Finite Group Databasebeta

← #13054889

Subgroups — #13054889

Order 1344.

340 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 100 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
5#10620529C211yesmaximal, characteristic, complemented
6#10620529C211yesmaximal, characteristic, complemented
7#10620529C211yesmaximal, characteristic, complemented
8#89725—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—
20C2×C7×C22⋊C4—31no—
21C2×C22⋊Dic7—31no—
22C2×C22⋊Dic7D311yescharacteristic
23C2×C4⋊Dic7—31no—
24C7⋊#95—31no—
25C7⋊#95—31no—
26C7⋊#95—31no—
27#4952—71nomaximal, Hall
28C23×C3×C7—21no—
29C23×C3×C7—21no—
30C23×C3×C7—21no—
31C23×C3×C7—21no—
32C23×C3×C7—21no—
33C23×C3×C7—21no—
34C23×C3×C7D412yes—
35C23×C3×C7C2311yescharacteristic
36C2×Dic21—21no—
37C2×Dic21—21no—
38C2×Dic21—21no—
39C2×Dic21—21no—
40C2×Dic21—21no—
41C2×Dic21—21no—
42C2×C7×Dic3—21no—
43C2×C7×Dic3—21no—
44C2×C7×Dic3—21no—
45C2×C7×Dic3—21no—
46C2×C7×Dic3—21no—
47C2×C7×Dic3D412yes—
48C2×C3×Dic7—21no—
49C2×C3×Dic7—21no—
50C2×C3×Dic7—21no—
51C2×C3×Dic7—21no—
52C2×C3×Dic7—21no—
53C2×C3×Dic7—21no—
54C24×C7C2×D311yescharacteristic
55C22×C4×C7—31no—
56C22×C4×C7—31no—
57C22×Dic7—31no—
58C22×Dic7—31no—
59C22×Dic7C2×D311yescharacteristic
60C22×Dic7C2×D311yescharacteristic
61C7×C22⋊C4—62no—
62C22⋊Dic7—22no—
63C22⋊Dic7—62no—
64C4⋊Dic7—62no—
65C2×C3×C22⋊C4—71no—
66C2×#217—71no—
67C2×#217D711yescharacteristic
68C2×Dic3⋊C4—71no—
69C3⋊#95—71no—
70C3⋊#95—71no—
71C3⋊#95—71no—
72C22×C3×C7—22no—
73C22×C3×C7—22no—
74C22×C3×C7—22no—
75C22×C3×C7—41no—
76C22×C3×C7—41no—
77C22×C3×C7—41no—
78C22×C3×C7—41no—
79C22×C3×C7C2×D411yescharacteristic
80C22×C3×C7C2×D411yescharacteristic
81C22×C3×C7#4111yescharacteristic
82C22×C3×C7#4111yescharacteristic
83C22×C3×C7#4111yescharacteristic
84C22×C3×C7#4111yescharacteristic
85C22×C3×C7#4111yescharacteristic
86Dic21—41no—
87Dic21—41no—
88C3×Dic7—41no—
89C3×Dic7—41no—
90C7×Dic3—22no—
91C7×Dic3—41no—
92#397—211noSylow
93C23×C7—21no—
94C23×C7—21no—
95C23×C7—21no—
96C23×C7—21no—
97C23×C7—21no—
98C23×C7—21no—
99C23×C7#6712yes—
100C23×C7C22×D311yescharacteristic
101C2×C4×C7—32no—
102C2×C4×C7—61no—
103C2×C4×C7—61no—
104C2×C4×C7—61no—
105C2×C4×C7—61no—
106C2×C4×C7—61no—
107C2×Dic7—21no—
108C2×Dic7—21no—
109C2×Dic7—21no—
110C2×Dic7—21no—
111C2×Dic7—21no—
112C2×Dic7—21no—
113C2×Dic7—61no—
114C2×Dic7—61no—
115C2×Dic7—61no—
116C2×Dic7—61no—
117C2×Dic7—61no—
118C2×Dic7—61no—
119C24×C3C2×D711yescharacteristic
120C22×C3×C4—71no—
121C22×C3×C4—71no—
122C22×Dic3—71no—
123C22×Dic3—71no—
124C22×Dic3C2×D711yescharacteristic
125C22×Dic3C2×D711yescharacteristic
126C3×C22⋊C4—142no—
127#217—22no—
128#217—142no—
129Dic3⋊C4—142no—
130C2×C3×C7—42no—
131C2×C3×C7#12111yescharacteristic
132C2×C3×C7#12311yescharacteristic
133C2×C3×C7#12311yescharacteristic
134C2×C3×C7#12311yescharacteristic
135C2×C3×C7#12411yescharacteristic
136C2×C3×C7#12611yescharacteristic
137C2×C3×C7#12611yescharacteristic
138C2×C4⋊C4—211no—
139C2×C22⋊C4—31no—
140C2×C22⋊C4—71no—
141C2×C22⋊C4—211no—
142#95—211no—
143#95—211no—
144#95—211no—
145C22×C7—22no—
146C22×C7—22no—
147C22×C7—22no—
148C22×C7—41no—
149C22×C7—41no—
150C22×C7—41no—
151C22×C7—41no—
152C22×C7#23511yescharacteristic
153C22×C7#23511yescharacteristic
154C22×C7D3×D411yescharacteristic
155C22×C7#23711yescharacteristic
