Finite Group Databasebeta

← #13060203

Subgroups — #13060203

Order 1344.

480 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 128 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#10621421C211yesmaximal, characteristic, complemented
2C7×#940C211yesmaximal, characteristic, complemented
3C3×D4×Dic7C211yesmaximal, characteristic, complemented
4#10621128C212yesmaximal, complemented
5#10621120C212yesmaximal, complemented
6#10621117C212yesmaximal, complemented
7#10621101C212yesmaximal, complemented
8#10621028C211yesmaximal, characteristic, complemented
9D12×Dic7C211yesmaximal, characteristic, complemented
10#10620948C211yesmaximal, characteristic, complemented
11#10620943C211yesmaximal, characteristic, complemented
12#90410—31nomaximal, Hall
13C2×C3×C7×D4C2211yescharacteristic
14C2×#4268C2212yes—
15C2×C4×D21C2211yescharacteristic
16C2×C7×#67C2212yes—
17C2×C7×#67C2212yes—
18C2×C7×D12C2211yescharacteristic
19C22×C3×Dic7C2212yes—
20C2×#4247C2212yes—
21C2×D3×Dic7C2212yes—
22#66636C2212yes—
23C4⋊Dic21C2211yescharacteristic
24C4×C7×Dic3C2211yescharacteristic
25C3×C22⋊Dic7C2212yes—
26C3×C4⋊Dic7C2211yescharacteristic
27C3×C4×Dic7C2211yescharacteristic
28#66577C2212yes—
29#66577C2211yescharacteristic
30#66577C2211yescharacteristic
31Dic7⋊Dic3C2212yes—
32#66575C2211yescharacteristic
33#66575C2211yescharacteristic
34(C2×D3)⋊Dic7C2212yes—
35(C2×D3)⋊Dic7C2212yes—
36C7×#127—31no—
37#6502—32no—
38#6502—32no—
39#6501—31no—
40#6501—31no—
41#6498—32no—
42#6498—32no—
43D4×Dic7—31no—
44D4×Dic7D311yescharacteristic, complemented
45#6435—31no—
46#6433—31no—
47#5562—71nomaximal, Hall
48C23×C3×C7C2312yes—
49C22×D21C2311yescharacteristic
50C22×D3×C7C2312yes—
51C3×C7×D4—22no—
52C2×C3×C4×C7C2311yescharacteristic
53#4268—42no—
54C2×Dic21C2311yescharacteristic
55C2×Dic21C2312yes—
56C4×D21—22no—
57C7×#67—42no—
58C7×#67—42no—
59C2×C7×Dic3C2311yescharacteristic
60C2×C7×Dic3C2311yescharacteristic
61C7×D12—22no—
62C2×C3×Dic7—42no—
63C2×C3×Dic7C2311yescharacteristic
64C2×C3×Dic7C2312yes—
65C2×C3×Dic7C2311yescharacteristic
66#4247—42no—
67D3×Dic7—42no—
68C2×C7×D4—32no—
69C2×C7×D4—32no—
70C2×C7×D4—31no—
71C2×C7×D4C2×D311yescharacteristic
72C2×C22⋊D7—32no—
73C2×C22⋊D7—32no—
74C22×Dic7—32no—
75C22×Dic7C2×D312yes—
76C2×C4×D7—31no—
77C42×C7—31no—
78C22⋊Dic7—32no—
79C22⋊Dic7—32no—
80C22⋊Dic7—32no—
81C22⋊Dic7C2×D312yes—
82(C2×D7)⋊C4—31no—
83(C2×D7)⋊C4—31no—
84C4⋊Dic7—31no—
85C4⋊Dic7C2×D311yescharacteristic
86Dic7⋊C4—32no—
87Dic7⋊C4—32no—
88Dic7⋊C4—31no—
89Dic7⋊C4—31no—
90C4×Dic7C2×D311yescharacteristic
91C3×C4×D4—71no—
