Finite Group Databasebeta

← #13054313

Subgroups — #13054313

Order 1344.

404 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 102 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×#4248C211yesmaximal, characteristic, complemented
2C2×C3×(C2×D7)⋊C4C211yesmaximal, characteristic
3C2×(C2×D7)⋊Dic3C211yesmaximal, characteristic
4C2×(C2×D7)⋊Dic3C211yesmaximal, characteristic
5#10620670C211yesmaximal, characteristic, complemented
6C7×C3⋊#95C211yesmaximal, characteristic, complemented
7#10620529C211yesmaximal, characteristic, complemented
8#89445—31nomaximal, Hall
9C23×C3×D7C2211yescharacteristic
10C22×C3×C4×C7C2211yescharacteristic
11C22×Dic21C2211yescharacteristic
12C22×Dic21C2211yescharacteristic
13C22×C7×Dic3C2211yescharacteristic
14C22×C7×Dic3C2211yescharacteristic
15C22×C3×Dic7C2211yescharacteristic
16C2×#4248—22no—
17C2×#4248—22no—
18C2×#4248—22no—
19C3×(C2×D7)⋊C4—22no—
20(C2×D7)⋊Dic3—22no—
21(C2×D7)⋊Dic3—22no—
22C22×Dic14—31no—
23C2×(C2×D7)⋊C4—31no—
24C2×(C2×D7)⋊C4—31no—
25C2×(C2×D7)⋊C4D311yescharacteristic
26C7×#95—31no—
27C7⋊#95—31no—
28C7⋊#95—31no—
29#4947—71nomaximal, Hall
30C23×C3×C7C2311yescharacteristic
31C22×C3×D7—21no—
32C22×C3×D7—21no—
33C22×C3×D7—21no—
34C22×C3×D7—21no—
35C22×C3×D7—21no—
36C22×C3×D7—21no—
37C22×C3×D7Q812yescomplemented
38C2×C3×C4×C7—21no—
39C2×C3×C4×C7—21no—
40C2×C3×C4×C7—21no—
41C2×Dic21—21no—
42C2×Dic21—21no—
43C2×Dic21—21no—
44C2×Dic21—21no—
45C2×Dic21—21no—
46C2×Dic21D412yes—
47C2×C7×Dic3—21no—
48C2×C7×Dic3—21no—
49C2×C7×Dic3—21no—
50C2×C7×Dic3—21no—
51C2×C7×Dic3—21no—
52C2×C7×Dic3D412yes—
53C2×C3×Dic7—21no—
54C2×C3×Dic7—21no—
55C2×C3×Dic7D412yes—
56#4248—28no—
57C23×D7C2×D311yescharacteristic
58C22×C4×C7—31no—
59C22×C4×C7—31no—
60C22×C4×C7C2×D311yescharacteristic
61C22×Dic7—31no—
62C22×Dic7—31no—
63C22×Dic7C2×D311yescharacteristic
64C2×Dic14—62no—
65C2×Dic14—62no—
66C2×Dic14—62no—
67(C2×D7)⋊C4—22no—
68(C2×D7)⋊C4—62no—
69(C2×D7)⋊C4—62no—
70C22×Dic6—71no—
71C2×C3×C22⋊C4—71no—
72C2×#217—71no—
73C2×#217—71no—
74C3⋊#95—71no—
75C3⋊#95—71no—
76C3⋊#95D711yescharacteristic, complemented
77C22×C3×C7C2×D411yescharacteristic
78C22×C3×C7C2×D411yescharacteristic
79C22×C3×C7C2×D411yescharacteristic
80C22×C3×C7C2×Q811yescharacteristic
81C22×C3×C7#4111yescharacteristic
82C22×C3×C7#4111yescharacteristic
83C22×C3×C7#4111yescharacteristic
84C2×C3×D7—22no—
85C2×C3×D7—22no—
86C2×C3×D7—22no—
87C2×C3×D7—41no—
88C2×C3×D7—41no—
89C2×C3×D7—41no—
90C2×C3×D7—41no—
91C3×C4×C7—41no—
92Dic21—22no—
93Dic21—41no—
94C3×Dic7—22no—
95C7×Dic3—22no—
96C7×Dic3—41no—
97#393—211noSylow
