Finite Group Databasebeta

← #13054007

Subgroups — #13054007

Order 1344.

380 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 112 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×C4×Dic21C211yesmaximal, characteristic, complemented
2C2×C7×Dic3⋊C4C211yesmaximal, characteristic, complemented
3C2×C3×(C2×D7)⋊C4C211yesmaximal, characteristic
4C2×(C2×D7)⋊Dic3C211yesmaximal, characteristic
5C2×(C2×D7)⋊Dic3C211yesmaximal, characteristic
6#10620529C211yesmaximal, characteristic, complemented
7#10620529C211yesmaximal, characteristic, complemented
8#89762—31nomaximal, Hall
9C23×C3×D7C2211yescharacteristic
10C22×C3×C4×C7C2211yescharacteristic
11C22×Dic21C2211yescharacteristic
12C22×Dic21C2211yescharacteristic
13C22×C7×Dic3C2211yescharacteristic
14C22×C7×Dic3C2211yescharacteristic
15C22×C3×Dic7C2211yescharacteristic
16C4×Dic21—22no—
17C7×Dic3⋊C4—22no—
18C3×(C2×D7)⋊C4—22no—
19(C2×D7)⋊Dic3—22no—
20(C2×D7)⋊Dic3C414yescomplemented
21C2×C7×C4⋊C4—31no—
22C2×(C2×D7)⋊C4—31no—
23C2×(C2×D7)⋊C4—31no—
24C2×(C2×D7)⋊C4D311yescharacteristic
25C2×C4×Dic7—31no—
26C7⋊#95—31no—
27C7⋊#95—31no—
28#4945—71nomaximal, Hall
29C23×C3×C7C2311yescharacteristic
30C22×C3×D7—21no—
31C22×C3×D7—21no—
32C22×C3×D7—21no—
33C22×C3×D7—21no—
34C22×C3×D7—21no—
35C22×C3×D7—21no—
36C22×C3×D7C2×C412yes—
37C2×C3×C4×C7—21no—
38C2×C3×C4×C7—21no—
39C2×C3×C4×C7D412yes—
40C2×Dic21—21no—
41C2×Dic21—21no—
42C2×Dic21—21no—
43C2×Dic21—21no—
44C2×Dic21C2×C412yes—
45C2×Dic21D412yes—
46C2×C7×Dic3—21no—
47C2×C7×Dic3—21no—
48C2×C7×Dic3—21no—
49C2×C7×Dic3—21no—
50C2×C7×Dic3—21no—
51C2×C7×Dic3C2×C412yes—
52C2×C3×Dic7—21no—
53C2×C3×Dic7—21no—
54C2×C3×Dic7—21no—
55C23×D7C2×D311yescharacteristic
56C22×C4×C7—31no—
57C22×C4×C7—31no—
58C22×C4×C7C2×D311yescharacteristic
59C22×Dic7—31no—
60C22×Dic7—31no—
61C22×Dic7C2×D311yescharacteristic
62C7×C4⋊C4—62no—
63(C2×D7)⋊C4—22no—
64(C2×D7)⋊C4—34no—
65(C2×D7)⋊C4—62no—
66C4×Dic7—62no—
67C2×C3×C22⋊C4—71no—
68C2×#217—71no—
69C2×#217—71no—
70C2×Dic3⋊C4D711yescharacteristic, complemented
71C2×C4×Dic3—71no—
72C3⋊#95—71no—
73C3⋊#95—71no—
74C22×C3×C7C22×C411yescharacteristic
75C22×C3×C7C2×D411yescharacteristic
76C22×C3×C7C2×D411yescharacteristic
77C22×C3×C7#4111yescharacteristic
78C22×C3×C7#4111yescharacteristic
79C22×C3×C7#4111yescharacteristic
80C22×C3×C7#4111yescharacteristic
81C2×C3×D7—22no—
82C2×C3×D7—22no—
83C2×C3×D7—22no—
84C2×C3×D7—41no—
85C2×C3×D7—41no—
86C2×C3×D7—41no—
87C2×C3×D7—41no—
88C3×C4×C7—22no—
89Dic21—22no—
90Dic21—22no—
91C3×Dic7—41no—
92C7×Dic3—22no—
93C7×Dic3—41no—
94#388—211noSylow
