Finite Group Databasebeta

← #13251648

Subgroups — #13251648

Order 1792.

542 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 127 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#11739944C211yesmaximal, characteristic
2#11739880C211yesmaximal, characteristic, complemented
3C7×#1801C211yesmaximal, characteristic, complemented
4#11728250C212yesmaximal
5#11725519C212yesmaximal, complemented
6C2×C4×C7×Q8C2211yescharacteristic
7C2×#6509—21no—
8C2×C4×D28C2211yescharacteristic
9C7×#423C2212yes—
10C2×C7×#101—21no—
11Q8×C7⋊C8—24no—
12#89763C412yes—
13#89759—21no—
14#89755C412yes—
15#89712—22no—
16#89708C2211yescharacteristic
17#89697—22no—
18#89697C2212yes—
19C7×#343—21no—
20#89278—24no—
21#9486—71nomaximal, Sylow
22C22×C7×Q8D411yescharacteristic
23C22×D28C2×C411yescharacteristic
24C22×C4×D7—21no—
25C2×C7×M16—22no—
26C4×C7×Q8—22no—
27C4×C7×Q8—22no—
28C2×C7×C4⋊C4C2×C412yes—
29C2×C7×C4⋊C4D411yescharacteristic
30C2×C42×C7D412yes—
31C2×C42×C7C2311yescharacteristic
32#6509—42no—
33#6509—42no—
34C2×(C2×D7)⋊C4—22no—
35C2×(C2×D7)⋊C4—22no—
36C2×(C2×D7)⋊C4—21no—
37C2×C4⋊Dic7—21no—
38C2×C4⋊Dic7C2×C411yescharacteristic
39C2×Dic7⋊C4—22no—
40C4×D28—22no—
41C4×D28—22no—
42C7×C4⋊C8—22no—
43C7×C4⋊C8—22no—
44C7×#101—42no—
45C7⋊#95—22no—
46C7⋊#95—21no—
47C7⋊#95C2×C412yes—
48C22⋊(C7⋊C8)—22no—
49C4⋊(C7⋊C8)—24no—
50C4⋊(C7⋊C8)—22no—
51C4×C7⋊C8—24no—
52C4×C7⋊C8D414yes—
53#2982—71no—
54#2536—71no—
55#1801D711yescharacteristic, complemented
56#1486—72no—
57#1286—72no—
58C23×D7—21no—
59C2×C7×Q8—22no—
60C2×C7×Q8—22no—
61C2×C7×Q8D812yes—
62C2×C7×Q8SD1612yes—
63C22×C4×C7C22⋊C412yes—
64C22×C4×C7C22⋊C412yes—
65C22×C4×C7C22×C411yescharacteristic
66C22×C4×C7C2×D411yescharacteristic
67C22×C4×C7C2×D411yescharacteristic
68C22×Dic7—22no—
69C22×Dic7—21no—
70C2×D28—22no—
71C2×D28—42no—
72C2×C4×D7—42no—
73C2×C4×D7—41no—
74C2×C4×D7—41no—
75C2×C4×D7—42no—
76C7×M16—42no—
77C7×M16—42no—
78C2×C7×C8—42no—
79C7×C4⋊C4—24no—
80C7×C4⋊C4—22no—
81C42×C7—22no—
82C42×C7—22no—
83C42×C7C2×D412yes—
84C42×C7#4112yes—
85(C2×D7)⋊C4—44no—
86(C2×D7)⋊C4—44no—
87(C2×D7)⋊C4—41no—
88(C2×D7)⋊C4—41no—
89C4⋊Dic7—22no—
90C4⋊Dic7—41no—
91C4⋊Dic7—41no—
92Dic7⋊C4—44no—
93C2×C7⋊C8—24no—
94C2×C22⋊Q8—141no—
95C2×C4×Q8C2×D711yescharacteristic
96C2×C4×D4—71no—
97C8×Q8—144no—
98#431—72no—
99#431—142no—
100#423C2×D712yes—
101C2×#101—21no—
102#398—72no—
103#394—72no—
104#388—142no—
105#382—71no—
106#380—141no—
107#343—21no—
108D4⋊C8—144no—
109C23×C7C2×C22⋊C411yescharacteristic
110C22×D7—22no—
111C22×D7—41no—
112C22×D7—41no—
113C22×D7—41no—
114C22×D7—41no—
115C22×D7—41no—
116C22×D7—41no—
117C7×Q8—24no—
118C7×Q8—24no—
119C2×C4×C7—22no—
120C2×C4×C7—22no—
121C2×C4×C7—22no—
122C2×C4×C7—41no—
123C2×C4×C7—41no—
124C2×C4×C7—42no—
125C2×C4×C7#10214yes—
126C2×C4×C7C4×D412yes—
127C2×C4×C7#12011yescharacteristic
128C2×C4×C7#12112yes—
129C2×C4×C7#12311yescharacteristic
130C2×C4×C7C2×D811yescharacteristic
131C2×C4×C7C2×SD1611yescharacteristic
132C2×C4×C7#13512yes—
133C2×Dic7—42no—
134C2×Dic7—42no—
135C2×Dic7—41no—
136C2×Dic7—41no—
137C2×Dic7—42no—
138C2×Dic7—41no—
139D28—24no—
140D28—42no—
141C4×D7—82no—
142C4×D7—82no—
143C7×C8—82no—
144C7⋊C8—44no—
145C22×Q8C22⋊D711yescharacteristic
146C22×D4—71no—
147C23×C4—141no—
148C2×M16—22no—
149C22⋊Q8—282no—
150C22⋊Q8—282no—
151C4×Q8—22no—
152C4×Q8—22no—
153C4×D4—142no—
154C4×D4—142no—
155C2×C4⋊C4—71no—