156C22×C7#23711yescharacteristic
157C22×C7#23711yescharacteristic
158C22×C7C2×#6711yescharacteristic
159C4×C7—62no—
160C4×C7—121no—
161Dic7—41no—
162Dic7—41no—
163Dic7—121no—
164Dic7—121no—
165C23×C3—21no—
166C23×C3—21no—
167C23×C3—21no—
168C23×C3—21no—
169C23×C3—21no—
170C23×C3—21no—
171C23×C3D2812yes—
172C23×C3C22×D711yescharacteristic
173C2×C3×C4—141no—
174C2×C3×C4—141no—
175C2×C3×C4—141no—
176C2×C3×C4—141no—
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—141no—
185C2×Dic3—141no—
186C2×Dic3—141no—
187C2×Dic3—141no—
188C2×Dic3—141no—
189C2×Dic3—141no—
190C2×Dic3C22⋊D712yes—
191C3×C7#39711yescharacteristic, complemented, Hall
192C24D3×D711yescharacteristic
193C22×C4—31no—
194C22×C4—31no—
195C22×C4—71no—
196C22×C4—71no—
197C22×C4—211no—
198C22×C4—211no—
199C4⋊C4—422no—
200C22⋊C4—62no—
201C22⋊C4—142no—
202C22⋊C4—422no—
203C2×C7—42no—
204C2×C7#87911yescharacteristic
205C2×C7#87911yescharacteristic
206C2×C7#88311yescharacteristic
207C2×C7#88511yescharacteristic
208C2×C7#92911yescharacteristic
209C2×C7#93511yescharacteristic
210C2×C7#93911yescharacteristic
211C22×C3—22no—
212C22×C3—22no—
213C22×C3—22no—
214C22×C3—41no—
215C22×C3—41no—
216C22×C3—41no—
217C22×C3—41no—
218C22×C3C2×D2811yescharacteristic
219C22×C3#115711yescharacteristic
220C22×C3#115911yescharacteristic
221C22×C3#115911yescharacteristic
222C22×C3#115911yescharacteristic
223C22×C3#115911yescharacteristic
224C22×C3C2×C22⋊D711yescharacteristic
225C3×C4—281no—
226C3×C4—281no—
227Dic3—22no—
228Dic3—41no—
229Dic3—281no—
230Dic3—281no—
231C23—21no—
232C23—21no—
233C23—21no—
234C23—21no—
235C23—21no—
236C23—21no—
237C23#424612yes—
238C23C2×D3×D711yescharacteristic
239C2×C4—32no—
240C2×C4—61no—
241C2×C4—61no—
242C2×C4—61no—
243C2×C4—61no—
244C2×C4—61no—
245C2×C4—141no—
246C2×C4—141no—
247C2×C4—141no—
248C2×C4—141no—
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#495211yescharacteristic, complemented, Sylow
258C2×C3—42no—
259C2×C3#644211yescharacteristic
260C2×C3#644211yescharacteristic
261C2×C3#644911yescharacteristic
262C2×C3#644911yescharacteristic
263C2×C3#649311yescharacteristic
264C2×C3#649811yescharacteristic
265C2×C3#649911yescharacteristic
266C22—22no—
267C22—22no—
268C22—22no—
269C22—41no—
270C22—41no—
271C22—41no—
272C22—41no—
273C22#6668311yescharacteristic
274C22#6668411yescharacteristic
275C22#6668411yescharacteristic
276C22#6668611yescharacteristic
277C22#6668611yescharacteristic
278C22C2×#424611yescharacteristic
279C22D3×C22⋊D711yescharacteristic
280C4—62no—
281C4—121no—
282C4—281no—
283C4—281no—
284C4—841no—
285C4—841no—
286C3#8972511yescharacteristic, complemented, Sylow
287C2—42no—
288C2#1062108711yescharacteristic
289C2#1062109211yescharacteristic
290C2#1062109311yescharacteristic
291C2#1062109911yescharacteristic
292C2#1062110811yescharacteristic
293C2#1062111711yescharacteristic
294C2#1062113111yescharacteristic
subgroupG/H
center, FrattiniC23C2×D3×D7
commutator, socle, cosocleC23×C3×C7C23
Fitting, prime radicalC24×C3×C7C22
hypercenterC24D3×D7
hypocenterC3×C7#397
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. #10620529C2semidirect
6. #10620529C2semidirect
7. #10620529C2semidirect
191. C3×C7#397semidirect
257. C7#4952semidirect
286. C3#89725semidirect

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
#106205292
#897253
#49527