Finite Group Databasebeta

← C3×#89720 #13055385

Subgroups — C3×#89720 #13055385

Order 1344.

372 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C7×C22⋊C4C211yesmaximal, characteristic
2C2×C3×C22⋊Dic7C211yesmaximal, characteristic
3C2×C3×C22⋊Dic7C211yesmaximal, characteristic
4C2×C3×C4⋊Dic7C211yesmaximal, characteristic, complemented
5C2×C3×Dic7⋊C4C211yesmaximal, characteristic, complemented
6C2×C3×Dic7⋊C4C211yesmaximal, characteristic, complemented
7C3×C7⋊#95C211yesmaximal, characteristic, complemented
8#89720C311yesmaximal, characteristic, direct product factor, Hall
9C24×C3×C7C2211yescharacteristic
10C22×C3×C4×C7C2211yescharacteristic
11C22×C3×C4×C7C2211yescharacteristic
12C22×C3×Dic7C2211yescharacteristic
13C22×C3×Dic7C2211yescharacteristic
14C22×C3×Dic7C2211yescharacteristic
15C22×C3×Dic7C2211yescharacteristic
16C3×C7×C22⋊C4—22no—
17C3×C22⋊Dic7—22no—
18C3×C22⋊Dic7—22no—
19C3×C4⋊Dic7—22no—
20C3×Dic7⋊C4—22no—
21C3×Dic7⋊C4—22no—
22C2×C7×C22⋊C4C2×C311yescharacteristic
23C2×C22⋊Dic7C2×C311yescharacteristic
24C2×C22⋊Dic7C2×C311yescharacteristic
25C2×C4⋊Dic7C2×C311yescharacteristic, complemented
26C2×Dic7⋊C4C2×C311yescharacteristic, complemented
27C2×Dic7⋊C4C2×C311yescharacteristic, complemented
28C7⋊#95C2×C311yescharacteristic, complemented
29C3×#396—71nomaximal, Hall
30C23×C3×C7—21no—
31C23×C3×C7—21no—
32C23×C3×C7—21no—
33C23×C3×C7—21no—
34C23×C3×C7—21no—
35C23×C3×C7—21no—
36C23×C3×C7Q812yes—
37C23×C3×C7C2311yescharacteristic
38C2×C3×C4×C7—21no—
39C2×C3×C4×C7—21no—
40C2×C3×C4×C7—21no—
41C2×C3×C4×C7—21no—
42C2×C3×C4×C7—21no—
43C2×C3×C4×C7D412yes—
44C2×C3×Dic7—21no—
45C2×C3×Dic7—21no—
46C2×C3×Dic7—21no—
47C2×C3×Dic7—21no—
48C2×C3×Dic7—21no—
49C2×C3×Dic7—21no—
50C2×C3×Dic7—21no—
51C2×C3×Dic7—21no—
52C2×C3×Dic7—21no—
53C2×C3×Dic7—21no—
54C2×C3×Dic7D412yes—
55C2×C3×Dic7D412yes—
56C24×C7C22×C311yescharacteristic
57C22×C4×C7C22×C311yescharacteristic
58C22×C4×C7C22×C311yescharacteristic
59C22×Dic7C22×C311yescharacteristic
60C22×Dic7C22×C311yescharacteristic
61C22×Dic7C22×C311yescharacteristic
62C22×Dic7C22×C311yescharacteristic
63C7×C22⋊C4—22no—
64C22⋊Dic7—22no—
65C22⋊Dic7—22no—
66C4⋊Dic7—22no—
67Dic7⋊C4—22no—
68Dic7⋊C4—22no—
69C2×C3×C4⋊C4—71no—
70C2×C3×C4⋊C4—71no—
71C2×C3×C4⋊C4—71no—
72C2×C3×C22⋊C4—71no—
73C2×C3×C22⋊C4—71no—
74C2×C3×C22⋊C4D711yescharacteristic
75C3×#95—71no—
76C22×C3×C7—22no—
77C22×C3×C7—22no—
78C22×C3×C7—22no—
79C22×C3×C7—41no—
80C22×C3×C7—41no—
81C22×C3×C7—41no—
82C22×C3×C7—41no—
83C22×C3×C7C2×D411yescharacteristic
84C22×C3×C7C2×D411yescharacteristic
85C22×C3×C7C2×D411yescharacteristic
86C22×C3×C7C2×Q811yescharacteristic
87C22×C3×C7#4111yescharacteristic
88C22×C3×C7#4111yescharacteristic
89C22×C3×C7#4111yescharacteristic
90C3×C4×C7—22no—
91C3×C4×C7—41no—
92C3×Dic7—22no—
