Finite Group Databasebeta

← #11735230

Subgroups — #11735230

Order 896.

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

normal subgroups only

This group has 540 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×#6451C212yesmaximal
2C2×#6451C211yesmaximal, characteristic
3C2×#6450C211yesmaximal, characteristic
4C7×#398C211yesmaximal, characteristic
5#89755C211yesmaximal, characteristic
6#89752C211yesmaximal, characteristic
7#89744C212yesmaximal
8#89433C211yesmaximal, characteristic
9#89421C211yesmaximal, characteristic
10#89419C212yesmaximal
11#89418C212yesmaximal
12C22×Dic14C2212yes—
13C22×Dic14C2211yescharacteristic
14C2×C7×C4⋊C4C2211yescharacteristic
15C2×C7×C4⋊C4C2212yes—
16C2×C7×C4⋊C4C2211yescharacteristic
17C2×C4⋊Dic7C2212yes—
18C2×C4⋊Dic7C2211yescharacteristic
19C2×Dic7⋊C4C2211yescharacteristic
20C2×Dic7⋊C4C2212yes—
21C2×Dic7⋊C4C2212yes—
22C2×Dic7⋊C4C2211yescharacteristic
23C2×Dic7⋊C4C2212yes—
24C2×Dic7⋊C4C2212yes—
25C2×C4×Dic7C2212yes—
26C2×C4×Dic7C2211yescharacteristic
27C2×C4×Dic7C2211yescharacteristic
28C2×C4×Dic7C2211yescharacteristic
29#6451—28no—
30#6451—24no—
31#6450—24no—
32C7×#95C2212yes—
33C7×#95C2211yescharacteristic
34C7⋊#95C2212yes—
35C7⋊#95C2212yes—
36C7⋊#95C2212yes—
37C7⋊#95C2211yescharacteristic
38#2546—71nomaximal, Sylow
39C22×C4×C7C2311yescharacteristic
40C22×C4×C7C2312yes—
41C22×C4×C7C2311yescharacteristic
42C22×C4×C7C2311yescharacteristic
43C22×C4×C7C2312yes—
44C22×Dic7C2311yescharacteristic
45C22×Dic7C2312yes—
46C22×Dic7C2311yescharacteristic
47C22×Dic7C2312yes—
48C22×Dic7C2311yescharacteristic
49C22×Dic7C2311yescharacteristic
50C2×Dic14—22no—
51C2×Dic14—22no—
52C2×Dic14—24no—
53C2×Dic14—24no—
54C2×Dic14—24no—
55C2×Dic14D414yes—
56C7×C4⋊C4—22no—
57C7×C4⋊C4—24no—
58C7×C4⋊C4—22no—
59C4⋊Dic7—22no—
60C4⋊Dic7—24no—
61Dic7⋊C4—22no—
62Dic7⋊C4—24no—
63Dic7⋊C4—22no—
64Dic7⋊C4—24no—
65Dic7⋊C4—24no—
66Dic7⋊C4Q818yescomplemented
67C4×Dic7—22no—
68C4×Dic7—22no—
69C4×Dic7—24no—
70C4×Dic7D414yes—
71C2×C4⋊Q8—71no—
72C2×C4⋊Q8—72no—
73C2×C4×Q8—71no—
74#398—71no—
75#398—71no—
76#398—71no—
77#398D711yescharacteristic
78#391—72no—
79#389—72no—
80#389—72no—
81C4×C4⋊C4—71no—
82C23×C7C2411yescharacteristic
83C2×C4×C7—22no—
84C2×C4×C7—21no—
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×D412yes—
95C2×C4×C7C2×Q814yes—
96C2×Dic7—22no—
97C2×Dic7—21no—
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×D412yes—
108C2×Dic7C2×D412yes—
109C2×Dic7C2×Q814yes—
110C2×Dic7C2×Q814yes—
111C2×Dic7#4112yes—
112Dic14—28no—
113Dic14—48no—
114C22×Q8—71no—
115C22×Q8—72no—
116C4⋊Q8—144no—
117C4⋊Q8—148no—
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—71no—
