Finite Group Databasebeta

← #13054873

Subgroups — #13054873

Order 1344.

476 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 106 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×C7×#67C211yesmaximal, characteristic
2C2×#66636C211yesmaximal, characteristic, complemented
3C2×C3×C22⋊Dic7C211yesmaximal, characteristic, complemented
4C2×#66577C211yesmaximal, characteristic, complemented
5C2×(C2×D3)⋊Dic7C211yesmaximal, characteristic, complemented
6C2×(C2×D3)⋊Dic7C211yesmaximal, characteristic, complemented
7#10620529C211yesmaximal, characteristic, complemented
8#89994—31nomaximal, Hall
9C24×C3×C7C2211yescharacteristic
10C23×D3×C7C2211yescharacteristic
11C22×Dic21C2211yescharacteristic
12C22×Dic21C2211yescharacteristic
13C2×C7×#67—22no—
14C2×C7×#67—22no—
15C2×C7×#67—22no—
16C22×C7×Dic3C2211yescharacteristic
17C22×C3×Dic7C2211yescharacteristic
18C22×C3×Dic7C2211yescharacteristic
19#66636—22no—
20C3×C22⋊Dic7—22no—
21#66577—22no—
22(C2×D3)⋊Dic7—22no—
23(C2×D3)⋊Dic7—22no—
24C22×C7×D4—31no—
25C2×C22⋊Dic7—31no—
26C2×C22⋊Dic7—31no—
27C2×C22⋊Dic7—31no—
28C2×C22⋊Dic7D311yescharacteristic, complemented
29C2×Dic7⋊C4—31no—
30C7⋊#95—31no—
31#4959—71nomaximal, Hall
32C23×C3×C7—21no—
33C23×C3×C7—21no—
34C23×C3×C7—21no—
35C23×C3×C7—21no—
36C23×C3×C7—21no—
37C23×C3×C7—21no—
38C23×C3×C7D412yes—
39C23×C3×C7C2311yescharacteristic
40C22×D3×C7—21no—
41C22×D3×C7—21no—
42C22×D3×C7—21no—
43C22×D3×C7—21no—
44C22×D3×C7—21no—
45C22×D3×C7—21no—
46C22×D3×C7D412yes—
47C2×Dic21—21no—
48C2×Dic21—21no—
49C2×Dic21—21no—
50C2×Dic21—21no—
51C2×Dic21—21no—
52C2×Dic21D412yescomplemented
53C7×#67—28no—
54C2×C7×Dic3—21no—
55C2×C7×Dic3—21no—
56C2×C7×Dic3D412yes—
57C2×C3×Dic7—21no—
58C2×C3×Dic7—21no—
59C2×C3×Dic7—21no—
60C2×C3×Dic7—21no—
61C2×C3×Dic7—21no—
62C2×C3×Dic7—21no—
63C24×C7—31no—
64C24×C7C2×D311yescharacteristic
65C2×C7×D4—62no—
66C2×C7×D4—62no—
67C2×C7×D4—62no—
68C22×C4×C7—31no—
69C22×Dic7—31no—
70C22×Dic7—31no—
71C22×Dic7C2×D311yescharacteristic
72C22×Dic7C2×D311yescharacteristic
73C22⋊Dic7—22no—
74C22⋊Dic7—62no—
75C22⋊Dic7—62no—
76C22⋊Dic7—62no—
77Dic7⋊C4—62no—
78C22×#67D711yescharacteristic
79C2×C3×C22⋊C4—71no—
80C2×#217—71no—
81C2×(C2×D3)⋊C4—71no—
82C2×(C2×D3)⋊C4—71no—
83C2×Dic3⋊C4—71no—
84C3⋊#95—71no—
85C22×C3×C7—22no—
86C22×C3×C7—22no—
87C22×C3×C7—22no—
88C22×C3×C7—41no—
89C22×C3×C7—41no—
90C22×C3×C7—41no—
91C22×C3×C7—41no—
92C22×C3×C7C2×D411yescharacteristic
93C22×C3×C7C2×D411yescharacteristic
94C22×C3×C7C2×D411yescharacteristic
95C22×C3×C7C2×D411yescharacteristic
96C22×C3×C7#4111yescharacteristic
97C22×C3×C7#4111yescharacteristic
98C22×C3×C7#4111yescharacteristic
99C2×D3×C7—22no—
100C2×D3×C7—22no—
101C2×D3×C7—22no—
102C2×D3×C7—41no—
103C2×D3×C7—41no—
104C2×D3×C7—41no—
105C2×D3×C7—41no—
106Dic21—22no—
107Dic21—41no—
108C3×Dic7—41no—
109C3×Dic7—41no—
110C7×Dic3—22no—
111#394—211noSylow
112C23×C7—21no—
113C23×C7—21no—
114C23×C7—21no—
