Finite Group Databasebeta

← #31901228

Subgroups — #31901228

Order 18176.

366 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 93 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#22598145C211yesmaximal, characteristic
2#22587110C211yesmaximal, characteristic
3#22586998C211yesmaximal, characteristic
4#22586903C211yesmaximal, characteristic
5#22586664C211yesmaximal, characteristic
6#22586198C211yesmaximal, characteristic
7C71×#1805C211yesmaximal, characteristic
8#17986556C2211yescharacteristic
9#17986306—22no—
10#17986294—21no—
11#17986284—21no—
12#17986052C2211yescharacteristic
13#17986052C2211yescharacteristic
14#17986039—21no—
15#17986023C2211yescharacteristic
16#17986008—21no—
17#17985749—22no—
18#17985747—21no—
19#17985744—21no—
20#17985735—22no—
21#17985725—22no—
22C2×C4×Q8×C71C2211yescharacteristic
23C2×C4⋊C8×C71C2211yescharacteristic
24C2×#103×C71—21no—
25#382×C71C2211yescharacteristic
26#340×C71—21no—
27C2×#11765849—21no—
28#14556128D411yescharacteristic
29#14556128D411yescharacteristic
30#14556120—21no—
31#14556107—21no—
32#14556107—21no—
33C4×Dic142—21no—
34C4×Dic142—21no—
35#14556087—21no—
36#14556061—22no—
37#14556059—22no—
38C4⋊(C71⋊C8)—21no—
39C4⋊(C71⋊C8)—21no—
40C4⋊(C71⋊C8)—21no—
41C4⋊(C71⋊C8)D412yes—
42#14556052—22no—
43#14556052—22no—
44C22⋊(C71⋊C8)—21no—
45C22⋊(C71⋊C8)D412yes—
46C4×C71⋊C8—21no—
47C4×C71⋊C8—21no—
48C22×Q8×C71D411yescharacteristic
49C22×C8×C71—21no—
50C4×Q8×C71—21no—
51C4×Q8×C71—21no—
52C4×Q8×C71—21no—
53C4×Q8×C71—21no—
54C2×C4⋊C4×C71—21no—
55C2×C4⋊C4×C71—21no—
56C2×C4⋊C4×C71D411yescharacteristic
57C2×C42×C71—21no—
58C2×C42×C71C2311yescharacteristic
59C4⋊C8×C71—21no—
60C4⋊C8×C71—21no—
61#103×C71—42no—
62#95×C71—21no—
63#95×C71—21no—
64C2×Dic142—21no—
65C2×Dic142—21no—
66#11765849—41no—
67#11765849—41no—
68C2×C71⋊C8—22no—
69C2×C71⋊C8—22no—
70C2×C71⋊C8—41no—
71C4⋊Dic71—21no—
72C4⋊Dic71—21no—
73C4⋊Dic71—21no—
74C4⋊Dic71—21no—
75Dic71⋊C4—41no—
76Dic71⋊C4—41no—
77C22⋊Dic71—41no—
78C4×Dic71—41no—
79C2×Q8×C71—21no—
80C2×Q8×C71—21no—
81C2×Q8×C71—21no—
82C2×Q8×C71—21no—
83C2×Q8×C71—21no—
84C2×Q8×C71SD1612yes—
85C22×C4×C71—21no—
86C22×C4×C71—21no—
87C22×C4×C71—41no—
88C22×C4×C71C2×D411yescharacteristic
89C22×C4×C71C2×D411yescharacteristic
90C22×C4×C71#4111yescharacteristic
91C2×C8×C71—41no—
92C2×C8×C71—41no—
93C2×C8×C71—41no—
94C4⋊C4×C71—21no—
95C4⋊C4×C71—21no—
96C4⋊C4×C71—41no—
97C4⋊C4×C71—41no—
98C4⋊C4×C71—41no—
99C4⋊C4×C71—41no—
100C42×C71—41no—
101C42×C71—41no—
102C42×C71C2×D411yescharacteristic
103C42×C71C2×D411yescharacteristic
104C42×C71#4111yescharacteristic
105C42×C71#4111yescharacteristic
106C23×C71#12111yescharacteristic
107Q8×C71—22no—
108Q8×C71—22no—
109Q8×C71—41no—
110Q8×C71—41no—
111C2×C4×C71—22no—
112C2×C4×C71—41no—
113C2×C4×C71—41no—
114C2×C4×C71—41no—
115C2×C4×C71—41no—
116C2×C4×C71—41no—
117C2×C4×C71—81no—
118C2×C4×C71—81no—
119C2×C4×C71—81no—
120C2×C4×C71—81no—
121C2×C4×C71#12011yescharacteristic
122C2×C4×C71#12111yescharacteristic
123C2×C4×C71#12111yescharacteristic
124C2×C4×C71#12311yescharacteristic
125C2×C4×C71#12311yescharacteristic
126C2×C4×C71#12411yescharacteristic
127C2×C4×C71C2×SD1611yescharacteristic
128C2×C4×C71#13511yescharacteristic
129C2×C4×C71#13711yescharacteristic
130C2×C4×C71#13711yescharacteristic
131C2×Dic71—41no—
132C2×Dic71—41no—
133Dic142—81no—
134Dic142—81no—
135C8×C71—81no—
136C71⋊C8—42no—
137C71⋊C8—42no—
138C71⋊C8—81no—
139C22×C71—21no—
140C22×C71—21no—
141C22×C71#39411yescharacteristic
142C22×C71#46511yescharacteristic
143C22×C71#47011yescharacteristic
144C4×C71—21no—
145C4×C71—21no—
146C4×C71—21no—
147C4×C71—21no—
148C4×C71—42no—
149C4×C71—81no—
150C4×C71—161no—
151C4×C71#44811yescharacteristic
152C4×C71#45211yescharacteristic
153C4×C71#48311yescharacteristic
154C4×C71#48511yescharacteristic
155Dic71—81no—
156Dic71—81no—
157#10470—711nomaximal, Sylow
158C2×C71—21no—