92#940D711yescharacteristic, complemented
93#938—71no—
94#930—72no—
95#929—72no—
96#895—71no—
97#890—71no—
98#884—72no—
99#883—72no—
100C4×D12—71no—
101#872—71no—
102C22×C3×C7—22no—
103C22×C3×C7—22no—
104C22×C3×C7—22no—
105C22×C3×C7C2411yescharacteristic
106C2×D21—21no—
107C2×D21—21no—
108C2×D21—21no—
109C2×D3×C7—22no—
110C2×D3×C7—22no—
111C2×D3×C7—22no—
112C3×C4×C7#4112yes—
113Dic21—21no—
114Dic21—22no—
115C3×Dic7—21no—
116C3×Dic7—22no—
117C3×Dic7—21no—
118C7×Dic3—21no—
119C7×Dic3—21no—
120#540—211noSylow
121C23×C7—32no—
122C23×C7C22×D312yes—
123C22×D7—31no—
124C7×D4—22no—
125C7×D4—62no—
126C7×D4—122no—
127C7×D4—122no—
128C2×C4×C7—31no—
129C2×C4×C7—31no—
130C2×C4×C7C22×D311yescharacteristic
131C22⋊D7—122no—
132C22⋊D7—122no—
133C2×Dic7—32no—
134C2×Dic7—31no—
135C2×Dic7—42no—
136C2×Dic7—122no—
137C2×Dic7C22×D311yescharacteristic
138C2×Dic7C22×D311yescharacteristic
139C2×Dic7C22×D312yes—
140C4×D7—62no—
141C2×C3×D4C2×D711yescharacteristic
142C22×C3×C4—72no—
143C2×#67—72no—
144C2×#67C2×D712yes—
145C2×#67C2×D712yes—
146C2×D12—72no—
147C2×D12C2×D711yescharacteristic
148C2×C4×D3—71no—
149C2×C4×D3—72no—
150C3×C4⋊C4—71no—
151C3×C22⋊C4—72no—
152C3×C42—71no—
153#217—72no—
154(C2×D3)⋊C4—72no—
155(C2×D3)⋊C4—72no—
156(C2×D3)⋊C4—71no—
157(C2×D3)⋊C4—71no—
158C4⋊Dic3—71no—
159C4⋊Dic3—72no—
160Dic3⋊C4—71no—
161Dic3⋊C4—71no—
162Dic3⋊C4—72no—
163C4×Dic3C2×D711yescharacteristic
164C2×C3×C7—42no—
165C2×C3×C7C2×#4111yescharacteristic
166C2×C3×C7H(5,𝔽2)11yescharacteristic
167C2×C3×C7H(5,𝔽2)11yescharacteristic
168D21—41no—
169D3×C7—42no—
170#127—31no—
171#125—211no—
172#123—212no—
173#123—212no—
174#121—212no—
175#121—211no—
176#121—212no—
177#121—211no—
178C4×D4—71no—
179C4×D4—211no—
180#117—211no—
181C22×C7—22no—
182C22×C7—22no—
183C22×C7—22no—
184C22×C7—62no—
185C22×C7—62no—
186C22×C7—62no—
187C22×C7C23×D311yescharacteristic
188C2×D7—61no—
189C2×D7—61no—
190C2×D7—61no—
191C4×C7—61no—
192C4×C7—61no—
193C4×C7#23512yes—
194Dic7—21no—
195Dic7—22no—
196Dic7—21no—
197Dic7—61no—
198Dic7—62no—
199C23×C3C22×D712yes—
200C22×D3—71no—
201C22×D3C22×D712yes—
202C3×D4—22no—
203C2×C3×C4—72no—
204C2×C3×C4—71no—
205C2×C3×C4—71no—
206C2×C3×C4—282no—
207C2×C3×C4C22×D711yescharacteristic
208#67—42no—
209#67—42no—
210#67—282no—
211C2×Dic3—71no—
212C2×Dic3—72no—
213C2×Dic3C22×D711yescharacteristic
214C2×Dic3C22×D711yescharacteristic
215D12—22no—
216D12—282no—
217C4×D3—142no—
218C4×D3—282no—
219C3×C7#54011yescharacteristic, complemented, Hall
220C2×D4—32no—
221C2×D4—32no—
222C2×D4—31no—
223C2×D4—212no—
224C2×D4—212no—
