Finite Group Databasebeta

← #11735626

Subgroups — #11735626

Order 896.

468 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 151 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×#6451C211yesmaximal, characteristic
2C2×#6450C211yesmaximal, characteristic
3C7×#400C211yesmaximal, characteristic
4#89758C211yesmaximal, characteristic
5#89754C211yesmaximal, characteristic
6#89431C211yesmaximal, characteristic
7#89422C212yesmaximal
8#89421C211yesmaximal, characteristic
9#89418C212yesmaximal
10#89418C212yesmaximal
11#89418C212yesmaximal
12C22×Dic14C2212yes—
13C22×Dic14C2211yescharacteristic
14C2×C7×C4⋊C4C2211yescharacteristic
15C2×C7×C4⋊C4C2211yescharacteristic
16C2×C4⋊Dic7C2211yescharacteristic
17C2×Dic7⋊C4C2212yes—
18C2×Dic7⋊C4C2212yes—
19C2×Dic7⋊C4C2211yescharacteristic
20C2×Dic7⋊C4C2211yescharacteristic
21C2×C4×Dic7C2211yescharacteristic
22C2×C4×Dic7C2211yescharacteristic
23C2×C4×Dic7C2212yes—
24C2×C4×Dic7C2211yescharacteristic
25#6451—24no—
26#6450—24no—
27C7×#95C2212yes—
28C7×#95C2212yes—
29C7×#95C2211yescharacteristic
30C7⋊#95C2212yes—
31C7⋊#95C2212yes—
32C7⋊#95C2212yes—
33C7⋊#95C2212yes—
34C7⋊#95C2212yes—
35C7⋊#95C2212yes—
36C7⋊#95C2211yescharacteristic
37#2568—71nomaximal, Sylow
38C22×C4×C7C2312yes—
39C22×C4×C7C2311yescharacteristic
40C22×C4×C7C2312yes—
41C22×C4×C7C2311yescharacteristic
42C22×C4×C7C2311yescharacteristic
43C22×Dic7C2312yes—
44C22×Dic7C2311yescharacteristic
45C22×Dic7C2312yes—
46C22×Dic7C2311yescharacteristic
47C22×Dic7C2311yescharacteristic
48C22×Dic7C2311yescharacteristic
49C2×Dic14—22no—
50C2×Dic14—22no—
51C2×Dic14—24no—
52C2×Dic14—28no—
53C2×Dic14Q814yescomplemented
54C7×C4⋊C4—22no—
55C7×C4⋊C4—22no—
56C4⋊Dic7—22no—
57Dic7⋊C4—22no—
58Dic7⋊C4—22no—
59Dic7⋊C4—24no—
60Dic7⋊C4—24no—
61C4×Dic7—22no—
62C4×Dic7—22no—
63C4×Dic7—24no—
64C4×Dic7D414yes—
65C2×C4⋊Q8—71no—
66C2×C4×Q8—71no—
67#400—71no—
68#400—71no—
69#400—71no—
70#400D711yescharacteristic
71#391—72no—
72#391—72no—
73#391—72no—
74#382—72no—
75C4×C4⋊C4—71no—
76C23×C7C2411yescharacteristic
77C2×C4×C7—21no—
78C2×C4×C7—21no—
79C2×C4×C7—22no—
80C2×C4×C7—21no—
81C2×C4×C7—22no—
82C2×C4×C7—22no—
83C2×C4×C7—22no—
84C2×C4×C7—24no—
85C2×C4×C7—22no—
86C2×C4×C7—22no—
87C2×C4×C7C2×D412yes—
88C2×C4×C7C2×Q812yes—
89C2×Dic7—21no—
90C2×Dic7—21no—
91C2×Dic7—21no—
92C2×Dic7—22no—
93C2×Dic7—22no—
94C2×Dic7—22no—
95C2×Dic7—24no—
96C2×Dic7—22no—
97C2×Dic7—22no—
98C2×Dic7C2×D412yes—
99C2×Dic7C2×D412yes—
100C2×Dic7C2×Q812yes—
101C2×Dic7C2×Q812yes—
102C2×Dic7#4112yes—
103C2×Dic7#4114yes—
104Dic14—28no—
105Dic14—48no—
106C22×Q8—71no—
107C22×Q8—72no—
108C4⋊Q8—144no—
109C4×Q8—144no—
110C2×C4⋊C4—71no—
111C2×C4⋊C4—71no—
