Finite Group Databasebeta

← #13272484

Subgroups — #13272484

Order 1792.

420 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 111 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×#2567C211yesmaximal, characteristic
2#11729391C211yesmaximal, characteristic
3C4×#6410C211yesmaximal, characteristic
4#11725450C214yesmaximal
5C2×C7×C4⋊Q8C411yescharacteristic
6C2×C4×C7×Q8C2211yescharacteristic
7C7×#395—21no—
8C7×#395—21no—
9C7×#391C2212yes—
10C7×#389—21no—
11C7×#389—21no—
12C7×#384C412yes—
13C4×C7×C4⋊C4C411yescharacteristic
14C2×#6410C2212yes—
15C2×#6410C2212yes—
16#89696—21no—
17C4×#1136—21no—
18#89325—21no—
19#89325—21no—
20#13302—71nomaximal, Sylow
21C22×C7×Q8—21no—
22C22×C7×Q8C2311yescharacteristic
23C7×C4⋊Q8—41no—
24C7×C4⋊Q8—41no—
25C4×C7×Q8—41no—
26C4×C7×Q8—41no—
27C2×C7×C4⋊C4—21no—
28C2×C7×C4⋊C4—21no—
29C2×C7×C4⋊C4—22no—
30C2×C7×C4⋊C4—22no—
31C2×C7×C4⋊C4C2×C412yes—
32C2×C7×C4⋊C4C2×C411yescharacteristic
33C2×C7×C4⋊C4D411yescharacteristic
34C2×C7×C4⋊C4D411yescharacteristic
35C2×C42×C7C2×C411yescharacteristic
36C2×C42×C7C2×C412yes—
37C2×C42×C7D411yescharacteristic
38C2×C42×C7D411yescharacteristic
39C2×#1136—22no—
40C2×#1136—22no—
41C7×#95—21no—
42C7×#95—22no—
43C7×#95—22no—
44#6410—22no—
45#6410—24no—
46#6410Q814yescomplemented
47C4⋊(C7⋊C8)—41no—
48C4⋊(C7⋊C8)—41no—
49#6377—41no—
50C4×C7⋊C8—41no—
51#2567D711yescharacteristic
52#1801—71no—
53C4×#101—71no—
54#1394—74no—
55C2×C7×Q8—41no—
56C2×C7×Q8—41no—
57C2×C7×Q8—42no—
58C2×C7×Q8—82no—
59C2×C7×Q8C2×Q812yes—
60C2×C7×Q8#4112yes—
61C22×C4×C7—22no—
62C22×C4×C7—22no—
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
70C7×C4⋊C4—24no—
71C7×C4⋊C4—41no—
72C7×C4⋊C4—41no—
73C7×C4⋊C4—41no—
74C7×C4⋊C4—42no—
75C7×C4⋊C4—41no—
76C7×C4⋊C4—41no—
77C7×C4⋊C4—82no—
78C7×C4⋊C4—82no—
79C42×C7—21no—
80C42×C7—21no—
81C42×C7—21no—
82C42×C7—21no—
83C42×C7—21no—
84C42×C7—21no—
85C42×C7—42no—
86#1136—24no—
87#1136—42no—
88#1136—42no—
89#1136—42no—
90C2×C7⋊C8—42no—
91C2×C7⋊C8—42no—
92C2×C4⋊Q8Dic711yescharacteristic
93C2×C4×Q8C2×D711yescharacteristic
94#423—141no—
95C2×#101—72no—
96C2×#101—72no—
97C4×M16—141no—
98#395—21no—
99#395—21no—
100#391C2×D712yes—
101#389—21no—
102#389—21no—
103#384Dic712yes—
104C4×C4⋊C4Dic711yescharacteristic
105#343—141no—
106#343—141no—
107C23×C7C2×C22⋊C411yescharacteristic
108C7×Q8—42no—
109C7×Q8—42no—
110C7×Q8—84no—
111C2×C4×C7—21no—
112C2×C4×C7—21no—
113C2×C4×C7—21no—
114C2×C4×C7—21no—
115C2×C4×C7—21no—
116C2×C4×C7—21no—
117C2×C4×C7—22no—
118C2×C4×C7—24no—
119C2×C4×C7—21no—
120C2×C4×C7—21no—
121C2×C4×C7—21no—
122C2×C4×C7—42no—
123C2×C4×C7—42no—
124C2×C4×C7—41no—
125C2×C4×C7—41no—
126C2×C4×C7—41no—
127C2×C4×C7—42no—
128C2×C4×C7—42no—
129C2×C4×C7—42no—
130C2×C4×C7#9911yescharacteristic
131C2×C4×C7#9911yescharacteristic
132C2×C4×C7C4×Q812yes—
133C2×C4×C7C22⋊Q812yes—
134C2×C4×C7#12411yescharacteristic
135C2×C4×C7C4⋊Q811yescharacteristic
136C7⋊C8—44no—
137C7⋊C8—82no—
138C22×Q8—21no—
139C22×Q8C22×D711yescharacteristic
140C2×M16—142no—
141C2×M16—142no—
142C4⋊Q8—41no—
