Finite Group Databasebeta

← #13054296

Subgroups — #13054296

Order 1344.

644 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 644 conjugacy classes of subgroups — too many to draw. Only its 108 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
1C22×#4247C211yesmaximal, characteristic, complemented
2C22×#4246C211yesmaximal, characteristic, complemented
3C22×#4245C211yesmaximal, characteristic, complemented
4C2×#66631C211yesmaximal, characteristic, complemented
5C2×C7×(C2×D3)⋊C4C211yesmaximal, characteristic, complemented
6C2×C3×(C2×D7)⋊C4C211yesmaximal, characteristic, complemented
7#10620529C211yesmaximal, characteristic, complemented
8#89732—31nomaximal, Hall
9C23×D21C2211yescharacteristic
10C23×D3×C7C2211yescharacteristic
11C23×C3×D7C2211yescharacteristic
12C22×C3×C4×C7C2211yescharacteristic
13C22×Dic21C2211yescharacteristic
14C22×C7×Dic3C2211yescharacteristic
15C22×C3×Dic7C2211yescharacteristic
16C2×#4247—22no—
17C2×#4247—22no—
18C2×#4247—22no—
19C2×#4246—22no—
20C2×#4246—22no—
21C2×#4246—22no—
22C2×#4245—22no—
23C2×#4245—22no—
24C2×#4245—22no—
25#66631—22no—
26C7×(C2×D3)⋊C4—22no—
27C3×(C2×D7)⋊C4—22no—
28C22×C22⋊D7—31no—
29C22×C22⋊D7—31no—
30C22×D28—31no—
31C2×C7×C22⋊C4—31no—
32C2×(C2×D7)⋊C4—31no—
33C2×(C2×D7)⋊C4D311yescharacteristic, complemented
34C7⋊#95—31no—
35#4960—71nomaximal, Hall
36C23×C3×C7C2311yescharacteristic
37C22×D21—21no—
38C22×D21—21no—
39C22×D21—21no—
40C22×D21—21no—
41C22×D21—21no—
42C22×D21—21no—
43C22×D21D412yescomplemented
44C22×D3×C7—21no—
45C22×D3×C7—21no—
46C22×D3×C7—21no—
47C22×D3×C7—21no—
48C22×D3×C7—21no—
49C22×D3×C7—21no—
50C22×D3×C7D412yescomplemented
51C22×C3×D7—21no—
52C22×C3×D7—21no—
53C22×C3×D7—21no—
54C22×C3×D7—21no—
55C22×C3×D7—21no—
56C22×C3×D7—21no—
57C22×C3×D7D412yescomplemented
58C2×C3×C4×C7—21no—
59C2×C3×C4×C7—21no—
60C2×C3×C4×C7—21no—
61C2×Dic21—21no—
62C2×Dic21—21no—
63C2×Dic21D412yescomplemented
64C2×C7×Dic3—21no—
65C2×C7×Dic3—21no—
66C2×C7×Dic3D412yescomplemented
67C2×C3×Dic7—21no—
68C2×C3×Dic7—21no—
69C2×C3×Dic7D412yescomplemented
70#4247—28no—
71#4246—28no—
72#4245—28no—
73C24×C7—31no—
74C23×D7—31no—
75C23×D7C2×D311yescharacteristic
76C22×C4×C7—31no—
77C22×C4×C7C2×D311yescharacteristic
78C2×C22⋊D7—62no—
79C2×C22⋊D7—62no—
80C2×C22⋊D7—62no—
81C2×C22⋊D7—62no—
82C2×C22⋊D7—62no—
83C2×C22⋊D7—62no—
84C22×Dic7—31no—
85C22×Dic7C2×D311yescharacteristic
86C2×D28—62no—
87C2×D28—62no—
88C2×D28—62no—
89C7×C22⋊C4—62no—
90(C2×D7)⋊C4—22no—
91(C2×D7)⋊C4—62no—
92C22×#67—71no—
93C22×#67—71no—
94C22×D12—71no—
95C2×C3×C22⋊C4—71no—
