Finite Group Databasebeta

← #13054052

Subgroups — #13054052

Order 1344.

340 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 96 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×C3×(C2×D7)⋊C4C211yesmaximal, characteristic
2C2×Dic7⋊Dic3C211yesmaximal, characteristic, complemented
3C2×(C2×D7)⋊Dic3C211yesmaximal, characteristic
4C2×(C2×D7)⋊Dic3C211yesmaximal, characteristic
5#10620670C211yesmaximal, characteristic, complemented
6C7×C3⋊#95C211yesmaximal, characteristic, complemented
7#10620529C211yesmaximal, characteristic, complemented
8#89447—31nomaximal, Hall
9C23×C3×D7C2211yescharacteristic
10C22×C3×C4×C7C2211yescharacteristic
11C22×Dic21C2211yescharacteristic
12C22×Dic21C2211yescharacteristic
13C22×C7×Dic3C2211yescharacteristic
14C22×C7×Dic3C2211yescharacteristic
15C22×C3×Dic7C2211yescharacteristic
16C3×(C2×D7)⋊C4—22no—
17Dic7⋊Dic3—22no—
18(C2×D7)⋊Dic3—22no—
19(C2×D7)⋊Dic3—22no—
20C2×(C2×D7)⋊C4—31no—
21C2×(C2×D7)⋊C4—31no—
22C2×(C2×D7)⋊C4D311yescharacteristic
23C2×Dic7⋊C4—31no—
24C7×#95—31no—
25C7⋊#95—31no—
26C7⋊#95—31no—
27#4953—71nomaximal, Hall
28C23×C3×C7C2311yescharacteristic
29C22×C3×D7—21no—
30C22×C3×D7—21no—
31C22×C3×D7—21no—
32C22×C3×D7—21no—
33C22×C3×D7—21no—
34C22×C3×D7—21no—
35C22×C3×D7D412yes—
36C2×C3×C4×C7—21no—
37C2×C3×C4×C7—21no—
38C2×C3×C4×C7—21no—
39C2×Dic21—21no—
40C2×Dic21—21no—
41C2×Dic21—21no—
42C2×Dic21—21no—
43C2×Dic21—21no—
44C2×Dic21—21no—
45C2×C7×Dic3—21no—
46C2×C7×Dic3—21no—
47C2×C7×Dic3—21no—
48C2×C7×Dic3—21no—
49C2×C7×Dic3—21no—
50C2×C7×Dic3—21no—
51C2×C3×Dic7—21no—
52C2×C3×Dic7—21no—
53C2×C3×Dic7D412yes—
54C23×D7C2×D311yescharacteristic
55C22×C4×C7—31no—
56C22×C4×C7—31no—
57C22×C4×C7C2×D311yescharacteristic
58C22×Dic7—31no—
59C22×Dic7—31no—
60C22×Dic7C2×D311yescharacteristic
61(C2×D7)⋊C4—22no—
62(C2×D7)⋊C4—62no—
63(C2×D7)⋊C4—62no—
64Dic7⋊C4—62no—
65C2×C3×C22⋊C4—71no—
66C2×#217—71no—
67C2×#217—71no—
68C2×C4⋊Dic3—71no—
69C3⋊#95—71no—
70C3⋊#95—71no—
71C3⋊#95D711yescharacteristic, complemented
72C22×C3×C7C2×D411yescharacteristic
73C22×C3×C7C2×D411yescharacteristic
74C22×C3×C7#4111yescharacteristic
75C22×C3×C7#4111yescharacteristic
76C22×C3×C7#4111yescharacteristic
77C22×C3×C7#4111yescharacteristic
78C22×C3×C7#4111yescharacteristic
79C2×C3×D7—22no—
80C2×C3×D7—22no—
81C2×C3×D7—22no—
82C2×C3×D7—41no—
83C2×C3×D7—41no—
84C2×C3×D7—41no—
85C2×C3×D7—41no—
86C3×C4×C7—41no—
87Dic21—41no—
88Dic21—41no—
89C3×Dic7—22no—
90C7×Dic3—41no—
91C7×Dic3—41no—
92#397—211noSylow
93C23×C7C22×D311yescharacteristic
94C22×D7—21no—
95C22×D7—21no—
96C22×D7—21no—
97C22×D7—21no—
98C22×D7—21no—
99C22×D7—21no—
100C22×D7D1212yes—
101C2×C4×C7—21no—
102C2×C4×C7—21no—
103C2×C4×C7—21no—
104C2×C4×C7—61no—
105C2×C4×C7—61no—
106C2×C4×C7—61no—
107C2×C4×C7—61no—
108C2×C4×C7—61no—
109C2×C4×C7—61no—
110C2×Dic7—21no—
111C2×Dic7—21no—
112C2×Dic7—61no—
113C2×Dic7—61no—
114C2×Dic7—61no—
115C2×Dic7—61no—
116C2×Dic7—61no—
117C2×Dic7—61no—
118C2×Dic7#6712yes—
119C24×C3—71no—
120C22×C3×C4—71no—
121C22×C3×C4C2×D711yescharacteristic
122C22×Dic3—71no—
123C22×Dic3—71no—
124C22×Dic3C2×D711yescharacteristic
125C22×Dic3C2×D711yescharacteristic
126C3×C22⋊C4—142no—
127#217—142no—
128#217—142no—
129C4⋊Dic3—142no—
130C2×C3×C7#12111yescharacteristic
131C2×C3×C7#12311yescharacteristic
132C2×C3×C7#12311yescharacteristic
133C2×C3×C7#12311yescharacteristic
134C2×C3×C7#12411yescharacteristic
135C2×C3×C7#12611yescharacteristic
136C2×C3×C7#12611yescharacteristic
137C3×D7—42no—
138C2×C4⋊C4—211no—
139C2×C22⋊C4—71no—
140C2×C22⋊C4—211no—
141C2×C22⋊C4—211no—
142#95—31no—
143#95—211no—
144#95—211no—
145C22×C7C2×D1211yescharacteristic
146C22×C7#23511yescharacteristic
147C22×C7#23711yescharacteristic
