Finite Group Databasebeta

← #11734616

Subgroups — #11734616

Order 896.

504 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 167 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×Q8×Dic7C211yesmaximal, characteristic
2C2×#6450C211yesmaximal, characteristic
3C7×#395C211yesmaximal, characteristic
4#90001C211yesmaximal, characteristic
5#89756C212yesmaximal
6#89755C211yesmaximal, characteristic
7C4⋊C4×Dic7C212yesmaximal
8#89433C211yesmaximal, characteristic
9#89428C212yesmaximal
10#89421C212yesmaximal
11#89420C211yesmaximal, characteristic
12C22×C7×Q8C2211yescharacteristic
13C22×Dic14C2211yescharacteristic
14C2×C7×C4⋊C4C2211yescharacteristic
15C2×C7×C4⋊C4C2212yes—
16Q8×Dic7—24no—
17C2×C4⋊Dic7C2212yes—
18C2×C4⋊Dic7C2212yes—
19C2×C4⋊Dic7C2211yescharacteristic
20C2×Dic7⋊C4C2211yescharacteristic
21C2×Dic7⋊C4C2211yescharacteristic
22C2×Dic7⋊C4C2212yes—
23C2×Dic7⋊C4C2212yes—
24C2×C4×Dic7C2211yescharacteristic
25C2×C4×Dic7C2211yescharacteristic
26C2×C4×Dic7C2212yes—
27C2×C4×Dic7C2211yescharacteristic
28C2×C4×Dic7C2212yes—
29#6450—24no—
30C7×#95C2211yescharacteristic
31C7×#95C2212yes—
32C7⋊#95C2212yes—
33C7⋊#95C2211yescharacteristic
34C7⋊#95C2212yes—
35C7⋊#95C2212yes—
36C7⋊#95C2212yes—
37#2399—71nomaximal, Sylow
38C2×C7×Q8—22no—
39C2×C7×Q8—24no—
40C22×C4×C7C2311yescharacteristic
41C22×C4×C7C2311yescharacteristic
42C22×C4×C7C2312yes—
43C22×C4×C7C2312yes—
44C22×C4×C7C2311yescharacteristic
45C22×Dic7C2311yescharacteristic
46C22×Dic7C2312yes—
47C22×Dic7C2311yescharacteristic
48C22×Dic7C2312yes—
49C22×Dic7C2311yescharacteristic
50C22×Dic7C2311yescharacteristic
51C2×Dic14—22no—
52C2×Dic14—22no—
53C2×Dic14—22no—
54C7×C4⋊C4—22no—
55C7×C4⋊C4—24no—
56C4⋊Dic7—22no—
57C4⋊Dic7—24no—
58C4⋊Dic7—24no—
59Dic7⋊C4—22no—
60Dic7⋊C4—24no—
61Dic7⋊C4—24no—
62Dic7⋊C4—22no—
63C4×Dic7—22no—
64C4×Dic7—22no—
65C4×Dic7—24no—
66C4×Dic7D414yes—
67C4×Dic7Q818yescomplemented
68C2×C4×Q8—71no—
69C2×C4×Q8—71no—
70#398—71no—
71#398—72no—
72#398—71no—
73#395—71no—
74#395—72no—
75#395D711yescharacteristic
76C4×C4⋊C4—72no—
77C4×C4⋊C4—72no—
78#376—71no—
79C23×C7C2411yescharacteristic
80C7×Q8—28no—
81C2×C4×C7—22no—
82C2×C4×C7—21no—
83C2×C4×C7—22no—
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×C7C2×D412yes—
91C2×C4×C7C2×Q814yes—
92C2×C4×C7#4114yes—
93C2×C4×C7#4112yes—
94C2×Dic7—21no—
95C2×Dic7—22no—
96C2×Dic7—22no—
97C2×Dic7—21no—
98C2×Dic7—21no—
99C2×Dic7—21no—
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×Q814yes—
109C2×Dic7#4112yes—
110Dic14—28no—
111C22×Q8—71no—
112C22×Q8C2×D711yescharacteristic
113C4×Q8—144no—
114C4×Q8—144no—
115C2×C4⋊C4—71no—
116C2×C4⋊C4—71no—
117C2×C4⋊C4—72no—
