Finite Group Databasebeta

← #13056617

Subgroups — #13056617

Order 1344.

372 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C7×C4⋊C4C211yesmaximal, characteristic, complemented
2C2×#66631C211yesmaximal, characteristic
3C2×#66631C211yesmaximal, characteristic
4C2×#66631C211yesmaximal, characteristic
5C2×C4⋊Dic21C211yesmaximal, characteristic, complemented
6C2×#66629C211yesmaximal, characteristic, complemented
7#10620670C211yesmaximal, characteristic, complemented
8#89766—31nomaximal, Hall
9C23×D21C2211yescharacteristic
10C22×C3×C4×C7C2211yescharacteristic
11C22×C3×C4×C7C2211yescharacteristic
12C22×C3×C4×C7C2211yescharacteristic
13C22×Dic21C2211yescharacteristic
14C22×Dic21C2211yescharacteristic
15C22×Dic21C2211yescharacteristic
16C3×C7×C4⋊C4—22no—
17#66631—22no—
18#66631—22no—
19#66631—22no—
20C4⋊Dic21—22no—
21#66629—22no—
22C2×C7×C4⋊C4D311yescharacteristic, complemented
23C2×(C2×D7)⋊C4—31no—
24C2×(C2×D7)⋊C4—31no—
25C2×(C2×D7)⋊C4—31no—
26C2×C4⋊Dic7—31no—
27C2×Dic7⋊C4—31no—
28C7⋊#95—31no—
29#4994—71nomaximal, Hall
30C23×C3×C7C2311yescharacteristic
31C22×D21—21no—
32C22×D21—21no—
33C22×D21—21no—
34C22×D21—21no—
35C22×D21—21no—
36C22×D21—21no—
37C22×D21Q812yes—
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×C3×C4×C7D412yes—
46C2×C3×C4×C7D412yes—
47C2×Dic21—21no—
48C2×Dic21—21no—
49C2×Dic21—21no—
50C2×Dic21—21no—
51C2×Dic21—21no—
52C2×Dic21—21no—
53C2×Dic21—21no—
54C2×Dic21—21no—
55C2×Dic21D412yes—
56C23×D7—31no—
57C22×C4×C7C2×D311yescharacteristic
58C22×C4×C7C2×D311yescharacteristic
59C22×C4×C7C2×D311yescharacteristic
60C22×Dic7—31no—
61C22×Dic7—31no—
62C22×Dic7—31no—
63C7×C4⋊C4—22no—
64(C2×D7)⋊C4—62no—
65(C2×D7)⋊C4—62no—
66(C2×D7)⋊C4—62no—
67C4⋊Dic7—62no—
68Dic7⋊C4—62no—
69C2×C3×C4⋊C4D711yescharacteristic, complemented
70C2×(C2×D3)⋊C4—71no—
71C2×(C2×D3)⋊C4—71no—
72C2×(C2×D3)⋊C4—71no—
73C2×C4⋊Dic3—71no—
74C2×Dic3⋊C4—71no—
75C3⋊#95—71no—
76C22×C3×C7C2×D411yescharacteristic
77C22×C3×C7C2×D411yescharacteristic
78C22×C3×C7C2×D411yescharacteristic
79C22×C3×C7C2×Q811yescharacteristic
80C22×C3×C7#4111yescharacteristic
81C22×C3×C7#4111yescharacteristic
82C22×C3×C7#4111yescharacteristic
83C2×D21—22no—
84C2×D21—22no—
85C2×D21—22no—
86C2×D21—41no—
87C2×D21—41no—
88C2×D21—41no—
89C2×D21—41no—
90C3×C4×C7—22no—
91C3×C4×C7—22no—
92C3×C4×C7—41no—
93Dic21—22no—
94Dic21—41no—
95Dic21—41no—
96#396—211noSylow
97C23×C7C22×D311yescharacteristic
98C22×D7—32no—
99C22×D7—61no—
100C22×D7—61no—
101C22×D7—61no—
102C22×D7—61no—
103C22×D7—61no—
104C22×D7—61no—
105C2×C4×C7—21no—
106C2×C4×C7—21no—
107C2×C4×C7—21no—
108C2×C4×C7—21no—
109C2×C4×C7—21no—
110C2×C4×C7—21no—
111C2×C4×C7—21no—
112C2×C4×C7D1212yes—
113C2×C4×C7#6712yes—
114C2×Dic7—32no—
115C2×Dic7—61no—
116C2×Dic7—61no—
117C2×Dic7—61no—
118C2×Dic7—61no—
119C2×Dic7—61no—
120C2×Dic7—61no—
121C2×Dic7—61no—
122C2×Dic7—61no—
123C23×D3—71no—
124C22×C3×C4C2×D711yescharacteristic
125C22×C3×C4C2×D711yescharacteristic
126C22×C3×C4C2×D711yescharacteristic
127C22×Dic3—71no—
128C22×Dic3—71no—
129C22×Dic3—71no—
130C3×C4⋊C4—22no—
131(C2×D3)⋊C4—142no—
132(C2×D3)⋊C4—142no—
133(C2×D3)⋊C4—142no—
134C4⋊Dic3—142no—
135Dic3⋊C4—142no—
136C2×C3×C7#12111yescharacteristic
137C2×C3×C7#12111yescharacteristic
138C2×C3×C7#12111yescharacteristic
139C2×C3×C7C22⋊Q811yescharacteristic
140C2×C3×C7C22⋊Q811yescharacteristic
141C2×C3×C7C22⋊Q811yescharacteristic
142C2×C3×C7#12611yescharacteristic
143D21—42no—
144C2×C4⋊C4—211no—
145C2×C4⋊C4—211no—
146C2×C4⋊C4D2111yescharacteristic, complemented
147C2×C22⋊C4—211no—
148C2×C22⋊C4—211no—
149C2×C22⋊C4—211no—
150#95—211no—
151C22×C7C2×D1211yescharacteristic
152C22×C7#23511yescharacteristic
153C22×C7D3×D411yescharacteristic
154C22×C7#23711yescharacteristic
155C22×C7D3×Q811yescharacteristic
