Finite Group Databasebeta

← #13249784

Subgroups — #13249784

Order 1792.

478 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 103 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, complemented
2#11739948C211yesmaximal, characteristic
3C4×#6410C211yesmaximal, characteristic, complemented
4#11728558C212yesmaximal
5#11725558C212yesmaximal, complemented
6C2×C4×C7×Q8C2211yescharacteristic
7C2×#6438C2211yescharacteristic
8C7×#446—24no—
9C7×#432—21no—
10C7×#432C2212yes—
11C7×#431—21no—
12#90003—21no—
13C2×#6410—21no—
14#89764—21no—
15#89763C411yescharacteristic
16#89763C411yescharacteristic
17#89751C411yescharacteristic
18#89751C411yescharacteristic
19#89712—21no—
20#89712—21no—
21#89709C2211yescharacteristic
22C4×#1136C2212yes—
23#89325—21no—
24#89259—24no—
25#9188—71nomaximal, Sylow
26C22×C7×Q8D411yescharacteristic
27C22×D28C2×C411yescharacteristic
28C22×Dic14C2×C411yescharacteristic
29C4×C7×Q8—22no—
30C4×C7×Q8—22no—
31C2×C7×C4⋊C4C2×C411yescharacteristic
32C2×C7×C4⋊C4C2×C411yescharacteristic
33C2×C7×C4⋊C4D411yescharacteristic
34C2×C42×C7D411yescharacteristic
35C2×C42×C7D411yescharacteristic
36C2×C42×C7C2311yescharacteristic
37C2×(C2×D7)⋊C4—21no—
38C2×(C2×D7)⋊C4—21no—
39C2×(C2×D7)⋊C4—21no—
40C2×(C2×D7)⋊C4—21no—
41C2×(C2×D7)⋊C4—21no—
42C2×(C2×D7)⋊C4—21no—
43C2×Dic7⋊C4—21no—
44C2×Dic7⋊C4—21no—
45C2×#1136—22no—
46#6438—22no—
47#6438—22no—
48C7×C4⋊C8—22no—
49C7×C4⋊C8—22no—
50C7×C4⋊C8—22no—
51C7×C22⋊C8—22no—
52C7×#97—22no—
53C7×#97D414yes—
54C4×C7×C8—22no—
55#6410—24no—
56C7⋊#95—21no—
57C7⋊#95—21no—
58C7⋊#95—21no—
59C7⋊#95—21no—
60C7⋊#95C2×C411yescharacteristic
61C7⋊#95C2×C411yescharacteristic
62#6377—22no—
63C4×C7⋊C8—22no—
64#2983D711yescharacteristic, complemented
65#2632—71no—
66C4×#101—71no—
67#1527—72no—
68#1288—72no—
69C23×D7—21no—
70C2×C7×Q8—22no—
71C2×C7×Q8—22no—
72C2×C7×Q8—41no—
73C22×C4×C7C22⋊C411yescharacteristic
74C22×C4×C7C22⋊C411yescharacteristic
75C22×C4×C7C22⋊C411yescharacteristic
76C22×C4×C7C22⋊C411yescharacteristic
77C22×C4×C7C22×C411yescharacteristic
78C22×C4×C7C2×D411yescharacteristic
79C22×C4×C7C2×D411yescharacteristic
80C22×Dic7—21no—
81C22×Dic7—21no—
82C22×Dic7—21no—
83C2×D28—22no—
84C2×D28—81no—
85C2×Dic14—22no—
86C2×Dic14—81no—
87C2×C7×C8—24no—
88C7×C4⋊C4—22no—
89C7×C4⋊C4—22no—
90C7×C4⋊C4—22no—
91C42×C7—21no—
92C42×C7—21no—
93C42×C7—21no—
94C42×C7—21no—
95C42×C7C2×D412yes—
96C42×C7#4112yes—
97(C2×D7)⋊C4—42no—
98(C2×D7)⋊C4—42no—
99(C2×D7)⋊C4—41no—
100(C2×D7)⋊C4—42no—
