Finite Group Databasebeta

← C2×C29×#2356 #20329019

Subgroups — C2×C29×#2356 #20329019

Order 7424.

2550 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×Q8×C29C211yesmaximal, characteristic
2C29×#2356C2116yesmaximal, direct product factor
3C2×C29×#391C212yesmaximal
4C2×C29×#387C214yesmaximal
5C2×C29×#383C212yesmaximal
6C2×C29×#382C211yesmaximal, characteristic
7C2×C4×C4⋊C4×C29C212yesmaximal
8C2×C4×C4⋊C4×C29C211yesmaximal, characteristic
9C2×C29×#376C212yesmaximal
10C23×Q8×C29C2211yescharacteristic
11C2×C4×Q8×C29C4116yescomplemented
12C2×C4×Q8×C29C2218yes—
13C22×C4⋊C4×C29C2212yes—
14C22×C4⋊C4×C29C2212yes—
15C22×C4⋊C4×C29C2211yescharacteristic
16C22×C4⋊C4×C29C2212yes—
17C22×C42×C29C2214yes—
18C22×C42×C29C2212yes—
19C22×C42×C29C2212yes—
20C22×C42×C29C2211yescharacteristic
21C29×#391C22116yes—
22C29×#387C22132yes—
23C29×#383C22116yes—
24C29×#382C2218yes—
25C4×C4⋊C4×C29C22116yes—
26C4×C4⋊C4×C29C2218yes—
27C29×#376C22116yes—
28C2×C29×#95C2211yescharacteristic
29C2×C29×#95C2212yes—
30C2×C29×#95C2214yes—
31C2×C29×#95C2211yescharacteristic
32C2×C29×#95C2214yes—
33C2×C29×#95C2212yes—
34C2×C29×#95C2214yes—
35C22×Q8×C29—24no—
36C22×Q8×C29—24no—
37C22×Q8×C29C2×C418yes—
38C22×Q8×C29C2314yes—
39C23×C4×C29C2314yes—
40C23×C4×C29C2312yes—
41C23×C4×C29C2312yes—
42C23×C4×C29C2311yescharacteristic
43C23×C4×C29C2311yescharacteristic
44C23×C4×C29C2312yes—
45C23×C4×C29C2312yes—
46C23×C4×C29C2311yescharacteristic
47C4×Q8×C29C2×C4164yescomplemented
48C2×C4⋊C4×C29—28no—
49C2×C4⋊C4×C29—28no—
50C2×C4⋊C4×C29C2×C4116yes—
51C2×C4⋊C4×C29C2×C418yes—
52C2×C4⋊C4×C29C2318yes—
53C2×C4⋊C4×C29C2318yes—
54C2×C4⋊C4×C29C2314yes—
55C2×C4⋊C4×C29C2318yes—
56C2×C42×C29—216no—
57C2×C42×C29—28no—
58C2×C42×C29C2×C4116yes—
59C2×C42×C29C2×C418yes—
60C2×C42×C29C23116yes—
61C2×C42×C29C2318yes—
62C2×C42×C29C2318yes—
63C2×C42×C29C2314yes—
64C29×#95C2314yes—
65C29×#95C2318yes—
66C29×#95C23116yes—
67C29×#95C2314yes—
68C29×#95C23116yes—
69C29×#95C2318yes—
70C29×#95C23116yes—
71C24×C29C2411yescharacteristic
72C2×Q8×C29—232no—
73C2×Q8×C29—28no—
74C2×Q8×C29—28no—
75C2×Q8×C29C22×C4116yes—
76C22×C4×C29—28no—
77C22×C4×C29—24no—
78C22×C4×C29—24no—
79C22×C4×C29—24no—
80C22×C4×C29—24no—
81C22×C4×C29—24no—
82C22×C4×C29—24no—
83C22×C4×C29—24no—
84C22×C4×C29—28no—
85C22×C4×C29—24no—
86C22×C4×C29—28no—
87C22×C4×C29—24no—
88C22×C4×C29—28no—
89C22×C4×C29C22×C414yes—
90C22×C4×C29C22×C418yes—
91C22×C4×C29C22×C414yes—
92C22×C4×C29C22×C418yes—
93C22×C4×C29C22×C414yes—
94C22×C4×C29#4118yes—
95C22×C4×C29#4114yes—
96C22×C4×C29#4114yes—
97C22×C4×C29C2418yes—
98C22×C4×C29C2414yes—
99C22×C4×C29C2414yes—
100C22×C4×C29C2412yes—
101C22×C4×C29C2412yes—
102C22×C4×C29C2414yes—
103C22×C4×C29C2414yes—
104C22×C4×C29C2412yes—
105C4⋊C4×C29—216no—
106C4⋊C4×C29—216no—
107C4⋊C4×C29C22×C4132yes—
108C4⋊C4×C29C22×C4116yes—
109C42×C29—232no—
110C42×C29—216no—
111C42×C29C22×C4132yes—
112C42×C29C22×C4116yes—
113C2×#2356C2911yesmaximal, characteristic, direct product factor, Sylow
114C23×C29C23×C412yes—
115C23×C29C2×#4112yes—
