Finite Group Databasebeta

← #11735242

Subgroups — #11735242

Order 896.

468 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 157 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×#6452C211yesmaximal, characteristic
2C2×#6450C211yesmaximal, characteristic
3C7×#398C211yesmaximal, characteristic
4#89752C211yesmaximal, characteristic
5#89751C211yesmaximal, characteristic
6#89745C212yesmaximal
7#89744C212yesmaximal
8#89428C211yesmaximal, characteristic
9#89421C211yesmaximal, characteristic
10#89419C212yesmaximal
11#89419C212yesmaximal
12C22×Dic14C2211yescharacteristic
13C2×C7×C4⋊C4C2211yescharacteristic
14C2×C7×C4⋊C4C2211yescharacteristic
15C2×C7×C4⋊C4C2212yes—
16C2×C4⋊Dic7C2211yescharacteristic
17C2×C4⋊Dic7C2212yes—
18C2×Dic7⋊C4C2211yescharacteristic
19C2×Dic7⋊C4C2212yes—
20C2×Dic7⋊C4C2212yes—
21C2×Dic7⋊C4C2211yescharacteristic
22C2×Dic7⋊C4C2211yescharacteristic
23C2×Dic7⋊C4C2211yescharacteristic
24C2×Dic7⋊C4C2212yes—
25C2×Dic7⋊C4C2212yes—
26C2×C4×Dic7C2212yes—
27C2×C4×Dic7C2212yes—
28C2×C4×Dic7C2211yescharacteristic
29#6452—24no—
30#6450—24no—
31C7×#95C2212yes—
32C7×#95C2211yescharacteristic
33C7⋊#95C2212yes—
34C7⋊#95C2212yes—
35C7⋊#95C2211yescharacteristic
36C7⋊#95C2212yes—
37#2584—71nomaximal, Sylow
38C22×C4×C7C2311yescharacteristic
39C22×C4×C7C2312yes—
40C22×C4×C7C2311yescharacteristic
41C22×C4×C7C2311yescharacteristic
42C22×C4×C7C2312yes—
43C22×Dic7C2311yescharacteristic
44C22×Dic7C2311yescharacteristic
45C22×Dic7C2311yescharacteristic
46C22×Dic7C2312yes—
47C22×Dic7C2312yes—
48C22×Dic7C2311yescharacteristic
49C2×Dic14—22no—
50C2×Dic14—24no—
51C7×C4⋊C4—22no—
52C7×C4⋊C4—22no—
53C7×C4⋊C4—42no—
54C4⋊Dic7—22no—
55C4⋊Dic7—24no—
56Dic7⋊C4—22no—
57Dic7⋊C4—24no—
58Dic7⋊C4—22no—
59Dic7⋊C4—22no—
60Dic7⋊C4—22no—
61Dic7⋊C4—24no—
62Dic7⋊C4—42no—
63Dic7⋊C4Q818yescomplemented
64C4×Dic7—24no—
65C4×Dic7—42no—
66C4×Dic7D414yes—
67C2×#125—71no—
68C2×C4×Q8—71no—
69#398—71no—
70#398D711yescharacteristic
71#395—71no—
72#395—71no—
73#389—72no—
74#389—72no—
75#389—72no—
76#389—72no—
77C4×C4⋊C4—71no—
78C23×C7C2411yescharacteristic
79C2×C4×C7—22no—
80C2×C4×C7—22no—
81C2×C4×C7—22no—
82C2×C4×C7—22no—
83C2×C4×C7—21no—
84C2×C4×C7—21no—
85C2×C4×C7—22no—
86C2×C4×C7—22no—
87C2×C4×C7—21no—
88C2×C4×C7—21no—
89C2×C4×C7—22no—
90C2×C4×C7C2×D412yes—
91C2×C4×C7C2×Q814yes—
92C2×Dic7—22no—
93C2×Dic7—22no—
94C2×Dic7—22no—
95C2×Dic7—21no—
96C2×Dic7—22no—
97C2×Dic7—21no—
98C2×Dic7—21no—
99C2×Dic7—21no—
100C2×Dic7—22no—
101C2×Dic7—21no—
102C2×Dic7—21no—
103C2×Dic7C2×D412yes—
104C2×Dic7C2×D412yes—
105C2×Dic7C2×Q814yes—
106C2×Dic7C2×Q814yes—
107C2×Dic7#4114yes—
108Dic14—28no—
109C22×Q8—71no—
110#125—144no—
111C4×Q8—144no—
112C2×C4⋊C4—72no—
