Finite Group Databasebeta

← #11734634

Subgroups — #11734634

Order 896.

500 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 171 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×#6507C211yesmaximal, characteristic
2C2×#6450C211yesmaximal, characteristic
3C7×#395C211yesmaximal, characteristic
4#90001C211yesmaximal, characteristic
5#89751C211yesmaximal, characteristic
6#89748C212yesmaximal
7#89744C212yesmaximal
8#89428C211yesmaximal, characteristic
9#89421C211yesmaximal, characteristic
10#89419C212yesmaximal
11#89418C212yesmaximal
12C22×C7×Q8C2211yescharacteristic
13C22×Dic14C2211yescharacteristic
14C22×Dic14C2211yescharacteristic
15C2×C7×C4⋊C4C2211yescharacteristic
16C2×C7×C4⋊C4C2212yes—
17#6507—24no—
18C2×C4⋊Dic7C2212yes—
19C2×Dic7⋊C4C2212yes—
20C2×Dic7⋊C4C2212yes—
21C2×Dic7⋊C4C2211yescharacteristic
22C2×Dic7⋊C4C2211yescharacteristic
23C2×Dic7⋊C4C2212yes—
24C2×Dic7⋊C4C2212yes—
25C2×Dic7⋊C4C2212yes—
26C2×C4×Dic7C2212yes—
27C2×C4×Dic7C2211yescharacteristic
28C2×C4×Dic7C2211yescharacteristic
29C2×C4×Dic7C2211yescharacteristic
30#6450—24no—
31C7×#95C2212yes—
32C7×#95C2211yescharacteristic
33C7⋊#95C2212yes—
34C7⋊#95C2212yes—
35C7⋊#95C2212yes—
36C7⋊#95C2211yescharacteristic
37#2567—71nomaximal, Sylow
38C2×C7×Q8—22no—
39C2×C7×Q8—42no—
40C22×C4×C7C2312yes—
41C22×C4×C7C2311yescharacteristic
42C22×C4×C7C2311yescharacteristic
43C22×C4×C7C2311yescharacteristic
44C22×C4×C7C2312yes—
45C22×Dic7C2312yes—
46C22×Dic7C2311yescharacteristic
47C22×Dic7C2311yescharacteristic
48C22×Dic7C2312yes—
49C22×Dic7C2311yescharacteristic
50C22×Dic7C2311yescharacteristic
51C2×Dic14—22no—
52C2×Dic14—22no—
53C2×Dic14—22no—
54C2×Dic14—42no—
55C2×Dic14Q814yescomplemented
56C7×C4⋊C4—22no—
57C7×C4⋊C4—42no—
58C4⋊Dic7—42no—
59Dic7⋊C4—22no—
60Dic7⋊C4—22no—
61Dic7⋊C4—24no—
62Dic7⋊C4—24no—
63Dic7⋊C4—42no—
64Dic7⋊C4—42no—
65Dic7⋊C4Q818yescomplemented
66C4×Dic7—22no—
67C4×Dic7—22no—
68C4×Dic7—24no—
69C4×Dic7D414yes—
70C2×C4⋊Q8—71no—
71C2×C4×Q8—71no—
72#395—71no—
73#395—71no—
74#395—71no—
75#395D711yescharacteristic
76#391—72no—
77#389—72no—
78#389—72no—
79#384—72no—
80C4×C4⋊C4—71no—
81C23×C7C2411yescharacteristic
82C7×Q8—44no—
83C2×C4×C7—21no—
84C2×C4×C7—22no—
85C2×C4×C7—21no—
86C2×C4×C7—22no—
87C2×C4×C7—22no—
88C2×C4×C7—21no—
89C2×C4×C7—22no—
90C2×C4×C7—22no—
91C2×C4×C7—22no—
92C2×C4×C7—22no—
93C2×C4×C7C2×D412yes—
94C2×C4×C7C2×Q814yes—
95C2×C4×C7C2×Q812yes—
96C2×Dic7—21no—
97C2×Dic7—22no—
98C2×Dic7—21no—
99C2×Dic7—22no—
100C2×Dic7—21no—
101C2×Dic7—22no—
102C2×Dic7—22no—
103C2×Dic7—22no—
104C2×Dic7—22no—
105C2×Dic7C2×D412yes—
106C2×Dic7C2×D412yes—
107C2×Dic7C2×Q814yes—
108C2×Dic7C2×Q812yes—
109C2×Dic7C2×Q812yes—
110C2×Dic7C2×Q814yes—
111C2×Dic7#4112yes—
112Dic14—28no—
113Dic14—44no—
114C22×Q8—71no—
115C22×Q8—71no—
116C22×Q8C2×D711yescharacteristic