101(C2×D7)⋊C4—41no—
102(C2×D7)⋊C4—81no—
103(C2×D7)⋊C4—81no—
104Dic7⋊C4—81no—
105Dic7⋊C4—81no—
106#1136—24no—
107#1136—42no—
108C2×C7⋊C8—42no—
109C2×#124—71no—
110C2×C4×Q8C2×D711yescharacteristic
111#446—24no—
112#432—21no—
113#432C2×D712yes—
114#431—21no—
115C2×#101—141no—
116C4×M16—72no—
117#399—141no—
118#395—71no—
119#395—71no—
120#394—71no—
121#394—71no—
122#393—141no—
123#388—141no—
124#388—141no—
125#383—71no—
126#343—141no—
127#329—144no—
128C23×C7C2×C22⋊C411yescharacteristic
129C22×D7—41no—
130C22×D7—41no—
131C22×D7—41no—
132C22×D7—41no—
133C22×D7—41no—
134C22×D7—41no—
135C22×D7—41no—
136C7×Q8—44no—
137C2×C4×C7—21no—
138C2×C4×C7—21no—
139C2×C4×C7—21no—
140C2×C4×C7—21no—
141C2×C4×C7—21no—
142C2×C4×C7—21no—
143C2×C4×C7—21no—
144C2×C4×C7—21no—
145C2×C4×C7—21no—
146C2×C4×C7—41no—
147C2×C4×C7—41no—
148C2×C4×C7—41no—
149C2×C4×C7—41no—
150C2×C4×C7#10412yes—
151C2×C4×C7C4×D412yes—
152C2×C4×C7#12011yescharacteristic
153C2×C4×C7#12112yes—
154C2×C4×C7#12311yescharacteristic
155C2×Dic7—41no—
156C2×Dic7—41no—
157C2×Dic7—41no—
158C2×Dic7—41no—
159C2×Dic7—41no—
160C2×Dic7—41no—
161C2×Dic7—41no—
162C2×Dic7—41no—
163C2×Dic7—41no—
164D28—82no—
165Dic14—82no—
166C7×C8—44no—
167C7⋊C8—44no—
168C22×Q8—71no—
169C22×Q8D2811yescharacteristic
170C22×D4—71no—
171C2×M16—142no—
172#124—142no—
173#124—142no—
174C4×Q8—22no—
175C4×Q8—22no—
176C2×C4⋊C4—141no—
177C2×C4⋊C4—141no—
178C2×C4⋊C4C4×D711yescharacteristic
179C2×C4⋊C4C4×D711yescharacteristic
180C2×C4⋊C4D2811yescharacteristic
181C2×C22⋊C4—141no—
182C2×C22⋊C4—141no—
183C2×C22⋊C4—141no—
184C2×C22⋊C4—141no—
185C2×C22⋊C4—141no—
186C2×C22⋊C4—141no—
187C2×C42C22⋊D711yescharacteristic
188C2×C42C22⋊D711yescharacteristic
189C2×C42C22×D711yescharacteristic
190C4⋊C8—22no—
191C4⋊C8—22no—
192C4⋊C8—22no—
193#101—144no—
194C22⋊C8—22no—
195#97—22no—
196#97—142no—
197#97C22⋊D714yes—
198C4×C8—22no—
199C4×C8—142no—
200#95—71no—
201#95—71no—
202#95—141no—
203#95—141no—
204#95—141no—
205#95—141no—
206C22×C7—21no—
207C22×C7—21no—
208C22×C7#38611yescharacteristic
209C22×C7C2×#10411yescharacteristic
210C22×C7#42111yescharacteristic
211C2×D7—81no—
212C2×D7—81no—
213C2×D7—81no—
214C2×D7—81no—
215C2×D7—81no—
216C2×D7—81no—
217C2×D7—81no—
218C4×C7—22no—
219C4×C7—22no—
220C4×C7—21no—
221C4×C7—21no—
222C4×C7—22no—
223C4×C7—41no—
224C4×C7—41no—
225C4×C7—41no—
226Dic7—81no—
227Dic7—81no—
228Dic7—81no—
229C24—141no—
230C2×Q8—22no—
231C2×Q8—22no—
232C2×Q8—41no—
233C2×Q8—142no—