95C23×C7C22×D311yescharacteristic
96C22×D7—21no—
97C22×D7—21no—
98C22×D7—21no—
99C22×D7—21no—
100C22×D7—21no—
101C22×D7—21no—
102C22×D7C4×D312yes—
103C2×C4×C7—21no—
104C2×C4×C7—21no—
105C2×C4×C7—32no—
106C2×C4×C7—61no—
107C2×C4×C7—61no—
108C2×C4×C7—61no—
109C2×C4×C7—61no—
110C2×C4×C7—61no—
111C2×C4×C7#6712yes—
112C2×Dic7—21no—
113C2×Dic7—21no—
114C2×Dic7—21no—
115C2×Dic7—32no—
116C2×Dic7—32no—
117C2×Dic7—61no—
118C2×Dic7—61no—
119C2×Dic7—61no—
120C2×Dic7—61no—
121C24×C3—71no—
122C22×C3×C4—71no—
123C22×C3×C4C2×D711yescharacteristic
124C22×Dic3—71no—
125C22×Dic3—71no—
126C22×Dic3C2×D711yescharacteristic
127C22×Dic3C2×D711yescharacteristic
128C3×C22⋊C4—142no—
129#217—74no—
130#217—142no—
131Dic3⋊C4—22no—
132C4×Dic3—142no—
133C2×C3×C7#11711yescharacteristic
134C2×C3×C7C4×D411yescharacteristic
135C2×C3×C7C4×D411yescharacteristic
136C2×C3×C7#12111yescharacteristic
137C2×C3×C7#12311yescharacteristic
138C2×C3×C7#12411yescharacteristic
139C2×C3×C7#12611yescharacteristic
140C3×D7—42no—
141C2×C4⋊C4—31no—
142C2×C22⋊C4—71no—
143C2×C22⋊C4—211no—
144C2×C22⋊C4—211no—
145C2×C42—211no—
146#95—211no—
147#95—211no—
148C22×C7C2×C4×D311yescharacteristic
149C22×C7#23511yescharacteristic
150C22×C7D3×D411yescharacteristic
151C22×C7#23711yescharacteristic
152C22×C7#23711yescharacteristic
153C22×C7#23711yescharacteristic
154C22×C7C2×#6711yescharacteristic
155C2×D7—22no—
156C2×D7—22no—
157C2×D7—22no—
158C2×D7—41no—
159C2×D7—41no—
160C2×D7—41no—
161C2×D7—41no—
162C4×C7—22no—
163C4×C7—62no—
164C4×C7—121no—
165Dic7—41no—
166Dic7—62no—
167Dic7—62no—
168C23×C3—72no—
169C23×C3—141no—
170C23×C3—141no—
171C23×C3—141no—
172C23×C3—141no—
173C23×C3—141no—
174C23×C3—141no—
175C23×C3C22×D711yescharacteristic
176C2×C3×C4—21no—
177C2×C3×C4—21no—
178C2×C3×C4—141no—
179C2×C3×C4—141no—
180C2×C3×C4—141no—
181C2×C3×C4C22⋊D712yes—
182C2×Dic3—21no—
183C2×Dic3—21no—
184C2×Dic3—21no—
185C2×Dic3—21no—
186C2×Dic3—21no—
187C2×Dic3—72no—
188C2×Dic3—72no—
189C2×Dic3—141no—
190C2×Dic3—141no—
191C2×Dic3—141no—
192C2×Dic3—141no—
193C2×Dic3C4×D712yes—
194C3×C7#38811yescharacteristic, complemented, Hall
195C24—71no—
196C22×C4—31no—
197C22×C4—31no—
198C22×C4—71no—
199C22×C4—211no—
200C22×C4—211no—
201C22×C4D3×D711yescharacteristic
202C4⋊C4—62no—
203C22⋊C4—142no—
204C22⋊C4—214no—
205C22⋊C4—422no—
206C42—422no—
207C2×C7#87711yescharacteristic
208C2×C7#87911yescharacteristic
209C2×C7#67⋊C411yescharacteristic
210C2×C7#88311yescharacteristic
211C2×C7C4×#6711yescharacteristic
212C2×C7#93611yescharacteristic
213C2×C7#93911yescharacteristic
214D7—42no—
215C22×C3—142no—