148C22×C7#23711yescharacteristic
149C22×C7#23711yescharacteristic
150C22×C7#23711yescharacteristic
151C22×C7C2×#6711yescharacteristic
152C2×D7—22no—
153C2×D7—22no—
154C2×D7—22no—
155C2×D7—41no—
156C2×D7—41no—
157C2×D7—41no—
158C2×D7—41no—
159C4×C7—41no—
160C4×C7—121no—
161C4×C7—121no—
162Dic7—22no—
163Dic7—121no—
164Dic7—121no—
165C23×C3—72no—
166C23×C3—141no—
167C23×C3—141no—
168C23×C3—141no—
169C23×C3—141no—
170C23×C3—141no—
171C23×C3—141no—
172C23×C3C22×D711yescharacteristic
173C2×C3×C4—21no—
174C2×C3×C4—21no—
175C2×C3×C4—21no—
176C2×C3×C4—72no—
177C2×C3×C4—141no—
178C2×C3×C4—141no—
179C2×Dic3—21no—
180C2×Dic3—21no—
181C2×Dic3—21no—
182C2×Dic3—21no—
183C2×Dic3—21no—
184C2×Dic3—21no—
185C2×Dic3—141no—
186C2×Dic3—141no—
187C2×Dic3—141no—
188C2×Dic3—141no—
189C2×Dic3—141no—
190C2×Dic3—141no—
191C3×C7#39711yescharacteristic, complemented, Hall
192C24—71no—
193C22×C4—31no—
194C22×C4—31no—
195C22×C4—71no—
196C22×C4—211no—
197C22×C4—211no—
198C22×C4D3×D711yescharacteristic
199C4⋊C4—422no—
200C22⋊C4—142no—
201C22⋊C4—422no—
202C22⋊C4—422no—
203C2×C7#87911yescharacteristic
204C2×C7#87911yescharacteristic
205C2×C7#88611yescharacteristic
206C2×C7#88611yescharacteristic
207C2×C7#93011yescharacteristic
208C2×C7#93511yescharacteristic
209C2×C7#93611yescharacteristic
210D7—42no—
211C22×C3—142no—
212C22×C3—142no—
213C22×C3—142no—
214C22×C3—281no—
215C22×C3—281no—
216C22×C3—281no—
217C22×C3—281no—
218C22×C3#115711yescharacteristic
219C22×C3#115711yescharacteristic
220C22×C3#115711yescharacteristic
221C22×C3D4×D711yescharacteristic
222C22×C3D4×D711yescharacteristic
223C22×C3#115911yescharacteristic
224C22×C3#116111yescharacteristic
225C3×C4—41no—
226C3×C4—142no—
227Dic3—41no—
228Dic3—41no—
229Dic3—281no—
230Dic3—281no—
231C23—72no—
232C23—141no—
233C23—141no—
234C23—141no—
235C23—141no—
236C23—141no—
237C23—141no—
238C23C2×D3×D711yescharacteristic
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—61no—
243C2×C4—61no—
244C2×C4—61no—
245C2×C4—61no—
246C2×C4—61no—
247C2×C4—61no—
248C2×C4—72no—
249C2×C4—141no—
250C2×C4—141no—
251C2×C4—421no—
252C2×C4—421no—
253C2×C4—421no—
254C2×C4—421no—
255C2×C4—421no—
256C2×C4—421no—
257C7#495311yescharacteristic, complemented, Sylow
258C2×C3—282no—
259C2×C3#643911yescharacteristic
260C2×C3#644611yescharacteristic
261C2×C3#644711yescharacteristic
262C2×C3#644811yescharacteristic
263C2×C3#645711yescharacteristic
264C2×C3#645711yescharacteristic
265C2×C3#646111yescharacteristic
266C22—142no—
267C22—142no—
268C22—142no—
269C22—281no—
270C22—281no—
271C22—281no—
272C22—281no—
273C22#6666911yescharacteristic
274C22#6667011yescharacteristic
275C22#6667511yescharacteristic
276C22D7×D1211yescharacteristic
277C22#6668311yescharacteristic
278C22#6668411yescharacteristic
279C22D7×#6711yescharacteristic
280C4—41no—
281C4—121no—
282C4—121no—
283C4—142no—
284C4—841no—
285C4—841no—
286C3#8944711yescharacteristic, complemented, Sylow
287C2—282no—
288C2#1062092511yescharacteristic
289C2#1062092811yescharacteristic
290C2#1062096911yescharacteristic
291C2#1062097511yescharacteristic
292C2#1062097611yescharacteristic
293C2#1062101011yescharacteristic
294C2#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#397
LCSG ▷ C23×C3×C7 ▷ C3×C7
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×Dic7⋊Dic3C2semidirect
5. #10620670C2semidirect
6. C7×C3⋊#95C2semidirect
7. #10620529C2semidirect
71. C3⋊#95D7semidirect
191. C3×C7#397semidirect
257. C7#4953semidirect
286. C3#89447semidirect

Maximal subgroups

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

subgroupindex
C2×C3×(C2×D7)⋊C42
C2×Dic7⋊Dic32
C2×(C2×D7)⋊Dic32
#106206702
C7×C3⋊#952
#106205292
#894473
#49537