113C2×C4⋊C4—72no—
114C2×C4⋊C4—71no—
115C2×C4⋊C4—71no—
116C2×C4⋊C4—71no—
117C2×C4⋊C4—71no—
118C2×C4⋊C4—71no—
119C2×C4⋊C4—72no—
120C2×C4⋊C4—72no—
121C2×C4⋊C4—72no—
122C2×C4⋊C4C2×D711yescharacteristic
123C2×C4⋊C4C2×D711yescharacteristic
124C2×C4⋊C4C2×D712yes—
125C2×C42—71no—
126C2×C42—72no—
127C2×C42—72no—
128#95—71no—
129#95—72no—
130#95—72no—
131#95—72no—
132#95C2×D711yescharacteristic
133#95C2×D712yes—
134C22×C7C22×D411yescharacteristic
135C22×C7C22×Q812yes—
136C22×C7C2×#4112yes—
137C22×C7H(5,𝔽2)11yescharacteristic
138C22×C72−1+411yescharacteristic
139C4×C7—22no—
140C4×C7—24no—
141C4×C7—41no—
142C4×C7—42no—
143C4×C7—41no—
144Dic7—24no—
145Dic7—22no—
146Dic7—24no—
147Dic7—41no—
148Dic7—41no—
149Dic7C4⋊Q814yes—
150C2×Q8—144no—
151C2×Q8—142no—
152C22×C4—71no—
153C22×C4—72no—
154C22×C4—72no—
155C22×C4—71no—
156C22×C4—71no—
157C22×C4—71no—
158C22×C4C22×D711yescharacteristic
159C22×C4C22×D712yes—
160C22×C4C22×D711yescharacteristic
161C22×C4C22×D711yescharacteristic
162C22×C4C22×D712yes—
163C4⋊C4—22no—
164C4⋊C4—22no—
165C4⋊C4—42no—
166C4⋊C4—78no—
167C4⋊C4—144no—
168C4⋊C4—144no—
169C4⋊C4—142no—
170C4⋊C4—142no—
171C4⋊C4—142no—
172C4⋊C4—144no—
173C4⋊C4—142no—
174C4⋊C4—142no—
175C4⋊C4—282no—
176C42—74no—
177C42—144no—
178C42—282no—
179C2×C7#52511yescharacteristic
180C2×C7C2×C4⋊Q811yescharacteristic
181C2×C7#53711yescharacteristic
182C2×C7#55412yes—
183C2×C7#55712yes—
184C23C23×D711yescharacteristic
185Q8—148no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—21no—
196C2×C4—22no—
197C2×C4—72no—
198C2×C4—74no—
199C2×C4—72no—
200C2×C4—74no—
201C2×C4—74no—
202C2×C4—142no—
203C2×C4—141no—
204C2×C4—142no—
205C2×C4—142no—
206C2×C4—142no—
207C2×C4—141no—
208C2×C4—141no—
209C2×C4—141no—
210C2×C4—141no—
211C2×C4—142no—
212C2×C4—141no—
213C2×C4D4×D712yes—
214C2×C4Q8×D714yes—
215C7#258411yescharacteristic, complemented, Sylow
216C22C2×D4×D711yescharacteristic
217C22#654811yescharacteristic
218C22C2×Q8×D712yes—
219C22D7×#4112yes—
220C22#655411yescharacteristic
221C4—24no—
222C4—22no—
223C4—42no—
224C4—41no—
225C4—41no—
226C4—74no—
227C4—142no—
228C4—144no—
229C4—144no—
230C4—281no—
231C4—281no—
232C2#9029911yescharacteristic
233C2#9032712yes—
234C2#9034711yescharacteristic
235C2#9037612yes—
236C2D7×C4⋊Q811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, prime radical, cosocleC23×C7C24
FittingC7×#398C2
hypercenter#398D7
hypocenterC7#2584
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
63. Dic7⋊C4Q8semidirect
215. C7#2584semidirect

Maximal subgroups

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

subgroupindex
C2×#64522
C2×#64502
C7×#3982
#897522
#897512
#897452
#897442
#894282
#894212
#894192
#25847