234C2×Q8—561no—
235C2×D4—142no—
236C2×D4—561no—
237C22×C4—141no—
238C22×C4—141no—
239C22×C4—141no—
240C22×C4(C2×D7)⋊C411yescharacteristic
241C22×C4(C2×D7)⋊C411yescharacteristic
242C22×C4(C2×D7)⋊C411yescharacteristic
243C22×C4(C2×D7)⋊C411yescharacteristic
244C22×C4C2×C4×D711yescharacteristic
245C22×C4C2×D2811yescharacteristic
246C22×C4C2×C22⋊D711yescharacteristic
247M16—144no—
248M16—282no—
249C2×C8—24no—
250C2×C8—282no—
251C4⋊C4—22no—
252C4⋊C4—22no—
253C4⋊C4—22no—
254C4⋊C4—561no—
255C4⋊C4—561no—
256C22⋊C4—282no—
257C22⋊C4—281no—
258C22⋊C4—282no—
259C22⋊C4—281no—
260C22⋊C4—282no—
261C22⋊C4—561no—
262C22⋊C4—561no—
263C42—21no—
264C42—21no—
265C42—21no—
266C42—21no—
267C42#115712yes—
268C42C2×C22⋊D712yes—
269C2×C7—21no—
270C2×C7—21no—
271C2×C7#187111yescharacteristic
272C2×C7#191211yescharacteristic
273C2×C7#191411yescharacteristic
274D7—161no—
275C23—281no—
276C23—281no—
277C23—281no—
278C23—281no—
279C23—281no—
280C23—281no—
281C23—281no—
282C23C2×(C2×D7)⋊C411yescharacteristic
283Q8—44no—
284Q8—562no—
285D4—562no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—281no—
300C2×C4—281no—
301C2×C4—281no—
302C2×C4—281no—
303C2×C4—281no—
304C2×C4—281no—
305C2×C4—281no—
306C2×C4—281no—
307C2×C4—281no—
308C2×C4#639912yes—
309C2×C4C4×C22⋊D712yes—
310C2×C4#649211yescharacteristic
311C2×C4#649312yes—
312C2×C4#651611yescharacteristic
313C8—44no—
314C8—284no—
315C7#918811yescharacteristic, complemented, Sylow
316C22—21no—
317C22—21no—
318C22—561no—
319C22—561no—
320C22—561no—
321C22—561no—
322C22—561no—
323C22—561no—
324C22—561no—
325C22C2×#639911yescharacteristic
326C22#8991111yescharacteristic
327C22#8998511yescharacteristic
328C4—21no—
329C4—21no—
330C4—22no—
331C4—22no—
332C4—22no—
333C4—41no—
334C4—41no—
335C4—41no—
336C4—561no—
337C4—561no—
338C4—561no—
339C2—21no—
340C2—21no—
341C2—1121no—
342C2#1172941711yescharacteristic
343C2#1172973811yescharacteristic
344C2#1172978411yescharacteristic
subgroupG/H
centerC22#89985
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#2983C2
socleC22×C7#386
prime radicalC22×D28C2×C4
cosocleC2×C42×C7C23
hypercenter#2983D7
hypocenterC7#9188
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
1. C7×#2983C2semidirect
3. C4×#6410C2semidirect
5. #11725558C2semidirect
64. #2983D7semidirect
315. C7#9188semidirect

Maximal subgroups

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

subgroupindex
C7×#29832
#117399482
C4×#64102
#117285582
#117255582
#91887