Finite Group Databasebeta

← #13053976

Subgroups — #13053976

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×#66629C211yesmaximal, characteristic, complemented
2C2×C7×C4⋊Dic3C211yesmaximal, characteristic, complemented
3C2×C3×(C2×D7)⋊C4C211yesmaximal, characteristic
4C2×Dic3⋊Dic7C211yesmaximal, characteristic, complemented
5C2×(C2×D7)⋊Dic3C211yesmaximal, characteristic
6C2×(C2×D7)⋊Dic3C211yesmaximal, characteristic
7#10620529C211yesmaximal, characteristic, complemented
8#89766—31nomaximal, Hall
9C23×C3×D7C2211yescharacteristic
10C22×C3×C4×C7C2211yescharacteristic
11C22×Dic21C2211yescharacteristic
12C22×Dic21C2211yescharacteristic
13C22×C7×Dic3C2211yescharacteristic
14C22×C7×Dic3C2211yescharacteristic
15C22×C3×Dic7C2211yescharacteristic
16#66629—22no—
17C7×C4⋊Dic3—22no—
18C3×(C2×D7)⋊C4—22no—
19Dic3⋊Dic7—22no—
20(C2×D7)⋊Dic3—22no—
21(C2×D7)⋊Dic3—22no—
22C2×C7×C4⋊C4—31no—
23C2×(C2×D7)⋊C4—31no—
24C2×(C2×D7)⋊C4—31no—
25C2×(C2×D7)⋊C4D311yescharacteristic
26C2×C4⋊Dic7—31no—
27C2×Dic7⋊C4—31no—
28C7⋊#95—31no—
29#4948—71nomaximal, Hall
30C23×C3×C7C2311yescharacteristic
31C22×C3×D7—21no—
32C22×C3×D7—21no—
33C22×C3×D7—21no—
34C22×C3×D7—21no—
35C22×C3×D7—21no—
36C22×C3×D7—21no—
37C22×C3×D7Q812yes—
38C2×C3×C4×C7—21no—
39C2×C3×C4×C7—21no—
40C2×C3×C4×C7D412yes—
41C2×Dic21—21no—
42C2×Dic21—21no—
43C2×Dic21—21no—
44C2×Dic21—21no—
45C2×Dic21—21no—
46C2×Dic21D412yes—
47C2×C7×Dic3—21no—
48C2×C7×Dic3—21no—
49C2×C7×Dic3—21no—
50C2×C7×Dic3—21no—
51C2×C7×Dic3—21no—
52C2×C7×Dic3D412yes—
53C2×C3×Dic7—21no—
54C2×C3×Dic7—21no—
55C2×C3×Dic7—21no—
56C23×D7C2×D311yescharacteristic
57C22×C4×C7—31no—
58C22×C4×C7—31no—
59C22×C4×C7C2×D311yescharacteristic
60C22×Dic7—31no—
61C22×Dic7—31no—
62C22×Dic7C2×D311yescharacteristic
63C7×C4⋊C4—62no—
64(C2×D7)⋊C4—22no—
65(C2×D7)⋊C4—62no—
66(C2×D7)⋊C4—62no—
67C4⋊Dic7—62no—
68Dic7⋊C4—62no—
69C2×C3×C22⋊C4—71no—
70C2×#217—71no—
71C2×#217—71no—
72C2×C4⋊Dic3D711yescharacteristic, complemented
73C2×Dic3⋊C4—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×C3×D7—22no—
84C2×C3×D7—22no—
85C2×C3×D7—22no—
86C2×C3×D7—41no—
87C2×C3×D7—41no—
88C2×C3×D7—41no—
89C2×C3×D7—41no—
90C3×C4×C7—22no—
91Dic21—22no—
92Dic21—41no—
93C3×Dic7—41no—
94C7×Dic3—22no—
95C7×Dic3—41no—
96#396—211noSylow
97C23×C7C22×D311yescharacteristic
98C22×D7—21no—
99C22×D7—21no—
100C22×D7—21no—
101C22×D7—21no—
102C22×D7—21no—
103C22×D7—21no—
104C22×D7Dic612yes—
105C2×C4×C7—21no—
106C2×C4×C7—21no—
107C2×C4×C7—32no—
108C2×C4×C7—61no—
109C2×C4×C7—61no—
110C2×C4×C7—61no—
111C2×C4×C7—61no—
112C2×C4×C7—61no—
113C2×C4×C7#6712yes—
114C2×Dic7—21no—
115C2×Dic7—21no—
116C2×Dic7—21no—
117C2×Dic7—32no—
118C2×Dic7—61no—
119C2×Dic7—61no—
120C2×Dic7—61no—
121C2×Dic7—61no—
122C2×Dic7—61no—
123C24×C3—71no—
124C22×C3×C4—71no—
125C22×C3×C4C2×D711yescharacteristic
126C22×Dic3—71no—
127C22×Dic3—71no—
128C22×Dic3C2×D711yescharacteristic
129C22×Dic3C2×D711yescharacteristic
130C3×C22⋊C4—142no—
131#217—142no—
132#217—142no—
133C4⋊Dic3—22no—
134Dic3⋊C4—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
143C3×D7—42no—
144C2×C4⋊C4—31no—
145C2×C4⋊C4—211no—
146C2×C4⋊C4—211no—
147C2×C22⋊C4—71no—
148C2×C22⋊C4—211no—
149C2×C22⋊C4—211no—
150#95—211no—
151C22×C7C2×Dic611yescharacteristic
152C22×C7#23511yescharacteristic
153C22×C7D3×D411yescharacteristic
154C22×C7D3×D411yescharacteristic
155C22×C7#23711yescharacteristic
156C22×C7#23711yescharacteristic
157C22×C7C2×#6711yescharacteristic
158C2×D7—22no—
159C2×D7—22no—
160C2×D7—22no—
161C2×D7—41no—
162C2×D7—41no—
163C2×D7—41no—
