Finite Group Databasebeta

← #13266817

Subgroups — #13266817

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#11731662C211yesmaximal, characteristic
2C7×#2061C211yesmaximal, characteristic
3#11729652C212yesmaximal
4#11725438C211yesmaximal, characteristic
5#11725301C212yesmaximal
6C7×#544C2211yescharacteristic
7C2×#6433—21no—
8C7×#428—21no—
9C7×#382C2211yescharacteristic
10#89815—22no—
11#89755—21no—
12#89755C2212yes—
13#89433—22no—
14#89431—22no—
15C7×#343C2211yescharacteristic
16C7×#339—21no—
17#89333—22no—
18#89323C2212yes—
19#12251—71nomaximal, Sylow
20C2×C7×M16—21no—
21C7×C4⋊Q8—41no—
22C7×#125—41no—
23C7×C22⋊Q8—22no—
24C7×#117—21no—
25C2×C7×C4⋊C4D411yescharacteristic
26C2×C7×C4⋊C4C2311yescharacteristic
27C2×C42×C7D411yescharacteristic
28C2×C4⋊Dic7—21no—
29C2×C4⋊Dic7—21no—
30C2×C4⋊Dic7—21no—
31C2×C4⋊Dic7D412yes—
32C2×Dic7⋊C4—22no—
33C2×Dic7⋊C4—22no—
34#6433—42no—
35#6433—42no—
36C7×#107—41no—
37C7×#106—41no—
38C7×#95—22no—
39#6409—42no—
40C7⋊#95—21no—
41C7⋊#95—22no—
42C7⋊#95—22no—
43C7⋊#95D412yes—
44C22⋊(C7⋊C8)—22no—
45#6382—42no—
46#2822—71no—
47#2061D711yescharacteristic
48#1611—72no—
49#1346—72no—
50#1324—71no—
51C2×C7×Q8—22no—
52C22×C4×C7—21no—
53C22×C4×C7—21no—
54C22×C4×C7C2×D411yescharacteristic
55C22×C4×C7C2×D412yes—
56C22×Dic7—22no—
57C22×Dic7—22no—
58C7×M16—22no—
59C7×M16—41no—
60C2×C7×C8—41no—
61C7×C4⋊C4—21no—
62C7×C4⋊C4—21no—
63C7×C4⋊C4—22no—
64C7×C4⋊C4—41no—
65C7×C4⋊C4—41no—
66C7×C4⋊C4—42no—
67C7×C22⋊C4—41no—
68C42×C7—21no—
69C42×C7—41no—
70C42×C7SD1612yes—
71C4⋊Dic7—42no—
72C4⋊Dic7—42no—
73C4⋊Dic7—42no—
74C4⋊Dic7—81no—
75Dic7⋊C4—44no—
76Dic7⋊C4—42no—
77Dic7⋊C4—42no—
78C2×C7⋊C8—42no—
79#544C2×D711yescharacteristic
80C2×#125—141no—
81#483—142no—
82#428—21no—
83#400—142no—
84#398—72no—
85#398—142no—
86#398—141no—
87#382C2×D711yescharacteristic
88#343C2×D711yescharacteristic
89#339—21no—
90#328—142no—
91#324—72no—
92C23×C7#12011yescharacteristic
93C7×Q8—82no—
94C2×C4×C7—21no—
95C2×C4×C7—21no—
96C2×C4×C7—22no—
97C2×C4×C7—22no—
98C2×C4×C7—41no—
99C2×C4×C7—41no—
100C2×C4×C7—42no—
101C2×C4×C7—42no—
102C2×C4×C7—41no—
103C2×C4×C7—41no—
104C2×C4×C7—41no—
105C2×C4×C7C2×SD1611yescharacteristic
106C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
107C2×Dic7—42no—
108C2×Dic7—42no—
109C2×Dic7—42no—
110C2×Dic7—42no—
111C2×Dic7—42no—
112C2×Dic7—42no—
113C7×C8—42no—
114C7⋊C8—82no—
115C2×M16—21no—
116C4⋊Q8—41no—
117#125—41no—
118#125—282no—
119#125—282no—
120C22⋊Q8—22no—
121#117—21no—
122C2×C4⋊C4—72no—
123C2×C4⋊C4—142no—
124C2×C4⋊C4—141no—
125C2×C4⋊C4—141no—
126C2×C4⋊C4—142no—
127C2×C4⋊C4—141no—
128C2×C4⋊C4C22⋊D711yescharacteristic
129C2×C4⋊C4C22×D711yescharacteristic
130C2×C42C22⋊D711yescharacteristic
131#107—41no—
132#106—41no—
133#106—282no—
134#103—282no—
135C22⋊C8—142no—
136#95—22no—
137#95—72no—
138#95—142no—
139#95—141no—
140#95—142no—
141C22×C7—21no—
142C22×C7—21no—
143C22×C7#45011yescharacteristic
144C22×C7#45611yescharacteristic
145C22×C7B(4,𝔽2)11yescharacteristic
146C4×C7—21no—
147C4×C7—21no—
148C4×C7—42no—
149C4×C7—42no—
150C4×C7—81no—
151C4×C7—81no—
152C4×C7—81no—
153Dic7—82no—
154Dic7—82no—
155C2×Q8—22no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—142no—
159C22×C4—142no—
160C22×C4D4×D712yes—
161C22×C4C2×C22⋊D711yescharacteristic
162M16—22no—
163M16—41no—
164C2×C8—41no—
165C2×C8—282no—
166C4⋊C4—21no—
167C4⋊C4—22no—
168C4⋊C4—21no—
169C4⋊C4—41no—
170C4⋊C4—41no—
171C4⋊C4—42no—
172C4⋊C4—282no—
173C4⋊C4—284no—
174C4⋊C4—282no—
175C4⋊C4—282no—
176C4⋊C4—282no—
177C4⋊C4—282no—
178C4⋊C4—561no—
179C22⋊C4—41no—
180C42—21no—
181C42—41no—
182C42#114312yes—
183C2×C7—21no—
184C2×C7—21no—
185C2×C7#160411yescharacteristic
186C2×C7#220111yescharacteristic
187C2×C7#220311yescharacteristic
188C23#650011yescharacteristic
189Q8—82no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—282no—
202C2×C4—282no—
203C2×C4—282no—
204C2×C4—282no—
205C2×C4—282no—
206C2×C4—282no—
207C2×C4C2×#114311yescharacteristic
208C2×C4#651211yescharacteristic
209C8—42no—
210C8—562no—
211C7#1225111yescharacteristic, complemented, Sylow
212C22—21no—
213C22—21no—
214C22#8980411yescharacteristic
215C22#8981811yescharacteristic
216C22#8986711yescharacteristic
217C4—21no—
218C4—21no—
219C4—42no—
220C4—42no—
221C4—81no—
222C4—81no—
223C4—81no—
224C4—562no—
225C4—562no—
226C2—21no—
227C2—21no—
228C2#1172746011yescharacteristic
229C2#1172874311yescharacteristic
230C2#1172874611yescharacteristic
subgroupG/H
centerC22#89804
commutator, cosocleC2×C7×C4⋊C4C23
FrattiniC2×C4⋊C4C22×D7
FittingC7×#2061C2
socleC22×C7B(4,𝔽2)
prime radicalC23×C7#120
hypercenter#2061D7
hypocenterC7#12251
LCSG ▷ C2×C7×C4⋊C4 ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2061

Complements

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

NH ≅ G/Ntype
211. C7#12251semidirect

Maximal subgroups

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

subgroupindex
#117316622
C7×#20612
#117296522
#117254382
#117253012
#122517