159C2×C71—21no—
160C2×C71#203511yescharacteristic
161C2×C71#204911yescharacteristic
162C2×C71#205311yescharacteristic
163#3078—711no—
164#1805D7111yescharacteristic
165#1570—711no—
166#1542—711no—
167#1518—711no—
168#1461—711no—
169#1286—711no—
170C71#1047011yescharacteristic, complemented, Sylow
171#533—711no—
172C2×C4×Q8C2×D7111yescharacteristic
173#479—1422no—
174#477—1421no—
175#475—1421no—
176#431—711no—
177#431—711no—
178#428—1421no—
179#423—711no—
180C2×C4⋊C8C2×D7111yescharacteristic
181#419—1421no—
182C2×#103—21no—
183#382C2×D7111yescharacteristic
184#340—21no—
185#334—1422no—
186#333—1421no—
187#332—1421no—
188#328—1422no—
189#324—1422no—
190C22×Q8C22⋊D7111yescharacteristic
191C2×M16—1421no—
192C22×C8—21no—
193C4⋊Q8—711no—
194C4⋊Q8—711no—
195#125—1421no—
196C22⋊Q8—1421no—
197C22⋊Q8—1421no—
198C4×Q8—21no—
199C4×Q8—21no—
200C4×Q8—21no—
201C4×Q8—21no—
202C4×Q8—1421no—
203C4×Q8—1421no—
204#117—1421no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4C22⋊D7111yescharacteristic
208C2×C42—21no—
209C2×C42C22×D7111yescharacteristic
210#107—1422no—
211#106—1422no—
212C4⋊C8—21no—
213C4⋊C8—21no—
214C4⋊C8—712no—
215C4⋊C8—1421no—
216C4⋊C8—1421no—
217C4⋊C8—1421no—
218#103—42no—
219#103—1422no—
220#103—1422no—
221C22⋊C8—712no—
222C22⋊C8—1421no—
223C4×C8—1421no—
224C4×C8—1421no—
225#95—21no—
226#95—21no—
227C2×Q8—21no—
228C2×Q8—21no—
229C2×Q8—21no—
230C2×Q8—21no—
231C2×Q8—21no—
232C2×Q8—1421no—
233C2×Q8—1421no—
234C2×Q8#1176585312yes—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—41no—
238C22×C4D4×D7111yescharacteristic
239C22×C4C2×C22⋊D7111yescharacteristic
240C22×C4#1176586411yescharacteristic
241M16—2841no—
242M16—2841no—
243C2×C8—41no—
244C2×C8—41no—
245C2×C8—41no—
246C2×C8—2841no—
247C2×C8—1422no—
248C2×C8—1422no—
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—2841no—
256C4⋊C4—2841no—
257C4⋊C4—1421no—
258C4⋊C4—1421no—
259C4⋊C4—1421no—
260C4⋊C4—1421no—
261C22⋊C4—2841no—
262C42—41no—
263C42—41no—
264C42—2841no—
265C42D4×D7111yescharacteristic
266C42D4×D7111yescharacteristic
267C42#1176586511yescharacteristic
268C42#1176586511yescharacteristic
269C23#1455610011yescharacteristic
270Q8—22no—
271Q8—22no—
272Q8—41no—
273Q8—41no—
274Q8—5681no—
275Q8—5681no—
276C2×C4—22no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—81no—
283C2×C4—81no—
284C2×C4—81no—
285C2×C4—81no—
286C2×C4—2841no—
287C2×C4—2841no—
288C2×C4#1455609611yescharacteristic
289C2×C4#1455610111yescharacteristic
290C2×C4#1455610111yescharacteristic
291C2×C4#1455611011yescharacteristic
292C2×C4#1455611111yescharacteristic
293C2×C4#1455611411yescharacteristic
294C2×C4C2×#1176585311yescharacteristic
295C2×C4#1455614811yescharacteristic
296C2×C4#1455615811yescharacteristic
297C2×C4#1455615911yescharacteristic
298C8—81no—
299C8—2842no—
300C8—2842no—
301C8—5681no—
302C22—21no—
303C22—21no—
304C22#1798591411yescharacteristic
305C22#1798622611yescharacteristic
306C22#1798625511yescharacteristic
307C4—21no—
308C4—21no—
309C4—21no—
310C4—21no—
311C4—42no—
312C4—81no—
313C4—161no—
314C4—5681no—
315C4—5681no—
316C4#1798613711yescharacteristic
317C4#1798615911yescharacteristic
318C4#1798632911yescharacteristic
319C4#1798634011yescharacteristic
320C2—21no—
321C2—21no—
322C2#2258953611yescharacteristic
323C2#2258961811yescharacteristic
324C2#2258964411yescharacteristic
subgroupG/H
centerC22#17985914
commutator, cosocleC2×C42×C71C23
FrattiniC2×C42C22×D71
FittingC71×#1805C2
socleC22×C71#394
prime radicalC23×C71#121
hypercenter#1805D71
hypocenterC71#10470
LCSG ▷ C2×C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1805

Complements

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

NH ≅ G/Ntype
170. C71#10470semidirect

Maximal subgroups

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

subgroupindex
#225981452
#225871102
#225869982
#225869032
#225866642
#225861982
C71×#18052
#1047071