Finite Group Databasebeta

← #13054894

Subgroups — #13054894

Order 1344.

380 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 112 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×#66636C211yesmaximal, characteristic
2C2×C7×#217C211yesmaximal, characteristic
3C2×C3×C22⋊Dic7C211yesmaximal, characteristic
4C2×#66577C211yesmaximal, characteristic, complemented
5C2×Dic3×Dic7C211yesmaximal, characteristic, complemented
6#10620529C211yesmaximal, characteristic, complemented
7#10620529C211yesmaximal, characteristic, complemented
8#89716—31nomaximal, Hall
9C24×C3×C7C2211yescharacteristic
10C22×Dic21C2211yescharacteristic
11C22×Dic21C2211yescharacteristic
12C22×C7×Dic3C2211yescharacteristic
13C22×C7×Dic3C2211yescharacteristic
14C22×C3×Dic7C2211yescharacteristic
15C22×C3×Dic7C2211yescharacteristic
16#66636C414yescomplemented
17C7×#217—22no—
18C3×C22⋊Dic7—22no—
19#66577—22no—
20Dic3×Dic7—22no—
21C2×C7×C22⋊C4—31no—
22C2×C22⋊Dic7—31no—
23C2×C22⋊Dic7D311yescharacteristic
24C2×Dic7⋊C4—31no—
25C2×C4×Dic7—31no—
26C7⋊#95—31no—
27C7⋊#95—31no—
28#4945—71nomaximal, Hall
29C23×C3×C7—21no—
30C23×C3×C7—21no—
31C23×C3×C7—21no—
32C23×C3×C7—21no—
33C23×C3×C7—21no—
34C23×C3×C7—21no—
35C23×C3×C7C2×C412yes—
36C23×C3×C7C2311yescharacteristic
37C2×Dic21—21no—
38C2×Dic21—21no—
39C2×Dic21—21no—
40C2×Dic21—21no—
41C2×Dic21C2×C412yes—
42C2×Dic21C2×C412yes—
43C2×C7×Dic3—21no—
44C2×C7×Dic3—21no—
45C2×C7×Dic3—21no—
46C2×C7×Dic3—21no—
47C2×C7×Dic3—21no—
48C2×C7×Dic3D412yes—
49C2×C3×Dic7—21no—
50C2×C3×Dic7—21no—
51C2×C3×Dic7—21no—
52C2×C3×Dic7—21no—
53C2×C3×Dic7—21no—
54C2×C3×Dic7D412yes—
55C24×C7C2×D311yescharacteristic
56C22×C4×C7—31no—
57C22×C4×C7—31no—
58C22×Dic7—31no—
59C22×Dic7—31no—
60C22×Dic7C2×D311yescharacteristic
61C22×Dic7C2×D311yescharacteristic
62C7×C22⋊C4—62no—
63C22⋊Dic7—22no—
64C22⋊Dic7—34no—
65Dic7⋊C4—62no—
66C4×Dic7—62no—
67C2×C3×C22⋊C4—71no—
68C2×#217—71no—
69C2×#217D711yescharacteristic
70C2×Dic3⋊C4—71no—
71C2×C4×Dic3—71no—
72C3⋊#95—71no—
73C3⋊#95—71no—
74C22×C3×C7—22no—
75C22×C3×C7—22no—
76C22×C3×C7—22no—
77C22×C3×C7—41no—
78C22×C3×C7—41no—
79C22×C3×C7—41no—
80C22×C3×C7—41no—
81C22×C3×C7C22×C411yescharacteristic
82C22×C3×C7C2×D411yescharacteristic
83C22×C3×C7C2×D411yescharacteristic
84C22×C3×C7#4111yescharacteristic
85C22×C3×C7#4111yescharacteristic
86C22×C3×C7#4111yescharacteristic
87C22×C3×C7#4111yescharacteristic
88Dic21—22no—
89Dic21—22no—
90C3×Dic7—22no—
91C3×Dic7—41no—
92C7×Dic3—22no—
93C7×Dic3—41no—
94#388—211noSylow
95C23×C7—21no—
96C23×C7—21no—
97C23×C7—21no—
