Finite Group Databasebeta

← #13054882

Subgroups — #13054882

Order 1344.

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

normal subgroups only

This group has 372 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
1C2×#66636C211yesmaximal, characteristic
2C2×C7×#217C211yesmaximal, characteristic
3C2×C3×C22⋊Dic7C211yesmaximal, characteristic
4C2×Dic3⋊Dic7C211yesmaximal, characteristic, complemented
5C2×#66577C211yesmaximal, characteristic, complemented
6C2×Dic7⋊Dic3C211yesmaximal, characteristic, complemented
7#10620529C211yesmaximal, characteristic, complemented
8#89721—31nomaximal, Hall
9C24×C3×C7C2211yescharacteristic
10C22×Dic21C2211yescharacteristic
11C22×Dic21C2211yescharacteristic
12C22×C7×Dic3C2211yescharacteristic
13C22×C7×Dic3C2211yescharacteristic
14C22×C3×Dic7C2211yescharacteristic
15C22×C3×Dic7C2211yescharacteristic
16#66636—22no—
17C7×#217—22no—
18C3×C22⋊Dic7—22no—
19Dic3⋊Dic7—22no—
20#66577—22no—
21Dic7⋊Dic3—22no—
22C2×C7×C22⋊C4—31no—
23C2×C22⋊Dic7—31no—
24C2×C22⋊Dic7D311yescharacteristic
25C2×C4⋊Dic7—31no—
26C2×Dic7⋊C4—31no—
27C2×Dic7⋊C4—31no—
28C7⋊#95—31no—
29#4949—71nomaximal, Hall
30C23×C3×C7—21no—
31C23×C3×C7—21no—
32C23×C3×C7—21no—
33C23×C3×C7—21no—
34C23×C3×C7—21no—
35C23×C3×C7—21no—
36C23×C3×C7Q812yes—
37C23×C3×C7C2311yescharacteristic
38C2×Dic21—21no—
39C2×Dic21—21no—
40C2×Dic21—21no—
41C2×Dic21—21no—
42C2×Dic21—21no—
43C2×Dic21D412yes—
44C2×C7×Dic3—21no—
45C2×C7×Dic3—21no—
46C2×C7×Dic3—21no—
47C2×C7×Dic3—21no—
48C2×C7×Dic3—21no—
49C2×C7×Dic3D412yes—
50C2×C3×Dic7—21no—
51C2×C3×Dic7—21no—
52C2×C3×Dic7—21no—
53C2×C3×Dic7—21no—
54C2×C3×Dic7—21no—
55C2×C3×Dic7D412yes—
56C24×C7C2×D311yescharacteristic
57C22×C4×C7—31no—
58C22×C4×C7—31no—
59C22×Dic7—31no—
60C22×Dic7—31no—
61C22×Dic7C2×D311yescharacteristic
62C22×Dic7C2×D311yescharacteristic
63C7×C22⋊C4—62no—
64C22⋊Dic7—22no—
65C22⋊Dic7—62no—
66C4⋊Dic7—62no—
67Dic7⋊C4—62no—
68Dic7⋊C4—62no—
69C2×C3×C22⋊C4—71no—
70C2×#217—71no—
71C2×#217D711yescharacteristic
72C2×C4⋊Dic3—71no—
73C2×Dic3⋊C4—71no—
74C2×Dic3⋊C4—71no—
75C3⋊#95—71no—
76C22×C3×C7—22no—
77C22×C3×C7—22no—
78C22×C3×C7—22no—
79C22×C3×C7—41no—
80C22×C3×C7—41no—
81C22×C3×C7—41no—
82C22×C3×C7—41no—
83C22×C3×C7C2×D411yescharacteristic
84C22×C3×C7C2×D411yescharacteristic
85C22×C3×C7C2×D411yescharacteristic
86C22×C3×C7C2×Q811yescharacteristic
87C22×C3×C7#4111yescharacteristic
88C22×C3×C7#4111yescharacteristic
89C22×C3×C7#4111yescharacteristic
90Dic21—22no—
91Dic21—41no—
92C3×Dic7—22no—
93C3×Dic7—41no—
94C7×Dic3—22no—
95C7×Dic3—41no—