156C2×C4⋊C4—142no—
157C2×C4⋊C4—141no—
158C2×C4⋊C4C4×D712yes—
159C2×C4⋊C4C22⋊D711yescharacteristic
160C2×C22⋊C4—142no—
161C2×C22⋊C4—142no—
162C2×C22⋊C4—141no—
163C2×C42D2812yes—
164C2×C42C22×D711yescharacteristic
165C4⋊C8—22no—
166C4⋊C8—22no—
167C4⋊C8—144no—
168C4⋊C8—142no—
169#101—42no—
170C22⋊C8—142no—
171C4×C8—74no—
172C4×C8—144no—
173#95—72no—
174#95—142no—
175#95—141no—
176C22×C7—21no—
177C22×C7—21no—
178C22×C7#38611yescharacteristic
179C22×C7C2×#10211yescharacteristic
180C22×C7#41611yescharacteristic
181C2×D7—42no—
182C2×D7—42no—
183C2×D7—42no—
184C2×D7—81no—
185C2×D7—81no—
186C2×D7—81no—
187C2×D7—81no—
188C4×C7—22no—
189C4×C7—24no—
190C4×C7—21no—
191C4×C7—21no—
192C4×C7—42no—
193C4×C7#46512yes—
194C4×C7#46612yes—
195Dic7—82no—
196Dic7—81no—
197C24—141no—
198C2×Q8—22no—
199C2×Q8—22no—
200C2×Q8#114112yes—
201C2×Q8#114312yes—
202C2×D4—142no—
203C2×D4—282no—
204C22×C4—142no—
205C22×C4—141no—
206C22×C4—281no—
207C22×C4—282no—
208C22×C4—281no—
209C22×C4—282no—
210C22×C4(C2×D7)⋊C412yes—
211C22×C4(C2×D7)⋊C412yes—
212C22×C4C2×C4×D711yescharacteristic
213C22×C4C2×D2811yescharacteristic
214C22×C4C2×C22⋊D711yescharacteristic
215M16—42no—
216M16—42no—
217C2×C8—42no—
218C2×C8—144no—
219C4⋊C4—24no—
220C4⋊C4—22no—
221C4⋊C4—142no—
222C4⋊C4—281no—
223C4⋊C4—281no—
224C4⋊C4—284no—
225C22⋊C4—284no—
226C22⋊C4—281no—
227C22⋊C4—281no—
228C22⋊C4—284no—
229C42—22no—
230C42—22no—
231C42D4×D712yes—
232C42#115912yes—
233C2×C7—21no—
234C2×C7—21no—
235C2×C7#189111yescharacteristic
236C2×C7#189311yescharacteristic
237C2×C7#191211yescharacteristic
238D7—82no—
239C23—142no—
240C23—281no—
241C23—281no—
242C23—281no—
243C23—281no—
244C23—281no—
245C23—281no—
246C23C2×(C2×D7)⋊C411yescharacteristic
247Q8—24no—
248Q8—24no—
249D4—144no—
250D4—282no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—42no—
257C2×C4—282no—
258C2×C4—282no—
259C2×C4—281no—
260C2×C4—282no—
261C2×C4—281no—
262C2×C4—281no—
263C2×C4—562no—
264C2×C4—562no—
265C2×C4#638314yes—
266C2×C4(C22⋊D7)⋊C412yes—
267C2×C4#644511yescharacteristic
268C2×C4#644911yescharacteristic
269C2×C4C2×#114111yescharacteristic
270C2×C4#650212yes—
271C2×C4C2×#114311yescharacteristic
272C2×C4#651312yes—
273C8—82no—
274C8—284no—
275C7#948611yescharacteristic, complemented, Sylow
276C22—21no—
277C22—21no—
278C22—282no—
279C22—282no—
280C22—282no—
281C22—561no—
282C22—561no—
283C22—561no—
284C22—561no—
285C22#8973011yescharacteristic
286C22C2×#638311yescharacteristic
287C22#8977411yescharacteristic
288C4—22no—
289C4—24no—
290C4—21no—
291C4—21no—
292C4—42no—
293C4—562no—
294C4—561no—
295C4#8981012yes—
296C4#8982012yes—
297C2—21no—
298C2—21no—
299C2—562no—
300C2#1172726911yescharacteristic
301C2#1172785911yescharacteristic
302C2#1172789011yescharacteristic
subgroupG/H
centerC22#89730
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#1801C2
socleC22×C7#386
prime radicalC22×D28C2×C4
cosocleC2×C42×C7C23
hypercenter#1801D7
hypocenterC7#9486
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1801

Complements

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

NH ≅ G/Ntype
2. #11739880C2semidirect
3. C7×#1801C2semidirect
5. #11725519C2semidirect
55. #1801D7semidirect
275. C7#9486semidirect

Maximal subgroups

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

subgroupindex
#117399442
#117398802
C7×#18012
#117282502
#117255192
#94867