Finite Group Databasebeta

← #31916542

Subgroups — #31916542

Order 18176.

370 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 97 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#22598864C211yesmaximal, characteristic
2#22590061C212yesmaximal
3#22589211C211yesmaximal, characteristic
4#22586446C212yesmaximal
5C71×#1377C211yesmaximal, characteristic
6C22×Dic284C2211yescharacteristic
7#17986556—21no—
8#17986408—24no—
9#17986037—21no—
10#17986037C2212yes—
11C2×#14556060C2211yescharacteristic
12C2×#14556053—21no—
13#17985905—21no—
14#17985804—24no—
15C2×M32×C71C2212yes—
16C2×#97×C71C2211yescharacteristic
17C22×Dic142—21no—
18C2×Dic284—22no—
19C2×Dic284—22no—
20C2×Dic284—21no—
21C2×Dic284Q812yescomplemented
22#14556128—22no—
23#14556120—22no—
24#14556107—42no—
25C4×Dic142—42no—
26#14556087—22no—
27C2×C4⋊Dic71—21no—
28#14556062—22no—
29#14556062D414yes—
30#14556060—22no—
31#14556060—22no—
32#14556060—22no—
33#14556053—42no—
34C71⋊#95—42no—
35C22×C8×C71D411yescharacteristic
36C22×C8×C71C2311yescharacteristic
37C2×C42×C71D411yescharacteristic
38M32×C71—22no—
39M32×C71—22no—
40C2×C16×C71—22no—
41#97×C71—22no—
42C2×Dic142—22no—
43C2×Dic142—41no—
44C2×Dic142—41no—
45C2×Dic142—41no—
46C2×Dic142—41no—
47C2×Dic142—42no—
48C2×Dic142—41no—
49C22×Dic71—41no—
50C22×Dic71—41no—
51Dic284—24no—
52Dic284—42no—
53C4⋊Dic71—24no—
54C4⋊Dic71—42no—
55C4⋊Dic71—41no—
56C4⋊Dic71—41no—
57Dic71⋊C4—44no—
58C22⋊Dic71—42no—
59C4×Dic71—42no—
60C22×C4×C71D812yes—
61C22×C4×C71C2×D411yescharacteristic
62C2×C8×C71—21no—
63C2×C8×C71—21no—
64C2×C8×C71SD1612yes—
65C2×C8×C71C2×D412yes—
66C2×C8×C71C2×Q811yescharacteristic
67C2×C8×C71#4111yescharacteristic
68C2×C8×C71#4112yes—
69C42×C71—21no—
70C42×C71—21no—
71C16×C71—42no—
72C23×C71C2×D811yescharacteristic
73C2×C4×C71—22no—
74C2×C4×C71—42no—
75C2×C4×C71#12011yescharacteristic
76C2×C4×C71C22⋊Q811yescharacteristic
77C2×C4×C71#12411yescharacteristic
78C2×C4×C71C2×SD1611yescharacteristic
79C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
80C2×C4×C71#13711yescharacteristic
81C2×Dic71—44no—
82C2×Dic71—42no—
83C2×Dic71—81no—
84C2×Dic71—81no—
85C2×Dic71—82no—
86C2×Dic71—81no—
87Dic142—42no—
88Dic142—42no—
89Dic142—84no—
90Dic142—81no—
91Dic142—81no—
92C8×C71—22no—
93C8×C71C22⋊Q812yes—
94C8×C71#12412yes—
95C22×C71—22no—
96C22×C71#44711yescharacteristic
97C22×C71#47411yescharacteristic
98C22×C71#48611yescharacteristic
99C4×C71—42no—
100C4×C71#39311yescharacteristic
101C4×C71#44811yescharacteristic
102C4×C71#47611yescharacteristic
103C4×C71#48911yescharacteristic
104Dic71—84no—
105Dic71—82no—
106Dic71—161no—
107#13368—711nomaximal, Sylow
108C2×C71—21no—
109C2×C71—21no—
110C2×C71#202311yescharacteristic
111C2×C71#219112yes—
112#3162—711no—
113#2141—712no—
114#1971—711no—
115#1386—712no—
116#1377D7111yescharacteristic
