Finite Group Databasebeta

← #13060130

Subgroups — #13060130

Order 1344.

588 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 588 conjugacy classes of subgroups — too many to draw. Only its 136 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×#66675C211yesmaximal, characteristic, complemented
3C7×#938C211yesmaximal, characteristic, complemented
4C3×#6501C211yesmaximal, characteristic, complemented
5#10621138C212yesmaximal, complemented
6#10621133C212yesmaximal, complemented
7#10621119C212yesmaximal, complemented
8#10621095C212yesmaximal, complemented
9#10620999C211yesmaximal, characteristic, complemented
10#10620927C211yesmaximal, characteristic, complemented
11#10620922C211yesmaximal, characteristic, complemented
12#90299—31nomaximal, Hall
13C2×C3×C7×D4C2211yescharacteristic
14C2×#4268C2212yes—
15C22×Dic21C2212yes—
16#66730—24no—
17C2×C4×D21C2211yescharacteristic
18C2×Dic42C2211yescharacteristic
19C2×C7×#67C2212yes—
20C2×C4×D3×C7C2211yescharacteristic
21C2×C3×C22⋊D7C2212yes—
22C2×C3×C4×D7C2211yescharacteristic
23C2×#4248C2211yescharacteristic
24C2×#4247C2211yescharacteristic
25C2×#4246C2211yescharacteristic
26C2×#4245C2211yescharacteristic
27C2×D21⋊C4C2212yes—
28#66675—24no—
29C7×#217C2212yes—
30C7×C4⋊Dic3C2211yescharacteristic
31C3×C22⋊Dic7C2212yes—
32C3×C4⋊Dic7C2211yescharacteristic
33Dic3⋊Dic7C2212yes—
34#66577C2212yes—
35Dic7⋊Dic3C2212yes—
36(C2×D3)⋊Dic7C2212yes—
37(C2×D7)⋊Dic3C2212yes—
38C2×#1159—31no—
39C2×#1157—31no—
40C7×#121—31no—
41#6502—32no—
42#6501—32no—
43#6501D311yescharacteristic, complemented
44#6487—31no—
45#6460—31no—
46D7×C4⋊C4—31no—
47#6447—32no—
48#6441—32no—
49#5608—71nomaximal, Hall
50C23×C3×C7C2312yes—
51C22×D21C2311yescharacteristic
52C22×D3×C7C2311yescharacteristic
53C22×C3×D7C2311yescharacteristic
54C3×C7×D4—22no—
55C2×C3×C4×C7C2311yescharacteristic
56#4268—24no—
57C2×Dic21—24no—
58C2×Dic21C2312yes—
59C2×Dic21C2311yescharacteristic
60C4×D21D414yescomplemented
61Dic42—22no—
62C7×#67—42no—
63C2×C7×Dic3C2312yes—
64C2×C7×Dic3C2311yescharacteristic
65C4×D3×C7—22no—
66C3×C22⋊D7—42no—
67C2×C3×Dic7C2312yes—
68C2×C3×Dic7C2311yescharacteristic
69C3×C4×D7—22no—
70#4248—22no—
71#4247—22no—
72#4246—22no—
73#4245—22no—
74D21⋊C4—42no—
75C2×C7×D4—32no—
76C2×C7×D4C2×D311yescharacteristic
77C22×C4×C7—31no—
78C2×C22⋊D7—32no—
79C2×C22⋊D7—31no—
80C2×C22⋊D7—31no—
81C2×C22⋊D7C2×D312yes—
82C22×Dic7—32no—
83#1159—64no—
84#1157—64no—
85C2×D28—31no—
86C2×C4×D7—31no—
87C2×C4×D7—32no—
88C2×C4×D7C2×D311yescharacteristic
89C2×Dic14—31no—
90C2×Dic14—31no—
91C7×C4⋊C4—31no—
92C7×C22⋊C4—32no—
93C22⋊Dic7—32no—
94C22⋊Dic7C2×D312yes—
95(C2×D7)⋊C4—32no—
96C4⋊Dic7—32no—
97C4⋊Dic7C2×D311yescharacteristic
98Dic7⋊C4—32no—
99Dic7⋊C4—32no—
100C2×#237—71no—
101C2×#235—71no—
102C3×#121—71no—
103#939—72no—
