Finite Group Databasebeta

← #11740119

Subgroups — #11740119

Order 896.

596 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 153 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×C7×#121C211yesmaximal, characteristic
2C22×Dic7⋊C4C211yesmaximal, characteristic, complemented
3#89994C212yesmaximal, complemented
4#89992C212yesmaximal, complemented
5#89992C212yesmaximal, complemented
6C4×C22⋊Dic7C211yesmaximal, characteristic, complemented
7#89754C212yesmaximal, complemented
8#89717C214yesmaximal, complemented
9C22×C7×D4C2212yes—
10C22×C7×D4C2211yescharacteristic
11C23×C4×C7C2211yescharacteristic
12C23×Dic7C2211yescharacteristic
13C23×Dic7C2211yescharacteristic
14C7×#121—24no—
15C2×C7×C4⋊C4C2211yescharacteristic
16C2×C7×C22⋊C4C2212yes—
17C2×C22⋊Dic7C2214yes—
18C2×C22⋊Dic7C2212yes—
19C2×C22⋊Dic7C2214yes—
20C2×Dic7⋊C4—24no—
21C2×Dic7⋊C4C2212yes—
22C2×Dic7⋊C4C2212yes—
23C2×C4×Dic7C2212yes—
24C7⋊#95C2214yes—
25C7⋊#95C2214yes—
26C7⋊#95C2212yes—
27#2591—71nomaximal, Sylow
28C24×C7C2312yes—
29C24×C7C2311yescharacteristic
30C2×C7×D4—24no—
31C2×C7×D4—28no—
32C2×C7×D4—22no—
33C2×C7×D4—42no—
34C22×C4×C7—42no—
35C22×C4×C7D414yes—
36C22×C4×C7C2312yes—
37C22×C4×C7C2311yescharacteristic
38C22×C4×C7C2311yescharacteristic
39C22×Dic7—22no—
40C22×Dic7—24no—
41C22×Dic7—22no—
42C22×Dic7—42no—
43C22×Dic7C2314yes—
44C22×Dic7C2312yes—
45C22×Dic7C2312yes—
46C7×C4⋊C4—22no—
47C7×C22⋊C4—24no—
48C22⋊Dic7—24no—
49C22⋊Dic7—28no—
50C22⋊Dic7—44no—
51Dic7⋊C4—24no—
52Dic7⋊C4—24no—
53C4×Dic7—42no—
54C2×#121D711yescharacteristic
55C22×C4⋊C4—71no—
56#400—72no—
57#394—72no—
58#388—74no—
59#386—72no—
60#386—72no—
61C4×C22⋊C4—71no—
62C23×C7—22no—
63C23×C7—22no—
64C23×C7—22no—
65C23×C7—24no—
66C23×C7—24no—
67C23×C7—24no—
68C23×C7C2×D412yes—
69C23×C7#4114yes—
70C23×C7C2411yescharacteristic
71C7×D4—44no—
72C7×D4—48no—
73C2×C4×C7—22no—
74C2×C4×C7—22no—
75C2×C4×C7—22no—
76C2×C4×C7—24no—
77C2×C4×C7—44no—
78C2×C4×C7C2×D412yes—
79C2×C4×C7C2×D412yes—
80C2×Dic7—28no—
81C2×Dic7—22no—
82C2×Dic7—22no—
83C2×Dic7—22no—
84C2×Dic7—24no—
85C2×Dic7—24no—
86C2×Dic7—24no—
87C2×Dic7—24no—
88C2×Dic7—44no—
89C2×Dic7#4114yes—
90C22×D4C2×D712yes—
91C22×D4C2×D711yescharacteristic
92C23×C4—71no—
93C23×C4—71no—
94C23×C4C2×D711yescharacteristic
95#121—24no—
96C2×C4⋊C4—72no—
97C2×C4⋊C4—72no—
98C2×C4⋊C4—144no—
99C2×C4⋊C4C2×D711yescharacteristic
100C2×C22⋊C4—72no—
101C2×C22⋊C4—74no—
102C2×C22⋊C4—74no—
103C2×C22⋊C4C2×D712yes—
104C2×C42—72no—
105#95—72no—
106#95—74no—
107#95—74no—
108C22×C7—22no—
109C22×C7—22no—
110C22×C7—28no—
111C22×C7—42no—
112C22×C7—42no—
113C22×C7—42no—
114C22×C7—48no—
115C22×C7#12414yes—
116C22×C7C22×D411yescharacteristic
117C22×C7C2×#4112yes—
118C22×C7C2×#4112yes—
119C22×C7H(5,𝔽2)12yes—
120C4×C7—22no—
121C4×C7—22no—
122C4×C7—42no—
123Dic7—24no—
124Dic7—42no—
125Dic7—44no—
126C24C22×D712yes—
127C24C22×D711yescharacteristic
128C2×D4—28no—
129C2×D4—24no—
130C2×D4—22no—
131C2×D4—42no—
132C22×C4—42no—
133C22×C4—72no—
134C22×C4—74no—
135C22×C4—72no—
136C22×C4—144no—
137C22×C4—142no—
138C22×C4—142no—
139C22×C4—282no—
140C22×C4C22⋊D714yes—
141C22×C4C22×D712yes—
142C22×C4C22×D711yescharacteristic
143C22×C4C22×D711yescharacteristic
144C4⋊C4—22no—
145C4⋊C4—144no—
146C4⋊C4—144no—
147C22⋊C4—24no—
148C22⋊C4—148no—
149C22⋊C4—144no—
150C22⋊C4—284no—
151C42—282no—
152C2×C7—24no—
153C2×C7—82no—
154C2×C7#52511yescharacteristic
155C2×C7C2×#12411yescharacteristic
156C2×C7#53411yescharacteristic
157C2×C7#55314yes—
158C23—22no—
159C23—22no—
160C23—24no—
161C23—24no—
162C23—24no—
163C23—22no—
164C23#115914yes—
165C23C2×C22⋊D712yes—
166C23C23×D711yescharacteristic
167D4—44no—
168D4—48no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—24no—
172C2×C4—22no—
173C2×C4—44no—
174C2×C4—74no—
175C2×C4—148no—
176C2×C4—142no—
177C2×C4—142no—
178C2×C4—144no—
179C2×C4—144no—
180C2×C4—144no—
181C2×C4—144no—
182C2×C4—142no—
183C2×C4—284no—
184C2×C4C2×C22⋊D712yes—
185C2×C4C2×C22⋊D712yes—
186C7#259111yescharacteristic, complemented, Sylow
187C22—22no—
188C22—22no—
189C22—28no—
190C22—42no—
191C22—42no—
192C22—42no—
193C22—48no—
194C22#649914yes—
195C22C2×#115912yes—
196C22#654812yes—
197C22D7×#4112yes—
198C22C22×C22⋊D711yescharacteristic
199C4—22no—
200C4—22no—
201C4—42no—
202C4—144no—
203C4—282no—
204C4—284no—
205C2—24no—
206C2—82no—
207C2#9029214yes—
208C2C2×#649911yescharacteristic
209C2#9049511yescharacteristic
210C2#9051911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
Fitting, prime radicalC2×C7×#121C2
hypercenterC2×#121D7
hypocenterC7#2591
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
2. C22×Dic7⋊C4C2semidirect
3. #89994C2semidirect
4. #89992C2semidirect
5. #89992C2semidirect
6. C4×C22⋊Dic7C2semidirect
7. #89754C2semidirect
8. #89717C2semidirect
186. C7#2591semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#1212
C22×Dic7⋊C42
#899942
#899922
C4×C22⋊Dic72
#897542
#897172
#25917