112C2×C4⋊C4—72no—
113C2×C4⋊C4—71no—
114C2×C4⋊C4—72no—
115C2×C4⋊C4C2×D711yescharacteristic
116C2×C4⋊C4C2×D711yescharacteristic
117C2×C42—71no—
118C2×C42—71no—
119C2×C42—71no—
120C2×C42—72no—
121#95—71no—
122#95—72no—
123#95—72no—
124#95—72no—
125#95—72no—
126#95—72no—
127#95—72no—
128#95C2×D711yescharacteristic
129#95C2×D712yes—
130#95C2×D712yes—
131C22×C7C22×D411yescharacteristic
132C22×C7C22×Q811yescharacteristic
133C22×C7C2×#4111yescharacteristic
134C22×C7C2×#4112yes—
135C22×C72−1+412yes—
136C4×C7—22no—
137C4×C7—22no—
138C4×C7—42no—
139C4×C7—41no—
140C4×C7—42no—
141Dic7—22no—
142Dic7—42no—
143Dic7—41no—
144Dic7—42no—
145Dic7C22⋊Q814yes—
146Dic7#12414yes—
147C2×Q8—74no—
148C2×Q8—142no—
149C2×Q8—142no—
150C2×Q8—144no—
151C2×Q8—148no—
152C22×C4—71no—
153C22×C4—71no—
154C22×C4—71no—
155C22×C4—72no—
156C22×C4—71no—
157C22×C4—72no—
158C22×C4C22×D712yes—
159C22×C4C22×D711yescharacteristic
160C22×C4C22×D712yes—
161C22×C4C22×D711yescharacteristic
162C22×C4C22×D711yescharacteristic
163C4⋊C4—22no—
164C4⋊C4—22no—
165C4⋊C4—144no—
166C4⋊C4—142no—
167C4⋊C4—144no—
168C4⋊C4—142no—
169C4⋊C4—142no—
170C42—74no—
171C42—144no—
172C42—142no—
173C42—142no—
174C2×C7C2×C22⋊Q811yescharacteristic
175C2×C7C2×#12411yescharacteristic
176C2×C7#53611yescharacteristic
177C2×C7#55612yes—
178C2×C7#55712yes—
179C23C23×D711yescharacteristic
180Q8—148no—
181Q8—288no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—22no—
185C2×C4—21no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—24no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—72no—
193C2×C4—72no—
194C2×C4—72no—
195C2×C4—72no—
196C2×C4—74no—
197C2×C4—72no—
198C2×C4—141no—
199C2×C4—141no—
200C2×C4—141no—
201C2×C4—142no—
202C2×C4—142no—
203C2×C4—142no—
204C2×C4—144no—
205C2×C4—142no—
206C2×C4—142no—
207C2×C4D4×D712yes—
208C2×C4Q8×D712yes—
209C7#256811yescharacteristic, complemented, Sylow
210C22C2×D4×D711yescharacteristic
211C22C2×Q8×D711yescharacteristic
212C22D7×#4112yes—
213C22D7×#4111yescharacteristic
214C22#655412yes—
215C4—22no—
216C4—22no—
217C4—42no—
218C4—41no—
219C4—42no—
220C4—74no—
221C4—74no—
222C4—142no—
223C4—282no—
224C4—281no—
225C4—282no—
226C2D7×C22⋊Q811yescharacteristic
227C2#9033911yescharacteristic
228C2D7×#12411yescharacteristic
229C2#9037612yes—
230C2#9039312yes—
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, prime radical, cosocleC23×C7C24
FittingC7×#400C2
hypercenter#400D7
hypocenterC7#2568
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
53. C2×Dic14Q8semidirect
209. C7#2568semidirect

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×#4002
#897582
#897542
#894312
#894222
#894212
#894182
#25687