Finite Group Databasebeta

← #13251500

Subgroups — #13251500

Order 1792.

432 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 107 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
1C7×#2983C211yesmaximal, characteristic
2#11739924C211yesmaximal, characteristic
3#11729379C211yesmaximal, characteristic
4#11728577C212yesmaximal
5#11725556C212yesmaximal
6C2×C4×C7×Q8C2211yescharacteristic
7C2×#6432C2211yescharacteristic
8C7×#446—24no—
9C7×#432—21no—
10C7×#432C2212yes—
11C7×#431—21no—
12#90002C412yes—
13#89758—21no—
14#89758—21no—
15#89749C412yes—
16C4×C4⋊Dic7C2211yescharacteristic
17#89701—21no—
18#89701—21no—
19#89698C2212yes—
20#89323—21no—
21#89323—21no—
22#89277—24no—
23#9456—71nomaximal, Sylow
24C22×C7×Q8C2×C411yescharacteristic
25C22×Dic14—21no—
26C4×C7×Q8—22no—
27C4×C7×Q8—22no—
28C2×C7×C4⋊C4C2×C411yescharacteristic
29C2×C7×C4⋊C4D411yescharacteristic
30C2×C7×C4⋊C4D411yescharacteristic
31C2×C42×C7D411yescharacteristic
32C2×C42×C7D411yescharacteristic
33C2×C42×C7C2311yescharacteristic
34C2×C4⋊Dic7—21no—
35C2×C4⋊Dic7—21no—
36C2×C4⋊Dic7C2×C412yes—
37C2×C4×Dic7C2×C412yes—
38#6432—22no—
39#6432—22no—
40C7×C4⋊C8—22no—
41C7×C4⋊C8—22no—
42C7×C4⋊C8D414yes—
43C7×C22⋊C8—22no—
44C7×#97—22no—
45C7×#97—22no—
46C4×C7×C8—22no—
47C7⋊#95—22no—
48C7⋊#95—22no—
49C7⋊#95—22no—
50C7⋊#95—22no—
51C7⋊#95—21no—
52C22⋊(C7⋊C8)—22no—
53C4⋊(C7⋊C8)—22no—
54#6377—22no—
55#2983D711yescharacteristic
56#2568—71no—
57#1556—72no—
58#1467—71no—
59#1285—72no—
60C2×C7×Q8—22no—
61C2×C7×Q8—22no—
62C2×C7×Q8—41no—
63C22×C4×C7C22⋊C411yescharacteristic
64C22×C4×C7C22⋊C411yescharacteristic
65C22×C4×C7C22⋊C411yescharacteristic
66C22×C4×C7C22⋊C411yescharacteristic
67C22×C4×C7C22×C411yescharacteristic
68C22×C4×C7C2×D411yescharacteristic
69C22×C4×C7C2×D411yescharacteristic
70C22×Dic7—22no—
71C22×Dic7—22no—
72C2×Dic14—44no—
73C2×Dic14—41no—
74C2×Dic14—41no—
75C2×C7×C8—24no—
76C7×C4⋊C4—22no—
77C7×C4⋊C4—22no—
78C7×C4⋊C4—22no—
79C42×C7—21no—
80C42×C7—21no—
81C42×C7—21no—
82C42×C7—21no—
83C42×C7C2×D412yes—
84C42×C7#4112yes—
85C4⋊Dic7—24no—
86C4⋊Dic7—42no—
87C4⋊Dic7—42no—
88C4×Dic7—42no—
89C2×C7⋊C8—42no—
90C2×C4⋊Q8—71no—
91C2×C4×Q8C2×D711yescharacteristic
92#446—24no—
93#432—21no—
94#432—72no—
95#432C2×D712yes—
96#431—21no—
97#400—141no—
98#400—141no—
99#391—72no—
100#391—141no—
101#391—141no—
102#382—72no—
103C4×C4⋊C4—71no—
104#332—144no—
105#324—141no—
106#324—141no—
107C23×C7C2×C22⋊C411yescharacteristic
108C7×Q8—44no—
109C2×C4×C7—21no—
110C2×C4×C7—21no—
111C2×C4×C7—21no—
112C2×C4×C7—21no—
113C2×C4×C7—22no—
114C2×C4×C7—21no—
115C2×C4×C7—21no—
116C2×C4×C7—21no—
117C2×C4×C7—21no—
118C2×C4×C7—21no—
119C2×C4×C7—41no—
120C2×C4×C7—41no—
121C2×C4×C7—41no—
122C2×C4×C7#10112yes—
123C2×C4×C7C4×D412yes—
124C2×C4×C7#12011yescharacteristic
125C2×C4×C7#12112yes—
126C2×C4×C7#12311yescharacteristic
127C2×Dic7—42no—
128C2×Dic7—44no—
129C2×Dic7—42no—
130C2×Dic7—42no—
131C2×Dic7—42no—
132Dic14—82no—
133Dic14—82no—
134C7×C8—44no—
135C7⋊C8—82no—
