Finite Group Databasebeta

← #13056594

Subgroups — #13056594

Order 1344.

572 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#4268C211yesmaximal, characteristic
2C22×C4×D21C211yesmaximal, characteristic, complemented
3C2×C3×C7×C22⋊C4C211yesmaximal, characteristic, complemented
4C2×#66636C211yesmaximal, characteristic, complemented
5C2×#66631C211yesmaximal, characteristic, complemented
6#10620670C211yesmaximal, characteristic, complemented
7#10620670C211yesmaximal, characteristic, complemented
8#89728—31nomaximal, Hall
9C24×C3×C7C2211yescharacteristic
10C23×D21C2211yescharacteristic
11C22×C3×C4×C7C2211yescharacteristic
12C22×C3×C4×C7C2211yescharacteristic
13C2×#4268—22no—
14C2×#4268—22no—
15C2×#4268C414yescomplemented
16C22×Dic21C2211yescharacteristic
17C22×Dic21C2211yescharacteristic
18C22×Dic21C2211yescharacteristic
19C2×C4×D21—22no—
20C2×C4×D21—22no—
21C2×C4×D21—22no—
22C3×C7×C22⋊C4—22no—
23#66636—22no—
24#66631—22no—
25C22×C22⋊D7—31no—
26C22×C4×D7—31no—
27C2×C7×C22⋊C4D311yescharacteristic, complemented
28C2×C22⋊Dic7—31no—
29C2×(C2×D7)⋊C4—31no—
30C7⋊#95—31no—
31C7⋊#95—31no—
32#4956—71nomaximal, Hall
33C23×C3×C7—21no—
34C23×C3×C7—21no—
35C23×C3×C7—21no—
36C23×C3×C7—21no—
37C23×C3×C7—21no—
38C23×C3×C7—21no—
39C23×C3×C7C2×C412yescomplemented
40C23×C3×C7C2311yescharacteristic
41C22×D21—21no—
42C22×D21—21no—
43C22×D21—21no—
44C22×D21—21no—
45C22×D21C2×C412yes—
46C22×D21D412yes—
47C22×D21D412yes—
48C2×C3×C4×C7—21no—
49C2×C3×C4×C7—21no—
50C2×C3×C4×C7—21no—
51C2×C3×C4×C7—21no—
52C2×C3×C4×C7—21no—
53C2×C3×C4×C7D412yescomplemented
54#4268—28no—
55C2×Dic21—21no—
56C2×Dic21—21no—
57C2×Dic21—21no—
58C2×Dic21—21no—
59C2×Dic21—21no—
60C2×Dic21—21no—
61C2×Dic21—21no—
62C2×Dic21C2×C412yescomplemented
63C2×Dic21D412yescomplemented
64C4×D21—28no—
65C24×C7C2×D311yescharacteristic
66C23×D7—31no—
67C22×C4×C7C2×D311yescharacteristic
68C22×C4×C7C2×D311yescharacteristic
69C2×C22⋊D7—34no—
70C2×C22⋊D7—62no—
71C2×C22⋊D7—62no—
72C22×Dic7—31no—
73C22×Dic7—31no—
74C22×Dic7—31no—
75C2×C4×D7—62no—
76C2×C4×D7—62no—
77C2×C4×D7—62no—
78C7×C22⋊C4—22no—
79C22⋊Dic7—62no—
80(C2×D7)⋊C4—62no—
81C22×#67—71no—
82C22×C4×D3—71no—
83C2×C3×C22⋊C4D711yescharacteristic, complemented
84C2×#217—71no—
85C2×(C2×D3)⋊C4—71no—
86C3⋊#95—71no—
87C3⋊#95—71no—
88C22×C3×C7—22no—
89C22×C3×C7—22no—
90C22×C3×C7—22no—
91C22×C3×C7—41no—
92C22×C3×C7—41no—
93C22×C3×C7—41no—
94C22×C3×C7—41no—
95C22×C3×C7C22×C411yescharacteristic
96C22×C3×C7C2×D411yescharacteristic
97C22×C3×C7C2×D411yescharacteristic
98C22×C3×C7C2×D411yescharacteristic
99C22×C3×C7C2×D411yescharacteristic
100C22×C3×C7#4111yescharacteristic
101C22×C3×C7#4111yescharacteristic
102C2×D21—22no—
103C2×D21—22no—
104C2×D21—22no—
105C2×D21—22no—
106C2×D21—22no—
107C2×D21—22no—
108C2×D21C22⋊C414yescomplemented
109C3×C4×C7—22no—
110C3×C4×C7—41no—
111Dic21—22no—
112Dic21—22no—
113Dic21—41no—
114#386—211noSylow
115C23×C7—21no—
116C23×C7—21no—
117C23×C7—21no—
118C23×C7—21no—
119C23×C7—21no—
120C23×C7—21no—
121C23×C7C4×D312yescomplemented
122C23×C7C22×D311yescharacteristic
123C22×D7—32no—
124C22×D7—32no—