96C2×(C2×D3)⋊C4—71no—
97C2×(C2×D3)⋊C4D711yescharacteristic, complemented
98C3⋊#95—71no—
99C22×C3×C7C2×D411yescharacteristic
100C22×C3×C7C2×D411yescharacteristic
101C22×C3×C7C2×D411yescharacteristic
102C22×C3×C7C2×D411yescharacteristic
103C22×C3×C7C2×D411yescharacteristic
104C22×C3×C7C2×D411yescharacteristic
105C22×C3×C7#4111yescharacteristic
106C2×D21—22no—
107C2×D21—22no—
108C2×D21—22no—
109C2×D21—41no—
110C2×D21—41no—
111C2×D21—41no—
112C2×D21—41no—
113C2×D3×C7—22no—
114C2×D3×C7—22no—
115C2×D3×C7—22no—
116C2×D3×C7—41no—
117C2×D3×C7—41no—
118C2×D3×C7—41no—
119C2×D3×C7—41no—
120C2×C3×D7—22no—
121C2×C3×D7—22no—
122C2×C3×D7—22no—
123C2×C3×D7—41no—
124C2×C3×D7—41no—
125C2×C3×D7—41no—
126C2×C3×D7—41no—
127C3×C4×C7—41no—
128Dic21—22no—
129C3×Dic7—22no—
130C7×Dic3—22no—
131#392—211noSylow
132C23×C7—32no—
133C23×C7—61no—
134C23×C7—61no—
135C23×C7—61no—
136C23×C7—61no—
137C23×C7—61no—
138C23×C7—61no—
139C23×C7C22×D311yescharacteristic
140C22×D7—21no—
141C22×D7—21no—
142C22×D7—21no—
143C22×D7—21no—
144C22×D7—21no—
145C22×D7—21no—
146C22×D7—32no—
147C22×D7—61no—
148C22×D7—61no—
149C22×D7—61no—
150C22×D7—61no—
151C22×D7—61no—
152C22×D7—61no—
153C22×D7D1212yescomplemented
154C2×C4×C7—21no—
155C2×C4×C7—21no—
156C2×C4×C7—21no—
157C2×C4×C7—32no—
158C2×C4×C7—61no—
159C2×C4×C7—61no—
160C22⋊D7—68no—
161C22⋊D7—68no—
162C2×Dic7—21no—
163C2×Dic7—21no—
164C2×Dic7—32no—
165C2×Dic7—61no—
166C2×Dic7—61no—
167C2×Dic7#6712yescomplemented
168D28—68no—
169C24×C3—71no—
170C23×D3—71no—
171C23×D3C2×D711yescharacteristic
172C22×C3×C4—71no—
173C22×C3×C4C2×D711yescharacteristic
174C2×#67—142no—
175C2×#67—142no—
176C2×#67—142no—
177C2×#67—142no—
178C2×#67—142no—
179C2×#67—142no—
180C22×Dic3—71no—
181C22×Dic3C2×D711yescharacteristic
182C2×D12—142no—
183C2×D12—142no—
184C2×D12—142no—
185C3×C22⋊C4—142no—
186(C2×D3)⋊C4—22no—
187(C2×D3)⋊C4—142no—
188C2×C3×C7#12011yescharacteristic
189C2×C3×C7#12011yescharacteristic
190C2×C3×C7#12011yescharacteristic
191C2×C3×C7#12111yescharacteristic
192C2×C3×C7#12111yescharacteristic
193C2×C3×C7#12111yescharacteristic
194C2×C3×C7#12711yescharacteristic
195D21—42no—
196C3×D7—42no—
197D3×C7—42no—
198C22×D4—211no—
199C22×D4—211no—
200C22×D4—211no—
201C2×C22⋊C4—31no—
202C2×C22⋊C4—71no—
203C2×C22⋊C4—211no—
204#95—211no—
205C22×C7—62no—
206C22×C7—62no—
207C22×C7—62no—
208C22×C7—121no—
209C22×C7—121no—
210C22×C7—121no—
211C22×C7—121no—
212C22×C7C2×D1211yescharacteristic
213C22×C7#23511yescharacteristic
214C22×C7D3×D411yescharacteristic
215C22×C7D3×D411yescharacteristic
216C22×C7D3×D411yescharacteristic
