Finite Group Databasebeta

← #31907356

Subgroups — #31907356

Order 18176.

322 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 91 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#22597303C211yesmaximal, characteristic
2#22590141C211yesmaximal, characteristic, complemented
3#22590094C211yesmaximal, characteristic, complemented
4#22586323C211yesmaximal, characteristic, complemented
5#22586321C211yesmaximal, characteristic, complemented
6#22586298C211yesmaximal, characteristic, complemented
7C71×#1330C211yesmaximal, characteristic, complemented
8C4×#11765864C411yescharacteristic
9C2×#14556067—21no—
10#17986129C2211yescharacteristic
11#17986126C411yescharacteristic
12#17986077C411yescharacteristic
13#17986073—21no—
14#17986068—21no—
15#17986051C411yescharacteristic
16#17986049—21no—
17M16×Dic71—21no—
18C4×C8×D71C2211yescharacteristic
19C4⋊(C71⋊C16)C2211yescharacteristic
20Dic71⋊C16—21no—
21Dic71⋊C16—21no—
22(C2×D71)⋊C16—21no—
23#17985775C2211yescharacteristic
24#436×C71C2211yescharacteristic
25C4⋊C16×C71C2211yescharacteristic
26#346×C71C2211yescharacteristic
27C2×#11765864C811yescharacteristic
28C2×#11765849—21no—
29C2×C8×D71D411yescharacteristic
30C2×C8×D71D411yescharacteristic
31C4×Dic142C2×C411yescharacteristic
32#14556093C811yescharacteristic
33D4×Dic71C811yescharacteristic
34C4×C22⋊D71—21no—
35(C22⋊D71)⋊C4—21no—
36#14556086C811yescharacteristic
37#14556084—21no—
38C2×C4×Dic71—21no—
39C42×D71C2×C411yescharacteristic
40#14556067—42no—
41C2×C71⋊C16—21no—
42C2×C71⋊C16—21no—
43C4⋊(C71⋊C8)C2×C411yescharacteristic
44Dic71⋊C8—21no—
45Dic71⋊C8—21no—
46C22⋊(C71⋊C8)—21no—
47(C2×D71)⋊C8—21no—
48#14556038—21no—
49#14556038—21no—
50C4×C71⋊C8C2×C411yescharacteristic
51C8×Dic71—21no—
52C8×Dic71D411yescharacteristic
53C8×Dic71D411yescharacteristic
54C2×M16×C71—21no—
55C4×D4×C71C2×C411yescharacteristic
56C2×C16×C71—21no—
57C2×C16×C71—21no—
58C4⋊C8×C71C2×C411yescharacteristic
59C22⋊C8×C71—21no—
60C4×C8×C71C2311yescharacteristic
61#11765864—42no—
62C2×Dic142C2×C811yescharacteristic
63C2×C22⋊D71—21no—
64C22×Dic71—21no—
65C22×Dic71—21no—
66C2×C4×D71C22⋊C411yescharacteristic
67C2×C4×D71C2×C811yescharacteristic
68C2×C4×D71M1611yescharacteristic
69#11765849—42no—
70C2×C71⋊C8—21no—
71C2×C71⋊C8C22⋊C411yescharacteristic
72C2×C71⋊C8C22⋊C411yescharacteristic
73C8×D71—21no—
74C8×D71—21no—
75C8×D71—41no—
76C4⋊Dic71C2×C811yescharacteristic
77Dic71⋊C4—21no—
78Dic71⋊C4—21no—
79C22⋊Dic71—21no—
80(C2×D71)⋊C4—21no—
81C4×Dic71—21no—
82C4×Dic71—21no—
83C4×Dic71—21no—
84C4×Dic71C22⋊C411yescharacteristic
85C4×Dic71C2×C811yescharacteristic
86C4×Dic71M1611yescharacteristic
87C71⋊C16—41no—
88C71⋊C16—41no—
89C2×D4×C71C2×C811yescharacteristic
90C22×C4×C71—21no—
91M16×C71—41no—
92M16×C71—41no—
93C2×C8×C71—21no—
94C2×C8×C71C2×D411yescharacteristic
95C2×C8×C71C2×D411yescharacteristic
96C4⋊C4×C71C2×C811yescharacteristic
97C22⋊C4×C71—21no—
98C42×C71C22×C411yescharacteristic
99C16×C71—41no—
100C16×C71—41no—
