Finite Group Databasebeta

← #13560219

Subgroups — #13560219

Order 1792.

1778 conjugacy classes of subgroups; only the 539 normal subgroups are shown.

normal subgroups only

This group has 1778 conjugacy classes of subgroups — too many to draw. Only its 539 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×#2840C211yesmaximal, characteristic
2#11734306C2112yesmaximal, complemented
3#11734223C2112yesmaximal, complemented
4#11731290C213yesmaximal
5#11731262C213yesmaximal
6C24×Dic7C2211yescharacteristic
7C2×Q8×Dic7C2214yescomplemented
8C2×#6440C2216yes—
9C7×#393C22112yes—
10C7×#379C2213yes—
11#90019C2213yes—
12#90001C22112yescomplemented
13#89721C22112yes—
14#89720C22124yes—
15#89715C2216yes—
16#89714C22112yes—
17C22⋊C4×Dic7C22112yes—
18#89431C22112yescomplemented
19#89429C22124yescomplemented
20#89421C22112yescomplemented
21#36674—71nomaximal, Sylow
22C25×C7C2311yescharacteristic
23C22×C7×Q8C2314yes—
24C23×Dic7—44no—
25C23×Dic7Q818yescomplemented
26C23×Dic7C2313yes—
27C23×Dic7C2313yes—
28C2×C7×C22⋊C4C2316yes—
29C2×C7×C22⋊C4C23112yes—
30C2×C22⋊Dic7C2316yes—
31C2×C22⋊Dic7C23112yes—
32Q8×Dic7—48no—
33C2×C4⋊Dic7C23112yes—
34C2×Dic7⋊C4C23112yes—
35C2×Dic7⋊C4C23124yes—
36C2×C4×Dic7C23112yes—
37#6440—412no—
38C7×#95C23112yes—
39C7⋊#95C23112yes—
40C7⋊#95C23124yes—
41#2842—73no—
42#2841—712no—
43#2840D711yescharacteristic
44#2396—712no—
45#2280—73no—
46C24×C7—42no—
47C24×C7—44no—
48C24×C7C2×Q814yes—
49C24×C7C2413yes—
50C2×C7×Q8—44no—
51C2×C7×Q8—48no—
52C22×C4×C7C24112yes—
53C22×Dic7—212no—
54C22×Dic7—416no—
55C22×Dic7—46no—
56C22×Dic7C2×Q8112yes—
57C22×Dic7C2×Q8112yes—
58C22×Dic7C24112yes—
59C22×Dic7C2413yes—
60C22×Dic7C2411yescharacteristic
61C7×C22⋊C4—46no—
62C7×C22⋊C4—412no—
63C22⋊Dic7—46no—
64C22⋊Dic7—412no—
65C4⋊Dic7—412no—
66Dic7⋊C4—412no—
67Dic7⋊C4—424no—
68C4×Dic7—224no—
69C24×C4—71no—
70C2×C4×Q8—74no—
71C2×#117—76no—
72#400—712no—
73#399—712no—
74#398—724no—
75#396—724no—
76#395—712no—
77#393C2×D7112yes—
78#381—76no—
79#380—712no—
80#379—73no—
81#379C2×D713yes—
82C4×C4⋊C4—712no—
83C4×C22⋊C4—712no—
84C23×C7—26no—
85C23×C7—212no—
86C23×C7—44no—
87C23×C7—48no—
88C23×C7—416no—
89C23×C7C22×Q816yes—
90C23×C7C2511yescharacteristic
91C7×Q8—416no—
92C2×C4×C7—212no—
93C2×C4×C7—224no—
94C2×Dic7—26no—
95C2×Dic7—224no—
96C2×Dic7—212no—
97C2×Dic7—224no—
98C2×Dic7—412no—
99C2×Dic7C22×Q816yes—
100C2×Dic7C22×Q812yes—
101C2×Dic7H(5,𝔽2)14yes—
102C25C22×D711yescharacteristic
103C22×Q8C22×D714yes—
104C23×C4—73no—
105C23×C4—78no—
106C23×C4—73no—
107C23×C4—284no—
108C4×Q8—288no—
109#117—2812no—
110C2×C4⋊C4—712no—
111C2×C4⋊C4—712no—
112C2×C4⋊C4—724no—
113C2×C22⋊C4—76no—
114C2×C22⋊C4—712no—
115C2×C22⋊C4C22×D716yes—
116C2×C22⋊C4C22×D7112yes—
117C2×C42—712no—
118#95—724no—
119#95—712no—
120#95C22×D7112yes—
121C22×C7—26no—
122C22×C7—212no—
123C22×C7—416no—
124C22×C7—412no—
125C22×C7C23×Q811yescharacteristic
126C22×C7C2×H(5,𝔽2)12yes—
127C22×C7#58514yes—
128C4×C7—412no—
129Dic7—26no—
130Dic7—412no—
131Dic7#54314yes—
132C24—42no—
133C24—44no—
134C24Q8×D714yes—
135C24C23×D713yes—
136C2×Q8—44no—
137C2×Q8—48no—
138C22×C4—71no—
139C22×C4—712no—
140C22×C4—73no—
141C22×C4—712no—
142C22×C4—712no—
143C22×C4—1412no—
144C22×C4—2816no—
145C22×C4—286no—
146C22×C4C23×D7112yes—
147C4⋊C4—2824no—
148C4⋊C4—2812no—
149C4⋊C4—2812no—
150C22⋊C4—46no—
151C22⋊C4—412no—
152C22⋊C4—286no—
153C22⋊C4—2812no—
154C42—1424no—
155C2×C7—46no—
156C2×C7C2×#54311yescharacteristic
157C2×C7#345812yes—
158C2×C7#352914yes—
159C23—26no—
160C23—212no—
161C23—44no—
162C23—48no—
163C23—416no—
164C23C2×Q8×D716yes—
165C23C24×D711yescharacteristic
166Q8—416no—
167C2×C4—212no—
168C2×C4—224no—
169C2×C4—72no—
170C2×C4—74no—
171C2×C4—76no—
172C2×C4—1424no—
173C2×C4—146no—
174C2×C4—1412no—
175C2×C4—1424no—
176C2×C4—2812no—
177C7#3667411yescharacteristic, complemented, Sylow
178C22—212no—
179C22—26no—
180C22—416no—
181C22—412no—
182C22C22×Q8×D711yescharacteristic
183C22D7×H(5,𝔽2)12yes—
184C22#9061814yes—
185C4—412no—
186C4—74no—
187C4—146no—
188C4—2812no—
189C2—46no—
190C2#1174363912yes—
191C2D7×#54311yescharacteristic
192C2#1174415314yes—
subgroupG/H
center, FrattiniC23C24×D7
commutator, socle, cosocleC23×C7C25
FittingC7×#2840C2
prime radicalC25×C7C23
hypercenter#2840D7
hypocenterC7#36674
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #2840

Complements

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

NH ≅ G/Ntype
2. #11734306C2semidirect
3. #11734223C2semidirect
7. C2×Q8×Dic7C22semidirect
12. #90001C22semidirect
18. #89431C22semidirect
19. #89429C22semidirect
20. #89421C22semidirect
25. C23×Dic7Q8semidirect
177. C7#36674semidirect

Maximal subgroups

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

subgroupindex
C7×#28402
#117343062
#117342232
#117312902
#117312622
#366747