Finite Group Databasebeta

← C2×#72222054 #72246552

Subgroups — C2×#72222054 #72246552

Order 7056.

671 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 671 conjugacy classes of subgroups — too many to draw. Only its 93 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
1C2×C3×#10596873C211yesmaximal, characteristic, complemented
2C22×C3⋊#64423C211yesmaximal, characteristic, complemented
3C2×#11924248C311yesmaximal, characteristic, complemented
4#72222054C214yesmaximal, direct product factor
5C22×C3⋊#64424C211yesmaximal, characteristic, complemented
6C2×#11924176—31nomaximal
7C22×C3×#3861C2211yescharacteristic
8C2×C3⋊#64423—22no—
9C2×C3⋊#64423C2212yescomplemented
10C2×C3⋊#64424—22no—
11C2×C3⋊#64424C2212yescomplemented
12C3×#10596873C2212yescomplemented
13C22×C3⋊(C7⋊D7)C2×C311yescharacteristic, complemented
14C22×C7×D21C2×C311yescharacteristic, complemented
15C22×#64423—31no—
16C22×#64424—31no—
17C2×C3×C7×Dic7C2×C311yescharacteristic, complemented
18#11924248C2×C314yescomplemented
19C2×#10596873—21no—
20C2×#10596873D311yescharacteristic, complemented
21#11924176—34no—
22C2×C4⋊(C3⋊AGL(1,𝔽7))—71nomaximal
23C2×#4247⋊C3—71nomaximal
24C2×C3×#3861D412yescomplemented
25C2×C3×#3861C2311yescharacteristic
26C3⋊#64424—22no—
27C3⋊#64423—22no—
28C2×#87641—31noHall
29C22×C3×C72C22×C311yescharacteristic
30C2×C3⋊(C7⋊D7)—22no—
31C2×C3⋊(C7⋊D7)C22×C312yescomplemented
32C2×C7×D21—22no—
33C2×C7×D21C22×C312yescomplemented
34C22×#3861—21no—
35C22×#3861C2×D311yescharacteristic
36C2×#64424—32no—
37C2×#64424—62no—
38C2×#64423—32no—
39C2×#64423—62no—
40C3×C7×Dic7C22×C312yescomplemented
41#10596873—22no—
42#10596873C2×D312yescomplemented
43C22×C3⋊AGL(1,𝔽7)—71no—
44C22×C3⋊AGL(1,𝔽7)—71no—
45C22×C3⋊AGL(1,𝔽7)—141no—
46C22×C3⋊AGL(1,𝔽7)—141no—
47C22×C3⋊AGL(1,𝔽7)—141no—
48C22×C3⋊AGL(1,𝔽7)—141no—
49C22×D3×ΓL(1,𝔽8)—71no—
50C2×C3×C4×ΓL(1,𝔽8)—71no—
51C4⋊(C3⋊AGL(1,𝔽7))—74no—
52C2×C3×ΓL(1,𝔽8)⋊C4—71no—
53#4247⋊C3—74no—
54C3×#3861C2×D411yescharacteristic, complemented, Hall
55C22×C7⋊D7—31no—
56C22×C7×D7—31no—
57C2×C7×Dic7C3×D311yescharacteristic, complemented
58#87641—34no—
59C2×D84—71no—
60C2×#4247—71no—
61C2×(C22⋊D7)⋊C3—211no—
62C2×C4⋊AGL(1,𝔽7)—211no—
63C2×C3×C72C3×D412yescomplemented
64C2×C3×C72C23×C311yescharacteristic
65C3⋊(C7⋊D7)—22no—
66C7×D21—22no—
67C2×#3861—22no—
68C2×#3861—21no—