127C2×C4⋊C4C2×D711yescharacteristic
128C2×C4⋊C4C2×D712yes—
129C2×C4⋊C4C2×D711yescharacteristic
130C2×C42—71no—
131C2×C42—71no—
132C2×C42—71no—
133C2×C42—72no—
134#95—71no—
135#95—72no—
136#95—72no—
137#95—72no—
138#95C2×D711yescharacteristic
139#95C2×D712yes—
140C22×C7C22×D411yescharacteristic
141C22×C7C22×D411yescharacteristic
142C22×C7C22×Q812yes—
143C22×C7C2×#4111yescharacteristic
144C22×C72−1+412yes—
145C4×C7—22no—
146C4×C7—22no—
147C4×C7—24no—
148C4×C7—42no—
149C4×C7—41no—
150Dic7—22no—
151Dic7—24no—
152Dic7—42no—
153Dic7—41no—
154Dic7#12114yes—
155Dic7C4⋊Q814yescomplemented
156C2×Q8—74no—
157C2×Q8—142no—
158C2×Q8—142no—
159C2×Q8—144no—
160C2×Q8—144no—
161C2×Q8—144no—
162C22×C4—71no—
163C22×C4—71no—
164C22×C4—72no—
165C22×C4—71no—
166C22×C4—71no—
167C22×C4—72no—
168C22×C4C22×D711yescharacteristic
169C22×C4C22×D712yes—
170C22×C4C22×D711yescharacteristic
171C22×C4C22×D711yescharacteristic
172C22×C4C22×D712yes—
173C4⋊C4—22no—
174C4⋊C4—24no—
175C4⋊C4—22no—
176C4⋊C4—78no—
177C4⋊C4—142no—
178C4⋊C4—144no—
179C4⋊C4—144no—
180C4⋊C4—144no—
181C4⋊C4—142no—
182C4⋊C4—144no—
183C4⋊C4—142no—
184C42—74no—
185C42—142no—
186C42—142no—
187C42—144no—
188C2×C7C2×#12111yescharacteristic
189C2×C7C2×C4⋊Q811yescharacteristic
190C2×C7#53611yescharacteristic
191C2×C7D4×Q812yes—
192C2×C7#55712yes—
193C23C23×D711yescharacteristic
194Q8—148no—
195Q8—288no—
196C2×C4—22no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—72no—
207C2×C4—72no—
208C2×C4—72no—
209C2×C4—74no—
210C2×C4—72no—
211C2×C4—72no—
212C2×C4—74no—
213C2×C4—142no—
214C2×C4—141no—
215C2×C4—141no—
216C2×C4—142no—
217C2×C4—141no—
218C2×C4—142no—
219C2×C4—142no—
220C2×C4—142no—
221C2×C4—142no—
222C2×C4D4×D712yes—
223C2×C4D4×D712yes—
224C2×C4Q8×D714yes—
225C7#254611yescharacteristic, complemented, Sylow
226C22C2×D4×D711yescharacteristic
227C22C2×D4×D711yescharacteristic
228C22C2×Q8×D712yes—
229C22D7×#4111yescharacteristic
230C22#655412yes—
231C4—22no—
232C4—24no—
233C4—22no—
234C4—42no—
235C4—41no—
236C4—74no—
237C4—74no—
238C4—142no—
239C4—144no—
240C4—282no—
241C4—281no—
242C2D7×#12111yescharacteristic
243C2#9032312yes—
244C2#9033911yescharacteristic
245C2#9037612yes—
246C2D7×C4⋊Q811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, prime radical, cosocleC23×C7C24
FittingC7×#398C2
hypercenter#398D7
hypocenterC7#2546
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
66. Dic7⋊C4Q8semidirect
155. Dic7C4⋊Q8semidirect
225. C7#2546semidirect

Maximal subgroups

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

subgroupindex
C2×#64512
C2×#64502
C7×#3982
#897552
#897522
#897442
#894332
#894212
#894192
#894182
#25467