104#938—72no—
105#938D711yescharacteristic, complemented
106#924—71no—
107#897—71no—
108D3×C4⋊C4—71no—
109#884—72no—
110#878—72no—
111C22×C3×C7—22no—
112C22×C3×C7—22no—
113C22×C3×C7—22no—
114C22×C3×C7C2411yescharacteristic
115C2×D21—21no—
116C2×D21—21no—
117C2×D21C2×D412yes—
118C2×D3×C7—21no—
119C2×D3×C7—21no—
120C2×D3×C7—21no—
121C2×C3×D7—21no—
122C2×C3×D7—21no—
123C2×C3×D7—21no—
124C3×C4×C7C2×D412yes—
125Dic21—22no—
126Dic21C2×D412yes—
127C3×Dic7—21no—
128C3×Dic7—22no—
129C7×Dic3—21no—
130C7×Dic3—22no—
131#537—211noSylow
132C23×C7—31no—
133C23×C7C22×D312yes—
134C22×D7—31no—
135C22×D7C22×D311yescharacteristic
136C7×D4—22no—
137C7×D4—122no—
138C2×C4×C7—31no—
139C2×C4×C7—32no—
140C2×C4×C7—62no—
141C2×C4×C7C22×D311yescharacteristic
142C22⋊D7—42no—
143C22⋊D7—62no—
144C22⋊D7—62no—
145C22⋊D7—64no—
146C2×Dic7—31no—
147C2×Dic7—32no—
148C2×Dic7—64no—
149C2×Dic7C22×D312yes—
150C2×Dic7C22×D311yescharacteristic
151D28—62no—
152C4×D7—22no—
153C4×D7—34no—
154C4×D7—122no—
155Dic14—62no—
156Dic14—62no—
157C2×C3×D4—72no—
158C2×C3×D4C2×D711yescharacteristic
159C22×C3×C4—71no—
160C2×#67—72no—
161C2×#67—71no—
162C2×#67—71no—
163C2×#67C2×D712yes—
164C22×Dic3—72no—
165#237—144no—
166#235—144no—
167C2×D12—71no—
168C2×C4×D3—71no—
169C2×C4×D3—72no—
170C2×C4×D3C2×D711yescharacteristic
171C2×Dic6—71no—
172C2×Dic6—71no—
173C3×C4⋊C4—71no—
174C3×C22⋊C4—72no—
175#217—72no—
176#217C2×D712yes—
177(C2×D3)⋊C4—72no—
178C4⋊Dic3—72no—
179C4⋊Dic3C2×D711yescharacteristic
180Dic3⋊C4—72no—
181Dic3⋊C4—72no—
182C2×C3×C7—42no—
183C2×C3×C7C22×D411yescharacteristic
184C2×C3×C7H(5,𝔽2)11yescharacteristic
185C2×C3×C72−1+411yescharacteristic
186D21—22no—
187C3×D7—41no—
188D3×C7—41no—
189C2×#41—211no—
190C2×#41—211no—
191C22⋊Q8—212no—
192C22⋊Q8—211no—
193C22⋊Q8—211no—
194#121—31no—
195#121—71no—
196#121—212no—
197#121—212no—
198#121—212no—
199C2×C4⋊C4—211no—
200C22×C7—22no—
201C22×C7—22no—
202C22×C7—22no—
203C22×C7—61no—
204C22×C7—61no—
205C22×C7—61no—
206C22×C7C23×D311yescharacteristic
207C2×D7—21no—
208C2×D7—21no—
209C2×D7—21no—
210C2×D7—32no—
211C2×D7—61no—
212C2×D7—61no—
213C4×C7—61no—
214C4×C7—62no—
215C4×C7D3×D412yes—
216Dic7—21no—
217Dic7—22no—
218Dic7—32no—
219Dic7—62no—
220C23×C3—71no—
221C23×C3C22×D712yes—
222C22×D3—71no—
223C22×D3C22×D711yescharacteristic
224C3×D4—22no—
225C3×D4—282no—
226C2×C3×C4—71no—
227C2×C3×C4—72no—
228C2×C3×C4—142no—
229C2×C3×C4C22×D711yescharacteristic
230#67—42no—
231#67—142no—
232#67—142no—
233#67—144no—
234C2×Dic3—71no—
235C2×Dic3—72no—
236C2×Dic3—144no—
237C2×Dic3C22×D712yes—
238C2×Dic3C22×D711yescharacteristic
239D12—142no—
240C4×D3—22no—
241C4×D3—74no—
242C4×D3—282no—
243Dic6—142no—
244Dic6—142no—
245C3×C7#53711yescharacteristic, complemented, Hall
246#41—424no—
247#41—424no—
248C2×Q8—211no—
