Finite Group Databasebeta

← C71×#12313 #31833185

Subgroups — C71×#12313 #31833185

Order 18176.

356 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 78 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×#3293C211yesmaximal, characteristic
2C71×#2214C212yesmaximal, complemented
3C71×#1713C211yesmaximal, characteristic, complemented
4C71×#1358C211yesmaximal, characteristic, complemented
5C71×#1335C212yesmaximal, complemented
6#578×C71—22no—
7#510×C71—22no—
8#500×C71—22no—
9#494×C71C2211yescharacteristic
10#494×C71C2211yescharacteristic
11#473×C71—22no—
12#456×C71—22no—
13#443×C71C2212yes—
14#364×C71C2212yes—
15#362×C71—22no—
16C82×C71C2211yescharacteristic
172−1+4×C71—22no—
18#137×C71—24no—
19#137×C71—44no—
20#135×C71—42no—
21#135×C71D412yes—
22C2×Q16×C71—22no—
23C2×Q16×C71—22no—
24C2×Q16×C71—42no—
25#131×C71—22no—
26C4⋊Q8×C71—21no—
27C4⋊Q8×C71D412yes—
28Q32×C71—42no—
29SD32×C71—42no—
30M32×C71—22no—
31#108×C71D412yes—
32#107×C71—44no—
33#104×C71—22no—
34#101×C71—22no—
35C4×C8×C71—21no—
36C4×C8×C71C2311yescharacteristic
37#41×C71—22no—
38#41×C71—44no—
39#41×C71—82no—
40C2×Q8×C71—22no—
41C2×Q8×C71—44no—
42C2×Q8×C71—41no—
43Q16×C71—44no—
44Q16×C71—42no—
45Q16×C71—42no—
46Q16×C71—42no—
47Q16×C71—42no—
48Q16×C71D812yes—
49SD16×C71—22no—
50SD16×C71—44no—
51D8×C71D812yes—
52M16×C71—22no—
53M16×C71—42no—
54C2×C8×C71—22no—
55C2×C8×C71—42no—
56C2×C8×C71C2×D412yes—
57C4⋊C4×C71—42no—
58C4⋊C4×C71—42no—
59C42×C71C2×D411yescharacteristic
60C16×C71—42no—
61Q8×C71—22no—
62Q8×C71—44no—
63Q8×C71—82no—
64Q8×C71—82no—
65D4×C71—22no—
66D4×C71—42no—
67D4×C71—82no—
68C2×C4×C71—21no—
69C2×C4×C71—42no—
70C2×C4×C71—42no—
71C2×C4×C71—84no—
72C2×C4×C71—81no—
73C2×C4×C71#12011yescharacteristic
74C8×C71—24no—
75C8×C71—42no—
76C8×C71C2×D812yes—
77C8×C71AGL(1,ℤ/8)12yes—
78C22×C71—42no—
79C22×C71#45311yescharacteristic
80C4×C71—22no—
81C4×C71—42no—
82C4×C71—82no—
83C4×C71—161no—
84C4×C71#44712yes—
85#12313C7111yesmaximal, characteristic, direct product factor, Sylow
86C2×C71—21no—
87C2×C71—82no—
88C2×C71#161911yescharacteristic
89#3293C2×C7111yescharacteristic
90#2214C2×C7112yescomplemented
91#1713C2×C7111yescharacteristic, complemented
92#1358C2×C7111yescharacteristic, complemented
93#1335C2×C7112yescomplemented
94C71#1231311yescharacteristic, direct product factor, Sylow
95#578—22no—
96#510—22no—
97#500—22no—
98#494C22×C7111yescharacteristic
99#494C22×C7111yescharacteristic
100#473—22no—
101#456—22no—
102#443C22×C7112yes—
103#364C22×C7112yes—
104#362—22no—
105C82C22×C7111yescharacteristic
1062−1+4—22no—
107#137—24no—
108#137—44no—
109#135—42no—
110#135D4×C7112yes—
111C2×Q16—22no—
112C2×Q16—22no—
113C2×Q16—42no—
114#131—22no—
115C4⋊Q8—21no—
116C4⋊Q8D4×C7112yes—
117Q32—42no—
118SD32—42no—
119M32—22no—
120#108D4×C7112yes—
121#107—44no—
122#104—22no—
123#101—22no—
124C4×C8—21no—
125C4×C8C23×C7111yescharacteristic
126#41—22no—
127#41—44no—
128#41—82no—
129C2×Q8—22no—
130C2×Q8—44no—
131C2×Q8—41no—
132Q16—44no—
133Q16—42no—
134Q16—42no—
135Q16—42no—
136Q16—42no—
137Q16D8×C7112yes—
138SD16—22no—
139SD16—44no—
140D8D8×C7112yes—
141M16—22no—
142M16—42no—
143C2×C8—22no—
144C2×C8—42no—
145C2×C8C2×D4×C7112yes—
146C4⋊C4—42no—
147C4⋊C4—42no—
148C42C2×D4×C7111yescharacteristic
149C16—42no—
150Q8—22no—
151Q8—44no—
152Q8—82no—
153Q8—82no—
154D4—22no—
155D4—42no—
156D4—82no—
157C2×C4—21no—
158C2×C4—42no—
159C2×C4—42no—
160C2×C4—84no—
161C2×C4—81no—
162C2×C4#120×C7111yescharacteristic
163C8—24no—
164C8—42no—
165C8C2×D8×C7112yes—
166C8AGL(1,ℤ/8)×C7112yes—
167C22—42no—
168C22#453×C7111yescharacteristic
169C4—22no—
170C4—42no—
171C4—82no—
172C4—161no—
173C4#447×C7112yes—
174C2—21no—
175C2—82no—
176C2C71×#161911yescharacteristic
subgroupG/H
center, socleC2×C71#1619
commutator, Frattini, cosocleC4×C8C23×C71
prime radicalC71×#3293C2
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C71 ◁ C2×C4×C71 ◁ C4×C8×C71 ◁ G

Complements

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

NH ≅ G/Ntype
2. C71×#2214C2semidirect
3. C71×#1713C2semidirect
4. C71×#1358C2semidirect
5. C71×#1335C2semidirect
85. #12313C71direct (N × H)
90. #2214C2×C71semidirect
91. #1713C2×C71semidirect
92. #1358C2×C71semidirect
93. #1335C2×C71semidirect
94. C71#12313direct (N × H)

Maximal subgroups

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

subgroupindex
C71×#32932
C71×#22142
C71×#17132
C71×#13582
C71×#13352
#1231371