98C23×C7C22×D311yescharacteristic
99C22×D7—21no—
100C22×D7—21no—
101C22×D7—21no—
102C22×D7—21no—
103C22×D7—21no—
104C22×D7—21no—
105C22×D7Dic612yescomplemented
106C2×C4×C7—21no—
107C2×C4×C7—21no—
108C2×C4×C7—21no—
109C2×C4×C7—32no—
110C2×C4×C7—61no—
111C2×C4×C7—61no—
112C2×C4×C7—61no—
113C2×C4×C7—61no—
114C2×C4×C7—61no—
115C2×Dic7—21no—
116C2×Dic7—21no—
117C2×Dic7—32no—
118C2×Dic7—61no—
119C2×Dic7—61no—
120C2×Dic7—61no—
121C2×Dic7—61no—
122C2×Dic7—61no—
123C2×Dic7#6712yes—
124Dic14—68no—
125C24×C3—71no—
126C22×C3×C4—71no—
127C22×C3×C4C2×D711yescharacteristic
128C22×Dic3—71no—
129C22×Dic3—71no—
130C22×Dic3C2×D711yescharacteristic
131C22×Dic3C2×D711yescharacteristic
132C2×Dic6—142no—
133C2×Dic6—142no—
134C2×Dic6—142no—
135C3×C22⋊C4—142no—
136#217—142no—
137#217—142no—
138C2×C3×C7#12011yescharacteristic
139C2×C3×C7C22⋊Q811yescharacteristic
140C2×C3×C7C22⋊Q811yescharacteristic
141C2×C3×C7C22⋊Q811yescharacteristic
142C2×C3×C7#12411yescharacteristic
143C2×C3×C7#12411yescharacteristic
144C2×C3×C7#12411yescharacteristic
145C3×D7—42no—
146C22×Q8—211no—
147C2×C22⋊C4—71no—
148C2×C22⋊C4—211no—
149C2×C22⋊C4—211no—
150#95—31no—
151#95—211no—
152#95—211no—
153C22×C7C2×Dic611yescharacteristic
154C22×C7#23511yescharacteristic
155C22×C7D3×D411yescharacteristic
156C22×C7D3×D411yescharacteristic
157C22×C7#23711yescharacteristic
158C22×C7#23711yescharacteristic
159C22×C7C2×#6711yescharacteristic
160C2×D7—22no—
161C2×D7—22no—
162C2×D7—22no—
163C2×D7—41no—
164C2×D7—41no—
165C2×D7—41no—
166C2×D7—41no—
167C4×C7—41no—
168C4×C7—62no—
169C4×C7—121no—
170Dic7—22no—
171Dic7—62no—
172Dic7—121no—
173C23×C3—72no—
174C23×C3—141no—
175C23×C3—141no—
176C23×C3—141no—
177C23×C3—141no—
178C23×C3—141no—
179C23×C3—141no—
180C23×C3C22×D711yescharacteristic
181C2×C3×C4—21no—
182C2×C3×C4—21no—
183C2×C3×C4—21no—
184C2×C3×C4—72no—
185C2×C3×C4—141no—
186C2×C3×C4—141no—
187C2×Dic3—21no—
188C2×Dic3—21no—
189C2×Dic3—21no—
190C2×Dic3—21no—
191C2×Dic3—21no—
192C2×Dic3—72no—
193C2×Dic3—141no—
194C2×Dic3—141no—
195C2×Dic3—141no—
196C2×Dic3—141no—
197C2×Dic3—141no—
198C2×Dic3D2812yes—
199Dic6—148no—
200C3×C7#39311yescharacteristic, complemented, Hall
201C24—71no—
202C2×Q8—422no—
203C2×Q8—422no—
204C2×Q8—422no—
205C22×C4—31no—
206C22×C4—31no—
207C22×C4—71no—
208C22×C4—211no—
209C22×C4—211no—
210C22×C4D3×D711yescharacteristic
211C22⋊C4—142no—
212C22⋊C4—422no—
213C22⋊C4—422no—
214C2×C7#87811yescharacteristic
215C2×C7#87811yescharacteristic
216C2×C7#88511yescharacteristic
217C2×C7#88511yescharacteristic
218C2×C7#92411yescharacteristic
219C2×C7#93611yescharacteristic
