Finite Group Databasebeta

← #31910719

Subgroups — #31910719

Order 18176.

310 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 67 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
1#22599685C211yesmaximal, characteristic
2#22590537C212yesmaximal
3#22586351C212yesmaximal
4#22586280C211yesmaximal, characteristic
5C71×#2086C211yesmaximal, characteristic
6#17986556—21no—
7#17986295—22no—
8#17986240—21no—
9#17986240C2212yes—
10#17986210—22no—
11#17985826—22no—
12#17985802—22no—
13C22⋊(C71⋊C16)C2212yes—
14C2×#107×C71C2211yescharacteristic
15C2×C4⋊C8×C71C2211yescharacteristic
16#389×C71—21no—
17#340×C71—21no—
18#340×C71C2211yescharacteristic
19C2×Dic284—21no—
20C2×Dic284D412yes—
21#14556128—21no—
22#14556128—21no—
23#14556120—41no—
24#14556107—22no—
25#14556107—41no—
26C4×Dic142—22no—
27C4×Dic142—41no—
28#14556087—41no—
29C2×C71⋊C16—22no—
30#14556062—21no—
31#14556062D412yes—
32#14556053—22no—
33#14556053—22no—
34C22×C8×C71D411yescharacteristic
35C22×C8×C71C2311yescharacteristic
36C2×C4⋊C4×C71—21no—
37C2×C4⋊C4×C71—21no—
38C2×C4⋊C4×C71—41no—
39C2×C42×C71D411yescharacteristic
40#107×C71—22no—
41C4⋊C8×C71—22no—
42#95×C71—41no—
43C2×Dic142—22no—
44C2×Dic142—41no—
45Dic284—42no—
46Dic284—81no—
47C4⋊Dic71—22no—
48C4⋊Dic71—41no—
49C4⋊Dic71—41no—
50C4⋊Dic71—41no—
51Dic71⋊C4—42no—
52Dic71⋊C4—42no—
53C22⋊Dic71—42no—
54C4×Dic71—42no—
55C71⋊C16—42no—
56C22×C4×C71—41no—
57C22×C4×C71—41no—
58C22×C4×C71D811yescharacteristic
59C22×C4×C71D811yescharacteristic
60C22×C4×C71C2×D411yescharacteristic
61C2×C8×C71—21no—
62C2×C8×C71—41no—
63C2×C8×C71—41no—
64C2×C8×C71C2×D412yes—
65C4⋊C4×C71—22no—
66C4⋊C4×C71—41no—
67C4⋊C4×C71—41no—
68C4⋊C4×C71—41no—
69C4⋊C4×C71—82no—
70C42×C71—21no—
71C42×C71—21no—
72C23×C71C2×D811yescharacteristic
73C2×C4×C71—21no—
74C2×C4×C71—21no—
75C2×C4×C71—21no—
76C2×C4×C71—21no—
77C2×C4×C71—21no—
78C2×C4×C71—41no—
79C2×C4×C71—42no—
80C2×C4×C71—81no—
81C2×C4×C71—81no—
82C2×C4×C71—81no—
83C2×C4×C71—81no—
84C2×C4×C71—81no—
85C2×C4×C71SD3212yes—
86C2×C4×C71#12011yescharacteristic
87C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
88C2×Dic71—42no—
89C2×Dic71—42no—
90Dic142—42no—
91Dic142—42no—
92Dic142—82no—
93C8×C71—22no—
94C8×C71—41no—
95C22×C71—21no—
96C22×C71—21no—
97C22×C71#44711yescharacteristic
98C22×C71C2×SD3211yescharacteristic
99C22×C71#51011yescharacteristic
100C4×C71—21no—
101C4×C71—22no—
102C4×C71—41no—
103C4×C71—82no—
104C4×C71—161no—
105C4×C71#45611yescharacteristic
106C4×C71#45811yescharacteristic
107Dic71—82no—
108Dic71—82no—
109#12289—711nomaximal, Sylow
110C2×C71—21no—
111C2×C71—21no—
112C2×C71#161011yescharacteristic
113C2×C71#218511yescharacteristic
114C2×C71#219111yescharacteristic
115#3249—711no—
116#2235—712no—
