Finite Group Databasebeta

← #31916538

Subgroups — #31916538

Order 18176.

370 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 101 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#22590059C212yesmaximal
2#22589211C211yesmaximal, characteristic
3#22586445C212yesmaximal
4#22586424C211yesmaximal, characteristic
5C71×#3162C211yesmaximal, characteristic
6C22×Dic284C2211yescharacteristic
7C2×#14556068C2212yes—
8C2×#14556060C2211yescharacteristic
9C2×#14556053—21no—
10#17985905—21no—
11#17985802—24no—
12#533×C71—21no—
13#501×C71—24no—
14#427×C71—21no—
15#427×C71C2212yes—
16C2×#97×C71C2211yescharacteristic
17C22×Dic142—21no—
18C2×Dic284—22no—
19C2×Dic284—22no—
20C2×Dic284—21no—
21C2×Dic284Q812yescomplemented
22C2×C4⋊Dic71—21no—
23#14556068—22no—
24#14556068—22no—
25C2×C71⋊C16—22no—
26#14556060—22no—
27#14556053—42no—
28C71⋊#95—42no—
29C22×C8×C71D411yescharacteristic
30C22×C8×C71C2311yescharacteristic
31C4⋊Q8×C71—22no—
32#125×C71—22no—
33C22⋊Q8×C71—42no—
34C4×Q8×C71—42no—
35#117×C71—22no—
36C2×C42×C71D411yescharacteristic
37#107×C71—22no—
38#107×C71D414yes—
39#106×C71—22no—
40#106×C71—22no—
41#97×C71—22no—
42C2×Dic142—22no—
43C2×Dic142—41no—
44C2×Dic142—41no—
45C2×Dic142—41no—
46C2×Dic142—41no—
47C2×Dic142—41no—
48C22×Dic71—41no—
49C22×Dic71—41no—
50Dic284—24no—
51Dic284—42no—
52C4⋊Dic71—41no—
53C4⋊Dic71—41no—
54C71⋊C16—42no—
55C2×Q8×C71—42no—
56C22×C4×C71D812yes—
57C22×C4×C71C2×D411yescharacteristic
58C2×C8×C71—21no—
59C2×C8×C71—21no—
60C2×C8×C71SD1612yes—
61C2×C8×C71C2×D412yes—
62C2×C8×C71C2×Q811yescharacteristic
63C2×C8×C71#4112yes—
64C2×C8×C71#4111yescharacteristic
65C4⋊C4×C71—24no—
66C4⋊C4×C71—44no—
67C4⋊C4×C71—42no—
68C22⋊C4×C71—42no—
69C42×C71—21no—
70C42×C71—21no—
71C42×C71—42no—
72C23×C71C2×D811yescharacteristic
73Q8×C71—84no—
74C2×C4×C71—22no—
75C2×C4×C71—42no—
76C2×C4×C71—44no—
77C2×C4×C71#12011yescharacteristic
78C2×C4×C71C22⋊Q811yescharacteristic
79C2×C4×C71#12411yescharacteristic
80C2×C4×C71C2×SD1611yescharacteristic
81C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
82C2×C4×C71#13711yescharacteristic
83C2×Dic71—42no—
84C2×Dic71—81no—
85C2×Dic71—81no—
86C2×Dic71—82no—
87C2×Dic71—81no—
88Dic142—42no—
89Dic142—42no—
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—84no—
101C4×C71#39311yescharacteristic
102C4×C71#44811yescharacteristic
103C4×C71#47611yescharacteristic
104C4×C71#48911yescharacteristic
105Dic71—82no—
106Dic71—161no—
107#13368—711nomaximal, Sylow
108C2×C71—21no—
109C2×C71—21no—
110C2×C71#202311yescharacteristic
111C2×C71#219112yes—
112#3162D7111yescharacteristic
113#2141—712no—
114#1971—711no—
115#1386—712no—
116#1377—711no—
117C71#1336811yescharacteristic, complemented, Sylow
118C22×Q16—711no—
119#533—21no—
120C2×M32—712no—
121#501—24no—
122#427—21no—
123#427C2×D7112yes—
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×C8C22⋊D7111yescharacteristic
135C22×C8C22×D7111yescharacteristic
136C4⋊Q8—22no—
137#125—22no—
138C22⋊Q8—42no—
139C4×Q8—42no—
140#117—22no—
141C2×C4⋊C4—1421no—
142C2×C42C22⋊D7111yescharacteristic
143M32—1422no—
144M32—1422no—
145C2×C16—1422no—
146#107—22no—
147#107C22⋊D7114yes—
148#106—22no—
149#106—22no—
150#106—1422no—
151#103—2842no—
152#97—22no—
153#95—2842no—
154C2×Q8—42no—
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×C4#1176585012yes—
164C22×C4C2×C22⋊D7111yescharacteristic
165Q16—2842no—
166Q16—1424no—
167C2×C8—21no—
168C2×C8—21no—
169C2×C8#1176585212yes—
170C2×C8C2×C22⋊D7112yes—
171C2×C8Q8×D7111yescharacteristic
172C2×C8#1176586612yes—
173C2×C8#1176586611yescharacteristic
174C4⋊C4—24no—
175C4⋊C4—44no—
176C4⋊C4—42no—
177C4⋊C4—2841no—
178C4⋊C4—2841no—
179C22⋊C4—42no—
180C42—21no—
181C42—21no—
182C42—42no—
183C16—2842no—
184C23C2×#1176585011yescharacteristic
185Q8—84no—
186Q8—2842no—
187Q8—2842no—
188Q8—5681no—
189Q8—5681no—
190C2×C4—22no—
191C2×C4—42no—
192C2×C4—44no—
193C2×C4—2842no—
194C2×C4—5681no—
195C2×C4—5681no—
196C2×C4—5682no—
197C2×C4—5681no—
198C2×C4#1455609711yescharacteristic
199C2×C4#1455610411yescharacteristic
200C2×C4#1455611711yescharacteristic
201C2×C4C2×#1176585211yescharacteristic
202C2×C4#1455615411yescharacteristic
203C2×C4#1455615911yescharacteristic
204C8—22no—
205C8#1455610412yes—
206C8#1455611712yes—
207C22—22no—
208C22#1798613311yescharacteristic
209C22#1798627611yescharacteristic
210C22#1798634711yescharacteristic
211C4—42no—
212C4—84no—
213C4—5682no—
214C4—11361no—
215C4#1798591011yescharacteristic
216C4#1798613811yescharacteristic
217C4#1798628611yescharacteristic
218C4#1798636111yescharacteristic
219C2—21no—
220C2—21no—
221C2#2258947311yescharacteristic
222C2#2259029112yes—
subgroupG/H
centerC22#17986133
commutator, cosocleC22×C8×C71C23
FrattiniC22×C8C22×D71
FittingC71×#3162C2
socleC22×C71#447
prime radicalC23×C71C2×D8
hypercenter#3162D71
hypocenterC71#13368
LCSG ▷ C22×C8×C71 ▷ C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#97 ◁ #3162

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
#225900592
#225892112
#225864452
#225864242
C71×#31622
#1336871