Finite Group Databasebeta

← #13061715

Subgroups — #13061715

Order 1344.

412 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 147 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×C22⋊Q8C211yesmaximal, characteristic
2#10621427C212yesmaximal, complemented
3#10621407C211yesmaximal, characteristic
4C2×#66629C211yesmaximal, characteristic
5#10621372C212yesmaximal, complemented
6#10621372C212yesmaximal, complemented
7#10621371C212yesmaximal, complemented
8#10621361C212yesmaximal
9#10621360C212yesmaximal
10#90314—31nomaximal, Hall
11C2×C3×C7×Q8C2211yescharacteristic
12C22×C3×C4×C7C2211yescharacteristic
13C22×Dic21C2211yescharacteristic
14C22×Dic21C2211yescharacteristic
15C2×Dic42C2212yes—
16C2×Dic42C2211yescharacteristic
17C3×C7×C4⋊C4C2212yes—
18C3×C7×C4⋊C4C2211yescharacteristic
19C3×C7×C22⋊C4C2212yes—
20#66636C2212yes—
21C4⋊Dic21C2212yes—
22C4⋊Dic21C2211yescharacteristic
23#66629C2212yes—
24#66629C2212yes—
25#66629C2212yes—
26#66629C2212yes—
27#66629C2212yes—
28#66629C2212yes—
29#66629C2212yes—
30C4×Dic21C2212yes—
31C4×Dic21C2212yes—
32C7×C22⋊Q8D311yescharacteristic
33#6507—32no—
34#6487—31no—
35C2×Dic7⋊C4—31no—
36#6452—32no—
37#6452—32no—
38#6451—32no—
39#6441—32no—
40#6440—32no—
41#5623—71nomaximal, Hall
42C23×C3×C7C2311yescharacteristic
43C3×C7×Q8—41no—
44C2×C3×C4×C7—41no—
45C2×C3×C4×C7C2312yes—
46C2×C3×C4×C7C2312yes—
47C2×C3×C4×C7C2311yescharacteristic
48C2×C3×C4×C7C2311yescharacteristic
49C2×Dic21—41no—
50C2×Dic21Q814yescomplemented
51C2×Dic21C2312yes—
52C2×Dic21C2312yes—
53C2×Dic21C2312yes—
54C2×Dic21C2312yes—
55Dic42—41no—
56Dic42—42no—
57C2×C7×Q8C2×D311yescharacteristic
58C22×C4×C7C2×D311yescharacteristic
59C22×Dic7—31no—
60C22×Dic7—31no—
61C2×Dic14—32no—
62C2×Dic14—31no—
63C7×C4⋊C4C2×D312yes—
64C7×C4⋊C4C2×D311yescharacteristic
65C7×C22⋊C4C2×D312yes—
66C22⋊Dic7—32no—
67C4⋊Dic7—32no—
68C4⋊Dic7—31no—
69Dic7⋊C4—32no—
70Dic7⋊C4—32no—
71Dic7⋊C4—32no—
72Dic7⋊C4—32no—
73Dic7⋊C4—32no—
74Dic7⋊C4—32no—
75Dic7⋊C4—32no—
76C4×Dic7—32no—
77C4×Dic7—32no—
78C3×C22⋊Q8D711yescharacteristic
79#944—72no—
80#924—71no—
81C2×Dic3⋊C4—71no—
82#889—72no—
83#889—72no—
84#888—72no—
85#878—72no—
86#877—72no—
87C22×C3×C7—21no—
88C22×C3×C7—21no—
89C22×C3×C7C2×Q812yes—
90C22×C3×C7C2411yescharacteristic
91C3×C4×C7—21no—
92C3×C4×C7—22no—
93C3×C4×C7—21no—
94C3×C4×C7—22no—
95Dic21—22no—
96Dic21—22no—
97Dic21—22no—
98Dic21C2×Q814yes—
99#544—211noSylow
100C23×C7C22×D311yescharacteristic
101C7×Q8—41no—
102C2×C4×C7—41no—
103C2×C4×C7C22×D312yes—
104C2×C4×C7C22×D312yes—
105C2×C4×C7C22×D311yescharacteristic
106C2×C4×C7C22×D311yescharacteristic
107C2×Dic7—34no—
108C2×Dic7—32no—
109C2×Dic7—32no—
110C2×Dic7—32no—
111C2×Dic7—32no—
112C2×Dic7—121no—
113Dic14—121no—
114Dic14—122no—
115C2×C3×Q8C2×D711yescharacteristic
116C22×C3×C4C2×D711yescharacteristic
117C22×Dic3—71no—
118C22×Dic3—71no—
119C2×Dic6—72no—
120C2×Dic6—71no—
121C3×C4⋊C4C2×D712yes—
122C3×C4⋊C4C2×D711yescharacteristic
123C3×C22⋊C4C2×D712yes—
124#217—72no—
125C4⋊Dic3—72no—
126C4⋊Dic3—71no—
127Dic3⋊C4—72no—
128Dic3⋊C4—72no—
129Dic3⋊C4—72no—
130Dic3⋊C4—72no—
131Dic3⋊C4—72no—
132Dic3⋊C4—72no—