93C3×Dic7—22no—
94C3×Dic7—41no—
95C3×Dic7—41no—
96#396—71noSylow
97C23×C7—21no—
98C23×C7—21no—
99C23×C7—21no—
100C23×C7—21no—
101C23×C7—21no—
102C23×C7—21no—
103C23×C7C3×Q812yes—
104C23×C7C23×C311yescharacteristic
105C2×C4×C7—21no—
106C2×C4×C7—21no—
107C2×C4×C7—21no—
108C2×C4×C7—21no—
109C2×C4×C7—21no—
110C2×C4×C7C3×D412yes—
111C2×Dic7—21no—
112C2×Dic7—21no—
113C2×Dic7—21no—
114C2×Dic7—21no—
115C2×Dic7—21no—
116C2×Dic7—21no—
117C2×Dic7—21no—
118C2×Dic7—21no—
119C2×Dic7—21no—
120C2×Dic7—21no—
121C2×Dic7C3×D412yes—
122C2×Dic7C3×D412yes—
123C24×C3C2×D711yescharacteristic
124C22×C3×C4—71no—
125C22×C3×C4—71no—
126C22×C3×C4—71no—
127C22×C3×C4—71no—
128C22×C3×C4C2×D711yescharacteristic
129C22×C3×C4C2×D711yescharacteristic
130C3×C4⋊C4—142no—
131C3×C4⋊C4—142no—
132C3×C4⋊C4—142no—
133C3×C22⋊C4—22no—
134C3×C22⋊C4—142no—
135C3×C22⋊C4—142no—
136C2×C3×C7—42no—
137C2×C3×C7#12111yescharacteristic
138C2×C3×C7#12111yescharacteristic
139C2×C3×C7#12111yescharacteristic
140C2×C3×C7C22⋊Q811yescharacteristic
141C2×C3×C7C22⋊Q811yescharacteristic
142C2×C3×C7C22⋊Q811yescharacteristic
143C2×C3×C7#12611yescharacteristic
144C2×C4⋊C4—71no—
145C2×C4⋊C4—71no—
146C2×C4⋊C4—71no—
147C2×C22⋊C4—71no—
148C2×C22⋊C4—71no—
149C2×C22⋊C4C3×D711yescharacteristic
150#95—71no—
151C22×C7—22no—
152C22×C7—22no—
153C22×C7—22no—
154C22×C7—41no—
155C22×C7—41no—
156C22×C7—41no—
157C22×C7—41no—
158C22×C7C2×C3×D411yescharacteristic
159C22×C7C2×C3×D411yescharacteristic
160C22×C7C2×C3×D411yescharacteristic
161C22×C7C2×C3×Q811yescharacteristic
162C22×C7C3×#4111yescharacteristic
163C22×C7C3×#4111yescharacteristic
164C22×C7C3×#4111yescharacteristic
165C4×C7—22no—
166C4×C7—41no—
167Dic7—22no—
168Dic7—22no—
169Dic7—41no—
170Dic7—41no—
171C23×C3—21no—
172C23×C3—21no—
173C23×C3—21no—
174C23×C3—21no—
175C23×C3—21no—
176C23×C3—21no—
177C23×C3Dic1412yes—
178C23×C3C22×D711yescharacteristic
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×C3×C4—21no—
183C2×C3×C4—21no—
184C2×C3×C4—72no—
185C2×C3×C4—72no—
186C2×C3×C4—141no—
187C2×C3×C4—141no—
188C2×C3×C4—141no—
189C2×C3×C4—141no—
190C2×C3×C4—141no—
191C2×C3×C4—141no—
192C2×C3×C4—141no—
193C2×C3×C4—141no—
194C2×C3×C4—141no—
195C2×C3×C4—141no—
196C2×C3×C4C22⋊D712yes—
197C3×C7#39611yescharacteristic, complemented, Hall
198C24C2×C3×D711yescharacteristic
199C22×C4—71no—
200C22×C4—71no—
201C22×C4—71no—
202C22×C4—71no—
203C22×C4C2×C3×D711yescharacteristic
204C22×C4C2×C3×D711yescharacteristic
205C4⋊C4—142no—
206C4⋊C4—142no—
207C4⋊C4—142no—
208C22⋊C4—22no—
209C22⋊C4—142no—
210C22⋊C4—142no—
211C2×C7—42no—
212C2×C7C3×#12111yescharacteristic
213C2×C7C3×#12111yescharacteristic
214C2×C7C3×#12111yescharacteristic
215C2×C7C3×C22⋊Q811yescharacteristic
216C2×C7C3×C22⋊Q811yescharacteristic
217C2×C7C3×C22⋊Q811yescharacteristic