164C2×D7—41no—
165C4×C7—22no—
166C4×C7—62no—
167C4×C7—121no—
168Dic7—41no—
169Dic7—62no—
170Dic7—121no—
171C23×C3—72no—
172C23×C3—141no—
173C23×C3—141no—
174C23×C3—141no—
175C23×C3—141no—
176C23×C3—141no—
177C23×C3—141no—
178C23×C3C22×D711yescharacteristic
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4—141no—
182C2×C3×C4—141no—
183C2×C3×C4—141no—
184C2×C3×C4D2812yes—
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#39611yescharacteristic, complemented, Hall
198C24—71no—
199C22×C4—31no—
200C22×C4—31no—
201C22×C4—71no—
202C22×C4—211no—
203C22×C4—211no—
204C22×C4D3×D711yescharacteristic
205C4⋊C4—62no—
206C4⋊C4—422no—
207C4⋊C4—422no—
208C22⋊C4—142no—
209C22⋊C4—422no—
210C22⋊C4—422no—
211C2×C7#87811yescharacteristic
212C2×C7#87811yescharacteristic
213C2×C7#87911yescharacteristic
214C2×C7#88411yescharacteristic
215C2×C7#92411yescharacteristic
216C2×C7#93811yescharacteristic
217C2×C7#93911yescharacteristic
218D7—42no—
219C22×C3—142no—
220C22×C3—142no—
221C22×C3—142no—
222C22×C3—281no—
223C22×C3—281no—
224C22×C3—281no—
225C22×C3—281no—
226C22×C3C2×D2811yescharacteristic
227C22×C3#115711yescharacteristic
228C22×C3D4×D711yescharacteristic
229C22×C3#115911yescharacteristic
230C22×C3Q8×D711yescharacteristic
231C22×C3#116111yescharacteristic
232C22×C3C2×C22⋊D711yescharacteristic
233C3×C4—22no—
234C3×C4—281no—
235Dic3—22no—
236Dic3—41no—
237Dic3—142no—
238Dic3—281no—
239C23—72no—
240C23—141no—
241C23—141no—
242C23—141no—
243C23—141no—
244C23—141no—
245C23—141no—
246C23C2×D3×D711yescharacteristic
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—32no—
250C2×C4—61no—
251C2×C4—61no—
252C2×C4—61no—
253C2×C4—61no—
254C2×C4—61no—
255C2×C4—141no—
256C2×C4—141no—
257C2×C4—141no—
258C2×C4—212no—
259C2×C4—421no—
260C2×C4—421no—
261C2×C4—421no—
262C2×C4—421no—
263C2×C4—421no—
264C2×C4#424612yes—
265C7#494811yescharacteristic, complemented, Sylow
266C2×C3—282no—
267C2×C3#645811yescharacteristic
268C2×C3#645911yescharacteristic
269C2×C3#646011yescharacteristic
270C2×C3#646111yescharacteristic
271C2×C3#649311yescharacteristic
272C2×C3#650211yescharacteristic
273C2×C3#650911yescharacteristic
274C22—142no—
275C22—142no—
276C22—142no—
277C22—281no—
278C22—281no—
279C22—281no—
280C22—281no—
281C22D7×Dic611yescharacteristic
282C22#6667011yescharacteristic
283C22#6667611yescharacteristic
284C22#6668111yescharacteristic
285C22#6668311yescharacteristic
286C22C2×#424611yescharacteristic
287C22D3×C22⋊D711yescharacteristic
288C4—22no—
289C4—62no—
290C4—121no—
291C4—281no—
292C4—422no—
293C4—841no—
294C3#8976611yescharacteristic, complemented, Sylow
295C2—282no—
296C2#1062092211yescharacteristic
297C2#1062092311yescharacteristic
298C2#1062098311yescharacteristic
299C2#1062099211yescharacteristic
300C2#1062101511yescharacteristic
301C2#1062102711yescharacteristic
302C2#1062113111yescharacteristic
subgroupG/H
center, FrattiniC23C2×D3×D7
commutator, socle, cosocleC23×C3×C7C23
FittingC22×C3×C4×C7C22
prime radicalC23×C3×D7C22
hypercenterC22×C4D3×D7
hypocenterC3×C7#396
LCSG ▷ C23×C3×C7 ▷ C3×C7
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×#66629C2semidirect
2. C2×C7×C4⋊Dic3C2semidirect
4. C2×Dic3⋊Dic7C2semidirect
7. #10620529C2semidirect
72. C2×C4⋊Dic3D7semidirect
197. C3×C7#396semidirect
265. C7#4948semidirect
294. C3#89766semidirect

Maximal subgroups

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

subgroupindex
C2×#666292
C2×C7×C4⋊Dic32
C2×C3×(C2×D7)⋊C42
C2×Dic3⋊Dic72
C2×(C2×D7)⋊Dic32
#106205292
#897663
#49487