117C4⋊Q8—144no—
118C4×Q8—144no—
119C2×C4⋊C4—72no—
120C2×C4⋊C4—71no—
121C2×C4⋊C4—71no—
122C2×C4⋊C4—72no—
123C2×C4⋊C4—72no—
124C2×C4⋊C4—72no—
125C2×C4⋊C4—72no—
126C2×C4⋊C4—72no—
127C2×C4⋊C4C2×D711yescharacteristic
128C2×C4⋊C4C2×D712yes—
129C2×C42—71no—
130C2×C42—71no—
131C2×C42—71no—
132C2×C42—72no—
133#95—71no—
134#95—72no—
135#95—72no—
136#95—72no—
137#95C2×D711yescharacteristic
138#95C2×D712yes—
139C22×C7C22×D411yescharacteristic
140C22×C7C22×Q811yescharacteristic
141C22×C7C22×Q812yes—
142C22×C7C2×#4111yescharacteristic
143C22×C7H(5,𝔽2)12yes—
144C4×C7—22no—
145C4×C7—22no—
146C4×C7—24no—
147C4×C7—41no—
148C4×C7—42no—
149Dic7—22no—
150Dic7—24no—
151Dic7—42no—
152Dic7—41no—
153Dic7C22⋊Q814yes—
154Dic7C4⋊Q814yes—
155C2×Q8—22no—
156C2×Q8—42no—
157C2×Q8—74no—
158C2×Q8—142no—
159C2×Q8—142no—
160C2×Q8—142no—
161C2×Q8—282no—
162C22×C4—71no—
163C22×C4—71no—
164C22×C4—71no—
165C22×C4—72no—
166C22×C4—72no—
167C22×C4—71no—
168C22×C4C22×D712yes—
169C22×C4C22×D711yescharacteristic
170C22×C4C22×D711yescharacteristic
171C22×C4C22×D711yescharacteristic
172C22×C4C22×D712yes—
173C4⋊C4—22no—
174C4⋊C4—42no—
175C4⋊C4—78no—
176C4⋊C4—142no—
177C4⋊C4—144no—
178C4⋊C4—144no—
179C4⋊C4—142no—
180C4⋊C4—282no—
181C4⋊C4—282no—
182C4⋊C4—282no—
183C42—74no—
184C42—144no—
185C42—142no—
186C42—142no—
187C2×C7C2×C22⋊Q811yescharacteristic
188C2×C7C2×C4⋊Q811yescharacteristic
189C2×C7#53511yescharacteristic
190C2×C7#55412yes—
191C2×C7Q8212yes—
192C23C23×D711yescharacteristic
193Q8—44no—
194Q8—148no—
195Q8—284no—
196C2×C4—21no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—72no—
207C2×C4—72no—
208C2×C4—74no—
209C2×C4—72no—
210C2×C4—72no—
211C2×C4—74no—
212C2×C4—72no—
213C2×C4—141no—
214C2×C4—142no—
215C2×C4—142no—
216C2×C4—141no—
217C2×C4—141no—
218C2×C4—142no—
219C2×C4—142no—
220C2×C4—142no—
221C2×C4—142no—
222C2×C4D4×D712yes—
223C2×C4Q8×D714yes—
224C2×C4Q8×D712yes—
225C7#256711yescharacteristic, complemented, Sylow
226C22C2×D4×D711yescharacteristic
227C22#654812yes—
228C22C2×Q8×D711yescharacteristic
229C22C2×Q8×D712yes—
230C22D7×#4111yescharacteristic
231C4—22no—
232C4—22no—
233C4—24no—
234C4—42no—
235C4—41no—
236C4—74no—
237C4—74no—
238C4—142no—
239C4—144no—
240C4—282no—
241C4—281no—
242C2#9028311yescharacteristic
243C2D7×C22⋊Q811yescharacteristic
244C2#9032712yes—
245C2#9041312yes—
246C2D7×C4⋊Q811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, prime radical, cosocleC23×C7C24
FittingC7×#395C2
hypercenter#395D7
hypocenterC7#2567
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
55. C2×Dic14Q8semidirect
65. Dic7⋊C4Q8semidirect
225. C7#2567semidirect

Maximal subgroups

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

subgroupindex
C2×#65072
C2×#64502
C7×#3952
#900012
#897512
#897482
#897442
#894282
#894212
#894192
#894182
#25677