Finite Group Databasebeta

← #13060132

Subgroups — #13060132

Order 1344.

624 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 624 conjugacy classes of subgroups — too many to draw. Only its 140 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×#66730C211yesmaximal, characteristic, complemented
2C2×#66676C211yesmaximal, characteristic, complemented
3C7×#940C211yesmaximal, characteristic, complemented
4C3×#6501C211yesmaximal, characteristic, complemented
5#10621133C212yesmaximal, complemented
6#10621129C212yesmaximal, complemented
7#67⋊Dic7C212yesmaximal, complemented
8#10621120C212yesmaximal, complemented
9#10621028C211yesmaximal, characteristic, complemented
10C4×#4246C211yesmaximal, characteristic, complemented
11#10620902C211yesmaximal, characteristic, complemented
12#90409—31nomaximal, Hall
13C2×C3×C7×D4C2211yescharacteristic
14C2×#4268C2212yes—
15C22×Dic21C2212yes—
16#66730—24no—
17C2×C4×D21C2211yescharacteristic
18C2×Dic42C2211yescharacteristic
19C2×C7×#67C2212yes—
20C2×C7×#67C2212yes—
21C2×C7×D12C2211yescharacteristic
22C2×C3×C22⋊D7C2212yes—
23C2×C3×C4×D7C2211yescharacteristic
24C2×#4247C2212yes—
25C2×#4246C2211yescharacteristic
26C2×#4245C2212yes—
27C2×D3×Dic7C2212yes—
28#66676—24no—
29C4×C7×Dic3C2211yescharacteristic
30C3×C22⋊Dic7C2212yes—
31C3×C4⋊Dic7C2211yescharacteristic
32Dic3⋊Dic7C2211yescharacteristic
33#66577C2212yes—
34#66575C2211yescharacteristic
35(C2×D3)⋊Dic7C2212yes—
36(C2×D7)⋊Dic3C2211yescharacteristic
37Dic3×Dic7C2212yes—
38C2×#1159—31no—
39C2×#1159—31no—
40C7×#127—31no—
41#6503—32no—
42#6502—32no—
43#6502—32no—
44#6501—31no—
45#6501D311yescharacteristic, complemented
46D4×Dic7—32no—
47C4×D28—31no—
48C4×Dic14—31no—
49#5599—71nomaximal, Hall
50C23×C3×C7C2312yes—
51C22×D21C2311yescharacteristic
52C22×D3×C7C2312yes—
53C22×C3×D7C2311yescharacteristic
54C3×C7×D4—22no—
55C2×C3×C4×C7C2311yescharacteristic
56#4268—24no—
57C2×Dic21—24no—
58C2×Dic21C2311yescharacteristic
59C2×Dic21C2312yes—
60C4×D21—22no—
61Dic42D414yescomplemented
62C7×#67—24no—
63C7×#67—42no—
64C2×C7×Dic3C2311yescharacteristic
65C2×C7×Dic3C2311yescharacteristic
66C7×D12—22no—
67C3×C22⋊D7—42no—
68C2×C3×Dic7C2312yes—
69C2×C3×Dic7C2311yescharacteristic
70C3×C4×D7—22no—
71#4247—42no—
72#4246—22no—
73#4245—24no—
74D3×Dic7—24no—
75C2×C7×D4—32no—
76C2×C7×D4—32no—
77C2×C7×D4—31no—
78C2×C7×D4C2×D311yescharacteristic
79C2×C22⋊D7—32no—
80C2×C22⋊D7—32no—
81C2×C22⋊D7—32no—
82C2×C22⋊D7C2×D312yes—
83C22×Dic7—32no—
84C22×Dic7—32no—
85#1159—64no—
86#1159—64no—
87C2×D28—31no—
88C2×C4×D7—31no—
89C2×C4×D7C2×D311yescharacteristic
90C2×Dic14—31no—
91C42×C7—31no—
92C22⋊Dic7—32no—
93C22⋊Dic7C2×D312yes—
94(C2×D7)⋊C4—31no—
95(C2×D7)⋊C4—31no—
96C4⋊Dic7—31no—
97C4⋊Dic7C2×D311yescharacteristic
98Dic7⋊C4—32no—
99C4×Dic7—32no—
100C2×#237—71no—
101C2×#235—71no—
102C3×#121—71no—