69C2×#3861D1212yescomplemented
70C2×#3861C22×D311yescharacteristic
71#64424—62no—
72#64423—62no—
73C22×C3×ΓL(1,𝔽8)—71no—
74C22×C3×ΓL(1,𝔽8)—141no—
75C22×C3×ΓL(1,𝔽8)—141no—
76C22×C3×ΓL(1,𝔽8)—281no—
77C22×C3×ΓL(1,𝔽8)—281no—
78C2×C3⋊AGL(1,𝔽7)—72no—
79C2×C3⋊AGL(1,𝔽7)—72no—
80C2×C3⋊AGL(1,𝔽7)—142no—
81C2×C3⋊AGL(1,𝔽7)—142no—
82C2×C3⋊AGL(1,𝔽7)—142no—
83C2×C3⋊AGL(1,𝔽7)—142no—
84C2×C3⋊AGL(1,𝔽7)—142no—
85C2×C3⋊AGL(1,𝔽7)—142no—
86C2×C3⋊AGL(1,𝔽7)—142no—
87C2×C3⋊AGL(1,𝔽7)—142no—
88C2×C3⋊AGL(1,𝔽7)—142no—
89C2×C3⋊AGL(1,𝔽7)—142no—
90C2×C3⋊AGL(1,𝔽7)—142no—
91C2×C3⋊AGL(1,𝔽7)—142no—
92C2×C3⋊AGL(1,𝔽7)—142no—
93C2×D3×ΓL(1,𝔽8)—72no—
94C2×D3×ΓL(1,𝔽8)—142no—
95C3×C4×ΓL(1,𝔽8)—72no—
96C3×ΓL(1,𝔽8)⋊C4—72no—
97C22×C72C2×C3×D311yescharacteristic
98C2×C7⋊D7—32no—
99C2×C7⋊D7—62no—
100C2×C7×D7—32no—
101C2×C7×D7—62no—
102C7×Dic7C2×C3×D312yescomplemented
103C22×D21—71no—
104C22×D21—71no—
105C22×D21—141no—
106C22×D21—141no—
107C22×D21—141no—
108C22×D21AGL(1,𝔽7)11yescharacteristic, complemented
109C22×D3×C7—71no—
110C23×ΓL(1,𝔽8)—211no—
111C22×AGL(1,𝔽7)—211no—
112C22×AGL(1,𝔽7)—211no—
113C22×AGL(1,𝔽7)—421no—
114C22×AGL(1,𝔽7)—421no—
115C22×AGL(1,𝔽7)—421no—
116C22×AGL(1,𝔽7)—421no—
117C2×C3×C4×C7—71no—
118D84—74no—
119C2×C3×Dic7AGL(1,𝔽7)11yescharacteristic, complemented
120C2×C4×ΓL(1,𝔽8)—71no—
121C2×C4×ΓL(1,𝔽8)—141no—
122#4247—74no—
123(C22⋊D7)⋊C3—214no—
124C2×ΓL(1,𝔽8)⋊C4—141no—
125C2×ΓL(1,𝔽8)⋊C4—141no—
126C4⋊AGL(1,𝔽7)—214no—
127C3×C72C2×C3×D411yescharacteristic, complemented
128#3861—21no—
129#3861C2×D1211yescharacteristic, complemented
130C2×C3×D12—491noHall
131C2×C3×ΓL(1,𝔽8)—72no—
132C2×C3×ΓL(1,𝔽8)—71no—
133C2×C3×ΓL(1,𝔽8)—142no—
134C2×C3×ΓL(1,𝔽8)—141no—
135C2×C3×ΓL(1,𝔽8)—142no—
136C2×C3×ΓL(1,𝔽8)—141no—
137C2×C3×ΓL(1,𝔽8)—282no—
138C2×C3×ΓL(1,𝔽8)—281no—
139C2×C3×ΓL(1,𝔽8)—282no—
140C2×C3×ΓL(1,𝔽8)—281no—
141C3⋊AGL(1,𝔽7)—142no—
142C3⋊AGL(1,𝔽7)—142no—
143C3⋊AGL(1,𝔽7)—142no—
144C3⋊AGL(1,𝔽7)—142no—
145C3⋊AGL(1,𝔽7)—142no—
146C3⋊AGL(1,𝔽7)—142no—
147C3⋊AGL(1,𝔽7)—142no—
148C3⋊AGL(1,𝔽7)—142no—
149C3⋊AGL(1,𝔽7)—142no—
150D3×ΓL(1,𝔽8)—142no—
151C2×C22⋊D7—211no—
152C2×D28—211no—
153C2×C72C3×D1212yescomplemented
154C2×C72C22×C3×D311yescharacteristic
155C7⋊D7—62no—
156C7×D7—62no—
157C22×C3×C7—21no—
158C22×C3×C7—21no—
159C22×C3×C7—21no—
160C22×C3×C7C2×AGL(1,𝔽7)11yescharacteristic
161C22×C3×C7C2×AGL(1,𝔽7)11yescharacteristic