216C22×C3—142no—
217C22×C3—142no—
218C22×C3—281no—
219C22×C3—281no—
220C22×C3—281no—
221C22×C3—281no—
222C22×C3C2×C4×D711yescharacteristic
223C22×C3#115711yescharacteristic
224C22×C3D4×D711yescharacteristic
225C22×C3#115911yescharacteristic
226C22×C3#116111yescharacteristic
227C22×C3#116111yescharacteristic
228C22×C3C2×C22⋊D711yescharacteristic
229C3×C4—22no—
230C3×C4—281no—
231Dic3—22no—
232Dic3—41no—
233Dic3—142no—
234Dic3—142no—
235C23—72no—
236C23—141no—
237C23—141no—
238C23—141no—
239C23—141no—
240C23—141no—
241C23—141no—
242C23C2×D3×D711yescharacteristic
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—32no—
246C2×C4—61no—
247C2×C4—61no—
248C2×C4—61no—
249C2×C4—61no—
250C2×C4—61no—
251C2×C4—141no—
252C2×C4—141no—
253C2×C4—141no—
254C2×C4—212no—
255C2×C4—212no—
256C2×C4—421no—
257C2×C4—421no—
258C2×C4—421no—
259C2×C4—421no—
260C2×C4#424512yes—
261C7#494511yescharacteristic, complemented, Sylow
262C2×C3—282no—
263C2×C3#645511yescharacteristic
264C2×C3#645611yescharacteristic
265C2×C3#645711yescharacteristic
266C2×C3#646111yescharacteristic
267C2×C3C4×C22⋊D711yescharacteristic
268C2×C3#650211yescharacteristic
269C2×C3#651011yescharacteristic
270C22—142no—
271C22—142no—
272C22—142no—
273C22—281no—
274C22—281no—
275C22—281no—
276C22—281no—
277C22#6667011yescharacteristic
278C22#6667011yescharacteristic
279C22#6667511yescharacteristic
280C22C4×D3×D711yescharacteristic
281C22#6668411yescharacteristic
282C22C2×#424511yescharacteristic
283C22#6669411yescharacteristic
284C4—22no—
285C4—62no—
286C4—121no—
287C4—281no—
288C4—422no—
289C4—422no—
290C3#8976211yescharacteristic, complemented, Sylow
291C2—282no—
292C2#1062091611yescharacteristic
293C2#1062092811yescharacteristic
294C2#1062093011yescharacteristic
295C2#1062095611yescharacteristic
296C2#1062095811yescharacteristic
297C2C4×#424511yescharacteristic
298C2#1062113311yescharacteristic
subgroupG/H
center, FrattiniC23C2×D3×D7
commutatorC22×C3×C7C22×C4
FittingC22×C3×C4×C7C22
socle, cosocleC23×C3×C7C23
prime radicalC23×C3×D7C22
hypercenterC22×C4D3×D7
hypocenterC3×C7#388
LCSG ▷ C22×C3×C7 ▷ C3×C7
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×C4×Dic21C2semidirect
2. C2×C7×Dic3⋊C4C2semidirect
6. #10620529C2semidirect
7. #10620529C2semidirect
20. (C2×D7)⋊Dic3C4semidirect
70. C2×Dic3⋊C4D7semidirect
194. C3×C7#388semidirect
261. C7#4945semidirect
290. C3#89762semidirect

Maximal subgroups

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

subgroupindex
C2×C4×Dic212
C2×C7×Dic3⋊C42
C2×C3×(C2×D7)⋊C42
C2×(C2×D7)⋊Dic32
#106205292
#897623
#49457