Finite Group Databasebeta

← #13052516

Subgroups — #13052516

Order 1344.

396 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 58 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
1#10621596C211yesmaximal, characteristic, complemented
2#10621365C211yesmaximal, characteristic, complemented
3#10621135C211yesmaximal, characteristic, complemented
4#10621134C211yesmaximal, characteristic, complemented
5C7×#806C211yesmaximal, characteristic, complemented
6C3×#6380C211yesmaximal, characteristic, complemented
7#10620532C211yesmaximal, characteristic, complemented
8#89513—31nomaximal, Hall
9C23×D21C2211yescharacteristic, complemented
10C2×#4268C2211yescharacteristic
11C2×D84—21no—
12C2×C7×#67C2211yescharacteristic
13C2×C3×C22⋊D7C2211yescharacteristic
14#66694—21no—
15D3×C22⋊D7—21no—
16D7×#67—21no—
17C2×#4247—21no—
18C2×#4246—21no—
19#66686—21no—
20#66684—21no—
21#66683—21no—
22C3×C7×C22⋊C4C2211yescharacteristic, complemented
23#66631—21no—
24C7×#217C2211yescharacteristic, complemented
25C3×C22⋊Dic7C2211yescharacteristic, complemented
26#66575—21no—
27#66575—21no—
28#6548—31no—
29#6500—31no—
30#6445—31no—
31#6445—31no—
32C7×#99—31no—
33#6380—31no—
34#6380D311yescharacteristic, complemented
35#4741—71nomaximal, Hall
36C23×C3×C7C2311yescharacteristic
37C22×D21—21no—
38C22×D21—21no—
39C22×D21—21no—
40C22×D21—42no—
41C22×D21D411yescharacteristic, complemented
42C22×D3×C7D411yescharacteristic, complemented
43C22×C3×D7D411yescharacteristic, complemented
44C2×D3×D7—21no—
45C2×C3×C4×C7—21no—
46#4268—21no—
47#4268—21no—
48C2×Dic21D411yescharacteristic, complemented
49D84—42no—
50C7×#67—21no—
51C7×#67—21no—
52C2×C7×Dic3—21no—
53C2×C7×Dic3D411yescharacteristic, complemented
54C3×C22⋊D7—21no—
55C3×C22⋊D7—21no—
56C2×C3×Dic7—21no—
57C2×C3×Dic7D411yescharacteristic, complemented
58#4248—21no—
59#4247—21no—
60#4247—42no—
61#4246—21no—
62#4246—42no—
63#4245—21no—
64D21⋊C4—21no—
65D3×Dic7—21no—
66Dic3×D7—21no—
67C23×D7—31no—
68C2×C7×D4—31no—
69C2×C22⋊D7—31no—
70C2×C22⋊D7—61no—
71C2×C22⋊D7—61no—
72C2×C22⋊D7C2×D311yescharacteristic
73#1159—61no—
74#1159—61no—
75D4×D7—61no—
76D4×D7—61no—
77#1157—61no—
78C2×D28—61no—
79C2×D28—61no—
80C7×C22⋊C4—31no—
81C7×C22⋊C4C2×D311yescharacteristic, complemented
82C22⋊Dic7C2×D311yescharacteristic, complemented
83(C2×D7)⋊C4—61no—
84(C2×D7)⋊C4—61no—
85(C2×D7)⋊C4—61no—
86#1004—71no—
87#937—71no—
88#882—71no—
89#882—71no—
90C3×#99—71no—
91#806—71no—
92#806D711yescharacteristic, complemented
93C22×C3×C7—41no—
94C22×C3×C7C2×D411yescharacteristic
95C22×C3×C7C2×D411yescharacteristic
96C22×C3×C7C2×D411yescharacteristic
97C2×D21—22no—
98C2×D21—22no—
99C2×D21—22no—
100C2×D21—21no—
101C2×D21—41no—
102C2×D21—41no—
103C2×D21—41no—
104C2×D21—41no—
105C2×D21—41no—
106C2×D3×C7—21no—
107C2×D3×C7—41no—
108C2×C3×D7—21no—
109C2×C3×D7—41no—
110D3×D7—81no—
111C3×C4×C7—41no—
112Dic21—21no—
113C3×Dic7—21no—