220C2×C7#93711yescharacteristic
221D7—42no—
222C22×C3—142no—
223C22×C3—142no—
224C22×C3—142no—
225C22×C3—281no—
226C22×C3—281no—
227C22×C3—281no—
228C22×C3—281no—
229C22×C3C2×D2811yescharacteristic
230C22×C3#115711yescharacteristic
231C22×C3#115711yescharacteristic
232C22×C3D4×D711yescharacteristic
233C22×C3D4×D711yescharacteristic
234C22×C3#115911yescharacteristic
235C22×C3Q8×D711yescharacteristic
236C3×C4—41no—
237C3×C4—142no—
238Dic3—22no—
239Dic3—41no—
240Dic3—142no—
241Dic3—281no—
242C23—72no—
243C23—141no—
244C23—141no—
245C23—141no—
246C23—141no—
247C23—141no—
248C23—141no—
249C23C2×D3×D711yescharacteristic
250Q8—428no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—32no—
255C2×C4—61no—
256C2×C4—61no—
257C2×C4—61no—
258C2×C4—61no—
259C2×C4—61no—
260C2×C4—72no—
261C2×C4—141no—
262C2×C4—141no—
263C2×C4—212no—
264C2×C4—421no—
265C2×C4—421no—
266C2×C4—421no—
267C2×C4—421no—
268C2×C4—421no—
269C7#494711yescharacteristic, complemented, Sylow
270C2×C3—282no—
271C2×C3#643811yescharacteristic
272C2×C3#644511yescharacteristic
273C2×C3#644811yescharacteristic
274C2×C3#644811yescharacteristic
275C2×C3#645911yescharacteristic
276C2×C3#645911yescharacteristic
277C2×C3#646011yescharacteristic
278C22—142no—
279C22—142no—
280C22—142no—
281C22—281no—
282C22—281no—
283C22—281no—
284C22—281no—
285C22D7×Dic611yescharacteristic
286C22#6667511yescharacteristic
287C22D3×D2811yescharacteristic
288C22#6668111yescharacteristic
289C22#6668311yescharacteristic
290C22#6668411yescharacteristic
291C22D7×#6711yescharacteristic
292C4—41no—
293C4—62no—
294C4—121no—
295C4—142no—
296C4—422no—
297C4—841no—
298C3#8944511yescharacteristic, complemented, Sylow
299C2—282no—
300C2#1062091111yescharacteristic
301C2#1062091511yescharacteristic
302C2#1062098411yescharacteristic
303C2#1062099211yescharacteristic
304C2#1062102611yescharacteristic
305C2#1062103611yescharacteristic
306C2#1062109611yescharacteristic
subgroupG/H
center, FrattiniC23C2×D3×D7
commutator, socle, cosocleC23×C3×C7C23
FittingC22×C3×C4×C7C22
prime radicalC23×C3×D7C22
hypercenterC22×C4D3×D7
hypocenterC3×C7#393
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×#4248C2semidirect
5. #10620670C2semidirect
6. C7×C3⋊#95C2semidirect
7. #10620529C2semidirect
37. C22×C3×D7Q8semidirect
76. C3⋊#95D7semidirect
105. C22×D7Dic6semidirect
200. C3×C7#393semidirect
269. C7#4947semidirect
298. C3#89445semidirect

Maximal subgroups

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

subgroupindex
C22×#42482
C2×C3×(C2×D7)⋊C42
C2×(C2×D7)⋊Dic32
#106206702
C7×C3⋊#952
#106205292
#894453
#49477