117#2086D7111yescharacteristic
118#1348—712no—
119#1322—711no—
120C71#1228911yescharacteristic, complemented, Sylow
121#533—1421no—
122#477—1422no—
123#467—712no—
124#467—1421no—
125#462—1422no—
126C2×#107C2×D7111yescharacteristic
127C2×C4⋊C8C2×D7111yescharacteristic
128#389—21no—
129#367—1422no—
130#358—1422no—
131C22⋊C16—712no—
132#340—21no—
133#340C2×D7111yescharacteristic
134C2×Q16—712no—
135C2×Q16—1421no—
136C22×C8C22⋊D7111yescharacteristic
137C22×C8C22×D7111yescharacteristic
138C4⋊Q8—1421no—
139C4⋊Q8—1421no—
140#125—2841no—
141C22⋊Q8—2841no—
142C22⋊Q8—1422no—
143C4×Q8—2841no—
144C4×Q8—1422no—
145#117—2841no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—41no—
149C2×C42C22⋊D7111yescharacteristic
150C2×C16—1422no—
151#107—22no—
152#107—712no—
153#107—1421no—
154C4⋊C8—22no—
155#103—1422no—
156#103—1422no—
157#95—41no—
158C2×Q8—2841no—
159C2×Q8—1422no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4#1176585011yescharacteristic
163C22×C4#1176585011yescharacteristic
164C22×C4C2×C22⋊D7111yescharacteristic
165Q16—2842no—
166Q16—5681no—
167C2×C8—21no—
168C2×C8—41no—
169C2×C8—41no—
170C2×C8D4×D7112yes—
171C4⋊C4—22no—
172C4⋊C4—41no—
173C4⋊C4—41no—
174C4⋊C4—41no—
175C4⋊C4—82no—
176C4⋊C4—2842no—
177C4⋊C4—2841no—
178C4⋊C4—2841no—
179C4⋊C4—2841no—
180C4⋊C4—2842no—
181C4⋊C4—1422no—
182C22⋊C4—2842no—
183C42—21no—
184C42—21no—
185C42—2842no—
186C16—2842no—
187C23C2×#1176585011yescharacteristic
188Q8—2842no—
189Q8—2842no—
190Q8—5682no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—41no—
197C2×C4—42no—
198C2×C4—81no—
199C2×C4—81no—
200C2×C4—81no—
201C2×C4—81no—
202C2×C4—81no—
203C2×C4—2842no—
204C2×C4—2842no—
205C2×C4#1455607212yes—
206C2×C4#1455609611yescharacteristic
207C2×C4#1455615311yescharacteristic
208C8—22no—
209C8—41no—
210C22—21no—
211C22—21no—
212C22#1798613211yescharacteristic
213C22C2×#1455607211yescharacteristic
214C22#1798643911yescharacteristic
215C4—22no—
216C4—21no—
217C4—41no—
218C4—82no—
219C4—161no—
220C4—5682no—
221C4—5682no—
222C4#1798618111yescharacteristic
223C4#1798618811yescharacteristic
224C2—21no—
225C2—21no—
226C2#2258734211yescharacteristic
227C2#2259025611yescharacteristic
228C2#2259029011yescharacteristic
subgroupG/H
centerC22#17986132
commutator, cosocleC22×C8×C71C23
FrattiniC22×C8C22×D71
FittingC71×#2086C2
socleC22×C71#447
prime radicalC23×C71C2×D8
hypercenter#2086D71
hypocenterC71#12289
LCSG ▷ C22×C8×C71 ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C8 ◁ #2086

Complements

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

NH ≅ G/Ntype
120. C71#12289semidirect

Maximal subgroups

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

subgroupindex
#225996852
#225905372
#225863512
#225862802
C71×#20862
#1228971