Finite Group Databasebeta

← #13052510

Subgroups — #13052510

Order 1344.

316 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 70 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
1C2×#66684C211yesmaximal, characteristic
2#10621360C211yesmaximal, characteristic, complemented
3#10621094C211yesmaximal, characteristic, complemented
4C7×#806C211yesmaximal, characteristic, complemented
5C3×#6380C211yesmaximal, characteristic, complemented
6#10620531C211yesmaximal, characteristic, complemented
7#10620530C211yesmaximal, characteristic, complemented
8#89512—31nomaximal, Hall
9C22×Dic21C2211yescharacteristic
10C2×C7×#67C2211yescharacteristic
11C2×C3×C22⋊D7C2211yescharacteristic
12C2×#4248C411yescharacteristic, complemented
13C2×#4245C411yescharacteristic, complemented
14C2×D3×Dic7C411yescharacteristic, complemented
15#66684—21no—
16#66684—21no—
17#66684—21no—
18#66684—21no—
19C2×Dic3×D7C411yescharacteristic, complemented
20C3×C7×C22⋊C4C2211yescharacteristic
21#66636C2211yescharacteristic
22#66629—21no—
23C4×Dic21—21no—
24C7×#217C2211yescharacteristic
25C3×C22⋊Dic7C2211yescharacteristic
26#66577—21no—
27Dic3×Dic7—21no—
28C2×#1159—31no—
29#6440—31no—
30#6440—31no—
31C7×#99—31no—
32#6408—31no—
33#6380—31no—
34#6380D311yescharacteristic, complemented
35#4740—71nomaximal, Hall
36C23×C3×C7C2311yescharacteristic
37C22×D3×C7C2×C411yescharacteristic
38C22×C3×D7C2×C411yescharacteristic
39C2×C3×C4×C7—21no—
40C2×Dic21—21no—
41C2×Dic21C2×C411yescharacteristic
42C2×Dic21C2×C411yescharacteristic
43C2×Dic21D411yescharacteristic
44C2×Dic21D411yescharacteristic
45C2×Dic21D411yescharacteristic
46C2×Dic21D411yescharacteristic
47C7×#67—21no—
48C7×#67—21no—
49C2×C7×Dic3—21no—
50C2×C7×Dic3C2×C411yescharacteristic
51C3×C22⋊D7—21no—
52C3×C22⋊D7—21no—
53C2×C3×Dic7—21no—
54C2×C3×Dic7C2×C411yescharacteristic
55#4248—21no—
56#4248—21no—
57#4245—21no—
58#4245—21no—
59D3×Dic7—21no—
60D3×Dic7—21no—
61Dic3×D7—21no—
62Dic3×D7—21no—
63C2×C7×D4—31no—
64C2×C22⋊D7—31no—
65C2×C22⋊D7C2×D311yescharacteristic
66C22×Dic7—31no—
67C22×Dic7—31no—
68#1159—61no—
69#1159—61no—
70#1159—61no—
71#1159—61no—
72C2×C4×D7—31no—
73C2×Dic14—31no—
74C7×C22⋊C4—31no—
75C7×C22⋊C4C2×D311yescharacteristic
76C22⋊Dic7—31no—
77C22⋊Dic7C2×D311yescharacteristic
78Dic7⋊C4—61no—
79Dic7⋊C4—61no—
80C4×Dic7—61no—
81C4×Dic7—61no—
82C2×#237—71no—
83#877—71no—
84#877—71no—
85C3×#99—71no—
86#834—71no—
87#806—71no—
88#806D711yescharacteristic, complemented
89C22×C3×C7—41no—
90C22×C3×C7C22×C411yescharacteristic
