Finite Group Databasebeta

← #13272479

Subgroups — #13272479

Order 1792.

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

normal subgroups only

This group has 404 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
1#11739924C211yesmaximal, characteristic
2C4×C7×#101C211yesmaximal, characteristic
3#11729391C211yesmaximal, characteristic
4#11725505C214yesmaximal
5C2×C4×C7×Q8C2211yescharacteristic
6C2×#6432C411yescharacteristic
7C2×C7×#101C2212yes—
8C4×C7×M16—21no—
9#90002C2212yes—
10C2×#6410C2212yes—
11#89758—21no—
12#89758—21no—
13#89749C412yes—
14C4×C4⋊Dic7C411yescharacteristic
15#89701—21no—
16#89701—21no—
17#89696—21no—
18C7×#343—21no—
19#89325—21no—
20#13301—71nomaximal, Sylow
21C22×C7×Q8C2311yescharacteristic
22C22×Dic14—21no—
23C2×C7×M16—22no—
24C4×C7×Q8—41no—
25C4×C7×Q8—41no—
26C2×C7×C4⋊C4C2×C411yescharacteristic
27C2×C7×C4⋊C4D411yescharacteristic
28C2×C7×C4⋊C4D411yescharacteristic
29C2×C42×C7C2×C411yescharacteristic
30C2×C42×C7D411yescharacteristic
31C2×C42×C7D411yescharacteristic
32C2×C4⋊Dic7—21no—
33C2×C4⋊Dic7—21no—
34C2×C4⋊Dic7C2×C412yes—
35C2×C4×Dic7C2×C412yes—
36C2×#1136—22no—
37#6432—41no—
38#6432—41no—
39C7×#101—22no—
40C7×#101Q814yescomplemented
41C7×#97—41no—
42C4×C7×C8—41no—
43#6410—24no—
44C7⋊#95—22no—
45C7⋊#95—22no—
46C7⋊#95—21no—
47C7⋊#95—22no—
48C7⋊#95—22no—
49C4⋊(C7⋊C8)—41no—
50C4⋊(C7⋊C8)—41no—
51#2568—71no—
52#1801—71no—
53C4×#101D711yescharacteristic
54#1394—74no—
55C2×C7×Q8—41no—
56C2×C7×Q8—41no—
57C2×C7×Q8C2×Q812yes—
58C2×C7×Q8#4112yes—
59C22×C4×C7C22⋊C411yescharacteristic
60C22×C4×C7C22⋊C411yescharacteristic
61C22×C4×C7C22⋊C411yescharacteristic
62C22×C4×C7C22⋊C411yescharacteristic
63C22×C4×C7C22×C411yescharacteristic
64C22×C4×C7C2×D411yescharacteristic
65C22×C4×C7C2×D411yescharacteristic
66C22×Dic7—22no—
67C22×Dic7—22no—
68C2×Dic14—44no—
69C2×Dic14—41no—
70C2×Dic14—41no—
71C7×M16—24no—
72C7×M16—42no—
73C2×C7×C8—42no—
74C7×C4⋊C4—41no—
75C7×C4⋊C4—41no—
76C7×C4⋊C4—41no—
77C42×C7—21no—
78C42×C7—21no—
79C42×C7—21no—
80C42×C7—21no—
81C42×C7—21no—
82C42×C7—21no—
83C4⋊Dic7—42no—
84C4⋊Dic7—42no—
85C4⋊Dic7—42no—
86C4×Dic7—42no—
87#1136—42no—
88#1136—42no—
89C2×C7⋊C8—42no—
90C2×C4⋊Q8—71no—
91C2×C4×Q8C2×D711yescharacteristic
92#423—141no—
93C2×#101—72no—
94C2×#101C2×D712yes—
95C4×M16—21no—
96#400—141no—
97#400—141no—
98#391—72no—
99#391—141no—
100#391—141no—
101#382—72no—
102C4×C4⋊C4—71no—
103#343—21no—
104#343—141no—
105C23×C7C2×C22⋊C411yescharacteristic
106C7×Q8—42no—
107C7×Q8—42no—
108C2×C4×C7—21no—
109C2×C4×C7—21no—
110C2×C4×C7—21no—
111C2×C4×C7—21no—
112C2×C4×C7—21no—
113C2×C4×C7—21no—
114C2×C4×C7—22no—
115C2×C4×C7—21no—
116C2×C4×C7—21no—
117C2×C4×C7—21no—
118C2×C4×C7—41no—
119C2×C4×C7—41no—
120C2×C4×C7—41no—
121C2×C4×C7#9911yescharacteristic
122C2×C4×C7#9911yescharacteristic
123C2×C4×C7C4×Q812yes—
124C2×C4×C7C22⋊Q812yes—
125C2×C4×C7#12411yescharacteristic
126C2×C4×C7C4⋊Q811yescharacteristic
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×Q8C22×D711yescharacteristic
138C2×M16—22no—
139C2×M16—142no—
140C4⋊Q8—281no—
141C4⋊Q8—281no—