96#399—211noSylow
97C23×C7—21no—
98C23×C7—21no—
99C23×C7—21no—
100C23×C7—21no—
101C23×C7—21no—
102C23×C7—21no—
103C23×C7Dic612yes—
104C23×C7C22×D311yescharacteristic
105C2×C4×C7—32no—
106C2×C4×C7—61no—
107C2×C4×C7—61no—
108C2×C4×C7—61no—
109C2×C4×C7—61no—
110C2×C4×C7—61no—
111C2×Dic7—21no—
112C2×Dic7—21no—
113C2×Dic7—21no—
114C2×Dic7—21no—
115C2×Dic7—21no—
116C2×Dic7—32no—
117C2×Dic7—61no—
118C2×Dic7—61no—
119C2×Dic7—61no—
120C2×Dic7—61no—
121C2×Dic7—61no—
122C2×Dic7#6712yes—
123C24×C3C2×D711yescharacteristic
124C22×C3×C4—71no—
125C22×C3×C4—71no—
126C22×Dic3—71no—
127C22×Dic3—71no—
128C22×Dic3C2×D711yescharacteristic
129C22×Dic3C2×D711yescharacteristic
130C3×C22⋊C4—142no—
131#217—22no—
132#217—142no—
133C4⋊Dic3—142no—
134Dic3⋊C4—142no—
135Dic3⋊C4—142no—
136C2×C3×C7—42no—
137C2×C3×C7C22⋊Q811yescharacteristic
138C2×C3×C7C22⋊Q811yescharacteristic
139C2×C3×C7C22⋊Q811yescharacteristic
140C2×C3×C7#12311yescharacteristic
141C2×C3×C7#12311yescharacteristic
142C2×C3×C7#12311yescharacteristic
143C2×C3×C7#12711yescharacteristic
144C2×C4⋊C4—211no—
145C2×C4⋊C4—211no—
146C2×C4⋊C4—211no—
147C2×C22⋊C4—31no—
148C2×C22⋊C4—71no—
149C2×C22⋊C4—211no—
150#95—211no—
151C22×C7—22no—
152C22×C7—22no—
153C22×C7—22no—
154C22×C7—41no—
155C22×C7—41no—
156C22×C7—41no—
157C22×C7—41no—
158C22×C7C2×Dic611yescharacteristic
159C22×C7#23511yescharacteristic
160C22×C7D3×D411yescharacteristic
161C22×C7D3×D411yescharacteristic
162C22×C7#23711yescharacteristic
163C22×C7#23711yescharacteristic
164C22×C7C2×#6711yescharacteristic
165C4×C7—62no—
166C4×C7—121no—
167Dic7—22no—
168Dic7—41no—
169Dic7—62no—
170Dic7—121no—
171C23×C3—21no—
172C23×C3—21no—
173C23×C3—21no—
174C23×C3—21no—
175C23×C3—21no—
176C23×C3—21no—
177C23×C3Dic1412yes—
178C23×C3C22×D711yescharacteristic
179C2×C3×C4—72no—
180C2×C3×C4—141no—
181C2×C3×C4—141no—
182C2×C3×C4—141no—
183C2×C3×C4—141no—
184C2×C3×C4—141no—
185C2×Dic3—21no—
186C2×Dic3—21no—
187C2×Dic3—21no—
188C2×Dic3—21no—
189C2×Dic3—21no—
190C2×Dic3—72no—
191C2×Dic3—141no—
192C2×Dic3—141no—
193C2×Dic3—141no—
194C2×Dic3—141no—
195C2×Dic3—141no—
196C2×Dic3C22⋊D712yes—
197C3×C7#39911yescharacteristic, complemented, Hall
198C24D3×D711yescharacteristic
199C22×C4—31no—
200C22×C4—31no—
201C22×C4—71no—
202C22×C4—71no—
203C22×C4—211no—
204C22×C4—211no—
205C4⋊C4—422no—
206C4⋊C4—422no—
207C4⋊C4—422no—
208C22⋊C4—62no—
209C22⋊C4—142no—
210C22⋊C4—422no—
211C2×C7—42no—
212C2×C7#87811yescharacteristic
213C2×C7#87811yescharacteristic
214C2×C7#88311yescharacteristic
215C2×C7#88311yescharacteristic
216C2×C7#92411yescharacteristic