115C23×C7—21no—
116C23×C7—21no—
117C23×C7—21no—
118C23×C7—32no—
119C23×C7—61no—
120C23×C7—61no—
121C23×C7—61no—
122C23×C7—61no—
123C23×C7—61no—
124C23×C7—61no—
125C23×C7#6712yes—
126C23×C7C22×D311yescharacteristic
127C7×D4—68no—
128C2×C4×C7—32no—
129C2×C4×C7—61no—
130C2×C4×C7—61no—
131C2×Dic7—21no—
132C2×Dic7—21no—
133C2×Dic7—21no—
134C2×Dic7—21no—
135C2×Dic7—21no—
136C2×Dic7—21no—
137C2×Dic7—32no—
138C2×Dic7—61no—
139C2×Dic7—61no—
140C2×Dic7—61no—
141C2×Dic7—61no—
142C2×Dic7—61no—
143C24×C3C2×D711yescharacteristic
144C23×D3C2×D711yescharacteristic
145C22×C3×C4—71no—
146C22×C3×C4—71no—
147C2×#67—22no—
148C2×#67—22no—
149C2×#67—22no—
150C22×Dic3—71no—
151C22×Dic3—71no—
152C22×Dic3C2×D711yescharacteristic
153C3×C22⋊C4—142no—
154#217—142no—
155(C2×D3)⋊C4—142no—
156(C2×D3)⋊C4—142no—
157Dic3⋊C4—142no—
158C2×C3×C7—42no—
159C2×C3×C7#12011yescharacteristic
160C2×C3×C7#12111yescharacteristic
161C2×C3×C7#12111yescharacteristic
162C2×C3×C7#12111yescharacteristic
163C2×C3×C7#12311yescharacteristic
164C2×C3×C7#12311yescharacteristic
165C2×C3×C7#12411yescharacteristic
166D3×C7—42no—
167C22×D4—31no—
168C2×C4⋊C4—211no—
169C2×C22⋊C4—71no—
170C2×C22⋊C4—211no—
171C2×C22⋊C4—211no—
172C2×C22⋊C4—211no—
173#95—211no—
174C22×C7—22no—
175C22×C7—22no—
176C22×C7—22no—
177C22×C7—41no—
178C22×C7—41no—
179C22×C7—41no—
180C22×C7—41no—
181C22×C7—62no—
182C22×C7—62no—
183C22×C7—62no—
184C22×C7—121no—
185C22×C7—121no—
186C22×C7—121no—
187C22×C7—121no—
188C22×C7#23511yescharacteristic
189C22×C7#23511yescharacteristic
190C22×C7D3×D411yescharacteristic
191C22×C7D3×D411yescharacteristic
192C22×C7D3×D411yescharacteristic
193C22×C7#23711yescharacteristic
194C22×C7C2×#6711yescharacteristic
195C4×C7—62no—
196Dic7—41no—
197Dic7—41no—
198Dic7—62no—
199Dic7—121no—
200C23×C3—21no—
201C23×C3—21no—
202C23×C3—21no—
203C23×C3—21no—
204C23×C3—21no—
205C23×C3—21no—
206C23×C3C22⋊D712yes—
207C23×C3C22×D711yescharacteristic
208C22×D3—21no—
209C22×D3—21no—
210C22×D3—21no—
211C22×D3—21no—
212C22×D3—21no—
213C22×D3—21no—
214C22×D3C22⋊D712yes—
215C2×C3×C4—141no—
216C2×C3×C4—141no—
217C2×C3×C4—141no—
218C2×C3×C4—141no—
219C2×C3×C4—141no—
220C2×C3×C4—141no—
221#67—28no—
222C2×Dic3—21no—
223C2×Dic3—21no—
224C2×Dic3—72no—
225C2×Dic3—141no—
226C2×Dic3—141no—
227C2×Dic3—141no—
228C2×Dic3—141no—
229C2×Dic3—141no—
230C2×Dic3C22⋊D712yes—
231C3×C7#39411yescharacteristic, complemented, Hall
232C24—31no—
233C24D3×D711yescharacteristic
234C2×D4—62no—
235C2×D4—62no—
236C2×D4—62no—
237C22×C4—31no—
238C22×C4—71no—
239C22×C4—71no—
240C22×C4—211no—
241C22×C4—211no—
242C4⋊C4—422no—
243C22⋊C4—142no—
244C22⋊C4—422no—
245C22⋊C4—422no—
246C22⋊C4—422no—
247C2×C7—42no—
248C2×C7—122no—
249C2×C7#88311yescharacteristic
250C2×C7#88411yescharacteristic
251C2×C7#88411yescharacteristic
252C2×C7#88511yescharacteristic
253C2×C7#92911yescharacteristic
254C2×C7#93711yescharacteristic
255C2×C7#93911yescharacteristic