91C22×C3×C7C2×D411yescharacteristic
92C22×C3×C7C2×D411yescharacteristic
93C2×D3×C7—21no—
94C2×D3×C7—41no—
95C2×C3×D7—21no—
96C2×C3×D7—41no—
97C3×C4×C7—41no—
98Dic21—41no—
99Dic21C22⋊C411yescharacteristic
100Dic21C22⋊C411yescharacteristic
101Dic21C22⋊C411yescharacteristic
102Dic21C22⋊C411yescharacteristic
103C3×Dic7—21no—
104C3×Dic7—41no—
105C7×Dic3—21no—
106C7×Dic3—41no—
107#410—211noSylow
108C23×C7—31no—
109C23×C7C22×D311yescharacteristic
110C22×D7C4×D311yescharacteristic
111C7×D4—61no—
112C7×D4—61no—
113C2×C4×C7—21no—
114C2×C4×C7—31no—
115C2×C4×C7—61no—
116C22⋊D7—21no—
117C22⋊D7—21no—
118C22⋊D7—61no—
119C22⋊D7—61no—
120C2×Dic7—21no—
121C2×Dic7—31no—
122C2×Dic7—31no—
123C2×Dic7—31no—
124C2×Dic7—31no—
125C2×Dic7—31no—
126C2×Dic7—31no—
127C2×Dic7—61no—
128C2×Dic7—61no—
129C2×Dic7—61no—
130C2×Dic7C4×D311yescharacteristic
131C4×D7—61no—
132C4×D7—61no—
133Dic14—61no—
134Dic14—61no—
135C2×C3×D4—71no—
136C2×#67—71no—
137C2×#67C2×D711yescharacteristic
138C22×Dic3—71no—
139C22×Dic3—71no—
140#237—141no—
141#237—141no—
142#237—141no—
143#237—141no—
144C2×C4×D3—71no—
145C2×Dic6—71no—
146C3×C22⋊C4—71no—
147C3×C22⋊C4C2×D711yescharacteristic
148#217—71no—
149#217C2×D711yescharacteristic
150Dic3⋊C4—141no—
151Dic3⋊C4—141no—
152C4×Dic3—141no—
153C4×Dic3—141no—
154C2×C3×C7—21no—
155C2×C3×C7—21no—
156C2×C3×C7—21no—
157C2×C3×C7C2×C22⋊C411yescharacteristic
158C3×D7—41no—
159D3×C7—41no—
160C2×#41—211no—
161#117—211no—
162#117—211no—
163#99—31no—
164#99—71no—
165#99—211no—
166#99—211no—
167C22×C7—41no—
168C22×C7—61no—
169C22×C7—121no—
170C22×C7C2×C4×D311yescharacteristic
171C22×C7D3×D411yescharacteristic
172C22×C7D3×D411yescharacteristic
173C2×D7—21no—
174C2×D7—41no—
175C4×C7—41no—
176C4×C7—61no—
177C4×C7—121no—
178Dic7—21no—
179Dic7—31no—
180Dic7—31no—
181Dic7—31no—
182Dic7—31no—
183Dic7—41no—
184Dic7—121no—
185C23×C3—71no—
186C23×C3C22×D711yescharacteristic
187C22×D3C4×D711yescharacteristic
188C3×D4—141no—
189C3×D4—141no—
190C2×C3×C4—21no—
191C2×C3×C4—71no—
192C2×C3×C4—141no—
193#67—21no—
194#67—21no—
195#67—141no—
196#67—141no—
197C2×Dic3—21no—
198C2×Dic3—71no—
199C2×Dic3—71no—
200C2×Dic3—71no—
201C2×Dic3—71no—
202C2×Dic3—71no—
203C2×Dic3—71no—
204C2×Dic3—141no—
205C2×Dic3—141no—
206C2×Dic3—141no—
207C2×Dic3C4×D711yescharacteristic
208C4×D3—141no—
209C4×D3—141no—
210Dic6—141no—
211Dic6—141no—
212C3×C7#41011yescharacteristic, complemented, Hall
213#41—421no—
214#41—421no—
215#41—421no—
216#41—421no—
217C2×Q8—211no—
218C2×D4—31no—
219C2×D4—71no—
220C2×D4—211no—