114C3×Dic7—41no—
115C7×Dic3—21no—
116C7×Dic3—41no—
117B(4,𝔽2)—211noSylow
118C23×C7—31no—
119C23×C7C22×D311yescharacteristic
120C22×D7—31no—
121C22×D7—61no—
122C22×D7—61no—
123C22×D7—61no—
124C22×D7—61no—
125C22×D7—122no—
126C22×D7D1211yescharacteristic, complemented
127C7×D4—61no—
128C7×D4—61no—
129C2×C4×C7—21no—
130C2×C4×C7—31no—
131C2×C4×C7—61no—
132C22⋊D7—21no—
133C22⋊D7—21no—
134C22⋊D7—61no—
135C22⋊D7—61no—
136C22⋊D7—61no—
137C22⋊D7—61no—
138C22⋊D7—122no—
139C2×Dic7—21no—
140C2×Dic7—31no—
141C2×Dic7—61no—
142C2×Dic7D1211yescharacteristic, complemented
143D28—61no—
144D28—122no—
145D28—122no—
146C4×D7—61no—
147C4×D7—61no—
148Dic14—61no—
149C23×D3—71no—
150C2×C3×D4—71no—
151C2×#67—71no—
152C2×#67—141no—
153C2×#67—141no—
154C2×#67C2×D711yescharacteristic
155#237—141no—
156#237—141no—
157D3×D4—141no—
158D3×D4—141no—
159#235—141no—
160C2×D12—141no—
161C2×D12—141no—
162C3×C22⋊C4—71no—
163C3×C22⋊C4C2×D711yescharacteristic, complemented
164#217C2×D711yescharacteristic, complemented
165(C2×D3)⋊C4—141no—
166(C2×D3)⋊C4—141no—
167(C2×D3)⋊C4—141no—
168C2×C3×C7—21no—
169C2×C3×C7—21no—
170C2×C3×C7—21no—
171C2×C3×C7#12011yescharacteristic
172D21—41no—
173D21—42no—
174C3×D7—41no—
175D3×C7—41no—
176H(5,𝔽2)—211no—
177#120—211no—
178#120—211no—
179#120—211no—
180#99—31no—
181#99—71no—
182#99—211no—
183C22×C7—41no—
184C22×C7—61no—
185C22×C7—121no—
186C22×C7C2×D1211yescharacteristic
187C22×C7D3×D411yescharacteristic
188C22×C7D3×D411yescharacteristic
189C2×D7—21no—
190C2×D7—41no—
191C2×D7—62no—
192C2×D7—62no—
193C2×D7—62no—
194C2×D7—61no—
195C2×D7—121no—
196C2×D7—121no—
197C2×D7—121no—
198C2×D7—121no—
199C2×D7—121no—
200C2×D7—241no—
201C4×C7—41no—
202C4×C7—61no—
203C4×C7—121no—
204Dic7—21no—
205Dic7—41no—
206Dic7—61no—
207C23×C3—71no—
208C23×C3C22×D711yescharacteristic
209C22×D3—71no—
210C22×D3—141no—
211C22×D3—141no—
212C22×D3—141no—
213C22×D3—141no—
214C22×D3—282no—
215C22×D3D2811yescharacteristic, complemented
216C3×D4—141no—
217C3×D4—141no—
218C2×C3×C4—21no—
219C2×C3×C4—71no—
220C2×C3×C4—141no—
221#67—21no—
222#67—21no—
223#67—141no—
224#67—141no—
225#67—141no—
226#67—141no—
227#67—282no—
228C2×Dic3—21no—
229C2×Dic3—71no—
230C2×Dic3—141no—
231C2×Dic3D2811yescharacteristic, complemented
232D12—141no—
233D12—282no—
234D12—282no—
235C4×D3—141no—
236C4×D3—141no—
237Dic6—141no—
238C3×C7B(4,𝔽2)11yescharacteristic, complemented, Hall
239C24—211no—
240#41—421no—
241#41—421no—
242#41—421no—
243C2×D4—31no—
244C2×D4—71no—
245C2×D4—211no—
246C2×D4—421no—
247C2×D4—421no—
248C2×D4—421no—
249C2×D4—421no—
250C2×D4—421no—
251C2×D4—421no—
252C22⋊C4—31no—
253C22⋊C4—71no—
254C22⋊C4—421no—
255C22⋊C4—421no—
256C22⋊C4—421no—
257C22⋊C4D3×D711yescharacteristic, complemented
258C2×C7—21no—
259C2×C7—21no—
260C2×C7—21no—
261C2×C7—121no—
262C2×C7#88211yescharacteristic
263D7—41no—
264D7—122no—
265D7—121no—
266C22×C3—41no—
267C22×C3—141no—
