Finite Group Databasebeta

← #13241007

Subgroups — #13241007

Order 1792.

388 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 75 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
1C4×C7×#100C211yesmaximal, characteristic
2C4×#6408C211yesmaximal, characteristic
3#11729365C211yesmaximal, characteristic
4C2×C4×C7×D4C2211yescharacteristic
5C22×#1136C411yescharacteristic, complemented
6C2×C7×#100—21no—
7C4×C7×M16—21no—
8C2×#6408—21no—
9C4×C22⋊Dic7—21no—
10C2×C22⋊(C7⋊C8)—21no—
11C2×C22⋊(C7⋊C8)C411yescharacteristic, complemented
12C7×#343C412yescomplemented
13#89321C412yes—
14#89319—21no—
15#7282—71nomaximal, Sylow
16C22×C7×D4D411yescharacteristic
17C23×C4×C7C2×C411yescharacteristic
18C23×C4×C7D411yescharacteristic
19C2×C7×M16—22no—
20C4×C7×D4—41no—
21C4×C7×D4—41no—
22C2×C7×C4⋊C4C2×C411yescharacteristic
23C2×C7×C22⋊C4C2×C411yescharacteristic
24C2×C7×C22⋊C4Q811yescharacteristic
25C2×C42×C7D411yescharacteristic
26C2×C22⋊Dic7—22no—
27C2×C4×Dic7—41no—
28C2×#1136—42no—
29C2×#1136—41no—
30C22×C7⋊C8—21no—
31C22×C7⋊C8—21no—
32C7×#100—24no—
33C7×#97—41no—
34C4×C7×C8—41no—
35#6408—24no—
36C7⋊#95—41no—
37C22⋊(C7⋊C8)—42no—
38C22⋊(C7⋊C8)—41no—
39#1870—71no—
40C4×#100D711yescharacteristic
41C4×#99—71no—
42C24×C7C22⋊C411yescharacteristic
43C24×C7C4⋊C411yescharacteristic
44C2×C7×D4—22no—
45C2×C7×D4—22no—
46C2×C7×D4—41no—
47C22×C4×C7—21no—
48C22×C4×C7—21no—
49C22×C4×C7—21no—
50C22×C4×C7—21no—
51C22×C4×C7—41no—
52C22×C4×C7—41no—
53C22×C4×C7—41no—
54C22×C4×C7—41no—
55C22×C4×C7C4211yescharacteristic
56C22×C4×C7C22⋊C411yescharacteristic
57C22×C4×C7C22⋊C411yescharacteristic
58C22×C4×C7C4⋊C411yescharacteristic
59C22×C4×C7C4⋊C411yescharacteristic
60C22×Dic7—42no—
61C7×M16—24no—
62C7×M16—42no—
63C2×C7×C8—42no—
64C7×C4⋊C4—41no—
65C7×C22⋊C4—41no—
66C7×C22⋊C4—41no—
67C42×C7—21no—
68C42×C7—21no—
69C22⋊Dic7—24no—
70C22⋊Dic7—42no—
71C4×Dic7—81no—
72C4×Dic7—81no—
73#1136—42no—
74#1136—42no—
75C2×C7⋊C8—22no—
76C2×C7⋊C8—41no—
77C2×C7⋊C8—41no—
78C2×C7⋊C8—41no—
79C2×C7⋊C8—41no—
80C2×C7⋊C8—41no—
81C22×M16—71no—
82C2×C4×D4C2×D711yescharacteristic
83C2×#100—21no—
84C2×#99—141no—
85C2×C22⋊C8—71no—
86C2×C22⋊C8—141no—
87C4×M16—21no—
88C4×C22⋊C4—141no—
89#343Dic712yescomplemented
90#342—72no—
91#336—141no—
92C23×C7—21no—
93C23×C7—21no—
94C23×C7—21no—
95C23×C7—21no—
96C23×C7—21no—
97C23×C7—41no—
98C23×C7—41no—
99C23×C7—41no—
100C23×C7—41no—
101C23×C7#9511yescharacteristic
102C23×C7#10412yes—
103C7×D4—44no—
104C2×C4×C7—21no—
105C2×C4×C7—21no—
106C2×C4×C7—21no—
107C2×C4×C7—21no—
108C2×C4×C7—21no—
109C2×C4×C7—21no—
110C2×C4×C7—21no—
111C2×C4×C7—42no—
112C2×C4×C7—41no—
113C2×C4×C7—41no—
114C2×C4×C7—41no—
115C2×C4×C7—41no—
116C2×C4×C7—41no—
117C2×C4×C7—41no—
118C2×C4×C7—41no—
119C2×C4×C7—81no—
120C2×C4×C7#9911yescharacteristic
121C2×C4×C7#9911yescharacteristic
122C2×C4×C7#10412yes—
123C2×Dic7—44no—
124C2×Dic7—82no—
125C2×Dic7—82no—
126C7×C8—44no—
127C7⋊C8—42no—
128C7⋊C8—81no—
129C22×D4C22⋊D711yescharacteristic
130C23×C4C4×D711yescharacteristic
131C23×C4D2811yescharacteristic
132C2×M16—22no—
133C2×M16—282no—
134C2×M16—281no—
135C22×C8—141no—
136C22×C8—141no—
137C4×D4—41no—
138C4×D4—41no—
139C2×C4⋊C4C2×Dic711yescharacteristic
140C2×C22⋊C4—142no—
