Finite Group Databasebeta

← #13061694

Subgroups — #13061694

Order 1344.

436 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 139 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
1C3×C7×#121C211yesmaximal, characteristic
2#10621418C212yesmaximal, complemented
3#10621418C212yesmaximal, complemented
4D4×Dic21C212yesmaximal, complemented
5#10621407C211yesmaximal, characteristic, complemented
6C2×#66629C211yesmaximal, characteristic, complemented
7#10621372C212yesmaximal, complemented
8#10621362C212yesmaximal, complemented
9#10621361C212yesmaximal, complemented
10#90293—31nomaximal, Hall
11C2×C3×C7×D4C2212yes—
12C2×C3×C7×D4C2211yescharacteristic
13C22×C3×C4×C7C2211yescharacteristic
14C22×Dic21C2211yescharacteristic
15C22×Dic21C2212yes—
16C22×Dic21C2211yescharacteristic
17C2×Dic42C2211yescharacteristic
18C3×C7×C4⋊C4C2211yescharacteristic
19C3×C7×C22⋊C4C2212yes—
20#66636C2212yes—
21#66636C2212yes—
22#66636C2212yes—
23#66636C2212yes—
24C4⋊Dic21C2212yes—
25C4⋊Dic21C2211yescharacteristic
26#66629C2212yes—
27#66629C2212yes—
28#66629C2212yes—
29#66629C2212yes—
30#66629C2212yes—
31C4×Dic21C2212yes—
32C7×#121D311yescharacteristic
33#6498—32no—
34#6498—32no—
35D4×Dic7—32no—
36#6487—31no—
37C2×Dic7⋊C4—31no—
38#6452—32no—
39#6442—32no—
40#6441—32no—
41#5602—71nomaximal, Hall
42C23×C3×C7C2312yes—
43C23×C3×C7C2311yescharacteristic
44C3×C7×D4—24no—
45C3×C7×D4—41no—
46C2×C3×C4×C7—41no—
47C2×C3×C4×C7C2312yes—
48C2×C3×C4×C7C2311yescharacteristic
49C2×C3×C4×C7C2311yescharacteristic
50C2×Dic21—22no—
51C2×Dic21—42no—
52C2×Dic21—41no—
53C2×Dic21C2312yes—
54C2×Dic21C2312yes—
55C2×Dic21C2312yes—
56C2×Dic21C2312yes—
57Dic42—41no—
58C2×C7×D4C2×D312yes—
59C2×C7×D4C2×D311yescharacteristic
60C22×C4×C7C2×D311yescharacteristic
61C22×Dic7—31no—
62C22×Dic7—32no—
63C22×Dic7—31no—
64C2×Dic14—31no—
65C7×C4⋊C4C2×D311yescharacteristic
66C7×C22⋊C4C2×D312yes—
67C22⋊Dic7—32no—
68C22⋊Dic7—32no—
69C22⋊Dic7—32no—
70C22⋊Dic7—32no—
71C4⋊Dic7—32no—
72C4⋊Dic7—31no—
73Dic7⋊C4—32no—
74Dic7⋊C4—32no—
75Dic7⋊C4—32no—
76Dic7⋊C4—32no—
77Dic7⋊C4—32no—
78C4×Dic7—32no—
79C3×#121D711yescharacteristic
80#935—72no—
81#935—72no—
82D4×Dic3—72no—
83#924—71no—
84C2×Dic3⋊C4—71no—
85#889—72no—
86#879—72no—
87#878—72no—
88C22×C3×C7—21no—
89C22×C3×C7—21no—
90C22×C3×C7—22no—
91C22×C3×C7—22no—
92C22×C3×C7—22no—
93C22×C3×C7#4112yes—
94C22×C3×C7C2411yescharacteristic
95C3×C4×C7—22no—
96C3×C4×C7—21no—
97C3×C4×C7—21no—
98Dic21—22no—
99Dic21—22no—
100Dic21—22no—
101Dic21—22no—
102#539—211noSylow
103C23×C7C22×D312yes—
104C23×C7C22×D311yescharacteristic
105C7×D4—24no—
106C7×D4—41no—
107C2×C4×C7—41no—
108C2×C4×C7C22×D312yes—
109C2×C4×C7C22×D311yescharacteristic
110C2×C4×C7C22×D311yescharacteristic
111C2×Dic7—32no—
112C2×Dic7—32no—
113C2×Dic7—32no—
114C2×Dic7—32no—
115C2×Dic7—62no—
116C2×Dic7—122no—
117C2×Dic7—121no—
118Dic14—121no—
119C2×C3×D4C2×D712yes—
120C2×C3×D4C2×D711yescharacteristic
121C22×C3×C4C2×D711yescharacteristic
122C22×Dic3—71no—
123C22×Dic3—72no—
124C22×Dic3—71no—
125C2×Dic6—71no—
126C3×C4⋊C4C2×D711yescharacteristic
127C3×C22⋊C4C2×D712yes—
128#217—72no—
129#217—72no—
130#217—72no—
131#217—72no—
132C4⋊Dic3—72no—
133C4⋊Dic3—71no—
134Dic3⋊C4—72no—
135Dic3⋊C4—72no—
136Dic3⋊C4—72no—
137Dic3⋊C4—72no—
138Dic3⋊C4—72no—
139C4×Dic3—72no—
140C2×C3×C7—22no—
