Finite Group Databasebeta

← #11729827

Subgroups — #11729827

Order 896.

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

normal subgroups only

This group has 370 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
1C22×C7×Q16C211yesmaximal, characteristic
2C2×#6507C211yesmaximal, characteristic
3C22×#1144C211yesmaximal, characteristic
4#90002C211yesmaximal, characteristic
5C2×#6409C211yesmaximal, characteristic
6C2×#6390C211yesmaximal, characteristic
7#89345C211yesmaximal, characteristic
8C22×C7×Q8C2211yescharacteristic
9C22×C7×Q8C2211yescharacteristic
10C22×Dic14C2211yescharacteristic
11C2×C7×Q16—22no—
12C2×C7×Q16—22no—
13C2×C7×Q16—22no—
14C22×C7×C8C2211yescharacteristic
15#6507—22no—
16#6507—22no—
17C2×#1144—22no—
18C2×#1144—22no—
19C2×#1144—22no—
20C2×C4⋊Dic7C2211yescharacteristic
21C2×Dic7⋊C4—22no—
22C2×C4×Dic7C2211yescharacteristic
23C22×C7⋊C8C2211yescharacteristic
24#6409—22no—
25C7⋊#95—22no—
26#6390—22no—
27#2016—71nomaximal, Sylow
28C2×C7×Q8—21no—
29C2×C7×Q8—21no—
30C2×C7×Q8—21no—
31C2×C7×Q8—21no—
32C2×C7×Q8—21no—
33C2×C7×Q8—21no—
34C2×C7×Q8—21no—
35C2×C7×Q8—21no—
36C2×C7×Q8—21no—
37C2×C7×Q8—21no—
38C2×C7×Q8D412yes—
39C2×C7×Q8D412yes—
40C22×C4×C7—21no—
41C22×C4×C7—21no—
42C22×C4×C7C2311yescharacteristic
43C22×Dic7—21no—
44C22×Dic7—21no—
45C22×Dic7D412yes—
46C2×Dic14—21no—
47C2×Dic14—21no—
48C2×Dic14—21no—
49C2×Dic14—21no—
50C2×Dic14—21no—
51C2×Dic14D412yes—
52C7×Q16—28no—
53C2×C7×C8—21no—
54C2×C7×C8—21no—
55C2×C7×C8D412yes—
56#1144—28no—
57C4⋊Dic7—21no—
58C4⋊Dic7—21no—
59Dic7⋊C4—28no—
60C4×Dic7—21no—
61C4×Dic7—21no—
62C2×C7⋊C8—21no—
63C2×C7⋊C8—21no—
64C2×C7⋊C8D412yes—
65C22×Q16—71no—
66C22×Q16D711yescharacteristic
67C2×C4⋊Q8—71no—
68C2×#103—71no—
69C2×#103—71no—
70#391—71no—
71#336—71no—
72C23×C7C2×D411yescharacteristic
73C7×Q8—22no—
74C7×Q8—22no—
75C7×Q8—22no—
76C7×Q8—22no—
77C7×Q8—41no—
78C7×Q8—41no—
79C7×Q8—41no—
80C7×Q8—41no—
81C2×C4×C7—22no—
82C2×C4×C7—22no—
83C2×C4×C7—41no—
84C2×C4×C7—41no—
85C2×C4×C7—41no—
86C2×C4×C7—41no—
87C2×C4×C7C2×D411yescharacteristic
88C2×C4×C7C2×D411yescharacteristic
89C2×C4×C7C2×D411yescharacteristic
90C2×C4×C7C2×D411yescharacteristic
91C2×C4×C7C2×D411yescharacteristic
92C2×C4×C7#4111yescharacteristic
93C2×Dic7—22no—
94C2×Dic7—22no—
95C2×Dic7—22no—
96C2×Dic7—41no—
97C2×Dic7—41no—
98C2×Dic7—41no—
99C2×Dic7—42no—
100C2×Dic7Q1614yescomplemented
101Dic14—22no—
102Dic14—22no—
103Dic14—41no—
104Dic14—41no—
105C7×C8—22no—
106C7⋊C8—22no—
107C22×Q8—71no—
108C22×Q8C2×D711yescharacteristic
109C22×Q8C2×D711yescharacteristic
110C2×Q16—22no—
111C2×Q16—22no—
112C2×Q16—22no—
113C2×Q16—142no—
114C2×Q16—142no—
115C2×Q16—142no—
116C22×C8—71no—
117C22×C8C2×D711yescharacteristic
118C4⋊Q8—142no—
119C4⋊Q8—142no—
120C2×C4⋊C4—71no—
121C2×C4⋊C4—142no—
122C2×C42—71no—