143C4⋊Q8—41no—
144C4×Q8—41no—
145C4×Q8—41no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4C2×Dic712yes—
151C2×C4⋊C4C2×Dic711yescharacteristic
152C2×C4⋊C4C22⋊D711yescharacteristic
153C2×C4⋊C4C22⋊D711yescharacteristic
154C2×C42C2×Dic711yescharacteristic
155C2×C42C2×Dic712yes—
156C2×C42C22⋊D711yescharacteristic
157C2×C42C22⋊D711yescharacteristic
158C4⋊C8—281no—
159C4⋊C8—281no—
160#101—74no—
161#101—142no—
162#101—144no—
163#97—281no—
164C4×C8—281no—
165#95—21no—
166#95—22no—
167#95—22no—
168C22×C7—21no—
169C22×C7—21no—
170C22×C7#39111yescharacteristic
171C22×C7C2×#9911yescharacteristic
172C22×C7#41011yescharacteristic
173C4×C7—21no—
174C4×C7—22no—
175C4×C7—21no—
176C4×C7—22no—
177C4×C7—21no—
178C4×C7—44no—
179C4×C7—41no—
180C4×C7—41no—
181C4×C7—41no—
182C4×C7—82no—
183C4×C7#35612yes—
184C2×Q8—41no—
185C2×Q8—41no—
186C2×Q8—42no—
187C2×Q8—82no—
188C2×Q8Q8×D712yes—
189C2×Q8#116112yes—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4C22⋊Dic711yescharacteristic
193C22×C4C22⋊Dic711yescharacteristic
194C22×C4C22⋊Dic711yescharacteristic
195C22×C4C22⋊Dic711yescharacteristic
196C22×C4C22×Dic711yescharacteristic
197C22×C4C2×C22⋊D711yescharacteristic
198C22×C4C2×C22⋊D711yescharacteristic
199M16—144no—
200M16—282no—
201M16—282no—
202M16—282no—
203C2×C8—282no—
204C2×C8—282no—
205C4⋊C4—24no—
206C4⋊C4—41no—
207C4⋊C4—41no—
208C4⋊C4—41no—
209C4⋊C4—42no—
210C4⋊C4—41no—
211C4⋊C4—41no—
212C4⋊C4—82no—
213C4⋊C4—82no—
214C42—21no—
215C42—21no—
216C42—21no—
217C42—21no—
218C42—21no—
219C42—21no—
220C42—42no—
221C2×C7—21no—
222C2×C7—21no—
223C2×C7#199611yescharacteristic
224C2×C7C2×#35611yescharacteristic
225C2×C7#213411yescharacteristic
226C23C2×C22⋊Dic711yescharacteristic
227Q8—42no—
228Q8—42no—
229Q8—84no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—24no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4#640811yescharacteristic
250C2×C4#640811yescharacteristic
251C2×C4#650711yescharacteristic
252C2×C4Q8×Dic712yes—
253C2×C4#650912yes—
254C2×C4#651011yescharacteristic
255C8—284no—
256C8—562no—
257C7#1330211yescharacteristic, complemented, Sylow
258C22—21no—
259C22—21no—
260C22C2×#640811yescharacteristic
261C22#9000211yescharacteristic
262C22#9001211yescharacteristic
263C4—22no—
264C4—21no—
265C4—21no—
266C4—22no—
267C4—21no—
268C4—44no—
269C4—41no—
270C4—41no—
271C4—41no—
272C4—82no—
273C4#8934312yes—
274C2—21no—
275C2—21no—
276C2#1172945511yescharacteristic
277C2C2×#8934311yescharacteristic
278C2#1172961611yescharacteristic
subgroupG/H
centerC22C2×#6408
commutatorC22×C4×C7C22×C4
FrattiniC22×Q8C22×D7
FittingC7×#2567C2
socleC22×C7C2×#99
prime radicalC23×C7C2×C22⋊C4
cosocleC22×C7×Q8C23
hypercenter#2567D7
hypocenterC7#13302
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #2567

Complements

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

NH ≅ G/Ntype
46. #6410Q8semidirect
257. C7#13302semidirect

Maximal subgroups

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

subgroupindex
C7×#25672
#117293912
C4×#64102
#117254502
#133027