Finite Group Databasebeta

← #13251632

Subgroups — #13251632

Order 1792.

542 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 113 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#11739946C211yesmaximal, characteristic
2#11736731C211yesmaximal, characteristic
3C7×#1487C211yesmaximal, characteristic, complemented
4#11729376C211yesmaximal, characteristic, complemented
5#11728281C211yesmaximal, characteristic
6#11725544C211yesmaximal, characteristic, complemented
7#11725519C211yesmaximal, characteristic, complemented
8C2×C4×C7×Q8C2211yescharacteristic
9C2×C4×D28C2211yescharacteristic
10C7×#431C2211yescharacteristic
11C7×#423C2211yescharacteristic
12#90003C412yes—
13C2×#6397—21no—
14#89764C412yes—
15#89761—22no—
16#89711—21no—
17#89708C2211yescharacteristic
18#89700—21no—
19#89697C2211yescharacteristic
20#89696C2211yescharacteristic
21#89606—22no—
22D7×C22⋊C8—22no—
23C8×D28—22no—
24C7×Q8⋊C8—22no—
25#89349—21no—
26Q8⋊(C7⋊C8)—22no—
27#9483—71nomaximal, Sylow
28C22×C7×Q8C2×C411yescharacteristic
29C22×D28D411yescharacteristic
30C22×C4×D7D412yes—
31C2×C7×M16—21no—
32C4×C7×Q8—21no—
33C4×C7×Q8—21no—
34C4×C7×Q8—21no—
35C4×C7×Q8—21no—
36C2×C7×C4⋊C4—21no—
37C2×C7×C4⋊C4C2×C411yescharacteristic
38C2×C42×C7—21no—
39C2×C42×C7C2311yescharacteristic
40C2×(C2×D7)⋊C4—22no—
41C2×(C2×D7)⋊C4—22no—
42C2×(C2×D7)⋊C4C2×C412yes—
43C2×C4⋊Dic7—21no—
44C2×C4⋊Dic7D411yescharacteristic
45C2×Dic7⋊C4—22no—
46C2×#1136—21no—
47C2×C8×D7—24no—
48C4×D28—21no—
49C4×D28—21no—
50C4×D28—21no—
51C4×D28—21no—
52C7×C4⋊C8—21no—
53C7×C4⋊C8—21no—
54C7×C4⋊C8—21no—
55C7×C22⋊C8—21no—
56C4×C7×C8D412yes—
57C7⋊#95—21no—
58C7⋊#95—22no—
59C7⋊#95C2×C412yes—
60C22⋊(C7⋊C8)—21no—
61#6397—42no—
62(C2×D7)⋊C8—24no—
63C4⋊(C7⋊C8)—21no—
64C4⋊(C7⋊C8)—21no—
65C4⋊(C7⋊C8)—21no—
66C4×C7⋊C8D412yes—
67#2972—71no—
68#2580—71no—
69#1800—71no—
70#1487—71no—
71#1487D711yescharacteristic, complemented
72#1286—71no—
73#1286—71no—
74C23×D7C22⋊C412yes—
75C2×C7×Q8—21no—
76C2×C7×Q8—21no—
77C2×C7×Q8—42no—
78C22×C4×C7—21no—
79C22×C4×C7—21no—
80C22×C4×C7C22×C411yescharacteristic
81C22×C4×C7C2×D411yescharacteristic
82C22×C4×C7C2×D411yescharacteristic
83C22×Dic7—22no—
84C22×Dic7C22⋊C412yes—
85C2×D28—21no—
86C2×D28—21no—
87C2×D28—21no—
88C2×D28—21no—
89C2×D28SD1612yes—
90C2×D28Q1612yes—
91C2×C4×D7—22no—
92C2×C4×D7—22no—
93C2×C4×D7—82no—
94C7×M16—41no—
95C7×M16—41no—
96C2×C7×C8—22no—
97C2×C7×C8—41no—
98C7×C4⋊C4—21no—
99C7×C4⋊C4—21no—
100C7×C4⋊C4—41no—
101C7×C4⋊C4—41no—
102C42×C7—41no—
103C42×C7—41no—
104C42×C7C2×D411yescharacteristic
105C42×C7C2×D411yescharacteristic
106C42×C7#4111yescharacteristic
107C42×C7#4111yescharacteristic
108(C2×D7)⋊C4—22no—
109(C2×D7)⋊C4—22no—
110(C2×D7)⋊C4—44no—
111(C2×D7)⋊C4—82no—
112C4⋊Dic7—21no—
113C4⋊Dic7—21no—
114C4⋊Dic7—41no—
115C4⋊Dic7—41no—
116Dic7⋊C4—82no—