218C2×C7C3×#12611yescharacteristic
219C22×C3—22no—
220C22×C3—22no—
221C22×C3—22no—
222C22×C3—41no—
223C22×C3—41no—
224C22×C3—41no—
225C22×C3—41no—
226C22×C3C2×Dic1411yescharacteristic
227C22×C3#115711yescharacteristic
228C22×C3D4×D711yescharacteristic
229C22×C3D4×D711yescharacteristic
230C22×C3#115911yescharacteristic
231C22×C3#115911yescharacteristic
232C22×C3C2×C22⋊D711yescharacteristic
233C3×C4—22no—
234C3×C4—41no—
235C3×C4—142no—
236C3×C4—142no—
237C3×C4—281no—
238C3×C4—281no—
239C23—21no—
240C23—21no—
241C23—21no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23C3×Dic1412yes—
246C23C22×C3×D711yescharacteristic
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—72no—
253C2×C4—72no—
254C2×C4—141no—
255C2×C4—141no—
256C2×C4—141no—
257C2×C4—141no—
258C2×C4—141no—
259C2×C4—141no—
260C2×C4—141no—
261C2×C4—141no—
262C2×C4—141no—
263C2×C4—141no—
264C2×C4C3×C22⋊D712yes—
265C7C3×#39611yescharacteristic, complemented, Sylow
266C2×C3—42no—
267C2×C3#644111yescharacteristic
268C2×C3#644111yescharacteristic
269C2×C3#644211yescharacteristic
270C2×C3#644711yescharacteristic
271C2×C3#648711yescharacteristic
272C2×C3#650111yescharacteristic
273C2×C3#650211yescharacteristic
274C22—22no—
275C22—22no—
276C22—22no—
277C22—41no—
278C22—41no—
279C22—41no—
280C22—41no—
281C22C2×C3×Dic1411yescharacteristic
282C22C3×#115711yescharacteristic
283C22C3×D4×D711yescharacteristic
284C22C3×D4×D711yescharacteristic
285C22C3×#115911yescharacteristic
286C22C3×#115911yescharacteristic
287C22C2×C3×C22⋊D711yescharacteristic
288C4—22no—
289C4—41no—
290C4—142no—
291C4—142no—
292C4—281no—
293C4—281no—
294C3#8972011yescharacteristic, direct product factor, Sylow
295C2—42no—
296C2C3×#644111yescharacteristic
297C2C3×#644111yescharacteristic
298C2C3×#644211yescharacteristic
299C2C3×#644711yescharacteristic
300C2C3×#648711yescharacteristic
301C2C3×#650111yescharacteristic
302C2C3×#650211yescharacteristic
subgroupG/H
centerC23×C3C22×D7
commutator, cosocleC23×C7C23×C3
FrattiniC23C22×C3×D7
FittingC2×C3×C7×C22⋊C4C2
socleC23×C3×C7C23
prime radicalC24×C3×C7C22
hypercenterC2×C3×C22⋊C4D7
hypocenterC7C3×#396
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23×C3 ◁ C2×C3×C22⋊C4

Complements

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

NH ≅ G/Ntype
4. C2×C3×C4⋊Dic7C2semidirect
5. C2×C3×Dic7⋊C4C2semidirect
6. C2×C3×Dic7⋊C4C2semidirect
7. C3×C7⋊#95C2semidirect
8. #89720C3direct (N × H)
25. C2×C4⋊Dic7C2×C3semidirect
26. C2×Dic7⋊C4C2×C3semidirect
27. C2×Dic7⋊C4C2×C3semidirect
28. C7⋊#95C2×C3semidirect
197. C3×C7#396semidirect
265. C7C3×#396semidirect
294. C3#89720direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C7×C22⋊C42
C2×C3×C22⋊Dic72
C2×C3×C4⋊Dic72
C2×C3×Dic7⋊C42
C3×C7⋊#952
#897203
C3×#3967