117C71#1336811yescharacteristic, complemented, Sylow
118C22×Q16—711no—
119#533—1421no—
120C2×M32C2×D7112yes—
121#501—1424no—
122#427—712no—
123#427—1421no—
124C2×#106—711no—
125C2×#103—1421no—
126C2×#97C2×D7111yescharacteristic
127#391—1421no—
128#358—1424no—
129C22×Q8—1421no—
130C2×Q16—712no—
131C2×Q16—1422no—
132C2×Q16—1421no—
133C2×Q16—1422no—
134C22×C8D28411yescharacteristic
135C22×C8C22×D7111yescharacteristic
136C4⋊Q8—1422no—
137#125—1422no—
138C22⋊Q8—2842no—
139C4×Q8—2842no—
140#117—1422no—
141C2×C4⋊C4—1421no—
142C2×C42D28411yescharacteristic
143M32—22no—
144M32—22no—
145C2×C16—22no—
146#107—714no—
147#107—1422no—
148#106—1422no—
149#106—1422no—
150#106—1422no—
151#103—2842no—
152#97—22no—
153#95—2842no—
154C2×Q8—2842no—
155C2×Q8—2841no—
156C2×Q8—2841no—
157C2×Q8—2841no—
158C2×Q8—2841no—
159C2×Q8—2841no—
160C2×Q8—1422no—
161C22×C4—2841no—
162C22×C4—2841no—
163C22×C4D56812yes—
164C22×C4C2×D28411yescharacteristic
165Q16—2842no—
166Q16—1424no—
167C2×C8—21no—
168C2×C8—21no—
169C2×C8#1176585412yes—
170C2×C8D4×D7112yes—
171C2×C8Q8×D7111yescharacteristic
172C2×C8#1176586411yescharacteristic
173C2×C8#1176586512yes—
174C4⋊C4—2844no—
175C4⋊C4—2842no—
176C4⋊C4—2841no—
177C4⋊C4—2841no—
178C4⋊C4—1424no—
179C22⋊C4—2842no—
180C42—21no—
181C42—21no—
182C42—2842no—
183C16—42no—
184C23C2×D56811yescharacteristic
185Q8—2842no—
186Q8—2842no—
187Q8—5684no—
188Q8—5681no—
189Q8—5681no—
190C2×C4—22no—
191C2×C4—42no—
192C2×C4—2844no—
193C2×C4—2842no—
194C2×C4—5681no—
195C2×C4—5681no—
196C2×C4—5682no—
197C2×C4—5681no—
198C2×C4#1455609811yescharacteristic
199C2×C4#1455610811yescharacteristic
200C2×C4#1455611611yescharacteristic
201C2×C4C2×#1176585411yescharacteristic
202C2×C4#1455615611yescharacteristic
203C2×C4#1455616111yescharacteristic
204C8—22no—
205C8#1455610612yes—
206C8#1455611412yes—
207C22—22no—
208C22#1798613511yescharacteristic
209C22#1798628011yescharacteristic
210C22#1798634611yescharacteristic
211C4—42no—
212C4—5682no—
213C4—5684no—
214C4—11361no—
215C4#1798591211yescharacteristic
216C4#1798614011yescharacteristic
217C4#1798629011yescharacteristic
218C4#1798635911yescharacteristic
219C2—21no—
220C2—21no—
221C2#2258947711yescharacteristic
222C2#2259029312yes—
subgroupG/H
centerC22#17986135
commutator, cosocleC22×C8×C71C23
FrattiniC22×C8C22×D71
FittingC71×#1377C2
socleC22×C71#447
prime radicalC23×C71C2×D8
hypercenter#1377D71
hypocenterC71#13368
LCSG ▷ C22×C8×C71 ▷ C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#97 ◁ #1377

Complements

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

NH ≅ G/Ntype
21. C2×Dic284Q8semidirect
117. C71#13368semidirect

Maximal subgroups

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

subgroupindex
#225988642
#225900612
#225892112
#225864462
C71×#13772
#1336871