117#1136—41no—
118#1136—41no—
119C2×C7⋊C8—22no—
120C2×C7⋊C8—41no—
121C8×D7—84no—
122C2×C4×Q8C2×D711yescharacteristic
123C2×C4×D4—71no—
124C8×D4—142no—
125C8×D4—142no—
126#431—71no—
127#431C2×D711yescharacteristic
128#423—71no—
129#423C2×D711yescharacteristic
130C2×#100—141no—
131C2×C22⋊C8—142no—
132#399—72no—
133#393—72no—
134#390—141no—
135#389—141no—
136#385—142no—
137#382—71no—
138#343—141no—
139Q8⋊C8—22no—
140Q8⋊C8—142no—
141C23×C7C2×C22⋊C411yescharacteristic
142C22×D7—22no—
143C22×D7—22no—
144C22×D7—42no—
145C22×D7—42no—
146C22×D7#10314yes—
147C7×Q8—22no—
148C7×Q8—22no—
149C7×Q8—41no—
150C7×Q8—41no—
151C2×C4×C7—22no—
152C2×C4×C7—41no—
153C2×C4×C7—41no—
154C2×C4×C7—41no—
155C2×C4×C7—41no—
156C2×C4×C7—41no—
157C2×C4×C7—41no—
158C2×C4×C7—41no—
159C2×C4×C7C4×D411yescharacteristic
160C2×C4×C7C4×D411yescharacteristic
161C2×C4×C7#12011yescharacteristic
162C2×C4×C7#12111yescharacteristic
163C2×C4×C7#12111yescharacteristic
164C2×C4×C7#12311yescharacteristic
165C2×C4×C7C2×SD1611yescharacteristic
166C2×C4×C7C2×Q1611yescharacteristic
167C2×C4×C7#13511yescharacteristic
168C2×C4×C7#13511yescharacteristic
169C2×Dic7—22no—
170C2×Dic7—42no—
171C2×Dic7—42no—
172C2×Dic7—42no—
173C2×Dic7—42no—
174D28—22no—
175D28—22no—
176D28—22no—
177D28—22no—
178C4×D7—44no—
179C4×D7—82no—
180C7×C8—42no—
181C7×C8—81no—
182C7⋊C8—42no—
183C7⋊C8—81no—
184C22×Q8C4×D711yescharacteristic
185C22×D4—71no—
186C23×C4—72no—
187C2×M16—21no—
188C2×M16—141no—
189C22×C8—144no—
190C4×Q8—21no—
191C4×Q8—21no—
192C4×Q8—21no—
193C4×Q8—21no—
194C4×D4—141no—
195C4×D4—141no—
196C4×D4—141no—
197C4×D4—141no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—71no—
200C2×C4⋊C4—141no—
201C2×C4⋊C4—142no—
202C2×C4⋊C4C4×D711yescharacteristic
203C2×C22⋊C4—72no—
204C2×C22⋊C4—142no—
205C2×C22⋊C4—142no—
206C2×C42—21no—
207C2×C42C22×D711yescharacteristic
208C4⋊C8—21no—
209C4⋊C8—21no—
210C4⋊C8—21no—
211C4⋊C8—141no—
212C4⋊C8—141no—
213C4⋊C8—141no—
214#100—282no—
215C22⋊C8—21no—
216C22⋊C8—144no—
217C22⋊C8—141no—
218C4×C8—72no—
219C4×C8C22⋊D712yes—
220#95—72no—
221#95—142no—
222#95—141no—
223C22×C7—21no—
224C22×C7—21no—
225C22×C7#38611yescharacteristic
226C22×C7C2×#10311yescharacteristic
227C22×C7#41611yescharacteristic
228C2×D7—24no—
229C2×D7—42no—
230C2×D7—42no—
231C2×D7—44no—
232C2×D7—82no—
233C4×C7—21no—
234C4×C7—21no—
235C4×C7—21no—
236C4×C7—21no—
237C4×C7—42no—
238C4×C7—81no—
239C4×C7#46511yescharacteristic
240C4×C7#46511yescharacteristic
241C4×C7#46711yescharacteristic
242C4×C7#46711yescharacteristic
243Dic7—42no—
244Dic7—82no—
245C24—72no—
246C2×Q8—21no—
247C2×Q8—21no—
248C2×Q8—42no—
249C2×D4—72no—
250C2×D4—72no—
251C2×D4—141no—
252C2×D4—141no—
253C2×D4—141no—
254C2×D4—141no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—72no—
258C22×C4—142no—
259C22×C4—142no—
260C22×C4—142no—
