Finite Group Databasebeta

← #13058060

Subgroups — #13058060

Order 1344.

660 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 660 conjugacy classes of subgroups — too many to draw. Only its 152 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×#66683C211yesmaximal, characteristic, complemented
2C22×Dic3×D7C211yesmaximal, characteristic, complemented
3#10621369C211yesmaximal, characteristic, complemented
4C7×#877C211yesmaximal, characteristic, complemented
5C3×#6445C211yesmaximal, characteristic, complemented
6#10621126C211yesmaximal, characteristic, complemented
7#10621099C211yesmaximal, characteristic, complemented
8Dic3×D28C212yesmaximal, complemented
9#10620984C212yesmaximal, complemented
10#10620971C212yesmaximal, complemented
11#10620957C212yesmaximal, complemented
12#90215—31nomaximal, Hall
13C23×C3×D7C2211yescharacteristic
14C2×#4268C2211yescharacteristic
15C22×Dic21C2211yescharacteristic
16C22×C7×Dic3C2211yescharacteristic
17C2×C3×C22⋊D7C2211yescharacteristic
18C2×C3×D28C2212yes—
19C2×#4248C2211yescharacteristic, complemented
20C2×#4246C2212yes—
21C2×#4246C2211yescharacteristic, complemented
22C2×D21⋊C4C2211yescharacteristic, complemented
23#66683—24no—
24C2×Dic3×D7—24no—
25C2×Dic3×D7C2212yes—
26C2×Dic3×D7C2212yes—
27C2×Dic3×D7C2211yescharacteristic, complemented
28C3×C7×C22⋊C4C2211yescharacteristic
29#66631C2212yescomplemented
30C4⋊Dic21C2212yescomplemented
31C7×#217C2211yescharacteristic
32C7×Dic3⋊C4C2212yescomplemented
33C4×C7×Dic3C2212yescomplemented
34C3×(C2×D7)⋊C4C2212yes—
35Dic3⋊Dic7C2212yescomplemented
36(C2×D7)⋊Dic3C2212yes—
37(C2×D7)⋊Dic3C2212yescomplemented
38C2×#1157—31no—
39C22×C4×D7—31no—
40C7×#117—31no—
41#6460—32no—
42#6458—32no—
43#6449—31no—
44#6449—31no—
45#6445—31no—
46#6445D311yescharacteristic, complemented
47C4×D28—32no—
48C4×D28—32no—
49#5586—71nomaximal, Hall
50C23×C3×C7C2311yescharacteristic
51C22×D21C2311yescharacteristic
52C22×C3×D7—22no—
53C22×C3×D7—24no—
54C22×C3×D7C2311yescharacteristic
55C22×C3×D7C2312yes—
56C2×C3×C4×C7C2312yes—
57#4268—22no—
58C2×Dic21—22no—
59C2×Dic21C2311yescharacteristic
60C2×Dic21C2312yes—
61C2×C7×Dic3D414yescomplemented
62C2×C7×Dic3C2312yes—
63C2×C7×Dic3C2311yescharacteristic
64C2×C7×Dic3C2311yescharacteristic
65C3×C22⋊D7—22no—
66C2×C3×Dic7C2311yescharacteristic
67C3×D28—24no—
68#4248—22no—
69#4246—24no—
70#4246—22no—
71D21⋊C4—22no—
72Dic3×D7—24no—
73Dic3×D7—24no—
74Dic3×D7—22no—
75C23×D7C2×D311yescharacteristic
76C22×C4×C7—31no—
77C2×C22⋊D7—31no—
78C2×C22⋊D7C2×D311yescharacteristic
79C22×Dic7—31no—
80#1157—64no—
81C2×D28—32no—
82C2×D28—31no—
83C2×D28C2×D312yes—
84C2×C4×D7—32no—
85C2×C4×D7—31no—
86C2×C4×D7—32no—
87C2×C4×D7—31no—
88C2×C4×D7—64no—
89C2×Dic14—31no—
90C7×C4⋊C4—32no—
91C7×C22⋊C4—31no—
92C7×C22⋊C4C2×D311yescharacteristic
93C42×C7—32no—
94(C2×D7)⋊C4—32no—
95(C2×D7)⋊C4—32no—
96(C2×D7)⋊C4—32no—
97(C2×D7)⋊C4C2×D312yes—
98C4⋊Dic7—32no—
99C4⋊Dic7—32no—
