Finite Group Databasebeta

← #13246813

Subgroups — #13246813

Order 1792.

404 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 101 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#11736486C211yesmaximal, characteristic
2#11728259C212yesmaximal
3#11728244C212yesmaximal, complemented
4C7×#1309C211yesmaximal, characteristic, complemented
5C7⋊#1309C211yesmaximal, characteristic, complemented
6C2×#6470C412yes—
7#90168C2211yescharacteristic
8C7×#423C2212yes—
9C7×#423C2211yescharacteristic
10#89901C412yes—
11#89696C2212yes—
12C2×C4×C7⋊C8C2211yescharacteristic
13#89610—22no—
14#89606—22no—
15#89280—22no—
16#89276—22no—
17#89275—22no—
18#89275—22no—
19#8636—71nomaximal, Sylow
20C2×#1157C2×C412yes—
21C2×C7×M16—22no—
22C2×C7×M16C2×C412yes—
23C2×C42×C7C2311yescharacteristic
24#6493—22no—
25C2×#1136—22no—
26C22×C7⋊C8C2×C411yescharacteristic
27C22×C7⋊C8C2×C411yescharacteristic
28#6470—44no—
29C2×#1131—22no—
30C2×C8×D7—22no—
31#6438Q812yes—
32#6437D411yescharacteristic
33C4×D28—22no—
34#6432D411yescharacteristic
35C7×C4⋊C8—22no—
36C7×C4⋊C8—22no—
37C7×C4⋊C8—22no—
38(C2×D7)⋊C8—24no—
39C4⋊(C7⋊C8)—22no—
40C4⋊(C7⋊C8)—22no—
41C4×C7⋊C8D412yes—
42C4×C7⋊C8D412yes—
43#2931—71no—
44#1566—72no—
45#1560—72no—
46#1309—71no—
47#1309D711yescharacteristic, complemented
48C22×C4×C7C22×C411yescharacteristic
49C22×C4×C7#4112yes—
50C2×C22⋊D7—22no—
51#1157—42no—
52#1157—42no—
53C2×D28—22no—
54C2×D28C4⋊C412yes—
55C2×C4×D7—22no—
56C2×Dic14C4⋊C412yes—
57C7×M16—42no—
58C7×M16—82no—
59C2×C7×C8—22no—
60C2×C7×C8—42no—
61C42×C7C2×D412yes—
62C42×C7C2×D411yescharacteristic
63C42×C7C2×Q811yescharacteristic
64(C2×D7)⋊C4—24no—
65C4⋊Dic7—22no—
66#1136—82no—
67C2×C7⋊C8—22no—
68C2×C7⋊C8—41no—
69C2×C7⋊C8—42no—
70C2×C7⋊C8C22⋊C411yescharacteristic
71C2×C7⋊C8C22⋊C412yes—
72C2×C7⋊C8C22⋊C411yescharacteristic
73#1131—44no—
74C8×D7—44no—
75C2×#131—72no—
76#532—71no—
77#436—142no—
78C8×D4—142no—
79#423—72no—
80#423C2×D712yes—
81#423C2×D711yescharacteristic
82#408—72no—
83C2×C4×C8—71no—
84#334—142no—
85#334—142no—
86#333—142no—
87#331—142no—
88C23×C7#11711yescharacteristic
89C22×D7—22no—
90C2×C4×C7—21no—
91C2×C4×C7—21no—
92C2×C4×C7—22no—
93C2×C4×C7—22no—
94C2×C4×C7C2×C22⋊C411yescharacteristic
95C2×C4×C7C2×C4⋊C411yescharacteristic
96C2×C4×C7#12112yes—
97C2×C4×C7C22⋊Q812yes—
98C22⋊D7—44no—
99C2×Dic7—22no—
100D28—42no—
101D28—42no—
102D28—42no—
103C4×D7—42no—
104C4×D7—42no—
105Dic14—42no—
106C7×C8—42no—
107C7×C8—82no—
108C7⋊C8—21no—
109C7⋊C8—22no—
110C7⋊C8—82no—
111C7⋊C8#10012yes—
112C2×#41—72no—
113#131—284no—