103#940—72no—
104#940D711yescharacteristic, complemented
105#939—72no—
106C4×#67—71no—
107#895—71no—
108#887—71no—
109#884—72no—
110#67⋊C4—72no—
111C22×C3×C7—22no—
112C22×C3×C7—22no—
113C22×C3×C7—22no—
114C22×C3×C7C2411yescharacteristic
115C2×D21—21no—
116C2×D21—21no—
117C2×D21—21no—
118C2×D3×C7—22no—
119C2×D3×C7—22no—
120C2×D3×C7—22no—
121C2×C3×D7—21no—
122C2×C3×D7—21no—
123C2×C3×D7—21no—
124C3×C4×C7C2×D412yes—
125Dic21—21no—
126Dic21C2×D414yes—
127C3×Dic7—21no—
128C3×Dic7—22no—
129C7×Dic3—21no—
130C7×Dic3#4112yescomplemented
131#550—211noSylow
132C23×C7—32no—
133C23×C7C22×D312yes—
134C22×D7—31no—
135C22×D7C22×D311yescharacteristic
136C7×D4—22no—
137C7×D4—64no—
138C7×D4—62no—
139C7×D4—122no—
140C2×C4×C7—31no—
141C2×C4×C7—31no—
142C2×C4×C7C22×D311yescharacteristic
143C22⋊D7—42no—
144C22⋊D7—64no—
145C22⋊D7—64no—
146C22⋊D7—122no—
147C2×Dic7—32no—
148C2×Dic7—31no—
149C2×Dic7—64no—
150C2×Dic7—64no—
151C2×Dic7C22×D312yes—
152C2×Dic7C22×D311yescharacteristic
153D28—62no—
154C4×D7—22no—
155C4×D7—62no—
156Dic14—34no—
157C2×C3×D4—72no—
158C2×C3×D4C2×D711yescharacteristic
159C22×C3×C4—71no—
160C2×#67—72no—
161C2×#67—72no—
162C2×#67—71no—
163C2×#67C2×D712yes—
164C2×#67C2×D712yes—
165C22×Dic3—72no—
166#237—144no—
167#235—144no—
168C2×D12—72no—
169C2×D12C2×D711yescharacteristic
170C2×C4×D3—72no—
171C2×C4×D3—71no—
172C2×Dic6—71no—
173C3×C4⋊C4—71no—
174C3×C22⋊C4—72no—
175#217—71no—
176(C2×D3)⋊C4—72no—
177(C2×D3)⋊C4—71no—
178Dic3⋊C4—72no—
179Dic3⋊C4—71no—
180C4×Dic3—72no—
181C4×Dic3C2×D711yescharacteristic
182C2×C3×C7—42no—
183C2×C3×C7C22×D411yescharacteristic
184C2×C3×C7C2×#4111yescharacteristic
185C2×C3×C7H(5,𝔽2)11yescharacteristic
186D21—41no—
187C3×D7—41no—
188D3×C7—42no—
189C2×#41—211no—
190C2×#41—211no—
191#127—31no—
192#127—212no—
193#121—71no—
194#121—212no—
195#121—211no—
196#121—212no—
197C4×Q8—211no—
198C4×D4—211no—
199C4×D4—212no—
200C22×C7—22no—
201C22×C7—22no—
202C22×C7—22no—
203C22×C7—62no—
204C22×C7—62no—
205C22×C7—62no—
206C22×C7C23×D311yescharacteristic
207C2×D7—21no—
208C2×D7—21no—
209C2×D7—21no—
210C2×D7—61no—
211C2×D7—61no—
212C2×D7—61no—
213C4×C7—32no—
214C4×C7—61no—
215C4×C7D3×D412yes—
216Dic7—21no—
217Dic7—22no—
218Dic7—34no—
219Dic7—61no—
220C23×C3—71no—
221C23×C3C22×D712yes—
222C22×D3—71no—
223C22×D3C22×D712yes—
224C3×D4—22no—
225C3×D4—282no—
226C2×C3×C4—72no—
227C2×C3×C4—71no—
228C2×C3×C4—142no—
229C2×C3×C4C22×D711yescharacteristic
230#67—24no—
231#67—42no—
232#67—144no—
233#67—144no—
234#67—142no—
235C2×Dic3—72no—
236C2×Dic3—71no—
237C2×Dic3—144no—
238C2×Dic3C22×D711yescharacteristic
239C2×Dic3C22×D711yescharacteristic
240D12—22no—
241D12—282no—
242C4×D3—144no—
243C4×D3—142no—
244Dic6—74no—
245C3×C7#55011yescharacteristic, complemented, Hall
246#41—424no—
247#41—424no—
248C2×Q8—211no—
249C2×D4—32no—
250C2×D4—32no—
251C2×D4—31no—