116C23×C29C2×#4112yes—
117C23×C29C2×#4114yes—
118C23×C292−1+414yes—
119C23×C29C2511yescharacteristic
120Q8×C29—232no—
121C2×C4×C29—216no—
122C2×C4×C29—216no—
123C2×C4×C29—28no—
124C2×C4×C29—28no—
125C2×C4×C29—216no—
126C2×C4×C29—28no—
127C2×C4×C29—24no—
128C2×C4×C29—24no—
129C2×C4×C29—24no—
130C2×C4×C29—24no—
131C2×C4×C29—24no—
132C2×C4×C29—24no—
133C2×C4×C29—24no—
134C2×C4×C29—28no—
135C2×C4×C29—24no—
136C2×C4×C29—28no—
137C2×C4×C29—24no—
138C2×C4×C29—28no—
139C2×C4×C29—48no—
140C2×C4×C29—416no—
141C2×C4×C29#117116yes—
142C2×C4×C29C23×C414yes—
143C2×C4×C29C23×C418yes—
144C2×C4×C29C23×C414yes—
145C2×C4×C29C23×C418yes—
146C2×C4×C29C23×C414yes—
147C2×C4×C29C2×#4118yes—
148C2×C4×C29C2×#4114yes—
149C2×C4×C29C2×#4114yes—
150C22×C4×Q8C2×C2911yescharacteristic
151#2356C2×C29116yesdirect product factor
152C2×#391C2×C2912yes—
153C2×#387C2×C2914yes—
154C2×#383C2×C2912yes—
155C2×#382C2×C2911yescharacteristic
156C2×C4×C4⋊C4C2×C2912yes—
157C2×C4×C4⋊C4C2×C2911yescharacteristic
158C2×#376C2×C2912yes—
159C22×C29C2×#11714yes—
160C22×C29C4×#4114yes—
161C22×C29#51914yes—
162C22×C29#55218yes—
163C22×C29#55618yes—
164C22×C29C24×C411yescharacteristic
165C22×C29C22×#4111yescharacteristic
166C22×C29C22×#4111yescharacteristic
167C22×C29C22×#4112yes—
168C22×C29C2×2−1+412yes—
169C4×C29—28no—
170C4×C29—28no—
171C4×C29—24no—
172C4×C29—24no—
173C4×C29—28no—
174C4×C29—44no—
175C4×C29—48no—
176C4×C29C2×#11718yes—
177C23×Q8C22×C2911yescharacteristic
178C2×C4×Q8C4×C29116yescomplemented
179C2×C4×Q8C22×C2918yes—
180C22×C4⋊C4C22×C2912yes—
181C22×C4⋊C4C22×C2912yes—
182C22×C4⋊C4C22×C2911yescharacteristic
183C22×C4⋊C4C22×C2912yes—
184C22×C42C22×C2914yes—
185C22×C42C22×C2912yes—
186C22×C42C22×C2912yes—
187C22×C42C22×C2911yescharacteristic
188#391C22×C29116yes—
189#387C22×C29132yes—
190#383C22×C29116yes—
191#382C22×C2918yes—
192C4×C4⋊C4C22×C2918yes—
193C4×C4⋊C4C22×C29116yes—
194#376C22×C29116yes—
195C2×#95C22×C2911yescharacteristic
196C2×#95C22×C2912yes—
197C2×#95C22×C2914yes—
198C2×#95C22×C2911yescharacteristic
199C2×#95C22×C2914yes—
200C2×#95C22×C2912yes—
201C2×#95C22×C2914yes—
202C2×C29#235618yesdirect product factor
203C2×C29C22×#11711yescharacteristic
204C2×C29C2×C4×#4111yescharacteristic
205C2×C29C2×#51911yescharacteristic
206C2×C29C2×#55212yes—
207C2×C29C2×#55612yes—
208C22×Q8—24no—
209C22×Q8—24no—
210C22×Q8C2×C4×C2918yes—
211C22×Q8C23×C2914yes—
212C23×C4C23×C2914yes—
213C23×C4C23×C2912yes—
214C23×C4C23×C2911yescharacteristic
215C23×C4C23×C2912yes—
216C23×C4C23×C2912yes—
217C23×C4C23×C2911yescharacteristic
218C23×C4C23×C2912yes—
219C23×C4C23×C2911yescharacteristic
220C4×Q8C2×C4×C29164yescomplemented
221C2×C4⋊C4—28no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4C2×C4×C29116yes—
224C2×C4⋊C4C2×C4×C2918yes—
225C2×C4⋊C4C23×C2918yes—
226C2×C4⋊C4C23×C2918yes—
227C2×C4⋊C4C23×C2914yes—
228C2×C4⋊C4C23×C2918yes—
229C2×C42—216no—
230C2×C42—28no—
231C2×C42C2×C4×C29116yes—
232C2×C42C2×C4×C2918yes—
233C2×C42C23×C29116yes—
234C2×C42C23×C2918yes—
235C2×C42C23×C2918yes—