123#103—142no—
124#103—142no—
125#95—142no—
126C22×C7#12011yescharacteristic
127C22×C7#12111yescharacteristic
128C22×C7#12711yescharacteristic
129C22×C7C2×Q1612yes—
130C22×C7#13712yes—
131C4×C7—42no—
132C4×C7—42no—
133C4×C7#12011yescharacteristic
134C4×C7#12011yescharacteristic
135C4×C7#12111yescharacteristic
136C4×C7#12111yescharacteristic
137Dic7—24no—
138Dic7—42no—
139Dic7—81no—
140C2×Q8—21no—
141C2×Q8—21no—
142C2×Q8—21no—
143C2×Q8—21no—
144C2×Q8—21no—
145C2×Q8—21no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—21no—
150C2×Q8—72no—
151C2×Q8—141no—
152C2×Q8—141no—
153C2×Q8—141no—
154C2×Q8—141no—
155C2×Q8—141no—
156C2×Q8C22⋊D712yes—
157C2×Q8C22⋊D712yes—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—72no—
161C22×C4—141no—
162C22×C4—141no—
163C22×C4C22×D711yescharacteristic
164Q16—28no—
165Q16—148no—
166C2×C8—21no—
167C2×C8—21no—
168C2×C8—72no—
169C2×C8—141no—
170C2×C8—141no—
171C2×C8C22⋊D712yes—
172C4⋊C4—148no—
173C4⋊C4—141no—
174C4⋊C4—141no—
175C42—141no—
176C42—141no—
177C2×C7#39211yescharacteristic
178C2×C7#45112yes—
179C2×C7#46212yes—
180C2×C7#49411yescharacteristic
181C2×C7#49711yescharacteristic
182C23D4×D711yescharacteristic
183Q8—22no—
184Q8—22no—
185Q8—22no—
186Q8—22no—
187Q8—41no—
188Q8—41no—
189Q8—41no—
190Q8—41no—
191Q8—142no—
192Q8—142no—
193Q8—281no—
194Q8—281no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—74no—
202C2×C4—142no—
203C2×C4—142no—
204C2×C4—142no—
205C2×C4—281no—
206C2×C4—281no—
207C2×C4—282no—
208C2×C4—281no—
209C2×C4D4×D711yescharacteristic
210C2×C4D4×D711yescharacteristic
211C2×C4#115911yescharacteristic
212C2×C4C2×C22⋊D711yescharacteristic
213C2×C4C2×C22⋊D711yescharacteristic
214C2×C4C2×C22⋊D711yescharacteristic
215C8—22no—
216C8—142no—
217C7#201611yescharacteristic, complemented, Sylow
218C22D7×Q1612yes—
219C22#648112yes—
220C22#650011yescharacteristic
221C22#650211yescharacteristic
222C22#650311yescharacteristic
223C4—42no—
224C4—42no—
225C4—144no—
226C4—282no—
227C4—561no—
228C4#650011yescharacteristic
229C4#650111yescharacteristic
230C4#650211yescharacteristic
231C4#651611yescharacteristic
232C2#8995311yescharacteristic
233C2#8995412yes—
234C2#8995812yes—
235C2#8996211yescharacteristic
236C2#8999511yescharacteristic
subgroupG/H
centerC23D4×D7
commutator, cosocleC22×C4×C7C23
FrattiniC22×C4C22×D7
FittingC22×C7×Q16C2
socle, prime radicalC23×C7C2×D4
hypercenterC22×Q16D7
hypocenterC7#2016
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×Q16

Complements

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

NH ≅ G/Ntype
100. C2×Dic7Q16semidirect
217. C7#2016semidirect

Maximal subgroups

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

subgroupindex
C22×C7×Q162
C2×#65072
C22×#11442
#900022
C2×#64092
C2×#63902
#893452
#20167