Finite Group Databasebeta

← #13056313

Subgroups — #13056313

Order 1344.

340 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 95 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×#66631C211yesmaximal, characteristic
2C2×#66631C211yesmaximal, characteristic
3C2×#66631C211yesmaximal, characteristic
4C2×#66629C211yesmaximal, characteristic, complemented
5C3×C7×#95C211yesmaximal, characteristic, complemented
6#10620670C211yesmaximal, characteristic, complemented
7#10620670C211yesmaximal, characteristic, complemented
8#89447—31nomaximal, Hall
9C23×D21C2211yescharacteristic
10C22×C3×C4×C7C2211yescharacteristic
11C22×C3×C4×C7C2211yescharacteristic
12C22×C3×C4×C7C2211yescharacteristic
13C22×Dic21C2211yescharacteristic
14C22×Dic21C2211yescharacteristic
15C22×Dic21C2211yescharacteristic
16#66631—22no—
17#66631—22no—
18#66631—22no—
19#66629—22no—
20C2×(C2×D7)⋊C4—31no—
21C2×(C2×D7)⋊C4—31no—
22C2×(C2×D7)⋊C4—31no—
23C2×Dic7⋊C4—31no—
24C7×#95D311yescharacteristic, complemented
25C7⋊#95—31no—
26C7⋊#95—31no—
27#4675—71nomaximal, Hall
28C23×C3×C7C2311yescharacteristic
29C22×D21—21no—
30C22×D21—21no—
31C22×D21—21no—
32C22×D21—21no—
33C22×D21—21no—
34C22×D21—21no—
35C22×D21D412yes—
36C2×C3×C4×C7—21no—
37C2×C3×C4×C7—21no—
38C2×C3×C4×C7—21no—
39C2×C3×C4×C7—21no—
40C2×C3×C4×C7—21no—
41C2×C3×C4×C7—21no—
42C2×C3×C4×C7—21no—
43C2×C3×C4×C7—21no—
44C2×C3×C4×C7—21no—
45C2×Dic21—21no—
46C2×Dic21—21no—
47C2×Dic21—21no—
48C2×Dic21—21no—
49C2×Dic21—21no—
50C2×Dic21—21no—
51C2×Dic21—21no—
52C2×Dic21—21no—
53C2×Dic21D412yes—
54C23×D7—31no—
55C22×C4×C7C2×D311yescharacteristic
56C22×C4×C7C2×D311yescharacteristic
57C22×C4×C7C2×D311yescharacteristic
58C22×Dic7—31no—
59C22×Dic7—31no—
60C22×Dic7—31no—
61(C2×D7)⋊C4—62no—
62(C2×D7)⋊C4—62no—
63(C2×D7)⋊C4—62no—
64Dic7⋊C4—62no—
65C2×(C2×D3)⋊C4—71no—
66C2×(C2×D3)⋊C4—71no—
67C2×(C2×D3)⋊C4—71no—
68C2×Dic3⋊C4—71no—
69C3×#95D711yescharacteristic, complemented
70C3⋊#95—71no—
71C3⋊#95—71no—
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×D21—22no—
80C2×D21—22no—
81C2×D21—22no—
82C2×D21—41no—
83C2×D21—41no—
84C2×D21—41no—
85C2×D21—41no—
86C3×C4×C7—41no—
87C3×C4×C7—41no—
88C3×C4×C7—41no—
89Dic21—22no—
90Dic21—41no—
91Dic21—41no—
92#397—211noSylow
93C23×C7C22×D311yescharacteristic
94C22×D7—32no—
95C22×D7—61no—
96C22×D7—61no—
97C22×D7—61no—
98C22×D7—61no—
99C22×D7—61no—
100C22×D7—61no—
101C2×C4×C7—21no—
102C2×C4×C7—21no—
103C2×C4×C7—21no—
104C2×C4×C7—21no—
105C2×C4×C7—21no—
106C2×C4×C7—21no—
107C2×C4×C7—21no—
108C2×C4×C7—21no—
109C2×C4×C7—21no—
110C2×Dic7—32no—
111C2×Dic7—61no—
112C2×Dic7—61no—
113C2×Dic7—61no—
114C2×Dic7—61no—
115C2×Dic7—61no—
116C2×Dic7—61no—
117C2×Dic7—61no—
118C2×Dic7—61no—
119C23×D3—71no—
120C22×C3×C4C2×D711yescharacteristic
121C22×C3×C4C2×D711yescharacteristic
122C22×C3×C4C2×D711yescharacteristic
123C22×Dic3—71no—
124C22×Dic3—71no—
125C22×Dic3—71no—
126(C2×D3)⋊C4—142no—
127(C2×D3)⋊C4—142no—
128(C2×D3)⋊C4—142no—
129Dic3⋊C4—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
137D21—42no—
138C2×C4⋊C4—211no—
139C2×C22⋊C4—211no—
140C2×C22⋊C4—211no—
141C2×C22⋊C4—211no—
142#95—211no—
143#95—211no—
144#95D2111yescharacteristic, complemented
145C22×C7#23511yescharacteristic
146C22×C7#23511yescharacteristic
147C22×C7#23511yescharacteristic
148C22×C7D3×D411yescharacteristic
149C22×C7D3×D411yescharacteristic
150C22×C7#23711yescharacteristic
151C22×C7#23911yescharacteristic
152C2×D7—62no—
153C2×D7—62no—
154C2×D7—62no—
155C2×D7—121no—
156C2×D7—121no—
157C2×D7—121no—
158C2×D7—121no—
159C4×C7—41no—