142C4×Q8—41no—
143C4×Q8—41no—
144C2×C4⋊C4—72no—
145C2×C4⋊C4—141no—
146C2×C4⋊C4—141no—
147C2×C4⋊C4C4×D711yescharacteristic
148C2×C4⋊C4D2811yescharacteristic
149C2×C4⋊C4C22⋊D711yescharacteristic
150C2×C42—72no—
151C2×C42C4×D711yescharacteristic
152C2×C42D2811yescharacteristic
153C2×C42C22⋊D711yescharacteristic
154C4⋊C8—281no—
155C4⋊C8—281no—
156#101—22no—
157#101—144no—
158#101Dic1414yescomplemented
159#97—41no—
160C4×C8—41no—
161#95—141no—
162#95—142no—
163#95—142no—
164#95—142no—
165#95—142no—
166C22×C7—21no—
167C22×C7—21no—
168C22×C7#39111yescharacteristic
169C22×C7C2×#9911yescharacteristic
170C22×C7#41011yescharacteristic
171C4×C7—21no—
172C4×C7—21no—
173C4×C7—22no—
174C4×C7—22no—
175C4×C7—21no—
176C4×C7—41no—
177C4×C7—41no—
178C4×C7—41no—
179C4×C7#35512yes—
180Dic7—82no—
181Dic7—82no—
182C2×Q8—41no—
183C2×Q8—41no—
184C2×Q8—284no—
185C2×Q8—281no—
186C2×Q8—281no—
187C2×Q8C2×Dic1412yes—
188C2×Q8#115712yes—
189C22×C4—142no—
190C22×C4—142no—
191C22×C4(C2×D7)⋊C411yescharacteristic
192C22×C4(C2×D7)⋊C411yescharacteristic
193C22×C4(C2×D7)⋊C411yescharacteristic
194C22×C4(C2×D7)⋊C411yescharacteristic
195C22×C4C2×C4×D711yescharacteristic
196C22×C4C2×D2811yescharacteristic
197C22×C4C2×C22⋊D711yescharacteristic
198M16—24no—
199M16—42no—
200M16—282no—
201M16—282no—
202C2×C8—42no—
203C2×C8—282no—
204C4⋊C4—41no—
205C4⋊C4—41no—
206C4⋊C4—41no—
207C4⋊C4—282no—
208C4⋊C4—282no—
209C4⋊C4—282no—
210C42—21no—
211C42—21no—
212C42—21no—
213C42—21no—
214C42—21no—
215C42—21no—
216C42—282no—
217C2×C7—21no—
218C2×C7—21no—
219C2×C7#199611yescharacteristic
220C2×C7C2×#35511yescharacteristic
221C2×C7#213511yescharacteristic
222C23C2×(C2×D7)⋊C411yescharacteristic
223Q8—42no—
224Q8—42no—
225Q8—562no—
226Q8—562no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—282no—
241C2×C4—282no—
242C2×C4—284no—
243C2×C4—282no—
244C2×C4—282no—
245C2×C4#638011yescharacteristic
246C2×C4#638011yescharacteristic
247C2×C4C4×Dic1412yes—
248C2×C4#643211yescharacteristic
249C2×C4#643811yescharacteristic
250C2×C4#648712yes—
251C8—44no—
252C8—562no—
253C7#1330111yescharacteristic, complemented, Sylow
254C22—21no—
255C22—21no—
256C22#8970111yescharacteristic
257C22C2×#638011yescharacteristic
258C22#8977811yescharacteristic
259C4—21no—
260C4—22no—
261C4—21no—
262C4—22no—
263C4—21no—
264C4—41no—
265C4—41no—
266C4—41no—
267C4—562no—
268C4—562no—
269C4#8927212yes—
270C2—21no—
271C2—21no—
272C2#1172721111yescharacteristic
273C2C2×#8927211yescharacteristic
274C2#1172790411yescharacteristic
subgroupG/H
centerC22C2×#6380
commutatorC22×C4×C7C22×C4
FrattiniC22×Q8C22×D7
FittingC4×C7×#101C2
socleC22×C7C2×#99
prime radicalC23×C7C2×C22⋊C4
cosocleC22×C7×Q8C23
hypercenterC4×#101D7
hypocenterC7#13301
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ C4×#101

Complements

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

NH ≅ G/Ntype
40. C7×#101Q8semidirect
158. #101Dic14semidirect
253. C7#13301semidirect

Maximal subgroups

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

subgroupindex
#117399242
C4×C7×#1012
#117293912
#117255052
#133017