141C2×C3×C7—42no—
142C2×C3×C7C2×#4111yescharacteristic
143C2×C3×C7H(5,𝔽2)11yescharacteristic
144C2×C3×C72−1+411yescharacteristic
145#126—212no—
146#125—212no—
147#123—212no—
148#123—212no—
149C22⋊Q8—211no—
150C22⋊Q8—212no—
151#121D2111yescharacteristic
152C4×D4—212no—
153C2×C4⋊C4—211no—
154C22×C7—21no—
155C22×C7—21no—
156C22×C7—22no—
157C22×C7—22no—
158C22×C7—22no—
159C22×C7#23712yes—
160C22×C7C23×D311yescharacteristic
161C4×C7—22no—
162C4×C7—21no—
163C4×C7—21no—
164Dic7—62no—
165Dic7—62no—
166Dic7—62no—
167Dic7—62no—
168C23×C3C22×D712yes—
169C23×C3C22×D711yescharacteristic
170C3×D4—24no—
171C3×D4—41no—
172C2×C3×C4—41no—
173C2×C3×C4C22×D712yes—
174C2×C3×C4C22×D711yescharacteristic
175C2×C3×C4C22×D711yescharacteristic
176C2×Dic3—72no—
177C2×Dic3—72no—
178C2×Dic3—72no—
179C2×Dic3—72no—
180C2×Dic3—142no—
181C2×Dic3—282no—
182C2×Dic3—281no—
183Dic6—281no—
184C3×C7#53911yescharacteristic, complemented, Hall
185C2×Q8—211no—
186C2×D4C2×D2112yes—
187C2×D4C2×D2111yescharacteristic
188C22×C4—211no—
189C22×C4—211no—
190C22×C4—212no—
191C22×C4C2×D2111yescharacteristic
192C4⋊C4—212no—
193C4⋊C4—212no—
194C4⋊C4—212no—
195C4⋊C4—212no—
196C4⋊C4—212no—
197C4⋊C4—212no—
198C4⋊C4—211no—
199C4⋊C4C2×D2111yescharacteristic
200C22⋊C4—212no—
201C22⋊C4—212no—
202C22⋊C4—212no—
203C22⋊C4—212no—
204C22⋊C4C2×D2112yes—
205C42—212no—
206C2×C7—22no—
207C2×C7—42no—
208C2×C7C2×#23711yescharacteristic
209C2×C7#100411yescharacteristic
210C2×C7#101011yescharacteristic
211C22×C3—21no—
212C22×C3—21no—
213C22×C3—22no—
214C22×C3—22no—
215C22×C3—22no—
216C22×C3#115912yes—
217C22×C3C23×D711yescharacteristic
218C3×C4—22no—
219C3×C4—21no—
220C3×C4—21no—
221Dic3—142no—
222Dic3—142no—
223Dic3—142no—
224Dic3—142no—
225C23C22×D2112yes—
226C23C22×D2111yescharacteristic
227Q8—841no—
228D4—24no—
229D4—41no—
230C2×C4—41no—
231C2×C4—212no—
232C2×C4—212no—
233C2×C4—212no—
234C2×C4—212no—
235C2×C4—422no—
236C2×C4—841no—
237C2×C4—842no—
238C2×C4C22×D2112yes—
239C2×C4C22×D2111yescharacteristic
240C2×C4C22×D2111yescharacteristic
241C7#560211yescharacteristic, complemented, Sylow
242C2×C3—22no—
243C2×C3—42no—
244C2×C3C2×#115911yescharacteristic
245C2×C3#654811yescharacteristic
246C2×C3#655411yescharacteristic
247C22—21no—
248C22—21no—
249C22—22no—
250C22—22no—
251C22—22no—
252C22#6673012yes—
253C22C23×D2111yescharacteristic
254C4—22no—
255C4—21no—
256C4—21no—
257C4—422no—
258C4—422no—
259C4—422no—
260C4—422no—
261C3#9029311yescharacteristic, complemented, Sylow
262C2—22no—
263C2—42no—
264C2C2×#6673011yescharacteristic
265C2#1062165711yescharacteristic
266C2#1062166311yescharacteristic
subgroupG/H
center, FrattiniC22C23×D21
commutator, socle, cosocleC22×C3×C7C24
Fitting, prime radicalC3×C7×#121C2
hypercenter#121D21
hypocenterC3×C7#539
LCSG ▷ C22×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
2. #10621418C2semidirect
3. #10621418C2semidirect
4. D4×Dic21C2semidirect
5. #10621407C2semidirect
6. C2×#66629C2semidirect
7. #10621372C2semidirect
8. #10621362C2semidirect
9. #10621361C2semidirect
184. C3×C7#539semidirect
241. C7#5602semidirect
261. C3#90293semidirect

Maximal subgroups

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

subgroupindex
C3×C7×#1212
#106214182
D4×Dic212
#106214072
C2×#666292
#106213722
#106213622
#106213612
#902933
#56027