160C4×C7—41no—
161C4×C7—41no—
162Dic7—62no—
163Dic7—121no—
164Dic7—121no—
165C23×C3C22×D711yescharacteristic
166C22×D3—72no—
167C22×D3—141no—
168C22×D3—141no—
169C22×D3—141no—
170C22×D3—141no—
171C22×D3—141no—
172C22×D3—141no—
173C2×C3×C4—21no—
174C2×C3×C4—21no—
175C2×C3×C4—21no—
176C2×C3×C4—21no—
177C2×C3×C4—21no—
178C2×C3×C4—21no—
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×Dic3—72no—
183C2×Dic3—141no—
184C2×Dic3—141no—
185C2×Dic3—141no—
186C2×Dic3—141no—
187C2×Dic3—141no—
188C2×Dic3—141no—
189C2×Dic3—141no—
190C2×Dic3—141no—
191C3×C7#39711yescharacteristic, complemented, Hall
192C24—211no—
193C22×C4—211no—
194C22×C4—211no—
195C22×C4—211no—
196C22×C4C2×D2111yescharacteristic
197C22×C4C2×D2111yescharacteristic
198C22×C4C2×D2111yescharacteristic
199C4⋊C4—422no—
200C22⋊C4—422no—
201C22⋊C4—422no—
202C22⋊C4—422no—
203C2×C7#87611yescharacteristic
204C2×C7#88311yescharacteristic
205C2×C7#88411yescharacteristic
206C2×C7#88511yescharacteristic
207C2×C7#89411yescharacteristic
208C2×C7#89411yescharacteristic
209C2×C7#89811yescharacteristic
210D7—122no—
211C22×C3#115711yescharacteristic
212C22×C3#115711yescharacteristic
213C22×C3#115711yescharacteristic
214C22×C3D4×D711yescharacteristic
215C22×C3D4×D711yescharacteristic
216C22×C3#115911yescharacteristic
217C22×C3#116111yescharacteristic
218C2×D3—142no—
219C2×D3—142no—
220C2×D3—142no—
221C2×D3—281no—
222C2×D3—281no—
223C2×D3—281no—
224C2×D3—281no—
225C3×C4—41no—
226C3×C4—41no—
227C3×C4—41no—
228Dic3—142no—
229Dic3—281no—
230Dic3—281no—
231C23—212no—
232C23—421no—
233C23—421no—
234C23—421no—
235C23—421no—
236C23—421no—
237C23—421no—
238C23C22×D2111yescharacteristic
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—212no—
249C2×C4—421no—
250C2×C4—421no—
251C2×C4—421no—
252C2×C4—421no—
253C2×C4—421no—
254C2×C4—421no—
255C2×C4—421no—
256C2×C4—421no—
257C7#467511yescharacteristic, complemented, Sylow
258C2×C3#643911yescharacteristic
259C2×C3#644611yescharacteristic
260C2×C3#644711yescharacteristic
261C2×C3#644811yescharacteristic
262C2×C3#645711yescharacteristic
263C2×C3#645711yescharacteristic
264C2×C3#646111yescharacteristic
265D3—282no—
266C22—422no—
267C22—422no—
268C22—422no—
269C22—841no—
270C22—841no—
271C22—841no—
272C22—841no—
273C22#6672811yescharacteristic
274C22#6672811yescharacteristic
275C22#6672811yescharacteristic
276C22D4×D2111yescharacteristic
277C22D4×D2111yescharacteristic
278C22#6673011yescharacteristic
279C22#6673211yescharacteristic
280C4—41no—
281C4—41no—
282C4—41no—
283C4—422no—
284C4—841no—
285C4—841no—
286C3#8944711yescharacteristic, complemented, Sylow
287C2—842no—
288C2#1062135911yescharacteristic
289C2#1062136611yescharacteristic
290C2#1062136711yescharacteristic
291C2#1062136811yescharacteristic
292C2#1062137711yescharacteristic
293C2#1062137711yescharacteristic
294C2#1062138111yescharacteristic
subgroupG/H
center, FrattiniC23C22×D21
commutator, socle, cosocleC23×C3×C7C23
FittingC3×C7×#95C2
prime radicalC23×D21C22
hypercenter#95D21
hypocenterC3×C7#397
LCSG ▷ C23×C3×C7 ▷ C3×C7
UCSC1 ◁ C23 ◁ #95

Complements

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

NH ≅ G/Ntype
4. C2×#66629C2semidirect
5. C3×C7×#95C2semidirect
6. #10620670C2semidirect
7. #10620670C2semidirect
24. C7×#95D3semidirect
69. C3×#95D7semidirect
144. #95D21semidirect
191. C3×C7#397semidirect
257. C7#4675semidirect
286. C3#89447semidirect

Maximal subgroups

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

subgroupindex
C2×#666312
C2×#666292
C3×C7×#952
#106206702
#894473
#46757