136C22×Q8—141no—
137C22×Q8C4×D711yescharacteristic
138C4⋊Q8—142no—
139C4⋊Q8—142no—
140C4×Q8—22no—
141C4×Q8—22no—
142C2×C4⋊C4—72no—
143C2×C4⋊C4—141no—
144C2×C4⋊C4—141no—
145C2×C4⋊C4C4×D711yescharacteristic
146C2×C4⋊C4D2811yescharacteristic
147C2×C4⋊C4D2811yescharacteristic
148C2×C42—72no—
149C2×C42C22⋊D711yescharacteristic
150C2×C42C22⋊D711yescharacteristic
151C2×C42C22×D711yescharacteristic
152C4⋊C8—22no—
153C4⋊C8—22no—
154C4⋊C8—142no—
155C4⋊C8C22⋊D714yes—
156C22⋊C8—22no—
157C22⋊C8—142no—
158#97—22no—
159#97—22no—
160#97—142no—
161C4×C8—22no—
162#95—142no—
163#95—142no—
164#95—142no—
165#95—142no—
166#95—141no—
167C22×C7—21no—
168C22×C7—21no—
169C22×C7#38611yescharacteristic
170C22×C7C2×#10111yescharacteristic
171C22×C7#41311yescharacteristic
172C4×C7—21no—
173C4×C7—22no—
174C4×C7—22no—
175C4×C7—22no—
176C4×C7—41no—
177C4×C7—41no—
178C4×C7—41no—
179C4×C7#45412yes—
180Dic7—82no—
181Dic7—82no—
182C2×Q8—22no—
183C2×Q8—22no—
184C2×Q8—41no—
185C2×Q8—284no—
186C2×Q8—281no—
187C2×Q8—281no—
188C22×C4—142no—
189C22×C4—142no—
190C22×C4(C2×D7)⋊C411yescharacteristic
191C22×C4(C2×D7)⋊C411yescharacteristic
192C22×C4(C2×D7)⋊C411yescharacteristic
193C22×C4(C2×D7)⋊C411yescharacteristic
194C22×C4C2×C4×D711yescharacteristic
195C22×C4C2×D2811yescharacteristic
196C22×C4C2×C22⋊D711yescharacteristic
197C2×C8—24no—
198C2×C8—282no—
199C4⋊C4—22no—
200C4⋊C4—22no—
201C4⋊C4—22no—
202C4⋊C4—144no—
203C4⋊C4—282no—
204C4⋊C4—282no—
205C42—21no—
206C42—21no—
207C42—21no—
208C42—21no—
209C42—282no—
210C42#115712yes—
211C42C2×C22⋊D712yes—
212C2×C7—21no—
213C2×C7—21no—
214C2×C7#188811yescharacteristic
215C2×C7#190111yescharacteristic
216C2×C7#190911yescharacteristic
217C23C2×(C2×D7)⋊C411yescharacteristic
218Q8—44no—
219Q8—562no—
220Q8—562no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—282no—
235C2×C4—282no—
236C2×C4—284no—
237C2×C4—282no—
238C2×C4—282no—
239C2×C4#639812yes—
240C2×C4C4×C22⋊D712yes—
241C2×C4#649211yescharacteristic
242C2×C4#649312yes—
243C2×C4#651611yescharacteristic
244C8—44no—
245C8—562no—
246C7#945611yescharacteristic, complemented, Sylow
247C22—21no—
248C22—21no—
249C22#8990411yescharacteristic
250C22C2×#639811yescharacteristic
251C22#8998511yescharacteristic
252C4—21no—
253C4—22no—
254C4—22no—
255C4—22no—
256C4—41no—
257C4—41no—
258C4—41no—
259C4—562no—
260C4—562no—
261C4#9001712yes—
262C2—21no—
263C2—21no—
264C2#1172941511yescharacteristic
265C2#1172975711yescharacteristic
266C2#1172977811yescharacteristic
subgroupG/H
centerC22#89985
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#2983C2
socleC22×C7#386
prime radicalC23×C7C2×C22⋊C4
cosocleC2×C42×C7C23
hypercenter#2983D7
hypocenterC7#9456
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2983

Complements

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

NH ≅ G/Ntype
246. C7#9456semidirect

Maximal subgroups

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

subgroupindex
C7×#29832
#117399242
#117293792
#117285772
#117255562
#94567