Finite Group Databasebeta

← C71×#7375 #31828247

Subgroups — C71×#7375 #31828247

Order 18176.

228 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 50 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×#1856C211yesmaximal, characteristic
2C71×#1318C211yesmaximal, characteristic
3C71×#1270C211yesmaximal, characteristic
4#518×C71C411yescharacteristic
5#428×C71—21no—
6C2×#100×C71C411yescharacteristic
7#390×C71C2211yescharacteristic
8#339×C71—21no—
9#336×C71—21no—
10#324×C71—21no—
11C22×D4×C71C2×C411yescharacteristic
12C2×M16×C71—21no—
13C22×C8×C71—41no—
14C4×D4×C71—41no—
15C4×D4×C71—41no—
16#117×C71—21no—
17C2×C4⋊C4×C71D411yescharacteristic
18C2×C22⋊C4×C71—41no—
19C2×C22⋊C4×C71—41no—
20C2×C42×C71D411yescharacteristic
21#107×C71—41no—
22#106×C71—41no—
23#100×C71—21no—
24#100×C71C812yescomplemented
25C22⋊C8×C71—41no—
26C24×C71—21no—
27C2×D4×C71—21no—
28C2×D4×C71—41no—
29C2×D4×C71—41no—
30C2×D4×C71C2×C811yescharacteristic
31C2×D4×C71M1611yescharacteristic
32C22×C4×C71—21no—
33C22×C4×C71—21no—
34C22×C4×C71—41no—
35C22×C4×C71—41no—
36C22×C4×C71C22⋊C411yescharacteristic
37M16×C71—22no—
38M16×C71—41no—
39C2×C8×C71—41no—
40C2×C8×C71—42no—
41C2×C8×C71—81no—
42C2×C8×C71—81no—
43C2×C8×C71—81no—
44C4⋊C4×C71—21no—
45C4⋊C4×C71—21no—
46C4⋊C4×C71—41no—
47C22⋊C4×C71—42no—
48C22⋊C4×C71—41no—
49C22⋊C4×C71—81no—
50C22⋊C4×C71—81no—
51C42×C71—41no—
52C42×C71—41no—
53C23×C71—21no—
54C23×C71—21no—
55C23×C71—41no—
56C23×C71—41no—
57C23×C71—81no—
58C23×C71—81no—
59C23×C71#9911yescharacteristic
60D4×C71—41no—
61D4×C71—41no—
62D4×C71—42no—
63C2×C4×C71—21no—
64C2×C4×C71—21no—
65C2×C4×C71—21no—
66C2×C4×C71—21no—
67C2×C4×C71—41no—
68C2×C4×C71—41no—
69C2×C4×C71—41no—
70C2×C4×C71—41no—
71C2×C4×C71—41no—
72C2×C4×C71—81no—
73C2×C4×C71—161no—
74C2×C4×C71—161no—
75C2×C4×C71C22⋊C811yescharacteristic
76C2×C4×C71#10111yescharacteristic
77C8×C71—42no—
78C8×C71—82no—
79C8×C71—161no—
80C22×C71—21no—
81C22×C71—21no—
82C22×C71—41no—
83C22×C71—41no—
84C22×C71—81no—
85C22×C71—81no—
86C22×C71—81no—
87C22×C71—81no—
88C22×C71—161no—
89C22×C71#32411yescharacteristic
90C22×C71AΣL(1,𝔽16)11yescharacteristic
91C22×C71#35211yescharacteristic
92C4×C71—21no—
93C4×C71—21no—
94C4×C71—41no—
95C4×C71—41no—
96C4×C71—81no—
97C4×C71—81no—
98C4×C71—81no—
99#7375C7111yesmaximal, characteristic, direct product factor, Sylow
100C2×C71—21no—
101C2×C71—21no—
102C2×C71—81no—
103C2×C71—81no—
104C2×C71#132611yescharacteristic
105C2×C71#140211yescharacteristic
106C2×C71#140311yescharacteristic
107#1856C2×C7111yescharacteristic
108#1318C2×C7111yescharacteristic
109#1270C2×C7111yescharacteristic
110C71#737511yescharacteristic, direct product factor, Sylow
111#518C4×C7111yescharacteristic
112#428—21no—
113C2×#100C4×C7111yescharacteristic
114#390C22×C7111yescharacteristic
115#339—21no—
116#336—21no—
117#324—21no—
118C22×D4C2×C4×C7111yescharacteristic
119C2×M16—21no—
120C22×C8—41no—
121C4×D4—41no—
122C4×D4—41no—
123#117—21no—
124C2×C4⋊C4D4×C7111yescharacteristic
125C2×C22⋊C4—41no—
126C2×C22⋊C4—41no—
127C2×C42D4×C7111yescharacteristic
128#107—41no—
129#106—41no—
130#100—21no—
131#100C8×C7112yescomplemented
132C22⋊C8—41no—
133C24—21no—
134C2×D4—21no—
135C2×D4—41no—
136C2×D4—41no—
137C2×D4C2×C8×C7111yescharacteristic
138C2×D4M16×C7111yescharacteristic
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—41no—
142C22×C4—41no—
143C22×C4C22⋊C4×C7111yescharacteristic
144M16—22no—
145M16—41no—
146C2×C8—41no—
147C2×C8—42no—
148C2×C8—81no—
149C2×C8—81no—
150C2×C8—81no—
151C4⋊C4—21no—
152C4⋊C4—21no—
153C4⋊C4—41no—
154C22⋊C4—42no—
155C22⋊C4—41no—
156C22⋊C4—81no—
157C22⋊C4—81no—
158C42—41no—
159C42—41no—
160C23—21no—
161C23—21no—
162C23—41no—
163C23—41no—
164C23—81no—
165C23—81no—
166C23#99×C7111yescharacteristic
167D4—41no—
168D4—41no—
169D4—42no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—81no—
180C2×C4—161no—
181C2×C4—161no—
182C2×C4C22⋊C8×C7111yescharacteristic
183C2×C4#101×C7111yescharacteristic
184C8—42no—
185C8—82no—
186C8—161no—
187C22—21no—
188C22—21no—
189C22—41no—
190C22—41no—
191C22—81no—
192C22—81no—
193C22—81no—
194C22—81no—
195C22—161no—
196C22#324×C7111yescharacteristic
197C22AΣL(1,𝔽16)×C7111yescharacteristic
198C22#352×C7111yescharacteristic
199C4—21no—
200C4—21no—
201C4—41no—
202C4—41no—
203C4—81no—
204C4—81no—
205C4—81no—
206C2—21no—
207C2—21no—
208C2—81no—
209C2—81no—
210C2C71×#132611yescharacteristic
211C2C71×#140211yescharacteristic
212C2C71×#140311yescharacteristic
subgroupG/H
center, socleC22×C71AΣL(1,𝔽16)
commutatorC2×D4C2×C8×C71
Frattini, cosocle#390C22×C71
prime radicalC22×D4×C71C2×C4
LCSG ▷ C2×D4 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C71 ◁ C23×C71 ◁ C22×C4×C71 ◁ #390×C71 ◁ G

Complements

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

NH ≅ G/Ntype
24. #100×C71C8semidirect
99. #7375C71direct (N × H)
110. C71#7375direct (N × H)
131. #100C8×C71semidirect

Maximal subgroups

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

subgroupindex
C71×#18562
C71×#13182
C71×#12702
#737571