217C2×C7#93511yescharacteristic
218C2×C7#94011yescharacteristic
219C22×C3—22no—
220C22×C3—22no—
221C22×C3—22no—
222C22×C3—41no—
223C22×C3—41no—
224C22×C3—41no—
225C22×C3—41no—
226C22×C3C2×Dic1411yescharacteristic
227C22×C3#115711yescharacteristic
228C22×C3D4×D711yescharacteristic
229C22×C3D4×D711yescharacteristic
230C22×C3#115911yescharacteristic
231C22×C3#115911yescharacteristic
232C22×C3C2×C22⋊D711yescharacteristic
233C3×C4—142no—
234C3×C4—281no—
235Dic3—22no—
236Dic3—41no—
237Dic3—142no—
238Dic3—281no—
239C23—21no—
240C23—21no—
241C23—21no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23#424812yes—
246C23C2×D3×D711yescharacteristic
247C2×C4—32no—
248C2×C4—61no—
249C2×C4—61no—
250C2×C4—61no—
251C2×C4—61no—
252C2×C4—61no—
253C2×C4—72no—
254C2×C4—141no—
255C2×C4—141no—
256C2×C4—141no—
257C2×C4—141no—
258C2×C4—141no—
259C2×C4—212no—
260C2×C4—421no—
261C2×C4—421no—
262C2×C4—421no—
263C2×C4—421no—
264C2×C4—421no—
265C7#494911yescharacteristic, complemented, Sylow
266C2×C3—42no—
267C2×C3#644111yescharacteristic
268C2×C3#644111yescharacteristic
269C2×C3#644611yescharacteristic
270C2×C3#644611yescharacteristic
271C2×C3#648711yescharacteristic
272C2×C3#649811yescharacteristic
273C2×C3#650311yescharacteristic
274C22—22no—
275C22—22no—
276C22—22no—
277C22—41no—
278C22—41no—
279C22—41no—
280C22—41no—
281C22#6668311yescharacteristic
282C22#6668411yescharacteristic
283C22#6668611yescharacteristic
284C22C2×#424811yescharacteristic
285C22D7×#6711yescharacteristic
286C22D3×C22⋊D711yescharacteristic
287C22#6669411yescharacteristic
288C4—62no—
289C4—121no—
290C4—142no—
291C4—281no—
292C4—422no—
293C4—841no—
294C3#8972111yescharacteristic, complemented, Sylow
295C2—42no—
296C2#1062111011yescharacteristic
297C2#1062111711yescharacteristic
298C2#1062112411yescharacteristic
299C2#1062112911yescharacteristic
300C2#1062113611yescharacteristic
301C2#1062113711yescharacteristic
302C2#1062113811yescharacteristic
subgroupG/H
center, FrattiniC23C2×D3×D7
commutator, socle, cosocleC23×C3×C7C23
Fitting, prime radicalC24×C3×C7C22
hypercenterC24D3×D7
hypocenterC3×C7#399
LCSG ▷ C23×C3×C7 ▷ C3×C7
UCSC1 ◁ C23 ◁ C24

Complements

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

NH ≅ G/Ntype
4. C2×Dic3⋊Dic7C2semidirect
5. C2×#66577C2semidirect
6. C2×Dic7⋊Dic3C2semidirect
7. #10620529C2semidirect
197. C3×C7#399semidirect
265. C7#4949semidirect
294. C3#89721semidirect

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×Dic3⋊Dic72
C2×#665772
C2×Dic7⋊Dic32
#106205292
#897213
#49497