221C22×C4—211no—
222C22×C4—211no—
223C22×C4—211no—
224C4⋊C4—421no—
225C4⋊C4—421no—
226C22⋊C4—31no—
227C22⋊C4—71no—
228C22⋊C4—211no—
229C22⋊C4D3×D711yescharacteristic
230C42—421no—
231C42—421no—
232C2×C7—21no—
233C2×C7—21no—
234C2×C7—21no—
235C2×C7—121no—
236C2×C7D3×C22⋊C411yescharacteristic
237D7—41no—
238C22×C3—41no—
239C22×C3—141no—
240C22×C3—281no—
241C22×C3C2×C4×D711yescharacteristic
242C22×C3D4×D711yescharacteristic
243C22×C3D4×D711yescharacteristic
244C2×D3—21no—
245C2×D3—41no—
246C3×C4—41no—
247C3×C4—141no—
248C3×C4—281no—
249Dic3—21no—
250Dic3—41no—
251Dic3—71no—
252Dic3—71no—
253Dic3—71no—
254Dic3—71no—
255Dic3—281no—
256C23—31no—
257C23—71no—
258C23C2×D3×D711yescharacteristic
259Q8—421no—
260Q8—421no—
261D4—61no—
262D4—61no—
263D4—141no—
264D4—141no—
265D4—421no—
266D4—421no—
267C2×C4—21no—
268C2×C4—31no—
269C2×C4—61no—
270C2×C4—71no—
271C2×C4—141no—
272C2×C4—211no—
273C2×C4—211no—
274C2×C4—211no—
275C2×C4—211no—
276C2×C4—211no—
277C2×C4—211no—
278C2×C4—421no—
279C2×C4—421no—
280C2×C4—421no—
281C2×C4—421no—
282C2×C4—421no—
283C7#474011yescharacteristic, complemented, Sylow
284C2×C3—21no—
285C2×C3—21no—
286C2×C3—21no—
287C2×C3—281no—
288C2×C3D7×C22⋊C411yescharacteristic
289D3—41no—
290C22—41no—
291C22—61no—
292C22—121no—
293C22—141no—
294C22—281no—
295C22C4×D3×D711yescharacteristic
296C22#6668111yescharacteristic
297C22#6669411yescharacteristic
298C4—41no—
299C4—61no—
300C4—121no—
301C4—141no—
302C4—211no—
303C4—211no—
304C4—211no—
305C4—211no—
306C4—281no—
307C4—841no—
308C3#8951211yescharacteristic, complemented, Sylow
309C2—21no—
310C2—21no—
311C2—21no—
312C2—121no—
313C2—281no—
314C2#1062103511yescharacteristic
subgroupG/H
centerC2#10621035
commutatorC22×C3×C7C22×C4
FrattiniC23C2×D3×D7
FittingC3×C7×C22⋊C4C22
socleC2×C3×C7C2×C22⋊C4
prime radicalC2×#66684C2
cosocleC23×C3×C7C23
hypercenterC22⋊C4D3×D7
hypocenterC3×C7#410
LCSG ▷ C22×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
2. #10621360C2semidirect
3. #10621094C2semidirect
4. C7×#806C2semidirect
5. C3×#6380C2semidirect
6. #10620531C2semidirect
7. #10620530C2semidirect
12. C2×#4248C4semidirect
13. C2×#4245C4semidirect
14. C2×D3×Dic7C4semidirect
19. C2×Dic3×D7C4semidirect
34. #6380D3semidirect
88. #806D7semidirect
212. C3×C7#410semidirect
283. C7#4740semidirect
308. C3#89512semidirect

Maximal subgroups

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

subgroupindex
C2×#666842
#106213602
#106210942
C7×#8062
C3×#63802
#106205312
#106205302
#895123
#47407