133Dic3⋊C4—72no—
134C4×Dic3—72no—
135C4×Dic3—72no—
136C2×C3×C7—22no—
137C2×C3×C7C22×Q811yescharacteristic
138C2×C3×C7H(5,𝔽2)11yescharacteristic
139C2×C3×C72−1+411yescharacteristic
140C4⋊Q8—212no—
141C4⋊Q8—212no—
142#125—212no—
143#125—212no—
144C22⋊Q8—211no—
145C22⋊Q8—212no—
146C22⋊Q8D2111yescharacteristic
147#117—212no—
148C2×C4⋊C4—211no—
149C22×C7—21no—
150C22×C7—21no—
151C22×C7D3×Q812yes—
152C22×C7C23×D311yescharacteristic
153C4×C7—21no—
154C4×C7—22no—
155C4×C7—21no—
156C4×C7—22no—
157Dic7—34no—
158Dic7—62no—
159Dic7—62no—
160Dic7—62no—
161C23×C3C22×D711yescharacteristic
162C3×Q8—41no—
163C2×C3×C4—41no—
164C2×C3×C4C22×D712yes—
165C2×C3×C4C22×D712yes—
166C2×C3×C4C22×D711yescharacteristic
167C2×C3×C4C22×D711yescharacteristic
168C2×Dic3—74no—
169C2×Dic3—72no—
170C2×Dic3—72no—
171C2×Dic3—72no—
172C2×Dic3—72no—
173C2×Dic3—281no—
174Dic6—281no—
175Dic6—282no—
176C3×C7#54411yescharacteristic, complemented, Hall
177C2×Q8—211no—
178C2×Q8—212no—
179C2×Q8C2×D2111yescharacteristic
180C22×C4—211no—
181C22×C4—211no—
182C22×C4C2×D2111yescharacteristic
183C4⋊C4—212no—
184C4⋊C4—212no—
185C4⋊C4—212no—
186C4⋊C4—212no—
187C4⋊C4—212no—
188C4⋊C4—212no—
189C4⋊C4—212no—
190C4⋊C4—211no—
191C4⋊C4—212no—
192C4⋊C4C2×D2111yescharacteristic
193C4⋊C4C2×D2112yes—
194C22⋊C4—212no—
195C22⋊C4C2×D2112yes—
196C42—212no—
197C42—212no—
198C2×C7—22no—
199C2×C7#100411yescharacteristic
200C2×C7C2×D3×Q811yescharacteristic
201C2×C7#101011yescharacteristic
202C22×C3—21no—
203C22×C3—21no—
204C22×C3Q8×D712yes—
205C22×C3C23×D711yescharacteristic
206C3×C4—21no—
207C3×C4—22no—
208C3×C4—21no—
209C3×C4—22no—
210Dic3—74no—
211Dic3—142no—
212Dic3—142no—
213Dic3—142no—
214C23C22×D2111yescharacteristic
215Q8—41no—
216Q8—841no—
217Q8—842no—
218C2×C4—41no—
219C2×C4—214no—
220C2×C4—212no—
221C2×C4—212no—
222C2×C4—212no—
223C2×C4—212no—
224C2×C4—841no—
225C2×C4C22×D2112yes—
226C2×C4C22×D2112yes—
227C2×C4C22×D2111yescharacteristic
228C2×C4C22×D2111yescharacteristic
229C7#562311yescharacteristic, complemented, Sylow
230C2×C3—22no—
231C2×C3#654811yescharacteristic
232C2×C3C2×Q8×D711yescharacteristic
233C2×C3#655411yescharacteristic
234C22—21no—
235C22—21no—
236C22Q8×D2112yes—
237C22C23×D2111yescharacteristic
238C4—21no—
239C4—21no—
240C4—22no—
241C4—22no—
242C4—214no—
243C4—422no—
244C4—422no—
245C4—422no—
246C3#9031411yescharacteristic, complemented, Sylow
247C2—22no—
248C2#1062165711yescharacteristic
249C2C2×Q8×D2111yescharacteristic
250C2#1062166311yescharacteristic
subgroupG/H
center, FrattiniC22C23×D21
commutator, socle, cosocleC22×C3×C7C24
FittingC3×C7×C22⋊Q8C2
prime radicalC23×C3×C7C23
hypercenterC22⋊Q8D21
hypocenterC3×C7#544
LCSG ▷ C22×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. #10621427C2semidirect
5. #10621372C2semidirect
6. #10621372C2semidirect
7. #10621371C2semidirect
50. C2×Dic21Q8semidirect
176. C3×C7#544semidirect
229. C7#5623semidirect
246. C3#90314semidirect

Maximal subgroups

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

subgroupindex
C3×C7×C22⋊Q82
#106214272
#106214072
C2×#666292
#106213722
#106213712
#106213612
#106213602
#903143
#56237