Finite Group Databasebeta

← #13062034

Subgroups — #13062034

Order 1344.

342 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 41 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
1C4×A4×D7C211yesmaximal, characteristic, complemented
2C4×C7⋊S4C211yesmaximal, characteristic, complemented
3C4×C7×S4C211yesmaximal, characteristic, complemented
4#10621483C211yesmaximal, characteristic, complemented
5#10621482C211yesmaximal, characteristic, complemented
6#10621480C211yesmaximal, characteristic, complemented
7#10621477C211yesmaximal, characteristic, complemented
8#90238—31nomaximal, Hall
9C2×A4×D7C2211yescharacteristic
10C2×C7⋊S4C2211yescharacteristic
11C2×C7×S4C2211yescharacteristic
12C4×C7×A4C2211yescharacteristic
13#66675—41nomaximal
14A4×Dic7C2211yescharacteristic
15C7⋊SL(2,ℤ/4)C2211yescharacteristic
16C7×SL(2,ℤ/4)C2211yescharacteristic
17C2×#1157—31no—
18C22×C4×D7D311yescharacteristic, complemented
19C4×C7×D4—31no—
20#6502—31no—
21#6500—31no—
22C4×C22⋊D7—31no—
23C4×C22⋊D7—31no—
24#6459—31no—
25#6457—31no—
26#6447—31no—
27#6446—31no—
28#6445—31no—
29#6441—31no—
30C4×D28—31no—
31#6435—31no—
32#5912—71nomaximal, Hall
33C2×C7×A4C2311yescharacteristic
34A4×D7—21no—
35C7⋊S4—21no—
36C7×S4—21no—
37C4×D21—41no—
38C4×D3×C7—41no—
39C3×C4×D7—41no—
40#4248—41no—
41#4247—41no—
42#4246—41no—
43#4245—41no—
44C23×D7C2×D311yescharacteristic
45C2×C7×D4—31no—
46C22×C4×C7—31no—
47C22×C4×C7C2×D311yescharacteristic
48C2×C22⋊D7—31no—
49C2×C22⋊D7—31no—
50C2×C22⋊D7—31no—
51C2×C22⋊D7—31no—
52C22×Dic7C2×D311yescharacteristic
53#1157—61no—
54#1157—61no—
55#1157—61no—
56#1157—61no—
57C2×D28—31no—
58C2×D28—31no—
59C2×C4×D7—31no—
60C2×C4×D7—31no—
61C2×C4×D7—31no—
62C2×C4×D7—31no—
63C2×C4×D7—31no—
64C2×Dic14—31no—
65C7×C4⋊C4—31no—
66C7×C22⋊C4—31no—
67C7×C22⋊C4—31no—
68C42×C7—31no—
69C22⋊Dic7—31no—
70C22⋊Dic7—31no—
71(C2×D7)⋊C4—31no—
72(C2×D7)⋊C4—31no—
73(C2×D7)⋊C4—31no—
74(C2×D7)⋊C4—31no—
75(C2×D7)⋊C4—31no—
76(C2×D7)⋊C4—31no—
77C4⋊Dic7—31no—
78Dic7⋊C4—31no—
79Dic7⋊C4—31no—
80Dic7⋊C4—31no—
81Dic7⋊C4—31no—
82C4×Dic7—31no—
83C2×C4×A4—71no—
84GL(2,ℤ/4)—71no—
85GL(2,ℤ/4)—71no—
86C4⋊S4—71no—
87C4×S4—71no—
88C4×S4D711yescharacteristic, complemented
89A4⋊Q8—71no—
90C2×D21—41no—
91C2×D3×C7—41no—
92C2×C3×D7—41no—
93C7×A4#4111yescharacteristic, complemented
94C3×C4×C7—41no—
95Dic21—41no—
96C3×Dic7—41no—
97C7×Dic3—41no—
98#525—211noSylow
99C23×C7—31no—
100C23×C7C22×D311yescharacteristic
101C22×D7—21no—
102C22×D7—31no—
103C22×D7—31no—
104C22×D7—31no—
105C22×D7—61no—
106C7×D4—61no—
107C7×D4—61no—
108C2×C4×C7—31no—
109C2×C4×C7—31no—
110C2×C4×C7—31no—
111C2×C4×C7—31no—
112C2×C4×C7—31no—
113C2×C4×C7—61no—
114C2×C4×C7—61no—
115C22⋊D7—61no—
116C22⋊D7—61no—
117C22⋊D7—62no—
118C22⋊D7—61no—
119C22⋊D7—61no—
120C22⋊D7—61no—
121C22⋊D7—61no—
122C2×Dic7—31no—
123C2×Dic7—31no—
124C2×Dic7—31no—
125C2×Dic7—31no—
126C2×Dic7—31no—
127D28—62no—
128D28—61no—
129D28—61no—
130C4×D7—31no—
131C4×D7—31no—
132C4×D7—31no—
133C4×D7—61no—
134C4×D7—61no—
135C4×D7—61no—
136C4×D7S411yescharacteristic, complemented
137Dic14—61no—
138Dic14—61no—
139C22×A4—71no—
140C2×S4—71no—
141C2×S4C2×D711yescharacteristic
142#235—281no—
143C4×A4—71no—
144C4×A4C2×D711yescharacteristic
145SL(2,ℤ/4)—71no—
146SL(2,ℤ/4)C2×D711yescharacteristic
147C2×C3×C7—41no—
148D21—81no—
149C3×D7—81no—
150D3×C7—81no—
151C2×#41—211no—
152C23×C4—71no—
153#123—211no—
154#123—211no—
155C22⋊Q8—211no—
156C22⋊Q8—211no—
157#121—211no—
158#121—211no—
159#120—211no—
160#120—211no—
161C4×D4—31no—
162C4×D4—211no—
163C4×D4—211no—
164C4×D4—211no—
165#117—211no—
166C22×C7—31no—
167C22×C7—31no—
