Finite Group Databasebeta

← #13267950

Subgroups — #13267950

Order 1792.

218 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 65 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#11729144C211yesmaximal, characteristic
2#11729114C211yesmaximal, characteristic
3#11727047C211yesmaximal, characteristic
4#11726311C211yesmaximal, characteristic
5C7×#1365C211yesmaximal, characteristic
6#11725360C211yesmaximal, characteristic
7#11725280C211yesmaximal, characteristic
8C7×#500C2211yescharacteristic
9#89853—21no—
10#89851—21no—
11C2×Dic56—21no—
12#89602—21no—
13#89471C2211yescharacteristic
14C4×Dic28C2211yescharacteristic
15C7×C4⋊C16C2211yescharacteristic
16C7×#335C2211yescharacteristic
17#89298—21no—
18#89287—21no—
19#89284—21no—
20#89279—21no—
21C4⋊(C7⋊C16)C2211yescharacteristic
22C8×C7⋊C8C2211yescharacteristic
23#12421—71nomaximal, Sylow
24C7×C4⋊Q8—21no—
25C2×Dic28—21no—
26C2×Dic28—21no—
27C2×Dic28D411yescharacteristic
28#6432—21no—
29C4×Dic14—21no—
30C2×C7×C16—21no—
31C7×#107—21no—
32C7×#107—21no—
33C7×C4⋊C8—21no—
34C4×C7×C8C2311yescharacteristic
35#6409—42no—
36#6392D411yescharacteristic
37#6390—21no—
38Dic7⋊C8—41no—
39C8×Dic7—41no—
40C2×C7⋊C16—21no—
41#6382—41no—
42#6382—41no—
43C4×C7⋊C8D411yescharacteristic
44C4×C7⋊C8D411yescharacteristic
45Dic56—42no—
46#2227—71no—
47#2209—71no—
48#1707—71no—
49C8×Q16—71no—
50#1367—71no—
51#1365D711yescharacteristic
52#1362—71no—
53C2×C7×Q8—41no—
54C2×Dic14—21no—
55C2×Dic14—41no—
56C2×C7×C8—41no—
57C2×C7×C8C2×D411yescharacteristic
58C2×C7×C8#4111yescharacteristic
59C7×C4⋊C4—41no—
60C7×C4⋊C4—41no—
61C42×C7C2×D411yescharacteristic
62C4⋊Dic7—21no—
63C4⋊Dic7—41no—
64C4⋊Dic7—41no—
65Dic7⋊C4—41no—
66C4×Dic7—41no—
67C2×C7⋊C8—21no—
68C2×C7⋊C8D811yescharacteristic
69C2×C7⋊C8D811yescharacteristic
70Dic28—21no—
71Dic28—41no—
72Dic28—41no—
73Dic28—81no—
74Dic28SD1612yes—
75C7×C16—41no—
76C7⋊C16—41no—
77C2×Q32—141no—
78#500C2×D711yescharacteristic
79#498—141no—
80#494—71no—
81#487—141no—
82C8×Q8—141no—
83C4×Q16—71no—
84#366—141no—
85C4⋊C16—71no—
86C4⋊C16C2×D711yescharacteristic
87#358—141no—
88#358—141no—
89#335C2×D711yescharacteristic
90Q8⋊C8—141no—
91C82—71no—
92C7×Q8—82no—
93C2×C4×C7—81no—
94C2×C4×C7#12111yescharacteristic
95C2×C4×C7C2×D811yescharacteristic
96C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
97C2×Dic7—41no—
98C2×Dic7—41no—
99C2×Dic7—81no—
100Dic14—22no—
101Dic14—41no—
102Dic14—81no—
103Dic14—81no—
104C7×C8—21no—
105C7×C8—81no—
106C7×C8C2×SD1611yescharacteristic
107C7×C8#13511yescharacteristic
108C7⋊C8—21no—
109C7⋊C8—41no—
110C7⋊C8Q3212yescomplemented
111C2×Q16—71no—
112C2×Q16—141no—
113C2×Q16—141no—
114C4⋊Q8—21no—
115C4⋊Q8—141no—
116C4×Q8—141no—
117Q32—282no—
118C2×C16—21no—
119C2×C16—141no—
120#107—21no—
121#107—21no—
122#107—71no—
123#106—281no—
124#106—281no—
125C4⋊C8—21no—
126C4⋊C8—281no—
127#103—141no—
128#103—282no—
129C4×C8—71no—
130C4×C8—71no—
131C4×C8—281no—
132C4×C8C22×D711yescharacteristic
133C22×C7#45911yescharacteristic
134C4×C7—21no—
135C4×C7—161no—
136C4×C7#46511yescharacteristic
137C4×C7#47111yescharacteristic
138C4×C7C2×Q3211yescharacteristic
139C4×C7#51011yescharacteristic
140Dic7—42no—
141Dic7—81no—
142Dic7—161no—
143C2×Q8—41no—
144C2×Q8—141no—
145C2×Q8—281no—
146Q16—72no—
147Q16—141no—
148Q16—281no—
149Q16—281no—
150Q16—561no—
151C2×C8—41no—
152C2×C8—71no—
153C2×C8—71no—
154C2×C8—141no—
155C2×C8#115711yescharacteristic
156C2×C8D4×D711yescharacteristic
157C4⋊C4—41no—
158C4⋊C4—41no—
159C4⋊C4—141no—
160C4⋊C4—281no—
161C4⋊C4—281no—
162C4⋊C4—281no—
163C42—281no—
164C42D4×D711yescharacteristic
165C16—41no—
166C16—281no—
167C2×C7#166511yescharacteristic
168C2×C7#220911yescharacteristic
169C2×C7#221211yescharacteristic
170Q8—82no—
171Q8—142no—
172Q8—281no—
173Q8—561no—
174Q8—561no—
175C2×C4—81no—
176C2×C4—281no—
177C2×C4—281no—
178C2×C4—561no—
179C2×C4#644711yescharacteristic
180C2×C4D7×D811yescharacteristic
181C2×C4#647711yescharacteristic
182C8—21no—
183C8—72no—
184C8—81no—
185C8—141no—
186C8—281no—
187C8D7×SD1611yescharacteristic
188C8#648211yescharacteristic
189C7#1242111yescharacteristic, complemented, Sylow
190C22#8955811yescharacteristic
191C4—21no—
192C4—161no—
193C4—282no—
194C4—561no—
195C4—1121no—
196C4#8952611yescharacteristic
197C4#8957911yescharacteristic
198C4#8968711yescharacteristic
199C4D7×Q3211yescharacteristic
200C2#1172619411yescharacteristic
201C2#1172699011yescharacteristic
202C2#1172701611yescharacteristic
subgroupG/H
centerC22#89558
commutator, cosocleC4×C7×C8C23
FrattiniC4×C8C22×D7
FittingC7×#1365C2
socle, prime radicalC22×C7#459
hypercenter#1365D7
hypocenterC7#12421
LCSG ▷ C4×C7×C8 ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1365

Complements

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

NH ≅ G/Ntype
110. C7⋊C8Q32semidirect
189. C7#12421semidirect

Maximal subgroups

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

subgroupindex
#117291442
#117291142
#117270472
#117263112
C7×#13652
#117253602
#117252802
#124217