristic
33#6433C2×C411yescharacteristic
34#6432C2×C411yescharacteristic
35C7×#102—22no—
36C7×#99—22no—
37C7×#97C2×C411yescharacteristic
38#6408—22no—
39#6406—22no—
40#6390—22no—
41C8×Dic7—21no—
42#6384—22no—
43#6380—42no—
44#6377C2×C411yescharacteristic
45#3142—71no—
46#2126—72no—
47#1412—71no—
48#1412—72no—
49#1412D711yescharacteristic, complemented
50C2×C7×D4—41no—
51C2×C7×D4C2×D412yes—
52C2×C22⋊D7—22no—
53C22×Dic7—22no—
54#1159—42no—
55#1159—42no—
56C2×C4×D7—21no—
57C2×C4×D7—41no—
58C2×C4×D7C22⋊C411yescharacteristic
59C2×Dic14C22⋊C412yes—
60C2×C7×C8—21no—
61C7×C4⋊C4—21no—
62C7×C4⋊C4—41no—
63C7×C22⋊C4—42no—
64C42×C7C22×C411yescharacteristic
65C22⋊Dic7—42no—
66C22⋊Dic7—42no—
67C4⋊Dic7—21no—
68C4⋊Dic7—41no—
69C4⋊Dic7—41no—
70Dic7⋊C4—41no—
71Dic7⋊C4—41no—
72Dic7⋊C4—41no—
73Dic7⋊C4—82no—
74C4×Dic7—21no—
75C4×Dic7—82no—
76C4×Dic7C22⋊C411yescharacteristic
77C2×C7⋊C8—21no—
78#1131—42no—
79#532—71no—
80C2×#125—71no—
81#491—71no—
82#491—71no—
83#490—71no—
84#490C2×D711yescharacteristic
85#488—142no—
86#417—142no—
87#410—142no—
88C4×M16—71no—
89#354—142no—
90#353—72no—
91#353C2×D712yes—
92C23×C7—22no—
93C22×D7#9911yescharacteristic
94C7×D4—42no—
95C7×D4—82no—
96C2×C4×C7—21no—
97C2×C4×C7—41no—
98C2×C4×C7—82no—
99C2×C4×C7C2×C22⋊C411yescharacteristic
100C22⋊D7—82no—
101C2×Dic7—21no—
102C2×Dic7—22no—
103C2×Dic7—42no—
104C2×Dic7—42no—
105C2×Dic7—41no—
106C2×Dic7—82no—
107C2×Dic7#9911yescharacteristic
108C4×D7—22no—
109C4×D7—81no—
110C4×D7—82no—
111Dic14—42no—
112C7×C8—41no—
113C7⋊C8—41no—
114C2×#41—72no—
115C2×M16—141no—
116C4⋊Q8—71no—
117#127C22×D711yescharacteristic
118#125—71no—
119#125—141no—
120#125—281no—
121#125C4×D711yescharacteristic
122#124—72no—
123#121—281no—
124C4×D4—281no—
125#117—282no—
126C2×C4⋊C4—141no—
127C2×C4⋊C4—281no—
128C2×C42—71no—
129#103—142no—
130#103—142no—
131#102—22no—
132#102—142no—
133#99—22no—
134#99—142no—
135#99—282no—
136#97—71no—
137#97C4×D711yescharacteristic
138C4×C8—141no—
139C22×C7—42no—
140C22×C7—82no—
141C22×C7C2×#9911yescharacteristic
142C2×D7—21no—
143C2×D7—41no—
144C4×C7—21no—
145C4×C7—41no—
146C4×C7—81no—
147C4×C7—162no—
148Dic7—41no—
149Dic7—42no—
150Dic7—81no—
151Dic7—162no—
152Dic7#35412yes—
153#41—282no—
154#41—282no—
155C2×Q8—72no—
156C2×D4—41no—
157C2×D4—142no—
158C2×D4D4×D712yes—
159C22×C4—71no—
160C22×C4—141no—
161C22×C4—142no—
162C22×C4—281no—
163M16—282no—
164C2×C8—21no—
165C2×C8—141no—
166C4⋊C4—21no—
167C4⋊C4—41no—
168C4⋊C4—141no—
169C4⋊C4—281no—
170C4⋊C4—281no—
171C4⋊C4—281no—
172C4⋊C4—281no—
173C4⋊C4—281no—
174C4⋊C4—562no—
175C22⋊C4—42no—
176C22⋊C4—282no—
177C22⋊C4—282no—
178C42—71no—
179C42—141no—
180C42—562no—
181C42C2×C4×D711yescharacteristic
182C2×C7—21no—
183C2×C7—82no—
184C2×C7C2×#35411yescharacteristic
185D7—41no—
186C23—22no—
187C23—71no—
188Q8—282no—
189D4—42no—
190D4—82no—
191D4—562no—
192C2×C4—21no—
193C2×C4—41no—
194C2×C4—71no—
195C2×C4—82no—
196C2×C4—142no—
197C2×C4—141no—
198C2×C4—142no—
199C2×C4—282no—
200C2×C4—282no—
201C2×C4—281no—
202C2×C4—561no—
203C2×C4—562no—
204C2×C4—562no—
205C2×C4D7×C22⋊C411yescharacteristic
206C8—41no—
207C8—281no—
208C7#1359611yescharacteristic, complemented, Sylow
209C22—42no—
210C22—82no—
211C22—141no—
212C22—281no—
213C22D7×#9911yescharacteristic
214C4—21no—
215C4—41no—
216C4—72no—
217C4—81no—
218C4—162no—
219C4—281no—
220C4—282no—
221C4—561no—
222C4—1122no—
223C2—21no—
224C2—82no—
225C2—281no—
226C2D7×#35411yescharacteristic
subgroupG/H
centerC2D7×#354
commutatorC42×C7C22×C4
Frattini#127C22×D7
FittingC7×#1412C2
socleC2×C7C2×#354
prime radical#90407C22
cosocleC7×#127C23
hypercenter#1412D7
hypocenterC7#13596
LCSG ▷ C42×C7 ▷ C2×C4×C7 ▷ C22×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ #127 ◁ #1412

Complements

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

NH ≅ G/Ntype
3. C7×#1412C2semidirect
4. #11725590C2semidirect
5. #11725345C2semidirect
7. D7×#125C4semidirect
10. #89841C4semidirect
14. #89479C4semidirect
15. #89475C4semidirect
49. #1412D7semidirect
208. C7#13596semidirect

Maximal subgroups

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

subgroupindex
#117391612
#117269052
C7×#14122
#117255902
#117253452
#135967