225C2×D4D3×D711yescharacteristic
226C22×C4—72no—
227C22×C4—212no—
228C22×C4—211no—
229C4⋊C4—71no—
230C4⋊C4—211no—
231C4⋊C4—211no—
232C4⋊C4—212no—
233C4⋊C4—212no—
234C4⋊C4—211no—
235C22⋊C4—72no—
236C22⋊C4—211no—
237C22⋊C4—211no—
238C22⋊C4—212no—
239C22⋊C4—212no—
240C22⋊C4—212no—
241C42—31no—
242C42—71no—
243C2×C7—42no—
244C2×C7—122no—
245C2×C7C2×#23511yescharacteristic
246C2×C7#100411yescharacteristic
247C2×C7#100911yescharacteristic
248D7—121no—
249C22×C3—22no—
250C22×C3—22no—
251C22×C3—22no—
252C22×C3C23×D711yescharacteristic
253C2×D3—22no—
254C2×D3—22no—
255C2×D3—22no—
256C2×D3—141no—
257C2×D3—141no—
258C2×D3—141no—
259C3×C4—141no—
260C3×C4—142no—
261C3×C4—141no—
262C3×C4#115912yes—
263Dic3—21no—
264Dic3—21no—
265Dic3—141no—
266Dic3—142no—
267C23—32no—
268C23—211no—
269C23C2×D3×D712yes—
270D4—22no—
271D4—62no—
272D4—122no—
273D4—122no—
274D4—842no—
275D4—842no—
276C2×C4—31no—
277C2×C4—31no—
278C2×C4—71no—
279C2×C4—71no—
280C2×C4—72no—
281C2×C4—211no—
282C2×C4—212no—
283C2×C4—282no—
284C2×C4—422no—
285C2×C4—842no—
286C2×C4C2×D3×D711yescharacteristic
287C7#556211yescharacteristic, complemented, Sylow
288C2×C3—42no—
289C2×C3C2×#115911yescharacteristic
290C2×C3#654811yescharacteristic
291C2×C3#654811yescharacteristic
292D3—42no—
293D3—281no—
294C22—22no—
295C22—22no—
296C22—22no—
297C22—62no—
298C22—62no—
299C22—62no—
300C22—421no—
301C22—421no—
302C22—421no—
303C22C22×D3×D711yescharacteristic
304C4—61no—
305C4—61no—
306C4—141no—
307C4—142no—
308C4—141no—
309C4—422no—
310C4—421no—
311C4#6668612yes—
312C3#9041011yescharacteristic, complemented, Sylow
313C2—42no—
314C2—122no—
315C2—841no—
316C2#1062157211yescharacteristic
317C2#1062157411yescharacteristic
318C2C2×#6668611yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D7
commutator, socle, cosocleC22×C3×C7C24
FittingC2×C3×C7×D4C22
hypercenterC2×D4D3×D7
hypocenterC3×C7#540
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. #10621421C2semidirect
2. C7×#940C2semidirect
3. C3×D4×Dic7C2semidirect
4. #10621128C2semidirect
5. #10621120C2semidirect
6. #10621117C2semidirect
7. #10621101C2semidirect
8. #10621028C2semidirect
9. D12×Dic7C2semidirect
10. #10620948C2semidirect
11. #10620943C2semidirect
44. D4×Dic7D3semidirect
92. #940D7semidirect
219. C3×C7#540semidirect
287. C7#5562semidirect
312. C3#90410semidirect

Maximal subgroups

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

subgroupindex
#106214212
C7×#9402
C3×D4×Dic72
#106211282
#106211202
#106211172
#106211012
#106210282
D12×Dic72
#106209482
#106209432
#904103
#55627