168C22×C7—61no—
169C22×C7—61no—
170C22×C7—61no—
171C22×C7#23511yescharacteristic, complemented
172C2×D7—32no—
173C2×D7—31no—
174C2×D7—61no—
175C2×D7—61no—
176C2×D7—61no—
177C2×D7—61no—
178C2×D7—61no—
179C2×D7—61no—
180C2×D7C2×S411yescharacteristic
181C4×C7—31no—
182C4×C7—61no—
183C4×C7—61no—
184C4×C7—61no—
185C4×C7C2×S411yescharacteristic
186Dic7—31no—
187Dic7—61no—
188Dic7—61no—
189Dic7—61no—
190Dic7C2×S411yescharacteristic
191C2×A4—141no—
192C2×A4C22×D711yescharacteristic
193S4—21no—
194S4—141no—
195C2×C3×C4—281no—
196#67—281no—
197#67—281no—
198D12—281no—
199C4×D3—41no—
200C4×D3—281no—
201Dic6—281no—
202C3×C7—41noHall
203C24—71no—
204#41—421no—
205#41—421no—
206#41—421no—
207#41—421no—
208C2×Q8—211no—
209C2×D4—31no—
210C2×D4—211no—
211C2×D4—211no—
212C2×D4—211no—
213C2×D4—211no—
214C2×D4—211no—
215C2×D4—211no—
216C22×C4—31no—
217C22×C4—71no—
218C22×C4—211no—
219C22×C4—211no—
220C22×C4—211no—
221C22×C4—211no—
222C22×C4—211no—
223C22×C4D3×D711yescharacteristic
224C4⋊C4—31no—
225C4⋊C4—211no—
226C4⋊C4—211no—
227C4⋊C4—211no—
228C4⋊C4—211no—
229C4⋊C4—211no—
230C22⋊C4—31no—
231C22⋊C4—31no—
232C22⋊C4—211no—
233C22⋊C4—211no—
234C22⋊C4—211no—
235C22⋊C4—211no—
236C22⋊C4—211no—
237C22⋊C4—211no—
238C22⋊C4—211no—
239C22⋊C4—211no—
240C42—31no—
241C42—211no—
242C2×C7—31no—
243C2×C7—31no—
244C2×C7—121no—
245C2×C7C22×S411yescharacteristic
246D7—21no—
247D7—61no—
248D7—121no—
249C22×C3—281no—
250C2×D3—41no—
251C2×D3—281no—
252A4#115711yescharacteristic, complemented
253C3×C4—41no—
254C3×C4—281no—
255Dic3—41no—
256Dic3—281no—
257C23—31no—
258C23—141no—
259C23—211no—
260C23—211no—
261C23—211no—
262C23—421no—
263C23C2×D3×D711yescharacteristic
264Q8—421no—
265Q8—421no—
266D4—61no—
267D4—61no—
268D4—421no—
269D4—422no—
270D4—421no—
271D4—421no—
272D4—421no—
273D4—421no—
274D4—421no—
275D4—422no—
276D4—421no—
277D4—421no—
278C2×C4—31no—
279C2×C4—31no—
280C2×C4—31no—
281C2×C4—31no—
282C2×C4—31no—
283C2×C4—61no—
284C2×C4—61no—
285C2×C4—71no—
286C2×C4—211no—
287C2×C4—211no—
288C2×C4—211no—
289C2×C4—211no—
290C2×C4—211no—
291C2×C4—211no—
292C2×C4—211no—
293C2×C4—211no—
294C2×C4—421no—
295C2×C4—421no—
296C2×C4—421no—
297C7#591211yescharacteristic, complemented, Sylow
298C2×C3—41no—
299C2×C3—561no—
300D3—81no—
301D3—561no—
302C22—31no—
303C22—31no—
304C22—61no—
305C22—61no—
306C22—61no—
307C22—71no—
308C22—212no—
309C22—211no—
310C22—421no—
311C22—421no—
312C22—421no—
313C22—421no—
314C22—421no—
315C22—421no—
316C22#6667511yescharacteristic, complemented
317C4—31no—
318C4—61no—
319C4—61no—
320C4—61no—
321C4—71no—
322C4—211no—
323C4—421no—
324C4—421no—
325C4—421no—
326C4D7×S411yescharacteristic
327C3—41noSylow
328C2—31no—
329C2—31no—
330C2—121no—
331C2—141no—
332C2—421no—
333C2—841no—
334C2C2×D7×S411yescharacteristic
subgroupG/H
center, hypercenterC4D7×S4
commutator, cosocleC2×C7×A4C23
FrattiniC2C2×D7×S4
FittingC22×C4×C7C2×D3
socleC23×C7C22×D3
hypocenterC7×A4#41
LCSG ▷ C2×C7×A4 ▷ C7×A4
UCSC1 ◁ C4

Complements

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

NH ≅ G/Ntype
1. C4×A4×D7C2semidirect
2. C4×C7⋊S4C2semidirect
3. C4×C7×S4C2semidirect
4. #10621483C2semidirect
5. #10621482C2semidirect
6. #10621480C2semidirect
7. #10621477C2semidirect
18. C22×C4×D7D3semidirect
88. C4×S4D7semidirect
93. C7×A4#41semidirect
136. C4×D7S4semidirect
171. C22×C7#235semidirect
252. A4#1157semidirect
297. C7#5912semidirect
316. C22#66675semidirect

Maximal subgroups

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

subgroupindex
C4×A4×D72
C4×C7⋊S42
C4×C7×S42
#106214832
#106214822
#106214802
#106214772
#902383
#666754
#59127