100C23×Dic3—71no—
101C2×#237—71no—
102C3×#120—71no—
103#953—71no—
104#939—72no—
105#935—71no—
106D4×Dic3—72no—
107#886—71no—
108#67⋊C4—72no—
109#878—72no—
110#877D711yescharacteristic, complemented
111C22×C3×C7—22no—
112C22×C3×C7C2×D412yes—
113C22×C3×C7C2411yescharacteristic
114C2×D21—21no—
115C2×D21—22no—
116C2×C3×D7—28no—
117C2×C3×D7—22no—
118C2×C3×D7—22no—
119C2×C3×D7—21no—
120C2×C3×D7—22no—
121C2×C3×D7#4114yescomplemented
122C3×C4×C7—22no—
123Dic21—22no—
124Dic21—21no—
125C3×Dic7—21no—
126C7×Dic3—22no—
127C7×Dic3C2×D412yes—
128C7×Dic3C2×D412yes—
129#525—211noSylow
130C23×C7C22×D311yescharacteristic
131C22×D7—22no—
132C22×D7—24no—
133C22×D7—31no—
134C22×D7C22×D311yescharacteristic
135C22×D7C22×D312yes—
136C2×C4×C7—34no—
137C2×C4×C7—31no—
138C2×C4×C7—31no—
139C2×C4×C7—32no—
140C2×C4×C7C22×D312yes—
141C22⋊D7—22no—
142C22⋊D7—62no—
143C2×Dic7—31no—
144C2×Dic7—32no—
145C2×Dic7—62no—
146C2×Dic7C22×D311yescharacteristic
147D28—24no—
148D28—62no—
149D28—64no—
150C4×D7—62no—
151C4×D7—64no—
152C4×D7—64no—
153C4×D7—62no—
154Dic14—62no—
155C24×C3—71no—
156C2×C3×D4—71no—
157C2×C3×D4—72no—
158C2×#67—71no—
159C2×#67—72no—
160C2×#67—71no—
161C22×Dic3—71no—
162C22×Dic3—72no—
163C22×Dic3—72no—
164C22×Dic3—71no—
165C22×Dic3—144no—
166C22×Dic3C2×D711yescharacteristic
167#237—144no—
168C2×C4×D3—71no—
169C2×Dic6—71no—
170C3×C22⋊C4—72no—
171C3×C22⋊C4C2×D711yescharacteristic
172#217—72no—
173#217—72no—
174#217C2×D711yescharacteristic
175(C2×D3)⋊C4—72no—
176C4⋊Dic3—72no—
177Dic3⋊C4—72no—
178Dic3⋊C4C2×D712yescomplemented
179C4×Dic3C2×D712yescomplemented
180C2×C3×C7—22no—
181C2×C3×C7C22×D411yescharacteristic
182C2×C3×C7C2×#4112yes—
183D21—41no—
184C3×D7—24no—
185C3×D7—41no—
186C2×#41—211no—
187C23×C4—211no—
188#123—211no—
189#123—211no—
190C22⋊Q8—212no—
191#121—212no—
192#120—71no—
193#120—211no—
194C4×D4—212no—
195C4×D4—212no—
196#117—31no—
197C22×C7—22no—
198C22×C7D3×D412yes—
199C22×C7C23×D311yescharacteristic
200C2×D7—28no—
201C2×D7—22no—
202C2×D7—22no—
203C2×D7—21no—
204C2×D7—22no—
205C2×D7—61no—
206C2×D7—62no—
207C2×D7#23714yescomplemented
208C4×C7—22no—
209C4×C7—32no—
210C4×C7—32no—
211C4×C7—62no—
212Dic7—21no—
213Dic7—61no—
214Dic7—62no—
215C23×C3—72no—
216C23×C3—71no—
217C23×C3—144no—
218C23×C3—142no—
219C23×C3C22×D711yescharacteristic
220C22×D3—71no—
221C3×D4—144no—
222C3×D4—142no—
223C2×C3×C4—71no—
224C2×C3×C4C22×D712yes—
225#67—144no—
226#67—142no—
227#67—142no—
228C2×Dic3—72no—
229C2×Dic3—71no—
230C2×Dic3—142no—
231C2×Dic3—142no—
232C2×Dic3—144no—
233C2×Dic3—144no—
234C2×Dic3D2814yescomplemented
235C2×Dic3C22×D712yes—
236C2×Dic3C22×D711yescharacteristic
237C2×Dic3C22×D711yescharacteristic
238C4×D3—142no—
239Dic6—142no—
240C3×C7#52511yescharacteristic, complemented, Hall
241C24—71no—
242#41—424no—
243C2×Q8—211no—
244C2×D4—71no—
245C2×D4—72no—
246C2×D4—212no—
247C2×D4—211no—
248C2×D4—211no—
249C22×C4—31no—
250C22×C4—211no—
251C22×C4—212no—
252C22×C4—211no—