252C2×D4—72no—
253C2×D4—211no—
254C2×D4—212no—
255C2×D4—212no—
256C2×D4—212no—
257C2×D4D3×D711yescharacteristic
258C22×C4—71no—
259C22×C4—212no—
260C22×C4—212no—
261C22×C4—211no—
262C4⋊C4—71no—
263C4⋊C4—212no—
264C4⋊C4—211no—
265C22⋊C4—72no—
266C22⋊C4—211no—
267C22⋊C4—211no—
268C22⋊C4—212no—
269C42—31no—
270C42—212no—
271C2×C7—42no—
272C2×C7—122no—
273C2×C7C2×D3×D411yescharacteristic
274C2×C7#100411yescharacteristic
275C2×C7D3×#4111yescharacteristic
276D7—41no—
277D7—121no—
278C22×C3—22no—
279C22×C3—22no—
280C22×C3—22no—
281C22×C3—141no—
282C22×C3—141no—
283C22×C3—141no—
284C22×C3C23×D711yescharacteristic
285C2×D3—22no—
286C2×D3—22no—
287C2×D3—22no—
288C2×D3—141no—
289C2×D3—141no—
290C2×D3—141no—
291C3×C4—141no—
292C3×C4—142no—
293C3×C4D4×D712yes—
294Dic3—21no—
295Dic3—74no—
296Dic3—141no—
297Dic3#115912yescomplemented
298C23—32no—
299C23—71no—
300C23—211no—
301C23C2×D3×D712yes—
302Q8—214no—
303D4—22no—
304D4—64no—
305D4—62no—
306D4—122no—
307D4—282no—
308D4—422no—
309D4—424no—
310D4—424no—
311D4—842no—
312C2×C4—31no—
313C2×C4—31no—
314C2×C4—72no—
315C2×C4—71no—
316C2×C4—142no—
317C2×C4—212no—
318C2×C4—211no—
319C2×C4—424no—
320C2×C4—424no—
321C2×C4—422no—
322C2×C4C2×D3×D711yescharacteristic
323C7#559911yescharacteristic, complemented, Sylow
324C2×C3—42no—
325C2×C3—281no—
326C2×C3C2×D4×D711yescharacteristic
327C2×C3C2×#115911yescharacteristic
328C2×C3#654811yescharacteristic
329D3—42no—
330D3—281no—
331C22—22no—
332C22—22no—
333C22—22no—
334C22—62no—
335C22—62no—
336C22—62no—
337C22—141no—
338C22—141no—
339C22—141no—
340C22—421no—
341C22—421no—
342C22—421no—
343C22C22×D3×D711yescharacteristic
344C4—32no—
345C4—61no—
346C4—142no—
347C4—141no—
348C4—214no—
349C4—421no—
350C4#6669412yes—
351C3#9040911yescharacteristic, complemented, Sylow
352C2—42no—
353C2—122no—
354C2—281no—
355C2—841no—
356C2D3×#115911yescharacteristic
357C2#1062157211yescharacteristic
358C2C2×#6669411yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D7
commutator, socle, cosocleC22×C3×C7C24
FittingC2×C3×C7×D4C22
hypercenterC2×D4D3×D7
hypocenterC3×C7#550
LCSG ▷ C22×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×#66730C2semidirect
2. C2×#66676C2semidirect
3. C7×#940C2semidirect
4. C3×#6501C2semidirect
5. #10621133C2semidirect
6. #10621129C2semidirect
7. #67⋊Dic7C2semidirect
8. #10621120C2semidirect
9. #10621028C2semidirect
10. C4×#4246C2semidirect
11. #10620902C2semidirect
45. #6501D3semidirect
61. Dic42D4semidirect
104. #940D7semidirect
130. C7×Dic3#41semidirect
245. C3×C7#550semidirect
297. Dic3#1159semidirect
323. C7#5599semidirect
351. C3#90409semidirect

Maximal subgroups

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

subgroupindex
C2×#667302
C2×#666762
C7×#9402
C3×#65012
#106211332
#106211292
#67⋊Dic72
#106211202
#106210282
C4×#42462
#106209022
#904093
#55997