98C23×C7—21no—
99C23×C7—21no—
100C23×C7—21no—
101C23×C7C4×D312yes—
102C23×C7C22×D311yescharacteristic
103C2×C4×C7—32no—
104C2×C4×C7—61no—
105C2×C4×C7—61no—
106C2×C4×C7—61no—
107C2×C4×C7—61no—
108C2×C4×C7—61no—
109C2×Dic7—21no—
110C2×Dic7—21no—
111C2×Dic7—21no—
112C2×Dic7—21no—
113C2×Dic7—21no—
114C2×Dic7—32no—
115C2×Dic7—32no—
116C2×Dic7—61no—
117C2×Dic7—61no—
118C2×Dic7—61no—
119C2×Dic7—61no—
120C2×Dic7#6712yes—
121C24×C3C2×D711yescharacteristic
122C22×C3×C4—71no—
123C22×C3×C4—71no—
124C22×Dic3—71no—
125C22×Dic3—71no—
126C22×Dic3C2×D711yescharacteristic
127C22×Dic3C2×D711yescharacteristic
128C3×C22⋊C4—142no—
129#217—22no—
130#217—74no—
131Dic3⋊C4—142no—
132C4×Dic3—142no—
133C2×C3×C7—42no—
134C2×C3×C7#11711yescharacteristic
135C2×C3×C7C4×D411yescharacteristic
136C2×C3×C7C4×D411yescharacteristic
137C2×C3×C7#12111yescharacteristic
138C2×C3×C7#12311yescharacteristic
139C2×C3×C7#12411yescharacteristic
140C2×C3×C7#12611yescharacteristic
141C2×C4⋊C4—211no—
142C2×C22⋊C4—31no—
143C2×C22⋊C4—71no—
144C2×C22⋊C4—211no—
145C2×C42—211no—
146#95—211no—
147#95—211no—
148C22×C7—22no—
149C22×C7—22no—
150C22×C7—22no—
151C22×C7—41no—
152C22×C7—41no—
153C22×C7—41no—
154C22×C7—41no—
155C22×C7C2×C4×D311yescharacteristic
156C22×C7#23511yescharacteristic
157C22×C7D3×D411yescharacteristic
158C22×C7#23711yescharacteristic
159C22×C7#23711yescharacteristic
160C22×C7#23711yescharacteristic
161C22×C7C2×#6711yescharacteristic
162C4×C7—62no—
163C4×C7—121no—
164Dic7—22no—
165Dic7—41no—
166Dic7—62no—
167Dic7—62no—
168C23×C3—21no—
169C23×C3—21no—
170C23×C3—21no—
171C23×C3—21no—
172C23×C3—21no—
173C23×C3—21no—
174C23×C3C4×D712yes—
175C23×C3C22×D711yescharacteristic
176C2×C3×C4—72no—
177C2×C3×C4—141no—
178C2×C3×C4—141no—
179C2×C3×C4—141no—
180C2×C3×C4—141no—
181C2×C3×C4—141no—
182C2×Dic3—21no—
183C2×Dic3—21no—
184C2×Dic3—21no—
185C2×Dic3—21no—
186C2×Dic3—21no—
187C2×Dic3—72no—
188C2×Dic3—72no—
189C2×Dic3—141no—
190C2×Dic3—141no—
191C2×Dic3—141no—
192C2×Dic3—141no—
193C2×Dic3C22⋊D712yes—
194C3×C7#38811yescharacteristic, complemented, Hall
195C24D3×D711yescharacteristic
196C22×C4—31no—
197C22×C4—31no—
198C22×C4—71no—
199C22×C4—71no—
200C22×C4—211no—
201C22×C4—211no—
202C4⋊C4—422no—
203C22⋊C4—62no—
204C22⋊C4—142no—
205C22⋊C4—214no—
206C42—422no—
207C2×C7—42no—
208C2×C7#87711yescharacteristic
209C2×C7#87911yescharacteristic
210C2×C7#67⋊C411yescharacteristic
211C2×C7#88311yescharacteristic
212C2×C7C4×#6711yescharacteristic
213C2×C7#93611yescharacteristic
214C2×C7#93911yescharacteristic
215C22×C3—22no—
216C22×C3—22no—
