Finite Group Databasebeta

← C71×#12261 #31833133

Subgroups — C71×#12261 #31833133

Order 18176.

330 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 74 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×#2844C211yesmaximal, characteristic
2C71×#2061C211yesmaximal, characteristic
3C71×#1617C212yesmaximal
4C71×#1353C212yesmaximal
5C71×#1324C211yesmaximal, characteristic
6#544×C71C2211yescharacteristic
7#484×C71—22no—
8#428×C71—21no—
9#400×C71—22no—
10#400×C71C2212yes—
11#395×C71—22no—
12#395×C71—21no—
13#382×C71—21no—
14#382×C71C2211yescharacteristic
15#343×C71C2211yescharacteristic
16#339×C71—21no—
17#328×C71—22no—
18#324×C71C2212yes—
19C22×Q8×C71—21no—
20C2×M16×C71—21no—
21C4⋊Q8×C71—41no—
22#125×C71—41no—
23C22⋊Q8×C71—22no—
24#117×C71—21no—
25C2×C4⋊C4×C71—21no—
26C2×C4⋊C4×C71—21no—
27C2×C4⋊C4×C71—22no—
28C2×C4⋊C4×C71D411yescharacteristic
29C2×C4⋊C4×C71D412yes—
30C2×C4⋊C4×C71C2311yescharacteristic
31C2×C42×C71D411yescharacteristic
32#107×C71—41no—
33#107×C71—42no—
34#106×C71—41no—
35#103×C71—42no—
36C22⋊C8×C71—22no—
37#95×C71—22no—
38#95×C71—21no—
39#95×C71—22no—
40#95×C71—22no—
41#95×C71—22no—
42#95×C71D412yes—
43C2×Q8×C71—22no—
44C2×Q8×C71—82no—
45C2×Q8×C71—81no—
46C22×C4×C71—21no—
47C22×C4×C71—22no—
48C22×C4×C71—22no—
49C22×C4×C71—21no—
50C22×C4×C71C2×D411yescharacteristic
51C22×C4×C71C2×D412yes—
52M16×C71—22no—
53M16×C71—41no—
54C2×C8×C71—41no—
55C2×C8×C71—42no—
56C4⋊C4×C71—21no—
57C4⋊C4×C71—21no—
58C4⋊C4×C71—22no—
59C4⋊C4×C71—41no—
60C4⋊C4×C71—42no—
61C4⋊C4×C71—41no—
62C4⋊C4×C71—42no—
63C4⋊C4×C71—81no—
64C4⋊C4×C71—81no—
65C4⋊C4×C71—82no—
66C22⋊C4×C71—41no—
67C42×C71—21no—
68C42×C71—41no—
69C42×C71Q1612yes—
70C23×C71#12011yescharacteristic
71Q8×C71—82no—
72Q8×C71—84no—
73C2×C4×C71—21no—
74C2×C4×C71—21no—
75C2×C4×C71—22no—
76C2×C4×C71—22no—
77C2×C4×C71—41no—
78C2×C4×C71—41no—
79C2×C4×C71—42no—
80C2×C4×C71—42no—
81C2×C4×C71—42no—
82C2×C4×C71—42no—
83C2×C4×C71—42no—
84C2×C4×C71—42no—
85C2×C4×C71—42no—
86C2×C4×C71—41no—
87C2×C4×C71—41no—
88C2×C4×C71—42no—
89C2×C4×C71—41no—
90C2×C4×C71C2×Q1611yescharacteristic
91C2×C4×C71#13711yescharacteristic
92C8×C71—42no—
93C8×C71—82no—
94C22×C71—21no—
95C22×C71—21no—
96C22×C71#45111yescharacteristic
97C22×C71#45611yescharacteristic
98C22×C71B(4,𝔽2)11yescharacteristic
99C4×C71—21no—
100C4×C71—21no—
101C4×C71—42no—
102C4×C71—42no—
103C4×C71—82no—
104C4×C71—81no—
105C4×C71—81no—
106C4×C71—82no—
107C4×C71—81no—
108#12261C7111yesmaximal, characteristic, direct product factor, Sylow
109C2×C71—21no—
110C2×C71—21no—
111C2×C71#160511yescharacteristic
112C2×C71#220111yescharacteristic
113C2×C71#220211yescharacteristic
114#2844C2×C7111yescharacteristic
115#2061C2×C7111yescharacteristic
116#1617C2×C7112yes—
117#1353C2×C7112yes—
118#1324C2×C7111yescharacteristic
119C71#1226111yescharacteristic, direct product factor, Sylow
120#544C22×C7111yescharacteristic
121#484—22no—
122#428—21no—
123#400—22no—
124#400C22×C7112yes—
125#395—22no—
126#395—21no—
127#382—21no—
128#382C22×C7111yescharacteristic
129#343C22×C7111yescharacteristic
130#339—21no—
131#328—22no—
132#324C22×C7112yes—
133C22×Q8—21no—
134C2×M16—21no—
135C4⋊Q8—41no—
136#125—41no—
137C22⋊Q8—22no—
138#117—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4D4×C7111yescharacteristic
143C2×C4⋊C4D4×C7112yes—
144C2×C4⋊C4C23×C7111yescharacteristic
145C2×C42D4×C7111yescharacteristic
146#107—41no—
147#107—42no—
148#106—41no—
149#103—42no—
150C22⋊C8—22no—
151#95—22no—
152#95—21no—
153#95—22no—
154#95—22no—
155#95—22no—
156#95D4×C7112yes—
157C2×Q8—22no—
158C2×Q8—82no—
159C2×Q8—81no—
160C22×C4—21no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—21no—
164C22×C4C2×D4×C7111yescharacteristic
165C22×C4C2×D4×C7112yes—
166M16—22no—
167M16—41no—
168C2×C8—41no—
169C2×C8—42no—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—22no—
173C4⋊C4—41no—
174C4⋊C4—42no—
175C4⋊C4—41no—
176C4⋊C4—42no—
177C4⋊C4—81no—
178C4⋊C4—81no—
179C4⋊C4—82no—
180C22⋊C4—41no—
181C42—21no—
182C42—41no—
183C42Q16×C7112yes—
184C23#120×C7111yescharacteristic
185Q8—82no—
186Q8—84no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—42no—
203C2×C4—41no—
204C2×C4C2×Q16×C7111yescharacteristic
205C2×C4#137×C7111yescharacteristic
206C8—42no—
207C8—82no—
208C22—21no—
209C22—21no—
210C22#451×C7111yescharacteristic
211C22#456×C7111yescharacteristic
212C22B(4,𝔽2)×C7111yescharacteristic
213C4—21no—
214C4—21no—
215C4—42no—
216C4—42no—
217C4—82no—
218C4—81no—
219C4—82no—
220C4—81no—
221C4—81no—
222C2—21no—
223C2—21no—
224C2C71×#160511yescharacteristic
225C2C71×#220111yescharacteristic
226C2C71×#220211yescharacteristic
subgroupG/H
center, socleC22×C71B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C71
prime radicalC23×C71#120
LCSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C71 ◁ C23×C71 ◁ C2×C4⋊C4×C71 ◁ G

Complements

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

NH ≅ G/Ntype
108. #12261C71direct (N × H)
119. C71#12261direct (N × H)

Maximal subgroups

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

subgroupindex
C71×#28442
C71×#20612
C71×#16172
C71×#13532
C71×#13242
#1226171