125C22×D7—32no—
126C22×D7—61no—
127C22×D7—61no—
128C22×D7—61no—
129C22×D7—61no—
130C2×C4×C7—21no—
131C2×C4×C7—21no—
132C2×C4×C7—21no—
133C2×C4×C7—21no—
134C2×C4×C7—21no—
135C2×C4×C7#6712yescomplemented
136C22⋊D7—68no—
137C2×Dic7—32no—
138C2×Dic7—32no—
139C2×Dic7—61no—
140C2×Dic7—61no—
141C2×Dic7—61no—
142C2×Dic7—61no—
143C2×Dic7—61no—
144C2×Dic7—61no—
145C2×Dic7—61no—
146C4×D7—68no—
147C24×C3C2×D711yescharacteristic
148C23×D3—71no—
149C22×C3×C4C2×D711yescharacteristic
150C22×C3×C4C2×D711yescharacteristic
151C2×#67—74no—
152C2×#67—142no—
153C2×#67—142no—
154C22×Dic3—71no—
155C22×Dic3—71no—
156C22×Dic3—71no—
157C2×C4×D3—142no—
158C2×C4×D3—142no—
159C2×C4×D3—142no—
160C3×C22⋊C4—22no—
161#217—142no—
162(C2×D3)⋊C4—142no—
163C2×C3×C7—42no—
164C2×C3×C7C2×C22⋊C411yescharacteristic
165C2×C3×C7C4×D411yescharacteristic
166C2×C3×C7C4×D411yescharacteristic
167C2×C3×C7#12011yescharacteristic
168C2×C3×C7#12111yescharacteristic
169C2×C3×C7#12111yescharacteristic
170C2×C3×C7#12311yescharacteristic
171D21—24no—
172C22×D4—211no—
173C23×C4—211no—
174C2×C22⋊C4—211no—
175C2×C22⋊C4—211no—
176C2×C22⋊C4D2111yescharacteristic, complemented
177#95—211no—
178#95—211no—
179C22×C7—22no—
180C22×C7—22no—
181C22×C7—22no—
182C22×C7—41no—
183C22×C7—41no—
184C22×C7—41no—
185C22×C7—41no—
186C22×C7C2×C4×D311yescharacteristic
187C22×C7#23511yescharacteristic
188C22×C7D3×D411yescharacteristic
189C22×C7D3×D411yescharacteristic
190C22×C7D3×D411yescharacteristic
191C22×C7#23711yescharacteristic
192C22×C7C2×#6711yescharacteristic
193C2×D7—34no—
194C2×D7—62no—
195C2×D7—62no—
196C2×D7—62no—
197C2×D7—62no—
198C2×D7—62no—
199C2×D7—62no—
200C4×C7—22no—
201C4×C7—41no—
202Dic7—62no—
203Dic7—62no—
204Dic7—121no—
205C23×C3—21no—
206C23×C3—21no—
207C23×C3—21no—
208C23×C3—21no—
209C23×C3—21no—
210C23×C3—21no—
211C23×C3C4×D712yescomplemented
212C23×C3C22×D711yescharacteristic
213C22×D3—72no—
214C22×D3—72no—
215C22×D3—72no—
216C22×D3—141no—
217C22×D3—141no—
218C22×D3—141no—
219C22×D3—141no—
220C2×C3×C4—21no—
221C2×C3×C4—21no—
222C2×C3×C4—21no—
223C2×C3×C4—21no—
224C2×C3×C4—21no—
225C2×C3×C4C22⋊D712yescomplemented
226#67—148no—
227C2×Dic3—72no—
228C2×Dic3—72no—
229C2×Dic3—141no—
230C2×Dic3—141no—
231C2×Dic3—141no—
232C2×Dic3—141no—
233C2×Dic3—141no—
234C2×Dic3—141no—
235C2×Dic3—141no—
236C4×D3—148no—
237C3×C7#38611yescharacteristic, complemented, Hall
238C24—211no—
239C24C2×D2111yescharacteristic
240C2×D4—214no—
241C2×D4—422no—
242C2×D4—422no—
243C22×C4—211no—
244C22×C4—211no—
245C22×C4—211no—
246C22×C4—422no—
247C22×C4—422no—
248C22×C4—422no—
249C22×C4C2×D2111yescharacteristic
250C22×C4C2×D2111yescharacteristic
251C22⋊C4—22no—
252C22⋊C4—422no—
253C22⋊C4—422no—
254C2×C7—42no—
255C2×C7D3×C22⋊C411yescharacteristic
256C2×C7#67⋊C411yescharacteristic
257C2×C7#88311yescharacteristic
258C2×C7#88411yescharacteristic
259C2×C7C4×#6711yescharacteristic
260C2×C7#93711yescharacteristic
261C2×C7#93811yescharacteristic
262D7—64no—
263C22×C3—22no—
264C22×C3—22no—
265C22×C3—22no—
266C22×C3—41no—
267C22×C3—41no—
268C22×C3—41no—
269C22×C3—41no—
270C22×C3C2×C4×D711yescharacteristic