261C22×C4—562no—
262C22×C4C2×C4×D711yescharacteristic
263C22×C4D4×D711yescharacteristic
264C22×C4D4×D711yescharacteristic
265M16—41no—
266M16—41no—
267M16—281no—
268M16—281no—
269C2×C8—22no—
270C2×C8—41no—
271C2×C8—142no—
272C2×C8—281no—
273C2×C8—564no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—141no—
279C4⋊C4—141no—
280C4⋊C4—281no—
281C4⋊C4—281no—
282C4⋊C4—562no—
283C22⋊C4—142no—
284C22⋊C4—142no—
285C22⋊C4—284no—
286C22⋊C4—562no—
287C42—41no—
288C42—41no—
289C42#115711yescharacteristic
290C42D4×D711yescharacteristic
291C42#115911yescharacteristic
292C42C2×C22⋊D711yescharacteristic
293C2×C7—21no—
294C2×C7—21no—
295C2×C7#189111yescharacteristic
296C2×C7#189211yescharacteristic
297C2×C7#191111yescharacteristic
298D7—44no—
299C23—74no—
300C23—142no—
301C23—142no—
302C23—282no—
303C23—282no—
304C23D7×C22⋊C411yescharacteristic
305Q8—22no—
306Q8—22no—
307Q8—41no—
308Q8—41no—
309D4—142no—
310D4—142no—
311D4—142no—
312D4—142no—
313C2×C4—22no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—142no—
322C2×C4—284no—
323C2×C4—282no—
324C2×C4—282no—
325C2×C4—282no—
326C2×C4—282no—
327C2×C4—562no—
328C2×C4(C22⋊D7)⋊C411yescharacteristic
329C2×C4#644611yescharacteristic
330C2×C4#644711yescharacteristic
331C2×C4D7×SD1611yescharacteristic
332C2×C4#647911yescharacteristic
333C2×C4D7×Q1611yescharacteristic
334C2×C4#648211yescharacteristic
335C2×C4C4×C22⋊D711yescharacteristic
336C2×C4#650011yescharacteristic
337C2×C4#650111yescharacteristic
338C8—42no—
339C8—81no—
340C8—282no—
341C8—561no—
342C7#948311yescharacteristic, complemented, Sylow
343C22—21no—
344C22—21no—
345C22—144no—
346C22—284no—
347C22—282no—
348C22—282no—
349C22—562no—
350C22D7×#10311yescharacteristic
351C22#8957611yescharacteristic
352C22#8972811yescharacteristic
353C4—21no—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—42no—
358C4—81no—
359C4—282no—
360C4—562no—
361C4#8957911yescharacteristic
362C4#8958511yescharacteristic
363C4#8994311yescharacteristic
364C4#8996011yescharacteristic
365C2—21no—
366C2—21no—
367C2—284no—
368C2#1172774611yescharacteristic
369C2#1172802411yescharacteristic
370C2#1172816211yescharacteristic
subgroupG/H
centerC22#89728
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#1487C2
socleC22×C7#386
prime radicalC22×D28D4
cosocleC2×C42×C7C23
hypercenter#1487D7
hypocenterC7#9483
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1487

Complements

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

NH ≅ G/Ntype
3. C7×#1487C2semidirect
4. #11729376C2semidirect
6. #11725544C2semidirect
7. #11725519C2semidirect
71. #1487D7semidirect
342. C7#9483semidirect

Maximal subgroups

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

subgroupindex
#117399462
#117367312
C7×#14872
#117293762
#117282812
#117255442
#117255192
#94837