268C22×C3—281no—
269C22×C3C2×D2811yescharacteristic
270C22×C3D4×D711yescharacteristic
271C22×C3D4×D711yescharacteristic
272C2×D3—21no—
273C2×D3—41no—
274C2×D3—142no—
275C2×D3—142no—
276C2×D3—142no—
277C2×D3—141no—
278C2×D3—281no—
279C2×D3—281no—
280C2×D3—281no—
281C2×D3—281no—
282C2×D3—281no—
283C2×D3—561no—
284C3×C4—41no—
285C3×C4—141no—
286C3×C4—281no—
287Dic3—21no—
288Dic3—41no—
289Dic3—141no—
290C23—31no—
291C23—71no—
292C23—211no—
293C23—421no—
294C23—421no—
295C23—421no—
296C23—421no—
297C23—842no—
298C23C2×D3×D711yescharacteristic
299Q8—421no—
300D4—61no—
301D4—61no—
302D4—141no—
303D4—141no—
304D4—421no—
305D4—421no—
306D4—421no—
307D4—421no—
308D4—421no—
309D4—842no—
310D4—842no—
311D4—842no—
312C2×C4—21no—
313C2×C4—31no—
314C2×C4—61no—
315C2×C4—71no—
316C2×C4—141no—
317C2×C4—211no—
318C2×C4—421no—
319C2×C4—421no—
320C2×C4—421no—
321C7#474111yescharacteristic, complemented, Sylow
322C2×C3—21no—
323C2×C3—21no—
324C2×C3—21no—
325C2×C3—281no—
326C2×C3#644511yescharacteristic
327D3—41no—
328D3—282no—
329D3—281no—
330C22—41no—
331C22—61no—
332C22—121no—
333C22—141no—
334C22—281no—
335C22—422no—
336C22—422no—
337C22—422no—
338C22—421no—
339C22—841no—
340C22—841no—
341C22—841no—
342C22—841no—
343C22—841no—
344C22—1681no—
345C22D7×D1211yescharacteristic
346C22D3×D2811yescharacteristic
347C22#6669411yescharacteristic
348C4—41no—
349C4—61no—
350C4—121no—
351C4—141no—
352C4—281no—
353C4—421no—
354C3#8951311yescharacteristic, complemented, Sylow
355C2—21no—
356C2—21no—
357C2—21no—
358C2—121no—
359C2—281no—
360C2—842no—
361C2—841no—
362C2#1062103811yescharacteristic
subgroupG/H
centerC2#10621038
commutator, cosocleC23×C3×C7C23
FrattiniC23C2×D3×D7
FittingC3×C7×C22⋊C4C22
socleC2×C3×C7#120
hypercenterC22⋊C4D3×D7
hypocenterC3×C7B(4,𝔽2)
LCSG ▷ C23×C3×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C2 ◁ C23 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
1. #10621596C2semidirect
2. #10621365C2semidirect
3. #10621135C2semidirect
4. #10621134C2semidirect
5. C7×#806C2semidirect
6. C3×#6380C2semidirect
7. #10620532C2semidirect
9. C23×D21C22semidirect
22. C3×C7×C22⋊C4C22semidirect
24. C7×#217C22semidirect
25. C3×C22⋊Dic7C22semidirect
34. #6380D3semidirect
41. C22×D21D4semidirect
42. C22×D3×C7D4semidirect
43. C22×C3×D7D4semidirect
48. C2×Dic21D4semidirect
53. C2×C7×Dic3D4semidirect
57. C2×C3×Dic7D4semidirect
81. C7×C22⋊C4C2×D3semidirect
82. C22⋊Dic7C2×D3semidirect
92. #806D7semidirect
126. C22×D7D12semidirect
142. C2×Dic7D12semidirect
163. C3×C22⋊C4C2×D7semidirect
164. #217C2×D7semidirect
215. C22×D3D28semidirect
231. C2×Dic3D28semidirect
238. C3×C7B(4,𝔽2)semidirect
257. C22⋊C4D3×D7semidirect
321. C7#4741semidirect
354. C3#89513semidirect

Maximal subgroups

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

subgroupindex
#106215962
#106213652
#106211352
#106211342
C7×#8062
C3×#63802
#106205322
#895133
#47417