249C2×Q8—211no—
250C2×D4—32no—
251C2×D4—72no—
252C2×D4—211no—
253C2×D4—211no—
254C2×D4—211no—
255C2×D4—212no—
256C2×D4D3×D711yescharacteristic
257C22×C4—31no—
258C22×C4—71no—
259C22×C4—212no—
260C22×C4—211no—
261C22×C4—212no—
262C4⋊C4—31no—
263C4⋊C4—71no—
264C4⋊C4—212no—
265C4⋊C4—212no—
266C4⋊C4—212no—
267C22⋊C4—32no—
268C22⋊C4—72no—
269C22⋊C4—212no—
270C22⋊C4—212no—
271C2×C7—42no—
272C2×C7—121no—
273C2×C7C2×D3×D411yescharacteristic
274C2×C7#100411yescharacteristic
275C2×C7#101011yescharacteristic
276D7—41no—
277D7—62no—
278C22×C3—22no—
279C22×C3—22no—
280C22×C3—22no—
281C22×C3—141no—
282C22×C3—141no—
283C22×C3—141no—
284C22×C3C23×D711yescharacteristic
285C2×D3—21no—
286C2×D3—21no—
287C2×D3—21no—
288C2×D3—72no—
289C2×D3—141no—
290C2×D3—141no—
291C3×C4—141no—
292C3×C4—142no—
293C3×C4D4×D712yes—
294Dic3—21no—
295Dic3—22no—
296Dic3—72no—
297Dic3—142no—
298C23—31no—
299C23—71no—
300C23—211no—
301C23C2×D3×D712yes—
302Q8—422no—
303Q8—422no—
304D4—22no—
305D4—122no—
306D4—282no—
307D4—422no—
308D4—422no—
309D4—422no—
310D4—424no—
311C2×C4—31no—
312C2×C4—32no—
313C2×C4—62no—
314C2×C4—71no—
315C2×C4—72no—
316C2×C4—142no—
317C2×C4—211no—
318C2×C4—214no—
319C2×C4—212no—
320C2×C4—424no—
321C2×C4—842no—
322C2×C4C2×D3×D711yescharacteristic
323C7#560811yescharacteristic, complemented, Sylow
324C2×C3—42no—
325C2×C3—281no—
326C2×C3C2×D4×D711yescharacteristic
327C2×C3#654811yescharacteristic
328C2×C3#655411yescharacteristic
329D3—41no—
330D3—142no—
331C22—22no—
332C22—22no—
333C22—22no—
334C22—61no—
335C22—61no—
336C22—61no—
337C22—141no—
338C22—141no—
339C22—141no—
340C22—212no—
341C22—421no—
342C22—421no—
343C22C22×D3×D711yescharacteristic
344C4—62no—
345C4—61no—
346C4—142no—
347C4—141no—
348C4—212no—
349C4—422no—
350C4#6669412yes—
351C3#9029911yescharacteristic, complemented, Sylow
352C2—42no—
353C2—121no—
354C2—281no—
355C2—422no—
356C2#1062156711yescharacteristic
357C2#1062157211yescharacteristic
358C2C2×#6669411yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D7
commutator, socle, cosocleC22×C3×C7C24
FittingC2×C3×C7×D4C22
hypercenterC2×D4D3×D7
hypocenterC3×C7#537
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×#66675C2semidirect
3. C7×#938C2semidirect
4. C3×#6501C2semidirect
5. #10621138C2semidirect
6. #10621133C2semidirect
7. #10621119C2semidirect
8. #10621095C2semidirect
9. #10620999C2semidirect
10. #10620927C2semidirect
11. #10620922C2semidirect
43. #6501D3semidirect
60. C4×D21D4semidirect
105. #938D7semidirect
245. C3×C7#537semidirect
323. C7#5608semidirect
351. C3#90299semidirect

Maximal subgroups

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

subgroupindex
C2×#667302
C2×#666752
C7×#9382
C3×#65012
#106211382
#106211332
#106211192
#106210952
#106209992
#106209272
#106209222
#902993
#56087