Finite Group Databasebeta

← #11726665

Subgroups — #11726665

Order 896.

258 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 83 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×#6453C211yesmaximal, characteristic
2C2×#6390C211yesmaximal, characteristic
3#89746C211yesmaximal, characteristic
4C2×#6384C211yesmaximal, characteristic
5C7×#340C211yesmaximal, characteristic
6#89345C211yesmaximal, characteristic
7#89324C211yesmaximal, characteristic
8C22×Dic14C2211yescharacteristic
9C22×C7×C8C2211yescharacteristic
10C2×C7×C4⋊C4C2211yescharacteristic
11C2×C7×C4⋊C4C2211yescharacteristic
12C2×C4⋊Dic7—21no—
13C2×C4⋊Dic7C2211yescharacteristic
14C2×Dic7⋊C4—21no—
15C2×C4×Dic7C2211yescharacteristic
16C22×C7⋊C8C2211yescharacteristic
17#6453—22no—
18#6453—22no—
19C7⋊#95—21no—
20C7⋊#95—21no—
21#6390—22no—
22#6384—22no—
23#2026—71nomaximal, Sylow
24C22×C4×C7—21no—
25C22×C4×C7—21no—
26C22×C4×C7C2311yescharacteristic
27C22×Dic7—21no—
28C22×Dic7—21no—
29C22×Dic7D411yescharacteristic
30C22×Dic7D411yescharacteristic
31C2×Dic14—21no—
32C2×Dic14—21no—
33C2×Dic14—21no—
34C2×Dic14—21no—
35C2×Dic14—21no—
36C2×Dic14Q812yes—
37C2×C7×C8—21no—
38C2×C7×C8—21no—
39C2×C7×C8—21no—
40C7×C4⋊C4—21no—
41C7×C4⋊C4—21no—
42C7×C4⋊C4—21no—
43C7×C4⋊C4D412yes—
44C4⋊Dic7—24no—
45C4⋊Dic7—21no—
46C4⋊Dic7D412yes—
47Dic7⋊C4—24no—
48C4×Dic7—21no—
49C4×Dic7—21no—
50C2×C7⋊C8—21no—
51C2×C7⋊C8—21no—
52C2×C7⋊C8—21no—
53C2×#125—71no—
54C2×#103—71no—
55C2×#103—71no—
56#391—71no—
57#340—71no—
58#340D711yescharacteristic
59#336—71no—
60C23×C7C2×D411yescharacteristic
61C2×C4×C7—22no—
62C2×C4×C7—41no—
63C2×C4×C7—41no—
64C2×C4×C7—41no—
65C2×C4×C7—41no—
66C2×C4×C7—41no—
67C2×C4×C7C2×D411yescharacteristic
68C2×C4×C7C2×D411yescharacteristic
69C2×C4×C7C2×Q811yescharacteristic
70C2×C4×C7#4111yescharacteristic
71C2×C4×C7#4111yescharacteristic
72C2×C4×C7#4111yescharacteristic
73C2×Dic7—21no—
74C2×Dic7—21no—
75C2×Dic7—22no—
76C2×Dic7—22no—
77C2×Dic7—21no—
78C2×Dic7—21no—
79C2×Dic7—21no—
80C2×Dic7—21no—
81C2×Dic7—41no—
82C2×Dic7—41no—
83C2×Dic7—41no—
84C2×Dic7—41no—
85Dic14—22no—
86Dic14—22no—
87Dic14—41no—
88Dic14—41no—
89C7×C8—41no—
90C7⋊C8—41no—
91C22×Q8—71no—
92C22×C8—71no—
93C22×C8C2×D711yescharacteristic
94#125—142no—
95#125—142no—
96C2×C4⋊C4—71no—
97C2×C4⋊C4—141no—
98C2×C4⋊C4—141no—
99C2×C4⋊C4C2×D711yescharacteristic
100C2×C4⋊C4C2×D711yescharacteristic
101C2×C42—71no—
102#103—142no—
103#103—142no—
104#95—141no—
105#95—141no—
106C22×C7#12011yescharacteristic
107C22×C7C22⋊Q811yescharacteristic
108C22×C7#12411yescharacteristic
109C22×C7#13511yescharacteristic
110C22×C7#13511yescharacteristic
111C22×C7#13711yescharacteristic
112C22×C7#13711yescharacteristic
