Finite Group Databasebeta

← C71×#12292 #31833164

Subgroups — C71×#12292 #31833164

Order 18176.

310 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 82 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×#3249C211yesmaximal, characteristic
2C71×#2236C212yesmaximal
3C71×#2087C211yesmaximal, characteristic
4C71×#1348C212yesmaximal
5C71×#1322C211yesmaximal, characteristic
6#533×C71—21no—
7#477×C71—22no—
8#467×C71—21no—
9#467×C71C2212yes—
10#462×C71—22no—
11C2×#107×C71C2211yescharacteristic
12C2×C4⋊C8×C71C2211yescharacteristic
13#389×C71—21no—
14#366×C71—22no—
15#358×C71—22no—
16C22⋊C16×C71C2212yes—
17#340×C71—21no—
18#340×C71C2211yescharacteristic
19C2×Q16×C71—21no—
20C2×Q16×C71D412yes—
21C22×C8×C71D411yescharacteristic
22C22×C8×C71C2311yescharacteristic
23C4⋊Q8×C71—21no—
24C4⋊Q8×C71—21no—
25#125×C71—41no—
26C22⋊Q8×C71—22no—
27C22⋊Q8×C71—41no—
28C4×Q8×C71—22no—
29C4×Q8×C71—41no—
30#117×C71—41no—
31C2×C4⋊C4×C71—21no—
32C2×C4⋊C4×C71—21no—
33C2×C4⋊C4×C71—41no—
34C2×C42×C71D411yescharacteristic
35C2×C16×C71—22no—
36#107×C71—21no—
37#107×C71—22no—
38#107×C71D412yes—
39C4⋊C8×C71—22no—
40#103×C71—22no—
41#103×C71—22no—
42#95×C71—41no—
43C2×Q8×C71—22no—
44C2×Q8×C71—41no—
45C22×C4×C71—41no—
46C22×C4×C71—41no—
47C22×C4×C71D811yescharacteristic
48C22×C4×C71D811yescharacteristic
49C22×C4×C71C2×D411yescharacteristic
50Q16×C71—42no—
51Q16×C71—81no—
52C2×C8×C71—21no—
53C2×C8×C71—41no—
54C2×C8×C71—41no—
55C2×C8×C71C2×D412yes—
56C4⋊C4×C71—22no—
57C4⋊C4×C71—22no—
58C4⋊C4×C71—41no—
59C4⋊C4×C71—41no—
60C4⋊C4×C71—41no—
61C4⋊C4×C71—42no—
62C4⋊C4×C71—41no—
63C4⋊C4×C71—42no—
64C4⋊C4×C71—41no—
65C4⋊C4×C71—41no—
66C4⋊C4×C71—82no—
67C22⋊C4×C71—42no—
68C42×C71—21no—
69C42×C71—21no—
70C42×C71—42no—
71C16×C71—42no—
72C23×C71C2×D811yescharacteristic
73Q8×C71—42no—
74Q8×C71—42no—
75Q8×C71—82no—
76C2×C4×C71—21no—
77C2×C4×C71—21no—
78C2×C4×C71—21no—
79C2×C4×C71—21no—
80C2×C4×C71—21no—
81C2×C4×C71—41no—
82C2×C4×C71—42no—
83C2×C4×C71—42no—
84C2×C4×C71—42no—
85C2×C4×C71—81no—
86C2×C4×C71—81no—
87C2×C4×C71—81no—
88C2×C4×C71—81no—
89C2×C4×C71—81no—
90C2×C4×C71Q3212yes—
91C2×C4×C71#12011yescharacteristic
92C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
93C8×C71—22no—
94C8×C71—41no—
95C22×C71—21no—
96C22×C71—21no—
97C22×C71#44711yescharacteristic
98C22×C71C2×Q3211yescharacteristic
99C22×C71#51011yescharacteristic
100C4×C71—21no—
101C4×C71—22no—
102C4×C71—41no—
103C4×C71—82no—
104C4×C71—82no—
105C4×C71—82no—
106C4×C71—161no—
107C4×C71#45611yescharacteristic
108C4×C71#45811yescharacteristic
109#12292C7111yesmaximal, characteristic, direct product factor, Sylow
110C2×C71—21no—
111C2×C71—21no—
112C2×C71#161011yescharacteristic
113C2×C71#218811yescharacteristic
114C2×C71#219011yescharacteristic