118C2×C4⋊C4—71no—
119C2×C4⋊C4—72no—
120C2×C4⋊C4—72no—
121C2×C4⋊C4—72no—
122C2×C4⋊C4C2×D711yescharacteristic
123C2×C4⋊C4C2×D712yes—
124C2×C42—71no—
125C2×C42—72no—
126C2×C42—71no—
127C2×C42—72no—
128C2×C42—71no—
129#95—71no—
130#95—72no—
131#95—72no—
132#95—72no—
133#95—72no—
134#95C2×D711yescharacteristic
135#95C2×D712yes—
136C22×C7C22×D411yescharacteristic
137C22×C7C22×Q812yes—
138C22×C7C2×#4112yes—
139C22×C7C2×#4111yescharacteristic
140C22×C7C2×#4111yescharacteristic
141C4×C7—22no—
142C4×C7—22no—
143C4×C7—24no—
144C4×C7—24no—
145C4×C7—41no—
146Dic7—22no—
147Dic7—22no—
148Dic7—24no—
149Dic7—41no—
150Dic7—42no—
151Dic7C4⋊Q814yes—
152C2×Q8—22no—
153C2×Q8—24no—
154C2×Q8—142no—
155C2×Q8—142no—
156C2×Q8—142no—
157C22×C4—71no—
158C22×C4—72no—
159C22×C4—71no—
160C22×C4—71no—
161C22×C4—71no—
162C22×C4—72no—
163C22×C4C22×D711yescharacteristic
164C22×C4C22×D711yescharacteristic
165C22×C4C22×D712yes—
166C22×C4C22×D712yes—
167C22×C4C22×D711yescharacteristic
168C4⋊C4—22no—
169C4⋊C4—24no—
170C4⋊C4—142no—
171C4⋊C4—144no—
172C4⋊C4—144no—
173C4⋊C4—142no—
174C4⋊C4—142no—
175C4⋊C4—144no—
176C4⋊C4—144no—
177C42—74no—
178C42—78no—
179C42—142no—
180C42—144no—
181C42—142no—
182C2×C7#52511yescharacteristic
183C2×C7#52511yescharacteristic
184C2×C7C2×C4⋊Q811yescharacteristic
185C2×C7#53312yes—
186C2×C7#53312yes—
187C23C23×D711yescharacteristic
188Q8—28no—
189Q8—148no—
190C2×C4—21no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—22no—
199C2×C4—72no—
200C2×C4—74no—
201C2×C4—72no—
202C2×C4—74no—
203C2×C4—72no—
204C2×C4—141no—
205C2×C4—142no—
206C2×C4—142no—
207C2×C4—141no—
208C2×C4—141no—
209C2×C4—141no—
210C2×C4—141no—
211C2×C4—142no—
212C2×C4—142no—
213C2×C4—142no—
214C2×C4—142no—
215C2×C4D4×D712yes—
216C2×C4#115914yes—
217C2×C4Q8×D714yes—
218C2×C4#116112yes—
219C7#239911yescharacteristic, complemented, Sylow
220C22C2×D4×D711yescharacteristic
221C22C2×#115912yes—
222C22C2×Q8×D712yes—
223C22C2×#116111yescharacteristic
224C22D7×#4111yescharacteristic
225C4—22no—
226C4—24no—
227C4—24no—
228C4—22no—
229C4—41no—
230C4—74no—
231C4—144no—
232C4—142no—
233C4—142no—
234C4—281no—
235C4—282no—
236C2#9027711yescharacteristic
237C2#9031312yes—
238C2#9031811yescharacteristic
239C2D7×C4⋊Q811yescharacteristic
240C2#9041912yes—
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, prime radical, cosocleC23×C7C24
FittingC7×#395C2
hypercenter#395D7
hypocenterC7#2399
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
67. C4×Dic7Q8semidirect
219. C7#2399semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×Dic72
C2×#64502
C7×#3952
#900012
#897562
#897552
C4⋊C4×Dic72
#894332
#894282
#894212
#894202
#23997