236C2×C42C23×C2914yes—
237#95C23×C2914yes—
238#95C23×C2918yes—
239#95C23×C29116yes—
240#95C23×C2914yes—
241#95C23×C29116yes—
242#95C23×C2918yes—
243#95C23×C29116yes—
244C29C2×#235611yescharacteristic, direct product factor, Sylow
245C24C24×C2911yescharacteristic
246C2×Q8—232no—
247C2×Q8—28no—
248C2×Q8—28no—
249C2×Q8C22×C4×C29116yes—
250C22×C4—28no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—28no—
259C22×C4—24no—
260C22×C4—28no—
261C22×C4—24no—
262C22×C4—28no—
263C22×C4C22×C4×C2914yes—
264C22×C4C22×C4×C2914yes—
265C22×C4C22×C4×C2918yes—
266C22×C4C22×C4×C2918yes—
267C22×C4C22×C4×C2914yes—
268C22×C4#41×C2918yes—
269C22×C4#41×C2914yes—
270C22×C4#41×C2914yes—
271C22×C4C24×C2918yes—
272C22×C4C24×C2914yes—
273C22×C4C24×C2912yes—
274C22×C4C24×C2914yes—
275C22×C4C24×C2914yes—
276C22×C4C24×C2912yes—
277C22×C4C24×C2914yes—
278C22×C4C24×C2912yes—
279C4⋊C4—216no—
280C4⋊C4—216no—
281C4⋊C4C22×C4×C29132yes—
282C4⋊C4C22×C4×C29116yes—
283C42—232no—
284C42—216no—
285C42C22×C4×C29132yes—
286C42C22×C4×C29116yes—
287C23C23×C4×C2912yes—
288C23C2×#41×C2912yes—
289C23C2×#41×C2912yes—
290C23C2×#41×C2914yes—
291C23C29×2−1+414yes—
292C23C25×C2911yescharacteristic
293Q8—232no—
294C2×C4—216no—
295C2×C4—216no—
296C2×C4—28no—
297C2×C4—28no—
298C2×C4—216no—
299C2×C4—24no—
300C2×C4—28no—
301C2×C4—24no—
302C2×C4—24no—
303C2×C4—24no—
304C2×C4—24no—
305C2×C4—24no—
306C2×C4—24no—
307C2×C4—28no—
308C2×C4—24no—
309C2×C4—28no—
310C2×C4—24no—
311C2×C4—28no—
312C2×C4—48no—
313C2×C4—416no—
314C2×C4C29×#117116yes—
315C2×C4C23×C4×C2914yes—
316C2×C4C23×C4×C2918yes—
317C2×C4C23×C4×C2914yes—
318C2×C4C23×C4×C2918yes—
319C2×C4C23×C4×C2914yes—
320C2×C4C2×#41×C2918yes—
321C2×C4C2×#41×C2914yes—
322C2×C4C2×#41×C2914yes—
323C22C2×C29×#11714yes—
324C22C4×#41×C2914yes—
325C22C29×#51914yes—
326C22C29×#55218yes—
327C22C29×#55618yes—
328C22C24×C4×C2911yescharacteristic
329C22C22×#41×C2911yescharacteristic
330C22C22×#41×C2912yes—
331C22C22×#41×C2911yescharacteristic
332C22C2×C29×2−1+412yes—
333C4—28no—
334C4—28no—
335C4—28no—
336C4—24no—
337C4—24no—
338C4—44no—
339C4—48no—
340C4C2×C29×#11718yes—
341C2C29×#235618yesdirect product factor
342C2C22×C29×#11711yescharacteristic
343C2C2×C4×#41×C2911yescharacteristic
344C2C2×C29×#51911yescharacteristic
345C2C2×C29×#55212yes—
346C2C2×C29×#55612yes—
subgroupG/H
center, socle, prime radicalC24×C29C24
commutatorC22C24×C4×C29
Frattini, cosocleC23C25×C29
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C24×C29 ◁ G

Complements

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

NH ≅ G/Ntype
2. C29×#2356C2direct (N × H)
11. C2×C4×Q8×C29C4semidirect
47. C4×Q8×C29C2×C4semidirect
113. C2×#2356C29direct (N × H)
151. #2356C2×C29direct (N × H)
178. C2×C4×Q8C4×C29semidirect
202. C2×C29#2356direct (N × H)
220. C4×Q8C2×C4×C29semidirect
244. C29C2×#2356direct (N × H)
341. C2C29×#2356direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×Q8×C292
C29×#23562
C2×C29×#3912
C2×C29×#3872
C2×C29×#3832
C2×C29×#3822
C2×C4×C4⋊C4×C292
C2×C29×#3762
C2×#235629