Finite Group Databasebeta

← #11731508

Subgroups — #11731508

Order 896.

864 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 864 conjugacy classes of subgroups — too many to draw. Only its 169 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
1C22×#1157C211yesmaximal, characteristic
2C22×Dic7⋊C4C211yesmaximal, characteristic, complemented
3C2×#6449C212yesmaximal, complemented
4C7×#381C211yesmaximal, characteristic, complemented
5#89730C212yesmaximal, complemented
6#89444C212yesmaximal, complemented
7#89440C214yesmaximal, complemented
8#89428C212yesmaximal, complemented
9C23×C4×C7C2211yescharacteristic
10C22×C22⋊D7C2212yes—
11C23×Dic7C2211yescharacteristic
12C23×Dic7C2211yescharacteristic
13C2×#1157—24no—
14C2×#1157—28no—
15C22×D28C2211yescharacteristic
16C22×C4×D7C2212yes—
17C22×Dic14C2211yescharacteristic
18C2×C7×C22⋊C4C2212yes—
19C2×(C2×D7)⋊C4C2214yes—
20C2×(C2×D7)⋊C4C2214yes—
21C2×C4⋊Dic7C2214yes—
22C2×Dic7⋊C4—24no—
23C2×Dic7⋊C4C2212yes—
24C2×Dic7⋊C4C2212yes—
25#6449—28no—
26C7×#95C2212yes—
27C7×#95C2212yes—
28C7⋊#95C2214yes—
29#2406—71nomaximal, Sylow
30C24×C7C2311yescharacteristic
31C23×D7C2312yes—
32C22×C4×C7—24no—
33C22×C4×C7D414yes—
34C22×C4×C7C2314yes—
35C22×C4×C7C2311yescharacteristic
36C22×C4×C7C2311yescharacteristic
37C2×C22⋊D7—24no—
38C2×C22⋊D7—24no—
39C2×C22⋊D7D418yes—
40C22×Dic7—22no—
41C22×Dic7—24no—
42C22×Dic7—24no—
43C22×Dic7—22no—
44C22×Dic7C2312yes—
45C22×Dic7C2312yes—
46C22×Dic7C2312yes—
47#1157—232no—
48C2×D28—24no—
49C2×D28—22no—
50C2×C4×D7—24no—
51C2×C4×D7—24no—
52C2×C4×D7—24no—
53C2×Dic14—22no—
54C2×Dic14—24no—
55C7×C22⋊C4—24no—
56(C2×D7)⋊C4—28no—
57(C2×D7)⋊C4—28no—
58C4⋊Dic7—28no—
59Dic7⋊C4—24no—
60Dic7⋊C4—24no—
61C22×#41—71no—
62C2×#123—72no—
63C22×C4⋊C4—71no—
64#395—72no—
65#394—72no—
66#386—72no—
67#385—74no—
68#381D711yescharacteristic, complemented
69C23×C7—22no—
70C23×C7—22no—
71C23×C7C2×D412yes—
72C23×C7C2×D414yes—
73C23×C7C2411yescharacteristic
74C22×D7—22no—
75C22×D7—22no—
76C22×D7—24no—
77C22×D7—24no—
78C22×D7C2×D414yes—
79C2×C4×C7—28no—
80C2×C4×C7—22no—
81C2×C4×C7—22no—
82C2×C4×C7—24no—
83C2×C4×C7—24no—
84C2×C4×C7—24no—
85C2×C4×C7C2×D412yes—
86C2×C4×C7C2×D412yes—
87C22⋊D7—216no—
88C2×Dic7—28no—
89C2×Dic7—22no—
90C2×Dic7—22no—
91C2×Dic7—22no—
92C2×Dic7—22no—
93C2×Dic7—22no—
94C2×Dic7—24no—
95C2×Dic7—44no—
96C2×Dic7C2×D414yes—
97C2×Dic7#4114yescomplemented
98D28—28no—
99C4×D7—216no—
100Dic14—28no—
101C2×#41—148no—
102C2×#41—144no—
103C22×Q8—71no—
104C22×D4—72no—
105C22×D4—71no—
106C23×C4—71no—
107C23×C4—71no—
108C23×C4—72no—
109C23×C4C2×D711yescharacteristic
110#123—148no—
111C2×C4⋊C4—72no—
112C2×C4⋊C4—72no—
113C2×C4⋊C4—74no—
114C2×C4⋊C4—144no—
115C2×C22⋊C4—74no—
116C2×C22⋊C4—74no—
117C2×C22⋊C4C2×D712yes—
118#95—74no—
119#95C2×D712yes—