253C22×C4—212no—
254C22×C4—211no—
255C22×C4—424no—
256C4⋊C4—32no—
257C4⋊C4—212no—
258C4⋊C4—212no—
259C22⋊C4—31no—
260C22⋊C4—72no—
261C22⋊C4—212no—
262C22⋊C4—212no—
263C22⋊C4—212no—
264C22⋊C4D3×D711yescharacteristic
265C42—32no—
266C2×C7—22no—
267C2×C7C2×D3×D411yescharacteristic
268C2×C7C2×#23712yes—
269D7—24no—
270D7—41no—
271D7—121no—
272C22×C3—22no—
273C22×C3—74no—
274C22×C3—148no—
275C22×C3—142no—
276C22×C3—142no—
277C22×C3—141no—
278C22×C3—142no—
279C22×C3C2×D2812yes—
280C22×C3C23×D711yescharacteristic
281C2×D3—141no—
282C2×D3—142no—
283C3×C4—22no—
284C3×C4—141no—
285Dic3—22no—
286Dic3—141no—
287Dic3—142no—
288Dic3C2×D2812yes—
289Dic3C2×D2812yes—
290C23—72no—
291C23—71no—
292C23—144no—
293C23—142no—
294C23—211no—
295C23C2×D3×D711yescharacteristic
296Q8—422no—
297D4—144no—
298D4—142no—
299D4—422no—
300D4—424no—
301D4—422no—
302C2×C4—34no—
303C2×C4—31no—
304C2×C4—31no—
305C2×C4—32no—
306C2×C4—71no—
307C2×C4—212no—
308C2×C4—211no—
309C2×C4—422no—
310C2×C4—424no—
311C2×C4—424no—
312C2×C4—422no—
313C2×C4—422no—
314C2×C4C2×D3×D712yes—
315C7#558611yescharacteristic, complemented, Sylow
316C2×C3—22no—
317C2×C3—144no—
318C2×C3—281no—
319C2×C3C22×D2811yescharacteristic
320C2×C3D7×#4112yes—
321D3—281no—
322C22—22no—
323C22—74no—
324C22—148no—
325C22—142no—
326C22—142no—
327C22—141no—
328C22—142no—
329C22—421no—
330C22—422no—
331C22D3×D2812yes—
332C22C22×D3×D711yescharacteristic
333C4—22no—
334C4—32no—
335C4—32no—
336C4—62no—
337C4—141no—
338C4—422no—
339C4—421no—
340C3#9021511yescharacteristic, complemented, Sylow
341C2—22no—
342C2—144no—
343C2—281no—
344C2—841no—
345C2C2×D3×D2811yescharacteristic
346C2D7×#23712yes—
subgroupG/H
center, FrattiniC22C22×D3×D7
commutator, socle, cosocleC22×C3×C7C24
FittingC3×C7×C22⋊C4C22
hypercenterC22⋊C4D3×D7
hypocenterC3×C7#525
LCSG ▷ C22×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#66683C2semidirect
2. C22×Dic3×D7C2semidirect
3. #10621369C2semidirect
4. C7×#877C2semidirect
5. C3×#6445C2semidirect
6. #10621126C2semidirect
7. #10621099C2semidirect
8. Dic3×D28C2semidirect
9. #10620984C2semidirect
10. #10620971C2semidirect
11. #10620957C2semidirect
19. C2×#4248C22semidirect
21. C2×#4246C22semidirect
22. C2×D21⋊C4C22semidirect
27. C2×Dic3×D7C22semidirect
29. #66631C22semidirect
30. C4⋊Dic21C22semidirect
32. C7×Dic3⋊C4C22semidirect
33. C4×C7×Dic3C22semidirect
35. Dic3⋊Dic7C22semidirect
37. (C2×D7)⋊Dic3C22semidirect
46. #6445D3semidirect
61. C2×C7×Dic3D4semidirect
110. #877D7semidirect
121. C2×C3×D7#41semidirect
178. Dic3⋊C4C2×D7semidirect
179. C4×Dic3C2×D7semidirect
207. C2×D7#237semidirect
234. C2×Dic3D28semidirect
240. C3×C7#525semidirect
315. C7#5586semidirect
340. C3#90215semidirect

Maximal subgroups

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

subgroupindex
C2×#666832
C22×Dic3×D72
#106213692
C7×#8772
C3×#64452
#106211262
#106210992
Dic3×D282
#106209842
#106209712
#106209572
#902153
#55867