Finite Group Databasebeta

← #11740683

Subgroups — #11740683

Order 896.

540 conjugacy classes of subgroups; only the 185 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 185 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×C4⋊Q8C211yesmaximal, characteristic
2C2×Q8×Dic7C211yesmaximal, characteristic
3C2×#6507C211yesmaximal, characteristic
4C2×#6507C211yesmaximal, characteristic
5#90002C211yesmaximal, characteristic
6#89756C214yesmaximal
7#89744C214yesmaximal
8C4×Dic7⋊C4C211yesmaximal, characteristic
9#89701C211yesmaximal, characteristic
10C22×C7×Q8C2211yescharacteristic
11C22×C7×Q8C2211yescharacteristic
12C22×Dic14C2211yescharacteristic
13C7×C4⋊Q8—24no—
14C2×C7×C4⋊C4C2214yes—
15C2×C42×C7C2211yescharacteristic
16Q8×Dic7—24no—
17#6507—24no—
18#6507—24no—
19C2×C4⋊Dic7C2211yescharacteristic
20C2×C4⋊Dic7C2212yes—
21C2×Dic7⋊C4C2212yes—
22C2×Dic7⋊C4C2214yes—
23C2×Dic7⋊C4C2212yes—
24C2×Dic7⋊C4C2212yes—
25C2×C4×Dic7C2211yescharacteristic
26C2×C4×Dic7C2212yes—
27C2×C4×Dic7C2211yescharacteristic
28C7⋊#95C2214yes—
29C7⋊#95C2212yes—
30C7⋊#95C2214yes—
31#2546—71nomaximal, Sylow
32C2×C7×Q8—22no—
33C2×C7×Q8—22no—
34C2×C7×Q8—22no—
35C2×C7×Q8—22no—
36C2×C7×Q8—22no—
37C2×C7×Q8D414yes—
38C22×C4×C7C2312yes—
39C22×C4×C7C2312yes—
40C22×C4×C7C2312yes—
41C22×C4×C7C2311yescharacteristic
42C22×Dic7C2312yes—
43C22×Dic7C2312yes—
44C22×Dic7C2311yescharacteristic
45C22×Dic7C2312yes—
46C22×Dic7C2311yescharacteristic
47C2×Dic14—22no—
48C2×Dic14—22no—
49C2×Dic14—22no—
50C7×C4⋊C4—28no—
51C42×C7—22no—
52C4⋊Dic7—22no—
53C4⋊Dic7—24no—
54Dic7⋊C4—28no—
55Dic7⋊C4—24no—
56Dic7⋊C4—24no—
57Dic7⋊C4Q818yescomplemented
58C4×Dic7—22no—
59C4×Dic7—24no—
60C4×Dic7D414yes—
61C2×C4⋊Q8—71no—
62C2×C4⋊Q8—71no—
63C2×C4⋊Q8D711yescharacteristic
64C2×C4×Q8—71no—
65#398—74no—
66#391—71no—
67#391—71no—
68#389—74no—
69C4×C4⋊C4—71no—
70C23×C7C2411yescharacteristic
71C7×Q8—28no—
72C7×Q8—44no—
73C2×C4×C7—22no—
74C2×C4×C7—22no—
75C2×C4×C7—22no—
76C2×C4×C7—22no—
77C2×C4×C7—22no—
78C2×C4×C7—22no—
79C2×C4×C7—22no—
80C2×C4×C7C2×D412yes—
81C2×C4×C7C2×D414yes—
82C2×C4×C7C2×D412yes—
83C2×C4×C7C2×Q814yes—
84C2×C4×C7#4112yes—
85C2×Dic7—22no—
86C2×Dic7—22no—
87C2×Dic7—22no—
88C2×Dic7—22no—
89C2×Dic7—22no—
90C2×Dic7—22no—
91C2×Dic7—22no—
92C2×Dic7—24no—
93C2×Dic7C2×D412yes—
94C2×Dic7C2×D412yes—
95C2×Dic7C2×Q814yes—
96C2×Dic7C2×Q814yes—
97Dic14—44no—
98C22×Q8—71no—
99C22×Q8C2×D711yescharacteristic
100C22×Q8C2×D711yescharacteristic
101C4⋊Q8—24no—
102C4⋊Q8—144no—
103C4⋊Q8—144no—
104C4×Q8—144no—