217C22×C7D3×D411yescharacteristic
218C22×C7C2×#6711yescharacteristic
219C2×D7—22no—
220C2×D7—22no—
221C2×D7—22no—
222C2×D7—41no—
223C2×D7—41no—
224C2×D7—41no—
225C2×D7—41no—
226C2×D7—62no—
227C2×D7—62no—
228C2×D7—62no—
229C2×D7—121no—
230C2×D7—121no—
231C2×D7—121no—
232C2×D7—121no—
233C4×C7—41no—
234C4×C7—62no—
235Dic7—22no—
236Dic7—62no—
237C23×C3—72no—
238C23×C3—141no—
239C23×C3—141no—
240C23×C3—141no—
241C23×C3—141no—
242C23×C3—141no—
243C23×C3—141no—
244C23×C3C22×D711yescharacteristic
245C22×D3—21no—
246C22×D3—21no—
247C22×D3—21no—
248C22×D3—21no—
249C22×D3—21no—
250C22×D3—21no—
251C22×D3—72no—
252C22×D3—141no—
253C22×D3—141no—
254C22×D3—141no—
255C22×D3—141no—
256C22×D3—141no—
257C22×D3—141no—
258C22×D3D2812yescomplemented
259C2×C3×C4—21no—
260C2×C3×C4—21no—
261C2×C3×C4—21no—
262C2×C3×C4—72no—
263C2×C3×C4—141no—
264C2×C3×C4—141no—
265#67—148no—
266#67—148no—
267C2×Dic3—21no—
268C2×Dic3—21no—
269C2×Dic3—72no—
270C2×Dic3—141no—
271C2×Dic3—141no—
272C2×Dic3C22⋊D712yescomplemented
273D12—148no—
274C3×C7#39211yescharacteristic, complemented, Hall
275C24—31no—
276C24—71no—
277C24—211no—
278C2×D4—422no—
279C2×D4—422no—
280C2×D4—422no—
281C2×D4—422no—
282C2×D4—422no—
283C2×D4—422no—
284C2×D4—422no—
285C2×D4—422no—
286C2×D4—422no—
287C22×C4—31no—
288C22×C4—71no—
289C22×C4—211no—
290C22×C4D3×D711yescharacteristic
291C22⋊C4—62no—
292C22⋊C4—142no—
293C22⋊C4—422no—
294C2×C7—122no—
295C2×C7#88211yescharacteristic
296C2×C7#88211yescharacteristic
297C2×C7#88411yescharacteristic
298C2×C7#88411yescharacteristic
299C2×C7#93011yescharacteristic
300C2×C7#93711yescharacteristic
301C2×C7#94011yescharacteristic
302D7—42no—
303D7—122no—
304C22×C3—142no—
305C22×C3—142no—
306C22×C3—142no—
307C22×C3—281no—
308C22×C3—281no—
309C22×C3—281no—
310C22×C3—281no—
311C22×C3C2×D2811yescharacteristic
312C22×C3#115711yescharacteristic
313C22×C3D4×D711yescharacteristic
314C22×C3D4×D711yescharacteristic
315C22×C3D4×D711yescharacteristic
316C22×C3D4×D711yescharacteristic
317C22×C3C2×C22⋊D711yescharacteristic
318C2×D3—22no—
319C2×D3—22no—
320C2×D3—22no—
321C2×D3—41no—
322C2×D3—41no—
323C2×D3—41no—
324C2×D3—41no—
325C2×D3—142no—
326C2×D3—142no—
327C2×D3—142no—
328C2×D3—281no—
329C2×D3—281no—
330C2×D3—281no—
331C2×D3—281no—
332C3×C4—41no—
333C3×C4—142no—
334Dic3—22no—
335Dic3—142no—
336C23—32no—
337C23—61no—
338C23—61no—
339C23—61no—
340C23—61no—
341C23—61no—
342C23—61no—
343C23—72no—
344C23—141no—
345C23—141no—
346C23—141no—
347C23—141no—
348C23—141no—
349C23—141no—
350C23—212no—
351C23—421no—
352C23—421no—