115#3249C2×C7111yescharacteristic
116#2236C2×C7112yes—
117#2087C2×C7111yescharacteristic
118#1348C2×C7112yes—
119#1322C2×C7111yescharacteristic
120C71#1229211yescharacteristic, direct product factor, Sylow
121#533—21no—
122#477—22no—
123#467—21no—
124#467C22×C7112yes—
125#462—22no—
126C2×#107C22×C7111yescharacteristic
127C2×C4⋊C8C22×C7111yescharacteristic
128#389—21no—
129#366—22no—
130#358—22no—
131C22⋊C16C22×C7112yes—
132#340—21no—
133#340C22×C7111yescharacteristic
134C2×Q16—21no—
135C2×Q16D4×C7112yes—
136C22×C8D4×C7111yescharacteristic
137C22×C8C23×C7111yescharacteristic
138C4⋊Q8—21no—
139C4⋊Q8—21no—
140#125—41no—
141C22⋊Q8—22no—
142C22⋊Q8—41no—
143C4×Q8—22no—
144C4×Q8—41no—
145#117—41no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—41no—
149C2×C42D4×C7111yescharacteristic
150C2×C16—22no—
151#107—21no—
152#107—22no—
153#107D4×C7112yes—
154C4⋊C8—22no—
155#103—22no—
156#103—22no—
157#95—41no—
158C2×Q8—22no—
159C2×Q8—41no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4D8×C7111yescharacteristic
163C22×C4D8×C7111yescharacteristic
164C22×C4C2×D4×C7111yescharacteristic
165Q16—42no—
166Q16—81no—
167C2×C8—21no—
168C2×C8—41no—
169C2×C8—41no—
170C2×C8C2×D4×C7112yes—
171C4⋊C4—22no—
172C4⋊C4—22no—
173C4⋊C4—41no—
174C4⋊C4—41no—
175C4⋊C4—41no—
176C4⋊C4—42no—
177C4⋊C4—41no—
178C4⋊C4—42no—
179C4⋊C4—41no—
180C4⋊C4—41no—
181C4⋊C4—82no—
182C22⋊C4—42no—
183C42—21no—
184C42—21no—
185C42—42no—
186C16—42no—
187C23C2×D8×C7111yescharacteristic
188Q8—42no—
189Q8—42no—
190Q8—82no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—41no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—81no—
201C2×C4—81no—
202C2×C4—81no—
203C2×C4—81no—
204C2×C4—81no—
205C2×C4Q32×C7112yes—
206C2×C4#120×C7111yescharacteristic
207C2×C4AGL(1,ℤ/8)×C7111yescharacteristic
208C8—22no—
209C8—41no—
210C22—21no—
211C22—21no—
212C22#447×C7111yescharacteristic
213C22C2×Q32×C7111yescharacteristic
214C22#510×C7111yescharacteristic
215C4—21no—
216C4—22no—
217C4—41no—
218C4—82no—
219C4—82no—
220C4—82no—
221C4—161no—
222C4#456×C7111yescharacteristic
223C4#458×C7111yescharacteristic
224C2—21no—
225C2—21no—
226C2C71×#161011yescharacteristic
227C2C71×#218811yescharacteristic
228C2C71×#219011yescharacteristic
subgroupG/H
center, socleC22×C71#447
commutator, Frattini, cosocleC22×C8C23×C71
prime radicalC23×C71C2×D8
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C71 ◁ C22×C4×C71 ◁ C2×C4⋊C8×C71 ◁ G

Complements

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

NH ≅ G/Ntype
109. #12292C71direct (N × H)
120. C71#12292direct (N × H)

Maximal subgroups

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

subgroupindex
C71×#32492
C71×#22362
C71×#20872
C71×#13482
C71×#13222
#1229271