162C2×D21—22no—
163C2×D21—72no—
164C2×D21—72no—
165C2×D21—142no—
166C2×D21—142no—
167C2×D21—142no—
168C2×D21—142no—
169C2×D21—142no—
170C2×D21—142no—
171C2×D21—142no—
172C2×D21—142no—
173C2×D21—142no—
174C2×D21—142no—
175C2×D21—142no—
176C2×D21C2×AGL(1,𝔽7)12yescomplemented
177C2×D3×C7—72no—
178C2×D3×C7—142no—
179C22×ΓL(1,𝔽8)—141no—
180C22×ΓL(1,𝔽8)—141no—
181C22×ΓL(1,𝔽8)—141no—
182C22×ΓL(1,𝔽8)—141no—
183C22×ΓL(1,𝔽8)—141no—
184C22×ΓL(1,𝔽8)—212no—
185C22×ΓL(1,𝔽8)—281no—
186C22×ΓL(1,𝔽8)—281no—
187C22×ΓL(1,𝔽8)—281no—
188C22×ΓL(1,𝔽8)—281no—
189C22×ΓL(1,𝔽8)—281no—
190C22×ΓL(1,𝔽8)—422no—
191C2×AGL(1,𝔽7)—212no—
192C2×AGL(1,𝔽7)—212no—
193C2×AGL(1,𝔽7)—422no—
194C2×AGL(1,𝔽7)—422no—
195C2×AGL(1,𝔽7)—422no—
196C2×AGL(1,𝔽7)—422no—
197C2×AGL(1,𝔽7)—422no—
198C2×AGL(1,𝔽7)—422no—
199C2×AGL(1,𝔽7)—422no—
200C2×AGL(1,𝔽7)—422no—
201C2×AGL(1,𝔽7)—422no—
202C2×AGL(1,𝔽7)—422no—
203C2×AGL(1,𝔽7)—422no—
204C2×AGL(1,𝔽7)—422no—
205C2×AGL(1,𝔽7)—422no—
206C3×C4×C7—72no—
207C3×Dic7C2×AGL(1,𝔽7)12yescomplemented
208C4×ΓL(1,𝔽8)—72no—
209C4×ΓL(1,𝔽8)—142no—
210ΓL(1,𝔽8)⋊C4—142no—
211ΓL(1,𝔽8)⋊C4—142no—
212C22×C3×D3—491no—
213C22×C3×D3—491no—
214C2×C32×C4—491no—
215C3×D12—494no—
216C3×ΓL(1,𝔽8)—71no—
217C3×ΓL(1,𝔽8)—141no—
218C3×ΓL(1,𝔽8)—141no—
219C3×ΓL(1,𝔽8)—281no—
220C3×ΓL(1,𝔽8)—281no—
221C23×C7—211no—
222C22×D7—31no—
223C22×D7—211no—
224C22×D7—211no—
225C22×D7—421no—
226C22×D7—421no—
227C22×D7—421no—
228C2×C4×C7—71no—
229C22⋊D7—214no—
230C2×Dic7C3⋊AGL(1,𝔽7)11yescharacteristic, complemented
231D28—214no—
232C72C2×C3×D1211yescharacteristic, complemented, Sylow
233C2×C3×D4—1471no—
234C2×D12—491no—
235C2×C3×C7—21no—
236C2×C3×C7—21no—
237C2×C3×C7—21no—
238C2×C3×C7—22no—
239C2×C3×C7—22no—
240C2×C3×C7—22no—
241C2×C3×C7C4⋊AGL(1,𝔽7)12yescomplemented
242C2×C3×C7(C22⋊D7)⋊C312yescomplemented
243C2×C3×C7C22×AGL(1,𝔽7)11yescharacteristic
244C2×C3×C7C22×AGL(1,𝔽7)11yescharacteristic
245D21—22no—
246D21—142no—
247D21—142no—
248D21—142no—
249D21—142no—
250D21—142no—
251D21—142no—
252D21—142no—
253D21—142no—
254D3×C7—142no—
255C2×ΓL(1,𝔽8)—142no—
256C2×ΓL(1,𝔽8)—141no—
257C2×ΓL(1,𝔽8)—141no—
258C2×ΓL(1,𝔽8)—141no—
259C2×ΓL(1,𝔽8)—141no—
260C2×ΓL(1,𝔽8)—141no—
261C2×ΓL(1,𝔽8)—142no—
262C2×ΓL(1,𝔽8)—142no—
263C2×ΓL(1,𝔽8)—142no—
264C2×ΓL(1,𝔽8)—142no—
265C2×ΓL(1,𝔽8)—282no—
266C2×ΓL(1,𝔽8)—281no—
267C2×ΓL(1,𝔽8)—282no—
268C2×ΓL(1,𝔽8)—282no—
