Finite Group Databasebeta

← #13056296

Subgroups — #13056296

Order 1344.

268 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 95 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
2C2×#66629C211yesmaximal, characteristic
3C3×C7×#95C211yesmaximal, characteristic
4#10620670C211yesmaximal, characteristic
5#10620670C211yesmaximal, characteristic
6#10620670C211yesmaximal, characteristic
7#10620670C211yesmaximal, characteristic
8#89430—31nomaximal, Hall
9C22×C3×C4×C7C2211yescharacteristic
10C22×C3×C4×C7C2211yescharacteristic
11C22×C3×C4×C7C2211yescharacteristic
12C22×Dic21C2211yescharacteristic
13C22×Dic21C2211yescharacteristic
14C22×Dic21C2211yescharacteristic
15C22×Dic21C2211yescharacteristic
16#66629—22no—
17#66629—22no—
18C2×Dic7⋊C4—31no—
19C2×Dic7⋊C4—31no—
20C7×#95D311yescharacteristic
21C7⋊#95—31no—
22C7⋊#95—31no—
23C7⋊#95—31no—
24C7⋊#95—31no—
25#4658—71nomaximal, Hall
26C23×C3×C7C2311yescharacteristic
27C2×C3×C4×C7—21no—
28C2×C3×C4×C7—21no—
29C2×C3×C4×C7—21no—
30C2×C3×C4×C7—21no—
31C2×C3×C4×C7—21no—
32C2×C3×C4×C7—21no—
33C2×C3×C4×C7—21no—
34C2×C3×C4×C7—21no—
35C2×C3×C4×C7—21no—
36C2×Dic21—21no—
37C2×Dic21—21no—
38C2×Dic21—21no—
39C2×Dic21—21no—
40C2×Dic21—21no—
41C2×Dic21—21no—
42C2×Dic21—21no—
43C2×Dic21—21no—
44C2×Dic21—21no—
45C2×Dic21—21no—
46C2×Dic21D412yes—
47C2×Dic21Q812yes—
48C22×C4×C7C2×D311yescharacteristic
49C22×C4×C7C2×D311yescharacteristic
50C22×C4×C7C2×D311yescharacteristic
51C22×Dic7—31no—
52C22×Dic7—31no—
53C22×Dic7—31no—
54C22×Dic7—31no—
55Dic7⋊C4—62no—
56Dic7⋊C4—62no—
57C2×Dic3⋊C4—71no—
58C2×Dic3⋊C4—71no—
59C3×#95D711yescharacteristic
60C3⋊#95—71no—
61C3⋊#95—71no—
62C3⋊#95—71no—
63C3⋊#95—71no—
64C22×C3×C7C2×D411yescharacteristic
65C22×C3×C7C2×Q811yescharacteristic
66C22×C3×C7#4111yescharacteristic
67C22×C3×C7#4111yescharacteristic
68C22×C3×C7#4111yescharacteristic
69C22×C3×C7#4111yescharacteristic
70C22×C3×C7#4111yescharacteristic
71C3×C4×C7—41no—
72C3×C4×C7—41no—
73C3×C4×C7—41no—
74Dic21—22no—
75Dic21—22no—
76Dic21—41no—
77Dic21—41no—
78#400—211noSylow
79C23×C7C22×D311yescharacteristic
80C2×C4×C7—21no—
81C2×C4×C7—21no—
82C2×C4×C7—21no—
83C2×C4×C7—21no—
84C2×C4×C7—21no—
85C2×C4×C7—21no—
86C2×C4×C7—21no—
87C2×C4×C7—21no—
88C2×C4×C7—21no—
89C2×Dic7—32no—
90C2×Dic7—32no—
91C2×Dic7—61no—
92C2×Dic7—61no—
93C2×Dic7—61no—
94C2×Dic7—61no—
95C2×Dic7—61no—
96C2×Dic7—61no—
97C2×Dic7—61no—
98C2×Dic7—61no—
99C2×Dic7—61no—
100C2×Dic7—61no—
101C22×C3×C4C2×D711yescharacteristic
102C22×C3×C4C2×D711yescharacteristic
103C22×C3×C4C2×D711yescharacteristic
104C22×Dic3—71no—
105C22×Dic3—71no—
106C22×Dic3—71no—
107C22×Dic3—71no—
108Dic3⋊C4—142no—
109Dic3⋊C4—142no—
110C2×C3×C7C22⋊Q811yescharacteristic
111C2×C3×C7#12311yescharacteristic
112C2×C3×C7#12411yescharacteristic
113C2×C3×C7#12511yescharacteristic
114C2×C3×C7#12511yescharacteristic
115C2×C3×C7#12611yescharacteristic
116C2×C3×C7#12611yescharacteristic
117C2×C4⋊C4—211no—
118C2×C4⋊C4—211no—
119#95—211no—
120#95—211no—
121#95—211no—
122#95—211no—
123#95D2111yescharacteristic
124C22×C7#23511yescharacteristic
125C22×C7#23511yescharacteristic
126C22×C7#23511yescharacteristic