156C22×C7#23911yescharacteristic
157C22×C7C2×#6711yescharacteristic
158C2×D7—62no—
159C2×D7—62no—
160C2×D7—62no—
161C2×D7—121no—
162C2×D7—121no—
163C2×D7—121no—
164C2×D7—121no—
165C4×C7—22no—
166C4×C7—22no—
167C4×C7—41no—
168Dic7—62no—
169Dic7—121no—
170Dic7—121no—
171C23×C3C22×D711yescharacteristic
172C22×D3—72no—
173C22×D3—141no—
174C22×D3—141no—
175C22×D3—141no—
176C22×D3—141no—
177C22×D3—141no—
178C22×D3—141no—
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×C3×C4—21no—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4—21no—
186C2×C3×C4D2812yes—
187C2×C3×C4C22⋊D712yes—
188C2×Dic3—72no—
189C2×Dic3—141no—
190C2×Dic3—141no—
191C2×Dic3—141no—
192C2×Dic3—141no—
193C2×Dic3—141no—
194C2×Dic3—141no—
195C2×Dic3—141no—
196C2×Dic3—141no—
197C3×C7#39611yescharacteristic, complemented, Hall
198C24—211no—
199C22×C4—211no—
200C22×C4—211no—
201C22×C4—211no—
202C22×C4C2×D2111yescharacteristic
203C22×C4C2×D2111yescharacteristic
204C22×C4C2×D2111yescharacteristic
205C4⋊C4—22no—
206C4⋊C4—422no—
207C4⋊C4—422no—
208C22⋊C4—422no—
209C22⋊C4—422no—
210C22⋊C4—422no—
211C2×C7#89511yescharacteristic
212C2×C7#89611yescharacteristic
213C2×C7#89711yescharacteristic
214C2×C7#89811yescharacteristic
215C2×C7#93011yescharacteristic
216C2×C7#93911yescharacteristic
217C2×C7#94611yescharacteristic
218D7—122no—
219C22×C3C2×D2811yescharacteristic
220C22×C3#115711yescharacteristic
221C22×C3D4×D711yescharacteristic
222C22×C3#115911yescharacteristic
223C22×C3Q8×D711yescharacteristic
224C22×C3#116111yescharacteristic
225C22×C3C2×C22⋊D711yescharacteristic
226C2×D3—142no—
227C2×D3—142no—
228C2×D3—142no—
229C2×D3—281no—
230C2×D3—281no—
231C2×D3—281no—
232C2×D3—281no—
233C3×C4—22no—
234C3×C4—22no—
235C3×C4—41no—
236Dic3—142no—
237Dic3—281no—
238Dic3—281no—
239C23—212no—
240C23—421no—
241C23—421no—
242C23—421no—
243C23—421no—
244C23—421no—
245C23—421no—
246C23C22×D2111yescharacteristic
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—212no—
255C2×C4—421no—
256C2×C4—421no—
257C2×C4—421no—
258C2×C4—421no—
259C2×C4—421no—
260C2×C4—421no—
261C2×C4—421no—
262C2×C4—421no—
263C2×C4D8412yes—
264C2×C4#426812yes—
265C7#499411yescharacteristic, complemented, Sylow
266C2×C3#645811yescharacteristic
267C2×C3#645911yescharacteristic
268C2×C3#646011yescharacteristic
269C2×C3#646111yescharacteristic
270C2×C3#649311yescharacteristic
271C2×C3#650211yescharacteristic
272C2×C3#650911yescharacteristic
273D3—282no—
274C22—422no—
275C22—422no—
276C22—422no—
277C22—841no—
278C22—841no—
279C22—841no—
280C22—841no—
281C22C2×D8411yescharacteristic
282C22#6672811yescharacteristic
283C22D4×D2111yescharacteristic
284C22#6673011yescharacteristic
285C22Q8×D2111yescharacteristic
286C22#6673211yescharacteristic
287C22C2×#426811yescharacteristic
288C4—22no—
289C4—22no—
290C4—41no—
291C4—422no—
292C4—841no—
293C4—841no—
294C3#8976611yescharacteristic, complemented, Sylow
295C2—842no—
296C2#1062137811yescharacteristic
297C2#1062137911yescharacteristic
298C2#1062138011yescharacteristic
299C2#1062138111yescharacteristic
300C2#1062141311yescharacteristic
301C2#1062142211yescharacteristic
302C2#1062142911yescharacteristic
subgroupG/H
center, FrattiniC23C22×D21
commutator, socle, cosocleC23×C3×C7C23
FittingC2×C3×C7×C4⋊C4C2
prime radicalC23×D21C22
hypercenterC2×C4⋊C4D21
hypocenterC3×C7#396
LCSG ▷ C23×C3×C7 ▷ C3×C7
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×C7×C4⋊C4C2semidirect
5. C2×C4⋊Dic21C2semidirect
6. C2×#66629C2semidirect
7. #10620670C2semidirect
22. C2×C7×C4⋊C4D3semidirect
69. C2×C3×C4⋊C4D7semidirect
146. C2×C4⋊C4D21semidirect
197. C3×C7#396semidirect
265. C7#4994semidirect
294. C3#89766semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C7×C4⋊C42
C2×#666312
C2×C4⋊Dic212
C2×#666292
#106206702
#897663
#49947