269C2×ΓL(1,𝔽8)—281no—
270C2×ΓL(1,𝔽8)—281no—
271C2×ΓL(1,𝔽8)—282no—
272C2×ΓL(1,𝔽8)—282no—
273C2×ΓL(1,𝔽8)—281no—
274C2×ΓL(1,𝔽8)—281no—
275C2×ΓL(1,𝔽8)—422no—
276AGL(1,𝔽7)—422no—
277AGL(1,𝔽7)—422no—
278AGL(1,𝔽7)—422no—
279AGL(1,𝔽7)—422no—
280AGL(1,𝔽7)—422no—
281AGL(1,𝔽7)—422no—
282AGL(1,𝔽7)—422no—
283AGL(1,𝔽7)—422no—
284AGL(1,𝔽7)—422no—
285C22×C32—491no—
286C2×C3×D3—492no—
287C2×C3×D3—492no—
288C2×C3×D3—982no—
289C2×C3×D3—982no—
290C32×C4—492no—
291C22×C7—21no—
292C22×C7—21no—
293C22×C7—21no—
294C22×C7—212no—
295C22×C7—422no—
296C22×C7D3×AGL(1,𝔽7)11yescharacteristic
297C22×C7C2×C3⋊AGL(1,𝔽7)11yescharacteristic
298C2×D7—32no—
299C2×D7—62no—
300C2×D7—212no—
301C2×D7—212no—
302C2×D7—422no—
303C2×D7—422no—
304C2×D7—422no—
305C2×D7—422no—
306C2×D7—422no—
307C2×D7—422no—
308C2×D7—422no—
309C2×D7—422no—
310C2×D7—422no—
311C2×D7—422no—
312C2×D7—422no—
313C4×C7—72no—
314Dic7C2×C3⋊AGL(1,𝔽7)12yescomplemented
315C23×C3—2941no—
316C23×C3—1471no—
317C22×D3—71no—
318C22×D3—491no—
319C3×D4—1474no—
320C2×C3×C4—71no—
321C2×C3×C4—981no—
322C2×C3×C4—981no—
323D12—494no—
324C3×C7—21no—
325C3×C7—21no—
326C3×C7—21no—
327C3×C7C2×C4⋊AGL(1,𝔽7)11yescharacteristic, complemented
328C3×C7C2×(C22⋊D7)⋊C311yescharacteristic, complemented
329ΓL(1,𝔽8)—141no—
330ΓL(1,𝔽8)—141no—
331ΓL(1,𝔽8)—141no—
332ΓL(1,𝔽8)—141no—
333ΓL(1,𝔽8)—141no—
334ΓL(1,𝔽8)—281no—
335ΓL(1,𝔽8)—281no—
336ΓL(1,𝔽8)—281no—
337ΓL(1,𝔽8)—281no—
338ΓL(1,𝔽8)—281no—
339C2×C32—491no—
340C2×C32—492no—
341C3×D3—982no—
342C3×D3—982no—
343C2×D4—1471noSylow
344C2×C7—22no—
345C2×C7—22no—
346C2×C7—22no—
347C2×C7—21no—
348C2×C7—21no—
349C2×C7—21no—
350C2×C7—422no—
351C2×C7#4247⋊C312yescomplemented
352C2×C7C4⋊(C3⋊AGL(1,𝔽7))12yescomplemented
353C2×C7C2×D3×AGL(1,𝔽7)11yescharacteristic
354C2×C7C22×C3⋊AGL(1,𝔽7)11yescharacteristic
355D7—62no—
356D7—422no—
357D7—422no—
358D7—422no—
359D7—422no—
360D7—422no—
361D7—422no—
362D7—422no—
363D7—422no—
364C22×C3—2942no—
365C22×C3—2942no—
366C22×C3—2942no—
367C22×C3—1472no—
368C22×C3—1961no—
369C22×C3—1961no—
370C22×C3#64423⋊C211yescharacteristic
371C2×D3—72no—
372C2×D3—142no—
373C2×D3—492no—
374C2×D3—982no—
375C3×C4—72no—
376C3×C4—982no—
377C3×C4—982no—
378C32—491noSylow
379C23—211no—
380C23—1471no—
381D4—1474no—
382C2×C4—71no—
383C7—21no—
384C7—21no—
385C7—21no—
386C7C2×#4247⋊C311yescharacteristic, complemented