271C22×C3#115711yescharacteristic
272C22×C3D4×D711yescharacteristic
273C22×C3D4×D711yescharacteristic
274C22×C3D4×D711yescharacteristic
275C22×C3#115911yescharacteristic
276C22×C3C2×C22⋊D711yescharacteristic
277C2×D3—74no—
278C2×D3—142no—
279C2×D3—142no—
280C2×D3—142no—
281C2×D3—142no—
282C2×D3—142no—
283C2×D3—142no—
284C3×C4—22no—
285C3×C4—41no—
286Dic3—142no—
287Dic3—142no—
288Dic3—281no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23—212no—
296C23—212no—
297C23—212no—
298C23—421no—
299C23—421no—
300C23—421no—
301C23—421no—
302C23C4×D2112yescomplemented
303C23C22×D2111yescharacteristic
304D4—428no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—212no—
311C2×C4—212no—
312C2×C4—428no—
313C2×C4—421no—
314C2×C4—421no—
315C2×C4—421no—
316C2×C4—421no—
317C2×C4—421no—
318C2×C4—421no—
319C2×C4—421no—
320C2×C4#426812yescomplemented
321C7#495611yescharacteristic, complemented, Sylow
322C2×C3—42no—
323C2×C3D7×C22⋊C411yescharacteristic
324C2×C3(C22⋊D7)⋊C411yescharacteristic
325C2×C3#644611yescharacteristic
326C2×C3#644711yescharacteristic
327C2×C3C4×C22⋊D711yescharacteristic
328C2×C3#650011yescharacteristic
329C2×C3#650111yescharacteristic
330D3—144no—
331C22—22no—
332C22—22no—
333C22—22no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—214no—
339C22—422no—
340C22—422no—
341C22—422no—
342C22—422no—
343C22—422no—
344C22—422no—
345C22C2×C4×D2111yescharacteristic
346C22#6672811yescharacteristic
347C22D4×D2111yescharacteristic
348C22D4×D2111yescharacteristic
349C22D4×D2111yescharacteristic
350C22#6673011yescharacteristic
351C22C2×#426811yescharacteristic
352C4—22no—
353C4—41no—
354C4—422no—
355C4—422no—
356C4—841no—
357C3#8972811yescharacteristic, complemented, Sylow
358C2—42no—
359C2—424no—
360C2C22⋊C4×D2111yescharacteristic
361C2#1062136411yescharacteristic
362C2#1062136611yescharacteristic
363C2#1062136711yescharacteristic
364C2C4×#426811yescharacteristic
365C2#1062142011yescharacteristic
366C2#1062142111yescharacteristic
subgroupG/H
center, FrattiniC23C22×D21
commutatorC22×C3×C7C22×C4
FittingC2×C3×C7×C22⋊C4C2
socle, cosocleC23×C3×C7C23
prime radicalC22×#4268C2
hypercenterC2×C22⋊C4D21
hypocenterC3×C7#386
LCSG ▷ C22×C3×C7 ▷ C3×C7
UCSC1 ◁ C23 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C22×C4×D21C2semidirect
3. C2×C3×C7×C22⋊C4C2semidirect
4. C2×#66636C2semidirect
5. C2×#66631C2semidirect
6. #10620670C2semidirect
7. #10620670C2semidirect
15. C2×#4268C4semidirect
27. C2×C7×C22⋊C4D3semidirect
39. C23×C3×C7C2×C4semidirect
53. C2×C3×C4×C7D4semidirect
62. C2×Dic21C2×C4semidirect
63. C2×Dic21D4semidirect
83. C2×C3×C22⋊C4D7semidirect
108. C2×D21C22⋊C4semidirect
121. C23×C7C4×D3semidirect
135. C2×C4×C7#67semidirect
176. C2×C22⋊C4D21semidirect
211. C23×C3C4×D7semidirect
225. C2×C3×C4C22⋊D7semidirect
237. C3×C7#386semidirect
302. C23C4×D21semidirect
320. C2×C4#4268semidirect
321. C7#4956semidirect
357. C3#89728semidirect

Maximal subgroups

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

subgroupindex
C22×#42682
C22×C4×D212
C2×C3×C7×C22⋊C42
C2×#666362
C2×#666312
#106206702
#897283
#49567