127C22×C7D3×D411yescharacteristic
128C22×C7#23711yescharacteristic
129C22×C7#23711yescharacteristic
130C22×C7D3×Q811yescharacteristic
131C4×C7—41no—
132C4×C7—41no—
133C4×C7—41no—
134Dic7—62no—
135Dic7—62no—
136Dic7—121no—
137Dic7—121no—
138C23×C3C22×D711yescharacteristic
139C2×C3×C4—21no—
140C2×C3×C4—21no—
141C2×C3×C4—21no—
142C2×C3×C4—21no—
143C2×C3×C4—21no—
144C2×C3×C4—21no—
145C2×C3×C4—21no—
146C2×C3×C4—21no—
147C2×C3×C4—21no—
148C2×Dic3—72no—
149C2×Dic3—72no—
150C2×Dic3—141no—
151C2×Dic3—141no—
152C2×Dic3—141no—
153C2×Dic3—141no—
154C2×Dic3—141no—
155C2×Dic3—141no—
156C2×Dic3—141no—
157C2×Dic3—141no—
158C2×Dic3—141no—
159C2×Dic3—141no—
160C3×C7#40011yescharacteristic, complemented, Hall
161C22×C4—211no—
162C22×C4—211no—
163C22×C4—211no—
164C22×C4—211no—
165C22×C4C2×D2111yescharacteristic
166C22×C4C2×D2111yescharacteristic
167C22×C4C2×D2111yescharacteristic
168C4⋊C4—422no—
169C4⋊C4—422no—
170C2×C7#87611yescharacteristic
171C2×C7#87911yescharacteristic
172C2×C7#88311yescharacteristic
173C2×C7#88511yescharacteristic
174C2×C7#88911yescharacteristic
175C2×C7#88911yescharacteristic
176C2×C7#89611yescharacteristic
177C22×C3#115711yescharacteristic
178C22×C3#115711yescharacteristic
179C22×C3#115711yescharacteristic
180C22×C3D4×D711yescharacteristic
181C22×C3#115911yescharacteristic
182C22×C3#115911yescharacteristic
183C22×C3Q8×D711yescharacteristic
184C3×C4—41no—
185C3×C4—41no—
186C3×C4—41no—
187Dic3—142no—
188Dic3—142no—
189Dic3—281no—
190Dic3—281no—
191C23C22×D2111yescharacteristic
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—212no—
202C2×C4—212no—
203C2×C4—421no—
204C2×C4—421no—
205C2×C4—421no—
206C2×C4—421no—
207C2×C4—421no—
208C2×C4—421no—
209C2×C4—421no—
210C2×C4—421no—
211C2×C4—421no—
212C2×C4—421no—
213C7#465811yescharacteristic, complemented, Sylow
214C2×C3#643911yescharacteristic
215C2×C3#644211yescharacteristic
216C2×C3#644611yescharacteristic
217C2×C3#644811yescharacteristic
218C2×C3#645211yescharacteristic
219C2×C3#645211yescharacteristic
220C2×C3#645911yescharacteristic
221C22#6672811yescharacteristic
222C22#6672811yescharacteristic
223C22#6672811yescharacteristic
224C22D4×D2111yescharacteristic
225C22#6673011yescharacteristic
226C22#6673011yescharacteristic
227C22Q8×D2111yescharacteristic
228C4—41no—
229C4—41no—
230C4—41no—
231C4—422no—
232C4—422no—
233C4—841no—
234C4—841no—
235C3#8943011yescharacteristic, complemented, Sylow
236C2#1062135911yescharacteristic
237C2#1062136211yescharacteristic
238C2#1062136611yescharacteristic
239C2#1062136811yescharacteristic
240C2#1062137211yescharacteristic
241C2#1062137211yescharacteristic
242C2#1062137911yescharacteristic
subgroupG/H
center, FrattiniC23C22×D21
commutator, socle, prime radical, cosocleC23×C3×C7C23
FittingC3×C7×#95C2
hypercenter#95D21
hypocenterC3×C7#400
LCSG ▷ C23×C3×C7 ▷ C3×C7
UCSC1 ◁ C23 ◁ #95

Complements

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

NH ≅ G/Ntype
160. C3×C7#400semidirect
213. C7#4658semidirect
235. C3#89430semidirect

Maximal subgroups

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

subgroupindex
C2×#666292
C3×C7×#952
#106206702
#894303
#46587