120#95C2×D712yes—
121C22×C7—22no—
122C22×C7—22no—
123C22×C7—28no—
124C22×C7#12014yes—
125C22×C7C22×D411yescharacteristic
126C22×C7C22×D412yes—
127C22×C7C2×#4112yes—
128C22×C72−1+412yes—
129C2×D7—24no—
130C2×D7—28no—
131C2×D7—42no—
132C2×D7—42no—
133C2×D7—44no—
134C4×C7—22no—
135C4×C7—22no—
136C4×C7—44no—
137Dic7—24no—
138Dic7—24no—
139Dic7—42no—
140C24—72no—
141C24C22×D711yescharacteristic
142#41—1432no—
143C2×Q8—144no—
144C2×Q8—142no—
145C2×D4—78no—
146C2×D4—144no—
147C2×D4—144no—
148C2×D4—144no—
149C2×D4—142no—
150C22×C4—24no—
151C22×C4—72no—
152C22×C4—72no—
153C22×C4—72no—
154C22×C4—144no—
155C22×C4—144no—
156C22×C4—144no—
157C22×C4—142no—
158C22×C4—144no—
159C22×C4—142no—
160C22×C4—144no—
161C22×C4D2814yes—
162C22×C4C22×D714yes—
163C22×C4C22×D711yescharacteristic
164C22×C4C22×D711yescharacteristic
165C4⋊C4—148no—
166C4⋊C4—144no—
167C4⋊C4—144no—
168C22⋊C4—24no—
169C22⋊C4—148no—
170C22⋊C4—148no—
171C2×C7—24no—
172C2×C7C2×#12011yescharacteristic
173C2×C7#52511yescharacteristic
174C2×C7#53611yescharacteristic
175C2×C7#54714yes—
176D7—44no—
177C23—22no—
178C23—22no—
179C23—74no—
180C23—142no—
181C23—142no—
182C23—144no—
183C23—144no—
184C23C2×D2812yes—
185C23D4×D714yes—
186C23C23×D711yescharacteristic
187Q8—148no—
188D4—148no—
189D4—1416no—
190C2×C4—28no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—24no—
194C2×C4—24no—
195C2×C4—24no—
196C2×C4—74no—
197C2×C4—74no—
198C2×C4—1416no—
199C2×C4—148no—
200C2×C4—142no—
201C2×C4—142no—
202C2×C4—142no—
203C2×C4—142no—
204C2×C4—142no—
205C2×C4—144no—
206C2×C4—284no—
207C2×C4C2×D2812yes—
208C2×C4C2×D2812yes—
209C7#240611yescharacteristic, complemented, Sylow
210C22—22no—
211C22—22no—
212C22—28no—
213C22—144no—
214C22—148no—
215C22—282no—
216C22—282no—
217C22—284no—
218C22#644514yes—
219C22C22×D2811yescharacteristic
220C22C2×D4×D712yes—
221C22D7×#4112yes—
222C22#655412yes—
223C4—22no—
224C4—22no—
225C4—44no—
226C4—144no—
227C4—144no—
228C4—282no—
229C2—24no—
230C2—284no—
231C2C2×#644511yescharacteristic
232C2#9021511yescharacteristic
233C2#9021711yescharacteristic
234C2#9034614yes—
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#381C2
prime radicalC22×#1157C2
hypercenter#381D7
hypocenterC7#2406
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #381

Complements

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

NH ≅ G/Ntype
2. C22×Dic7⋊C4C2semidirect
3. C2×#6449C2semidirect
4. C7×#381C2semidirect
5. #89730C2semidirect
6. #89444C2semidirect
7. #89440C2semidirect
8. #89428C2semidirect
68. #381D7semidirect
97. C2×Dic7#41semidirect
209. C7#2406semidirect

Maximal subgroups

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

subgroupindex
C22×#11572
C22×Dic7⋊C42
C2×#64492
C7×#3812
#897302
#894442
#894402
#894282
#24067