114C2×M16—22no—
115C2×M16—142no—
116C2×M16—142no—
117C2×M16C4×D712yes—
118C22×C8—71no—
119C22×C8—71no—
120C22×C8—142no—
121C4⋊Q8—71no—
122#127—71no—
123#124—72no—
124#121—142no—
125C4×D4—142no—
126C2×C42C22×D711yescharacteristic
127C4⋊C8—22no—
128C4⋊C8—22no—
129C4⋊C8—22no—
130C4⋊C8—142no—
131C4⋊C8—142no—
132C22⋊C8—144no—
133C4×C8—72no—
134C4×C8—72no—
135C22×C7—21no—
136C22×C7—22no—
137C22×C7#38011yescharacteristic
138C2×D7—42no—
139C2×D7—44no—
140C4×C7—21no—
141C4×C7—22no—
142C4×C7C2×#10011yescharacteristic
143C4×C7#41311yescharacteristic
144C4×C7#47112yes—
145C4×C7#47112yes—
146Dic7—42no—
147#41—282no—
148#41—282no—
149C2×Q8—72no—
150C2×D4—72no—
151C2×D4—142no—
152C2×D4—142no—
153C22×C4—142no—
154C22×C4C2×C4×D711yescharacteristic
155C22×C4#115712yes—
156M16—42no—
157M16—82no—
158M16—284no—
159M16—562no—
160C2×C8—22no—
161C2×C8—42no—
162C2×C8—71no—
163C2×C8—72no—
164C2×C8—71no—
165C2×C8—142no—
166C2×C8—284no—
167C2×C8—281no—
168C2×C8—282no—
169C4⋊C4—142no—
170C22⋊C4—144no—
171C42D4×D712yes—
172C42D4×D711yescharacteristic
173C42Q8×D711yescharacteristic
174C2×C7—41no—
175C2×C7#180011yescharacteristic
176C2×C7#181512yes—
177D7—82no—
178C23—142no—
179C23#643511yescharacteristic
180Q8—282no—
181D4—282no—
182D4—284no—
183D4—282no—
184D4—282no—
185C2×C4—21no—
186C2×C4—22no—
187C2×C4—21no—
188C2×C4—22no—
189C2×C4—142no—
190C2×C4—282no—
191C2×C4—282no—
192C2×C4D7×C22⋊C411yescharacteristic
193C2×C4#644712yes—
194C2×C4D7×C4⋊C411yescharacteristic
195C2×C4#645912yes—
196C8—42no—
197C8—72no—
198C8—82no—
199C8—141no—
200C8—142no—
201C8—562no—
202C7#863611yescharacteristic, complemented, Sylow
203C22—21no—
204C22—22no—
205C22—282no—
206C22—284no—
207C22#8943911yescharacteristic
208C4—21no—
209C4—22no—
210C4—282no—
211C4D7×#10011yescharacteristic
212C4#8952012yes—
213C4#8952311yescharacteristic
214C4#8952612yes—
215C2—41no—
216C2—562no—
217C2#1172674811yescharacteristic
218C2#1172675412yes—
subgroupG/H
centerC22#89439
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#1309C2
socleC22×C7#380
prime radical#90168C22
cosocleC2×C42×C7C23
hypercenter#1309D7
hypocenterC7#8636
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1309

Complements

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

NH ≅ G/Ntype
3. #11728244C2semidirect
4. C7×#1309C2semidirect
5. C7⋊#1309C2semidirect
47. #1309D7semidirect
202. C7#8636semidirect

Maximal subgroups

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

subgroupindex
#117364862
#117282592
#117282442
C7×#13092
C7⋊#13092
#86367