Finite Group Databasebeta

← C71×#7349 #31828221

Subgroups — C71×#7349 #31828221

Order 18176.

302 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 46 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
1C71×#1819C211yesmaximal, characteristic
2C71×#1316C212yesmaximal
3C22×C22⋊C4×C71C411yescharacteristic
4#379×C71C411yescharacteristic
5#379×C71C2211yescharacteristic
6#342×C71—22no—
7#323×C71—22no—
8C25×C71C2×C411yescharacteristic
9C23×C4×C71—21no—
10C2×C22⋊C4×C71—21no—
11C2×C22⋊C4×C71—21no—
12C2×C22⋊C4×C71—41no—
13C2×C22⋊C4×C71—41no—
14C2×C22⋊C4×C71—41no—
15C2×C22⋊C4×C71—41no—
16C2×C22⋊C4×C71C812yescomplemented
17C2×C22⋊C4×C71D412yes—
18C22⋊C8×C71—42no—
19C24×C71—21no—
20C24×C71—21no—
21C24×C71—42no—
22C24×C71—82no—
23C24×C71C22⋊C411yescharacteristic
24C24×C71C2×C811yescharacteristic
25C24×C71M1611yescharacteristic
26C22×C4×C71—22no—
27C22×C4×C71—22no—
28C22×C4×C71—41no—
29C22×C4×C71—41no—
30C22×C4×C71—41no—
31C22×C4×C71—82no—
32C2×C8×C71—82no—
33C22⋊C4×C71—41no—
34C22⋊C4×C71—41no—
35C22⋊C4×C71—42no—
36C22⋊C4×C71—42no—
37C22⋊C4×C71—42no—
38C22⋊C4×C71—82no—
39C22⋊C4×C71—82no—
40C23×C71—21no—
41C23×C71—22no—
42C23×C71—21no—
43C23×C71—42no—
44C23×C71—41no—
45C23×C71—42no—
46C23×C71—42no—
47C23×C71—42no—
48C23×C71—41no—
49C23×C71—42no—
50C23×C71—81no—
51C23×C71—81no—
52C23×C71—81no—
53C23×C71—81no—
54C23×C71—84no—
55C23×C71—84no—
56C23×C71C22⋊C811yescharacteristic
57C23×C71#9911yescharacteristic
58C23×C71#10011yescharacteristic
59C2×C4×C71—42no—
60C2×C4×C71—42no—
61C2×C4×C71—82no—
62C2×C4×C71—82no—
63C2×C4×C71—82no—
64C2×C4×C71—82no—
65C2×C4×C71—81no—
66C2×C4×C71—82no—
67C8×C71—162no—
68C22×C71—22no—
69C22×C71—21no—
70C22×C71—21no—
71C22×C71—41no—
72C22×C71—41no—
73C22×C71—41no—
74C22×C71—41no—
75C22×C71—41no—
76C22×C71—41no—
77C22×C71—42no—
78C22×C71—42no—
79C22×C71—41no—
80C22×C71—41no—
81C22×C71—82no—
82C22×C71—84no—
83C22×C71—84no—
84C22×C71—82no—
85C22×C71#32311yescharacteristic
86C22×C71#35212yes—
87C4×C71—82no—
88C4×C71—82no—
89C4×C71—161no—
90#7349C7111yesmaximal, characteristic, direct product factor, Sylow
91C2×C71—21no—
92C2×C71—22no—
93C2×C71—41no—
94C2×C71—42no—
95C2×C71—41no—
96C2×C71—81no—
97C2×C71#131711yescharacteristic
98#1819C2×C7111yescharacteristic
99#1316C2×C7112yes—
100C71#734911yescharacteristic, direct product factor, Sylow
101C22×C22⋊C4C4×C7111yescharacteristic
102#379C4×C7111yescharacteristic
103#379C22×C7111yescharacteristic
104#342—22no—
105#323—22no—
106C25C2×C4×C7111yescharacteristic
107C23×C4—21no—
108C2×C22⋊C4—21no—
109C2×C22⋊C4—21no—
110C2×C22⋊C4—41no—
111C2×C22⋊C4—41no—
112C2×C22⋊C4—41no—
113C2×C22⋊C4—41no—
114C2×C22⋊C4C8×C7112yescomplemented
115C2×C22⋊C4D4×C7112yes—
116C22⋊C8—42no—
117C24—21no—
118C24—21no—
119C24—42no—
120C24—82no—
121C24C22⋊C4×C7111yescharacteristic
122C24C2×C8×C7111yescharacteristic
123C24M16×C7111yescharacteristic
124C22×C4—22no—
125C22×C4—22no—
126C22×C4—41no—
127C22×C4—41no—
128C22×C4—41no—
129C22×C4—82no—
130C2×C8—82no—
131C22⋊C4—41no—
132C22⋊C4—41no—
133C22⋊C4—42no—
134C22⋊C4—42no—
135C22⋊C4—42no—
136C22⋊C4—82no—
137C22⋊C4—82no—
138C23—21no—
139C23—22no—
140C23—21no—
141C23—42no—
142C23—41no—
143C23—42no—
144C23—42no—
145C23—42no—
146C23—41no—
147C23—42no—
148C23—81no—
149C23—81no—
150C23—81no—
151C23—81no—
152C23—84no—
153C23—84no—
154C23C22⋊C8×C7111yescharacteristic
155C23#99×C7111yescharacteristic
156C23#100×C7111yescharacteristic
157C2×C4—42no—
158C2×C4—42no—
159C2×C4—82no—
160C2×C4—82no—
161C2×C4—82no—
162C2×C4—82no—
163C2×C4—81no—
164C2×C4—82no—
165C8—162no—
166C22—22no—
167C22—21no—
168C22—21no—
169C22—41no—
170C22—41no—
171C22—41no—
172C22—41no—
173C22—41no—
174C22—41no—
175C22—42no—
176C22—42no—
177C22—41no—
178C22—41no—
179C22—82no—
180C22—84no—
181C22—84no—
182C22—82no—
183C22#323×C7111yescharacteristic
184C22#352×C7112yes—
185C4—82no—
186C4—82no—
187C4—161no—
188C2—21no—
189C2—22no—
190C2—41no—
191C2—42no—
192C2—41no—
193C2—81no—
194C2C71×#131711yescharacteristic
subgroupG/H
center, socleC2×C71#1317
commutatorC24C2×C8×C71
Frattini, cosocle#379C22×C71
prime radicalC25×C71C2×C4
LCSG ▷ C24 ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C71 ◁ C23×C71 ◁ C24×C71 ◁ #379×C71 ◁ G

Complements

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

NH ≅ G/Ntype
16. C2×C22⋊C4×C71C8semidirect
90. #7349C71direct (N × H)
100. C71#7349direct (N × H)
114. C2×C22⋊C4C8×C71semidirect

Maximal subgroups

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

subgroupindex
C71×#18192
C71×#13162
#734971