217C22×C3—22no—
218C22×C3—41no—
219C22×C3—41no—
220C22×C3—41no—
221C22×C3—41no—
222C22×C3C2×C4×D711yescharacteristic
223C22×C3#115711yescharacteristic
224C22×C3D4×D711yescharacteristic
225C22×C3#115911yescharacteristic
226C22×C3#115911yescharacteristic
227C22×C3#115911yescharacteristic
228C22×C3C2×C22⋊D711yescharacteristic
229C3×C4—142no—
230C3×C4—281no—
231Dic3—22no—
232Dic3—41no—
233Dic3—142no—
234Dic3—142no—
235C23—21no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—21no—
240C23—21no—
241C23D21⋊C412yes—
242C23C2×D3×D711yescharacteristic
243C2×C4—32no—
244C2×C4—61no—
245C2×C4—61no—
246C2×C4—61no—
247C2×C4—61no—
248C2×C4—61no—
249C2×C4—72no—
250C2×C4—141no—
251C2×C4—141no—
252C2×C4—141no—
253C2×C4—141no—
254C2×C4—141no—
255C2×C4—212no—
256C2×C4—212no—
257C2×C4—421no—
258C2×C4—421no—
259C2×C4—421no—
260C2×C4—421no—
261C7#494511yescharacteristic, complemented, Sylow
262C2×C3—42no—
263C2×C3#644011yescharacteristic
264C2×C3#644211yescharacteristic
265C2×C3(C22⋊D7)⋊C411yescharacteristic
266C2×C3#644811yescharacteristic
267C2×C3C4×C22⋊D711yescharacteristic
268C2×C3#649811yescharacteristic
269C2×C3#650111yescharacteristic
270C22—22no—
271C22—22no—
272C22—22no—
273C22—41no—
274C22—41no—
275C22—41no—
276C22—41no—
277C22#6668311yescharacteristic
278C22#6668411yescharacteristic
279C22#6668411yescharacteristic
280C22#6668611yescharacteristic
281C22C2×D21⋊C411yescharacteristic
282C22D7×#6711yescharacteristic
283C22D3×C22⋊D711yescharacteristic
284C4—62no—
285C4—121no—
286C4—142no—
287C4—281no—
288C4—422no—
289C4—422no—
290C3#8971611yescharacteristic, complemented, Sylow
291C2—42no—
292C2#1062109311yescharacteristic
293C2#1062109411yescharacteristic
294C2#1062109611yescharacteristic
295C2#1062111111yescharacteristic
296C2#1062111411yescharacteristic
297C2#1062111711yescharacteristic
298C2#1062111811yescharacteristic
subgroupG/H
center, FrattiniC23C2×D3×D7
commutatorC22×C3×C7C22×C4
Fitting, prime radicalC24×C3×C7C22
socle, cosocleC23×C3×C7C23
hypercenterC24D3×D7
hypocenterC3×C7#388
LCSG ▷ C22×C3×C7 ▷ C3×C7
UCSC1 ◁ C23 ◁ C24

Complements

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

NH ≅ G/Ntype
4. C2×#66577C2semidirect
5. C2×Dic3×Dic7C2semidirect
6. #10620529C2semidirect
7. #10620529C2semidirect
16. #66636C4semidirect
194. C3×C7#388semidirect
261. C7#4945semidirect
290. C3#89716semidirect

Maximal subgroups

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

subgroupindex
C2×#666362
C2×C7×#2172
C2×C3×C22⋊Dic72
C2×#665772
C2×Dic3×Dic72
#106205292
#897163
#49457