353C23—421no—
354C23—421no—
355C23—421no—
356C23—421no—
357C23C2×D3×D711yescharacteristic
358D4—428no—
359D4—428no—
360D4—428no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—32no—
365C2×C4—61no—
366C2×C4—61no—
367C2×C4—72no—
368C2×C4—141no—
369C2×C4—141no—
370C2×C4—212no—
371C2×C4—421no—
372C2×C4—421no—
373C7#496011yescharacteristic, complemented, Sylow
374C2×C3—282no—
375C2×C3#644511yescharacteristic
376C2×C3#644511yescharacteristic
377C2×C3#644711yescharacteristic
378C2×C3#644711yescharacteristic
379C2×C3#649311yescharacteristic
380C2×C3#650011yescharacteristic
381C2×C3#650311yescharacteristic
382D3—42no—
383D3—282no—
384C22—62no—
385C22—62no—
386C22—62no—
387C22—121no—
388C22—121no—
389C22—121no—
390C22—121no—
391C22—142no—
392C22—142no—
393C22—142no—
394C22—281no—
395C22—281no—
396C22—281no—
397C22—281no—
398C22—422no—
399C22—422no—
400C22—422no—
401C22—841no—
402C22—841no—
403C22—841no—
404C22—841no—
405C22#6667511yescharacteristic
406C22D7×D1211yescharacteristic
407C22D3×D2811yescharacteristic
408C22#6668111yescharacteristic
409C22D7×#6711yescharacteristic
410C22D3×C22⋊D711yescharacteristic
411C22#6669411yescharacteristic
412C4—41no—
413C4—62no—
414C4—142no—
415C4—422no—
416C3#8973211yescharacteristic, complemented, Sylow
417C2—122no—
418C2—282no—
419C2—842no—
420C2#1062101011yescharacteristic
421C2#1062101611yescharacteristic
422C2#1062102311yescharacteristic
423C2#1062103611yescharacteristic
424C2#1062103711yescharacteristic
425C2#1062103811yescharacteristic
426C2#1062112911yescharacteristic
subgroupG/H
center, FrattiniC23C2×D3×D7
commutator, socle, cosocleC23×C3×C7C23
FittingC22×C3×C4×C7C22
hypercenterC22×C4D3×D7
hypocenterC3×C7#392
LCSG ▷ C23×C3×C7 ▷ C3×C7
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C22×#4247C2semidirect
2. C22×#4246C2semidirect
3. C22×#4245C2semidirect
4. C2×#66631C2semidirect
5. C2×C7×(C2×D3)⋊C4C2semidirect
6. C2×C3×(C2×D7)⋊C4C2semidirect
7. #10620529C2semidirect
33. C2×(C2×D7)⋊C4D3semidirect
43. C22×D21D4semidirect
50. C22×D3×C7D4semidirect
57. C22×C3×D7D4semidirect
63. C2×Dic21D4semidirect
66. C2×C7×Dic3D4semidirect
69. C2×C3×Dic7D4semidirect
97. C2×(C2×D3)⋊C4D7semidirect
153. C22×D7D12semidirect
167. C2×Dic7#67semidirect
258. C22×D3D28semidirect
272. C2×Dic3C22⋊D7semidirect
274. C3×C7#392semidirect
373. C7#4960semidirect
416. C3#89732semidirect

Maximal subgroups

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

subgroupindex
C22×#42472
C22×#42462
C22×#42452
C2×#666312
C2×C7×(C2×D3)⋊C42
C2×C3×(C2×D7)⋊C42
#106205292
#897323
#49607