256C22×C3—22no—
257C22×C3—22no—
258C22×C3—22no—
259C22×C3—41no—
260C22×C3—41no—
261C22×C3—41no—
262C22×C3—41no—
263C22×C3D4×D711yescharacteristic
264C22×C3#115911yescharacteristic
265C22×C3#115911yescharacteristic
266C22×C3#115911yescharacteristic
267C22×C3C2×C22⋊D711yescharacteristic
268C22×C3C2×C22⋊D711yescharacteristic
269C22×C3C2×C22⋊D711yescharacteristic
270C2×D3—22no—
271C2×D3—22no—
272C2×D3—22no—
273C2×D3—41no—
274C2×D3—41no—
275C2×D3—41no—
276C2×D3—41no—
277C3×C4—281no—
278C3×C4—281no—
279Dic3—22no—
280Dic3—142no—
281Dic3—281no—
282C23—21no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—32no—
289C23—61no—
290C23—61no—
291C23—61no—
292C23—61no—
293C23—61no—
294C23—61no—
295C23#424512yes—
296C23C2×D3×D711yescharacteristic
297D4—68no—
298C2×C4—32no—
299C2×C4—61no—
300C2×C4—61no—
301C2×C4—141no—
302C2×C4—141no—
303C2×C4—141no—
304C2×C4—141no—
305C2×C4—141no—
306C2×C4—141no—
307C2×C4—212no—
308C2×C4—421no—
309C2×C4—421no—
310C2×C4—421no—
311C2×C4—421no—
312C2×C4—421no—
313C7#495911yescharacteristic, complemented, Sylow
314C2×C3—42no—
315C2×C3#649811yescharacteristic
316C2×C3#649811yescharacteristic
317C2×C3#649911yescharacteristic
318C2×C3#650111yescharacteristic
319C2×C3#650211yescharacteristic
320C2×C3#650211yescharacteristic
321C2×C3#651611yescharacteristic
322D3—42no—
323C22—22no—
324C22—22no—
325C22—22no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—62no—
331C22—62no—
332C22—62no—
333C22—121no—
334C22—121no—
335C22—121no—
336C22—121no—
337C22#6668411yescharacteristic
338C22#6668611yescharacteristic
339C22#6668611yescharacteristic
340C22C2×#424511yescharacteristic
341C22D3×C22⋊D711yescharacteristic
342C22D3×C22⋊D711yescharacteristic
343C22#6669411yescharacteristic
344C4—62no—
345C4—281no—
346C4—281no—
347C4—422no—
348C4—841no—
349C3#8999411yescharacteristic, complemented, Sylow
350C2—42no—
351C2—122no—
352C2#1062108711yescharacteristic
353C2#1062110111yescharacteristic
354C2#1062111711yescharacteristic
355C2#1062111911yescharacteristic
356C2#1062112011yescharacteristic
357C2#1062113011yescharacteristic
358C2#1062113311yescharacteristic
subgroupG/H
center, FrattiniC23C2×D3×D7
commutator, socle, cosocleC23×C3×C7C23
FittingC24×C3×C7C22
prime radicalC22×C7×#67C2
hypercenterC24D3×D7
hypocenterC3×C7#394
LCSG ▷ C23×C3×C7 ▷ C3×C7
UCSC1 ◁ C23 ◁ C24

Complements

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

NH ≅ G/Ntype
2. C2×#66636C2semidirect
3. C2×C3×C22⋊Dic7C2semidirect
4. C2×#66577C2semidirect
5. C2×(C2×D3)⋊Dic7C2semidirect
6. C2×(C2×D3)⋊Dic7C2semidirect
7. #10620529C2semidirect
28. C2×C22⋊Dic7D3semidirect
52. C2×Dic21D4semidirect
231. C3×C7#394semidirect
313. C7#4959semidirect
349. C3#89994semidirect

Maximal subgroups

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

subgroupindex
C22×C7×#672
C2×#666362
C2×C3×C22⋊Dic72
C2×#665772
C2×(C2×D3)⋊Dic72
#106205292
#899943
#49597