387C7C2×C4⋊(C3⋊AGL(1,𝔽7))11yescharacteristic, complemented
388C2×C3—2942no—
389C2×C3—2942no—
390C2×C3—1962no—
391C2×C3—1962no—
392C2×C3—1961no—
393C2×C3—1961no—
394C2×C3#1192417612yescomplemented
395C2×C3C2×#64423⋊C211yescharacteristic
396D3—142no—
397D3—982no—
398C22—212no—
399C22—2942no—
400C22—422no—
401C22—1472no—
402C22#64423⋊D311yescharacteristic
403C4—72no—
404C3—1961no—
405C3—1961no—
406C3C2×#1192417611yescharacteristic, complemented
407C2—2942no—
408C2—422no—
409C2#7222205412yesdirect product factor
410C2C2×#64423⋊D311yescharacteristic
subgroupG/H
center, hypercenterC22#64423⋊D3
commutator, cosocleC2×C3×C72C23×C3
FrattiniC2C2×#64423⋊D3
Fitting, socleC22×C3×C72C22×C3
hypocenterC3×C72C2×C3×D4
LCSG ▷ C2×C3×C72 ▷ C3×C72
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C2×C3×#10596873C2semidirect
2. C22×C3⋊#64423C2semidirect
3. C2×#11924248C3semidirect
4. #72222054C2direct (N × H)
5. C22×C3⋊#64424C2semidirect
9. C2×C3⋊#64423C22semidirect
11. C2×C3⋊#64424C22semidirect
12. C3×#10596873C22semidirect
13. C22×C3⋊(C7⋊D7)C2×C3semidirect
14. C22×C7×D21C2×C3semidirect
17. C2×C3×C7×Dic7C2×C3semidirect
18. #11924248C2×C3semidirect
20. C2×#10596873D3semidirect
24. C2×C3×#3861D4semidirect
31. C2×C3⋊(C7⋊D7)C22×C3semidirect
33. C2×C7×D21C22×C3semidirect
40. C3×C7×Dic7C22×C3semidirect
42. #10596873C2×D3semidirect
54. C3×#3861C2×D4semidirect
57. C2×C7×Dic7C3×D3semidirect
63. C2×C3×C72C3×D4semidirect
69. C2×#3861D12semidirect
102. C7×Dic7C2×C3×D3semidirect
108. C22×D21AGL(1,𝔽7)semidirect
119. C2×C3×Dic7AGL(1,𝔽7)semidirect
127. C3×C72C2×C3×D4semidirect
129. #3861C2×D12semidirect
153. C2×C72C3×D12semidirect
176. C2×D21C2×AGL(1,𝔽7)semidirect
207. C3×Dic7C2×AGL(1,𝔽7)semidirect
230. C2×Dic7C3⋊AGL(1,𝔽7)semidirect
232. C72C2×C3×D12semidirect
241. C2×C3×C7C4⋊AGL(1,𝔽7)semidirect
242. C2×C3×C7(C22⋊D7)⋊C3semidirect
314. Dic7C2×C3⋊AGL(1,𝔽7)semidirect
327. C3×C7C2×C4⋊AGL(1,𝔽7)semidirect
328. C3×C7C2×(C22⋊D7)⋊C3semidirect
351. C2×C7#4247⋊C3semidirect
352. C2×C7C4⋊(C3⋊AGL(1,𝔽7))semidirect
386. C7C2×#4247⋊C3semidirect
387. C7C2×C4⋊(C3⋊AGL(1,𝔽7))semidirect
394. C2×C3#11924176semidirect
406. C3C2×#11924176semidirect
409. C2#72222054direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#105968732
C22×C3⋊#644232
C2×#119242483
#722220542
C22×C3⋊#644242
C2×#119241763
C2×C4⋊(C3⋊AGL(1,𝔽7))7
C2×#4247⋊C37