141C2×C22⋊C4Dic1411yescharacteristic
142C2×C22⋊C4C4×D711yescharacteristic
143C2×C42—281no—
144C2×C42C22⋊D711yescharacteristic
145#100—24no—
146#99—144no—
147C22⋊C8—282no—
148C22⋊C8—281no—
149#97—41no—
150C4×C8—41no—
151#95—281no—
152C22×C7—21no—
153C22×C7—21no—
154C22×C7—22no—
155C22×C7—42no—
156C22×C7—41no—
157C22×C7—41no—
158C22×C7—41no—
159C22×C7—41no—
160C22×C7—41no—
161C22×C7—81no—
162C22×C7—81no—
163C22×C7—81no—
164C22×C7#33911yescharacteristic
165C22×C7#33911yescharacteristic
166C22×C7#34211yescharacteristic
167C4×C7—22no—
168C4×C7—21no—
169C4×C7—41no—
170C4×C7—41no—
171C4×C7—41no—
172C4×C7AΣL(1,𝔽16)11yescharacteristic
173C4×C7#35211yescharacteristic
174Dic7—84no—
175C24Dic7⋊C411yescharacteristic
176C24(C2×D7)⋊C411yescharacteristic
177C2×D4—22no—
178C2×D4—22no—
179C2×D4—41no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—282no—
189C22×C4C4×Dic711yescharacteristic
190C22×C4Dic7⋊C411yescharacteristic
191C22×C4C4⋊Dic711yescharacteristic
192C22×C4(C2×D7)⋊C411yescharacteristic
193C22×C4C22⋊Dic711yescharacteristic
194M16—24no—
195M16—42no—
196M16—282no—
197M16—282no—
198C2×C8—42no—
199C2×C8—142no—
200C2×C8—281no—
201C2×C8—281no—
202C2×C8—281no—
203C2×C8—281no—
204C2×C8—281no—
205C4⋊C4—41no—
206C22⋊C4—41no—
207C22⋊C4—41no—
208C22⋊C4—144no—
209C22⋊C4—282no—
210C42—21no—
211C42—21no—
212C42—561no—
213C42—561no—
214C2×C7—21no—
215C2×C7—21no—
216C2×C7—42no—
217C2×C7—81no—
218C2×C7#129711yescharacteristic
219C2×C7#139011yescharacteristic
220C2×C7#139511yescharacteristic
221C23—21no—
222C23—21no—
223C23—21no—
224C23—21no—
225C23—21no—
226C23—41no—
227C23—41no—
228C23—41no—
229C23—41no—
230C23#637912yes—
231C23C7⋊#9511yescharacteristic
232D4—44no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—42no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—81no—
249C2×C4—284no—
250C2×C4—562no—
251C2×C4—562no—
252C2×C4#638011yescharacteristic
253C2×C4#638011yescharacteristic
254C2×C4#641112yes—
255C8—44no—
256C8—282no—
257C8—561no—
258C7#728211yescharacteristic, complemented, Sylow
259C22—21no—
260C22—21no—
261C22—22no—
262C22—42no—
263C22—41no—
264C22—41no—
265C22—41no—
266C22—41no—
267C22—41no—
268C22—81no—
269C22—81no—
270C22—81no—
271C22#8931811yescharacteristic
272C22#8932211yescharacteristic
273C22#8932711yescharacteristic
274C4—22no—
275C4—21no—
276C4—41no—
277C4—41no—
278C4—41no—
279C4—564no—
280C4#8927011yescharacteristic
281C4#8927111yescharacteristic
282C2—21no—
283C2—21no—
284C2—42no—
285C2—81no—
286C2#1172542411yescharacteristic
287C2#1172550411yescharacteristic
288C2#1172550711yescharacteristic
subgroupG/H
centerC2×C4#6380
commutatorC22×C4×C7C42
FrattiniC2×C4×D4C2×D7
FittingC4×C7×#100C2
socleC22×C7#342
prime radicalC22×C7×D4D4
cosocleC2×C4×C7×D4C22
hypercenterC4×#100D7
hypocenterC7#7282
LCSG ▷ C22×C4×C7 ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×D4 ◁ C4×#100

Complements

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

NH ≅ G/Ntype
5. C22×#1136C4semidirect
11. C2×C22⋊(C7⋊C8)C4semidirect
12. C7×#343C4semidirect
89. #343Dic7semidirect
258. C7#7282semidirect

Maximal subgroups

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

subgroupindex
C4×C7×#1002
C4×#64082
#117293652
#72827