105C2×C4⋊C4—72no—
106C2×C4⋊C4—71no—
107C2×C4⋊C4—72no—
108C2×C4⋊C4—72no—
109C2×C4⋊C4—74no—
110C2×C4⋊C4—72no—
111C2×C4⋊C4C2×D714yes—
112C2×C42—71no—
113C2×C42—72no—
114C2×C42—71no—
115C2×C42C2×D711yescharacteristic
116#95—72no—
117#95—74no—
118#95—74no—
119C22×C7C22×D411yescharacteristic
120C22×C7C22×D411yescharacteristic
121C22×C7C22×Q812yes—
122C22×C7C2×#4111yescharacteristic
123C22×C72−1+412yes—
124C4×C7—24no—
125C4×C7—24no—
126C4×C7—42no—
127C4×C7#12114yes—
128Dic7—22no—
129Dic7—24no—
130Dic7—42no—
131Dic7—42no—
132Dic7C4⋊Q814yescomplemented
133C2×Q8—22no—
134C2×Q8—22no—
135C2×Q8—22no—
136C2×Q8—22no—
137C2×Q8—22no—
138C2×Q8—142no—
139C2×Q8—142no—
140C2×Q8—142no—
141C2×Q8C22⋊D714yes—
142C22×C4—71no—
143C22×C4—72no—
144C22×C4—71no—
145C22×C4—72no—
146C22×C4—72no—
147C22×C4C22×D712yes—
148C22×C4C22×D712yes—
149C22×C4C22×D712yes—
150C22×C4C22×D711yescharacteristic
151C4⋊C4—28no—
152C4⋊C4—78no—
153C4⋊C4—144no—
154C4⋊C4—144no—
155C4⋊C4—144no—
156C4⋊C4—142no—
157C4⋊C4—148no—
158C42—22no—
159C42—74no—
160C42—144no—
161C42—142no—
162C2×C7C2×#12111yescharacteristic
163C2×C7C2×C4⋊Q811yescharacteristic
164C2×C7#53611yescharacteristic
165C2×C7D4×Q812yes—
166C2×C7#55712yes—
167C23C23×D711yescharacteristic
168Q8—28no—
169Q8—44no—
170Q8—284no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—72no—
179C2×C4—74no—
180C2×C4—72no—
181C2×C4—74no—
182C2×C4—142no—
183C2×C4—142no—
184C2×C4—142no—
185C2×C4—142no—
186C2×C4—142no—
187C2×C4—142no—
188C2×C4—144no—
189C2×C4—142no—
190C2×C4D4×D712yes—
191C2×C4#115912yes—
192C2×C4Q8×D714yes—
193C2×C4C2×C22⋊D714yes—
194C2×C4C2×C22⋊D712yes—
195C7#254611yescharacteristic, complemented, Sylow
196C22C2×D4×D711yescharacteristic
197C22C2×#115911yescharacteristic
198C22C2×Q8×D712yes—
199C22#655112yes—
200C22C22×C22⋊D711yescharacteristic
201C4—24no—
202C4—24no—
203C4—42no—
204C4—74no—
205C4—142no—
206C4—144no—
207C4—282no—
208C4—282no—
209C4#650214yes—
210C2#9041212yes—
211C2D7×C4⋊Q811yescharacteristic
212C2#9041511yescharacteristic
213C2C2×#650211yescharacteristic
214C2Q8×C22⋊D712yes—
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, prime radical, cosocleC23×C7C24
FittingC2×C7×C4⋊Q8C2
hypercenterC2×C4⋊Q8D7
hypocenterC7#2546
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
57. Dic7⋊C4Q8semidirect
132. Dic7C4⋊Q8semidirect
195. C7#2546semidirect

Maximal subgroups

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

subgroupindex
C2×C7×C4⋊Q82
C2×Q8×Dic72
C2×#65072
#900022
#897562
#897442
C4×Dic7⋊C42
#897012
#25467