Finite Group Databasebeta

← #13266855

Subgroups — #13266855

Order 1792.

342 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 59 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
1#11731787C211yesmaximal, characteristic
2C7×#2060C211yesmaximal, characteristic
3#11729653C212yesmaximal
4#11725441C211yesmaximal, characteristic
5#11725302C212yesmaximal
6C7×#544C2211yescharacteristic
7C7×#428—21no—
8C7×#384C2211yescharacteristic
9#89816—22no—
10#89758—21no—
11#89758C2212yes—
12#89757—22no—
13#89749—22no—
14C4×C4⋊Dic7—21no—
15C7×#343C2211yescharacteristic
16C7×#339—21no—
17#89333—22no—
18#89323C2212yes—
19#12257—71nomaximal, Sylow
20C2×C7×M16—21no—
21C7×C4⋊Q8—41no—
22C7×#125—41no—
23C7×C22⋊Q8—22no—
24C7×#117—21no—
25C2×C7×C4⋊C4—22no—
26C2×C7×C4⋊C4D411yescharacteristic
27C2×C7×C4⋊C4C2311yescharacteristic
28C2×C42×C7D411yescharacteristic
29C2×C4⋊Dic7—21no—
30C2×C4⋊Dic7—21no—
31C2×C4⋊Dic7—21no—
32C2×C4⋊Dic7D412yes—
33C2×C4×Dic7—22no—
34C7×#107—41no—
35C7×#106—41no—
36#6409—42no—
37C7⋊#95—21no—
38C7⋊#95—22no—
39C7⋊#95—22no—
40C7⋊#95—22no—
41C7⋊#95D412yes—
42C22⋊(C7⋊C8)—22no—
43#6381—42no—
44#2571—71no—
45#2060D711yescharacteristic
46#1617—72no—
47#1353—72no—
48#1325—71no—
49C2×C7×Q8—22no—
50C22×C4×C7—21no—
51C22×C4×C7—21no—
52C22×C4×C7C2×D411yescharacteristic
53C22×C4×C7C2×D412yes—
54C22×Dic7—22no—
55C22×Dic7—22no—
56C7×M16—22no—
57C7×M16—41no—
58C2×C7×C8—41no—
59C7×C4⋊C4—21no—
60C7×C4⋊C4—21no—
61C7×C4⋊C4—22no—
62C7×C4⋊C4—41no—
63C7×C4⋊C4—41no—
64C7×C4⋊C4—44no—
65C7×C4⋊C4—42no—
66C7×C22⋊C4—41no—
67C42×C7—21no—
68C42×C7—41no—
69C42×C7Q1612yes—
70C4⋊Dic7—42no—
71C4⋊Dic7—42no—
72C4⋊Dic7—42no—
73C4⋊Dic7—81no—
74C4×Dic7—44no—
75C2×C7⋊C8—42no—
76#544C2×D711yescharacteristic
77#484—142no—
78#428—21no—
79#400—72no—
80#400—141no—
81#389—142no—
82#384C2×D711yescharacteristic
83#382—142no—
84C4×C4⋊C4—141no—
85#343C2×D711yescharacteristic
86#339—21no—
87#328—142no—
88#324—72no—
89C23×C7#12011yescharacteristic
90C7×Q8—82no—
91C2×C4×C7—22no—
92C2×C4×C7—22no—
93C2×C4×C7—22no—
94C2×C4×C7—21no—
95C2×C4×C7—21no—
96C2×C4×C7—22no—
97C2×C4×C7—42no—
98C2×C4×C7—42no—
99C2×C4×C7—41no—
100C2×C4×C7—41no—
101C2×C4×C7—41no—
102C2×C4×C7C2×Q1611yescharacteristic
103C2×C4×C7#13711yescharacteristic
104C2×Dic7—42no—
105C2×Dic7—42no—
106C2×Dic7—42no—
107C2×Dic7—42no—
108C2×Dic7—42no—
109C2×Dic7—42no—
110C7×C8—42no—
111C7⋊C8—82no—
112C2×M16—21no—
113C4⋊Q8—41no—
114#125—41no—
115C22⋊Q8—22no—
116#117—21no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—72no—
119C2×C4⋊C4—141no—
120C2×C4⋊C4—141no—
121C2×C4⋊C4—141no—
122C2×C4⋊C4C22⋊D711yescharacteristic
123C2×C4⋊C4C22×D711yescharacteristic
124C2×C42—142no—
125C2×C42C22⋊D711yescharacteristic
126#107—41no—
127#107—282no—
128#106—41no—
129#103—282no—
130C22⋊C8—142no—
131#95—72no—
132#95—141no—
133#95—142no—
134#95—142no—
135#95—142no—
136C22×C7—21no—
137C22×C7—21no—
138C22×C7#45111yescharacteristic
139C22×C7#45611yescharacteristic
140C22×C7B(4,𝔽2)11yescharacteristic
141C4×C7—24no—
142C4×C7—21no—
143C4×C7—21no—
144C4×C7—42no—
145C4×C7—81no—
146C4×C7—81no—
147C4×C7—81no—
148Dic7—82no—
149Dic7—82no—
150C2×Q8—22no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—142no—
154C22×C4—142no—
155C22×C4D4×D712yes—
156C22×C4C2×C22⋊D711yescharacteristic
157M16—22no—
158M16—41no—
159C2×C8—41no—
160C2×C8—282no—
161C4⋊C4—21no—
162C4⋊C4—21no—
163C4⋊C4—22no—
164C4⋊C4—41no—
165C4⋊C4—41no—
166C4⋊C4—44no—
167C4⋊C4—42no—
168C4⋊C4—282no—
169C4⋊C4—282no—
170C4⋊C4—282no—
171C4⋊C4—561no—
172C22⋊C4—41no—
173C42—21no—
174C42—41no—
175C42—284no—
176C42#114412yes—
177C2×C7—21no—
178C2×C7—21no—
179C2×C7#160511yescharacteristic
180C2×C7#219811yescharacteristic
181C2×C7#220411yescharacteristic
182C23#650011yescharacteristic
183Q8—82no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—282no—
196C2×C4—282no—
197C2×C4—282no—
198C2×C4—282no—
199C2×C4—282no—
200C2×C4—282no—
201C2×C4C2×#114411yescharacteristic
202C2×C4#651411yescharacteristic
203C8—42no—
204C8—562no—
205C7#1225711yescharacteristic, complemented, Sylow
206C22—21no—
207C22—21no—
208C22#8980411yescharacteristic
209C22#8982211yescharacteristic
210C22#8986711yescharacteristic
211C4—24no—
212C4—21no—
213C4—21no—
214C4—42no—
215C4—81no—
216C4—81no—
217C4—81no—
218C4—562no—
219C4—562no—
220C2—21no—
221C2—21no—
222C2#1172746111yescharacteristic
223C2#1172874011yescharacteristic
224C2#1172874811yescharacteristic
subgroupG/H
centerC22#89804
commutator, cosocleC2×C7×C4⋊C4C23
FrattiniC2×C4⋊C4C22×D7
FittingC7×#2060C2
socleC22×C7B(4,𝔽2)
prime radicalC23×C7#120
hypercenter#2060D7
hypocenterC7#12257
LCSG ▷ C2×C7×C4⋊C4 ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2060

Complements

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

NH ≅ G/Ntype
205. C7#12257semidirect

Maximal subgroups

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

subgroupindex
#117317872
C7×#20602
#117296532
#117254412
#117253022
#122577