Finite Group Databasebeta

← #13266644

Subgroups — #13266644

Order 1792.

336 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 63 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
1C7×#2809C211yesmaximal, characteristic
2#11727473C212yesmaximal
3#11727427C211yesmaximal, characteristic
4#11725458C212yesmaximal
5#11725449C211yesmaximal, characteristic
6#90191C2211yescharacteristic
7C7×#400C2212yes—
8C7×#398—22no—
9C7×#391—21no—
10C7×#391C2211yescharacteristic
11C7×#389—21no—
12C7×#387—22no—
13#89997—21no—
14#89742—22no—
15#89325C2211yescharacteristic
16#89323C2212yes—
17#89318—21no—
18#89247—22no—
19#12225—71nomaximal, Sylow
20C22×C7×Q8D411yescharacteristic
21C2×C7×C4⋊C4—22no—
22C2×C7×C4⋊C4—21no—
23C2×C7×C4⋊C4—21no—
24C2×C7×C4⋊C4—21no—
25C2×C7×C4⋊C4D412yes—
26C2×C7×C4⋊C4C2311yescharacteristic
27C2×C42×C7—21no—
28C2×C42×C7D411yescharacteristic
29#6489—21no—
30#6487—22no—
31C2×#1136—21no—
32#6453—41no—
33#6451—41no—
34C7×#95—22no—
35C7×#95—22no—
36C7×#95—22no—
37C7×#95—22no—
38C7×#95D412yes—
39#6409—41no—
40#6409—41no—
41C22⋊(C7⋊C8)—22no—
42#6384—42no—
43#6381—42no—
44#2809D711yescharacteristic
45#2028—71no—
46#1617—72no—
47#1353—72no—
48#1319—71no—
49C2×C7×Q8—21no—
50C2×C7×Q8—41no—
51C2×C7×Q8—41no—
52C2×C7×Q8Q1612yes—
53C22×C4×C7—21no—
54C22×C4×C7—21no—
55C22×C4×C7—22no—
56C22×C4×C7—22no—
57C22×C4×C7C2×D411yescharacteristic
58C22×C4×C7C2×D412yes—
59C2×Dic14—22no—
60C7×C4⋊C4—41no—
61C7×C4⋊C4—42no—
62C7×C4⋊C4—44no—
63C7×C4⋊C4—42no—
64C7×C4⋊C4—81no—
65C7×C4⋊C4—81no—
66C42×C7—41no—
67C42×C7—81no—
68C22⋊Dic7—41no—
69C4⋊Dic7—21no—
70C4⋊Dic7—21no—
71C4⋊Dic7—22no—
72Dic7⋊C4—42no—
73C4×Dic7—41no—
74#1136—41no—
75#1136—41no—
76C2×C7⋊C8—42no—
77C2×C7⋊C8—41no—
78#544—71no—
79#484—142no—
80#419—141no—
81#400C2×D712yes—
82#398—22no—
83#391—21no—
84#391C2×D711yescharacteristic
85#389—21no—
86#387—22no—
87#343—71no—
88#339—141no—
89#328—142no—
90#324—72no—
91C23×C7#12011yescharacteristic
92C7×Q8—42no—
93C7×Q8—81no—
94C2×C4×C7—21no—
95C2×C4×C7—21no—
96C2×C4×C7—22no—
97C2×C4×C7—22no—
98C2×C4×C7—22no—
99C2×C4×C7—41no—
100C2×C4×C7—41no—
101C2×C4×C7—41no—
102C2×C4×C7—41no—
103C2×C4×C7—42no—
104C2×C4×C7—42no—
105C2×C4×C7—42no—
106C2×C4×C7—42no—
107C2×C4×C7—42no—
108C2×C4×C7—42no—
109C2×C4×C7—42no—
110C2×C4×C7C2×Q1611yescharacteristic
111C2×C4×C7#13711yescharacteristic
112C2×Dic7—41no—
113C2×Dic7—41no—
114Dic14—82no—
115C7⋊C8—81no—
116C7⋊C8—82no—
117C22×Q8C22⋊D711yescharacteristic
118C2×M16—141no—
119C4⋊Q8—281no—
120#125—281no—
121C22⋊Q8—142no—
122#117—141no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4C22⋊D712yes—
128C2×C4⋊C4C22×D711yescharacteristic
129C2×C42—21no—
130C2×C42C22⋊D711yescharacteristic
131#107—282no—
132#103—282no—
133#103—281no—
134#103—281no—
135C22⋊C8—142no—
136#95—22no—
137#95—22no—
138#95—22no—
139#95—22no—
140#95C22⋊D712yes—
141C22×C7—21no—
142C22×C7—21no—
143C22×C7#45111yescharacteristic
144C22×C7#45511yescharacteristic
145C22×C7B(4,𝔽2)11yescharacteristic
146C4×C7—21no—
147C4×C7—21no—
148C4×C7—42no—
149C4×C7—42no—
150C4×C7—42no—
151C4×C7—82no—
152C4×C7—82no—
153Dic7—81no—
154Dic7—81no—
155C2×Q8—21no—
156C2×Q8—41no—
157C2×Q8—41no—
158C2×Q8—142no—
159C2×Q8#114412yes—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4C2×C22⋊D711yescharacteristic
165C22×C4C2×C22⋊D712yes—
166M16—281no—
167M16—281no—
168C2×C8—282no—
169C2×C8—281no—
170C4⋊C4—41no—
171C4⋊C4—42no—
172C4⋊C4—44no—
173C4⋊C4—42no—
174C4⋊C4—81no—
175C4⋊C4—81no—
176C4⋊C4—141no—
177C4⋊C4—141no—
178C4⋊C4—142no—
179C4⋊C4—282no—
180C22⋊C4—281no—
181C42—41no—
182C42—81no—
183C42—281no—
184C2×C7—21no—
185C2×C7—21no—
186C2×C7#160211yescharacteristic
187C2×C7#220111yescharacteristic
188C2×C7#220411yescharacteristic
189C23#651611yescharacteristic
190Q8—42no—
191Q8—81no—
192Q8—562no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—281no—
210C2×C4—281no—
211C2×C4#650511yescharacteristic
212C2×C4C2×#114411yescharacteristic
213C8—561no—
214C8—562no—
215C7#1222511yescharacteristic, complemented, Sylow
216C22—21no—
217C22—21no—
218C22#9000011yescharacteristic
219C22#9001411yescharacteristic
220C22#9001611yescharacteristic
221C4—21no—
222C4—21no—
223C4—42no—
224C4—42no—
225C4—42no—
226C4—82no—
227C4—82no—
228C4—561no—
229C4—561no—
230C2—21no—
231C2—21no—
232C2#1172964711yescharacteristic
233C2#1172968411yescharacteristic
234C2#1172971211yescharacteristic
subgroupG/H
centerC22#90016
commutator, cosocleC2×C7×C4⋊C4C23
FrattiniC2×C4⋊C4C22×D7
FittingC7×#2809C2
socleC22×C7B(4,𝔽2)
prime radicalC23×C7#120
hypercenter#2809D7
hypocenterC7#12225
LCSG ▷ C2×C7×C4⋊C4 ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2809

Complements

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

NH ≅ G/Ntype
215. C7#12225semidirect

Maximal subgroups

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

subgroupindex
C7×#28092
#117274732
#117274272
#117254582
#117254492
#122257