113C4×C7—42no—
114C4×C7—81no—
115C4×C7C22⋊Q811yescharacteristic
116C4×C7C22⋊Q811yescharacteristic
117C4×C7#12411yescharacteristic
118C4×C7#12411yescharacteristic
119Dic7—42no—
120Dic7—41no—
121Dic7—41no—
122Dic7—42no—
123C2×Q8—72no—
124C2×Q8—141no—
125C2×Q8—141no—
126C2×Q8—141no—
127C2×Q8—141no—
128C2×Q8—141no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—71no—
132C22×C4—71no—
133C22×C4—141no—
134C22×C4—141no—
135C22×C4C22×D711yescharacteristic
136C2×C8—21no—
137C2×C8—21no—
138C2×C8—21no—
139C2×C8—141no—
140C2×C8—141no—
141C2×C8—141no—
142C4⋊C4—21no—
143C4⋊C4—21no—
144C4⋊C4—21no—
145C4⋊C4—72no—
146C4⋊C4—144no—
147C4⋊C4—144no—
148C4⋊C4—141no—
149C4⋊C4D2812yes—
150C42—141no—
151C42—141no—
152C2×C7#39311yescharacteristic
153C2×C7#45211yescharacteristic
154C2×C7#45211yescharacteristic
155C2×C7#47911yescharacteristic
156C2×C7#47911yescharacteristic
157C2×C7#48811yescharacteristic
158C2×C7#49111yescharacteristic
159C23D4×D711yescharacteristic
160Q8—142no—
161Q8—142no—
162Q8—281no—
163Q8—281no—
164C2×C4—22no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—141no—
171C2×C4—141no—
172C2×C4—142no—
173C2×C4—142no—
174C2×C4—141no—
175C2×C4—141no—
176C2×C4—141no—
177C2×C4—141no—
178C2×C4—281no—
179C2×C4—281no—
180C2×C4—281no—
181C2×C4—281no—
182C2×C4C2×D2811yescharacteristic
183C2×C4#115711yescharacteristic
184C2×C4#115711yescharacteristic
185C2×C4D4×D711yescharacteristic
186C2×C4#115911yescharacteristic
187C2×C4Q8×D711yescharacteristic
188C8—41no—
189C8—281no—
190C7#202611yescharacteristic, complemented, Sylow
191C22#644511yescharacteristic
192C22#644811yescharacteristic
193C22#645911yescharacteristic
194C22#647511yescharacteristic
195C22#647811yescharacteristic
196C22#647911yescharacteristic
197C22#648111yescharacteristic
198C4—42no—
199C4—81no—
200C4—282no—
201C4—281no—
202C4—281no—
203C4—282no—
204C4#643811yescharacteristic
205C4#644811yescharacteristic
206C4#645911yescharacteristic
207C4#646011yescharacteristic
208C2#8944511yescharacteristic
209C2#8954011yescharacteristic
210C2#8955511yescharacteristic
211C2#8956911yescharacteristic
212C2#8958211yescharacteristic
213C2#8962911yescharacteristic
214C2#8964911yescharacteristic
subgroupG/H
centerC23D4×D7
commutator, cosocleC22×C4×C7C23
FrattiniC22×C4C22×D7
FittingC7×#340C2
socle, prime radicalC23×C7C2×D4
hypercenter#340D7
hypocenterC7#2026
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×C4 ◁ #340

Complements

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

NH ≅ G/Ntype
190. C7#2026semidirect

Maximal subgroups

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

subgroupindex
C2×#64532
C2×#63902
#897462
C2×#63842
C7×#3402
#893452
#893242
#20267