101C23×C71—21no—
102C22×D71C22⋊C811yescharacteristic
103D4×C71—41no—
104C2×C4×C71—21no—
105C2×C4×C71—41no—
106C2×C4×C71—41no—
107C2×C4×C71C2×C22⋊C411yescharacteristic
108C2×C4×C71C22×C811yescharacteristic
109C2×C4×C71C2×M1611yescharacteristic
110C22⋊D71—42no—
111C2×Dic71—21no—
112C2×Dic71—21no—
113C2×Dic71—42no—
114C2×Dic71—42no—
115C2×Dic71C22⋊C811yescharacteristic
116C2×Dic71C22⋊C811yescharacteristic
117C2×Dic71C22⋊C811yescharacteristic
118C4×D71—21no—
119C4×D71—21no—
120C4×D71—22no—
121C4×D71—41no—
122Dic142—41no—
123C8×C71—21no—
124C8×C71—41no—
125C8×C71AGL(1,ℤ/8)11yescharacteristic
126C8×C71#13711yescharacteristic
127C71⋊C8—21no—
128C71⋊C8—21no—
129C71⋊C8—41no—
130C22×C71—41no—
131C22×C71—41no—
132C22×C71—41no—
133C22×C71C2×C22⋊C811yescharacteristic
134C2×D71—21no—
135C2×D71—21no—
136C2×D71—21no—
137C4×C71—21no—
138C4×C71—21no—
139C4×C71—41no—
140C4×C71#41711yescharacteristic
141C4×C71#42111yescharacteristic
142Dic71—21no—
143Dic71—21no—
144Dic71—41no—
145Dic71—41no—
146Dic71#34912yes—
147#11628—711nomaximal, Sylow
148C2×C71—81no—
149C2×C71#147711yescharacteristic
150C2×C71C2×#34911yescharacteristic
151C2×C71#211711yescharacteristic
152D71—41no—
153#2965—711no—
154#2163—711no—
155#2150—711no—
156#1338—711no—
157#1338—711no—
158#1330—711no—
159#1330D7111yescharacteristic, complemented
160C71#1162811yescharacteristic, complemented, Sylow
161C4×#41—711no—
162C2×M32—1421no—
163#446—711no—
164#446—711no—
165#436—711no—
166#436C2×D7111yescharacteristic
167#435—1421no—
168#435—1421no—
169#431—711no—
170#431—1421no—
171C4×M16—1421no—
172C2×C4×C8—711no—
173C4⋊C16—711no—
174C4⋊C16—1421no—
175C4⋊C16—1421no—
176C4⋊C16C2×D7111yescharacteristic
177C22⋊C16—1421no—
178#346—711no—
179#346C2×D7111yescharacteristic
180C2×#41—711no—
181C2×M16—21no—
182C2×M16—1421no—
183C22×C8—711no—
184C22×C8—711no—
185C4×Q8—711no—
186C4×Q8—711no—
187C4×D4—711no—
188C4×D4—1421no—
189C4×D4—1421no—
190C4×D4C4×D7111yescharacteristic
191#117—711no—
192#117—1421no—
193C2×C42—711no—
194C2×C42—1421no—
195M32—2842no—
196C2×C16—21no—
197C2×C16—21no—
198C2×C16—1421no—
199C2×C16—1421no—
200C4⋊C8—711no—
201C4⋊C8—1421no—
202C4⋊C8—1421no—
203C4⋊C8C4×D7111yescharacteristic
204C22⋊C8—21no—
205C22⋊C8—1421no—
206C22⋊C8—1421no—
207#97—1421no—
208#97—1421no—
209C4×C8—711no—
210C4×C8—711no—
211C4×C8—711no—
212C4×C8—1421no—
213C4×C8C22×D7111yescharacteristic
214#41—2842no—
215C2×Q8—711no—
216C2×D4—1421no—
217C2×D4C8×D7111yescharacteristic
218C22×C4—21no—
219C22×C4—711no—
220C22×C4—711no—
221C22×C4—711no—
222C22×C4—1421no—
223C22×C4—1421no—
224M16—41no—
225M16—41no—
226M16—2842no—
227C2×C8—21no—
228C2×C8—2841no—
229C2×C8—711no—
230C2×C8—711no—
231C2×C8—1421no—
232C2×C8—1421no—
233C2×C8—1421no—
234C2×C8D4×D7111yescharacteristic
235C2×C8D4×D7111yescharacteristic
236C4⋊C4—711no—
237C4⋊C4—1421no—
238C4⋊C4—1421no—
239C4⋊C4C8×D7111yescharacteristic
240C22⋊C4—21no—
241C22⋊C4—1421no—
242C22⋊C4—1421no—
243C42—711no—
244C42—711no—
245C42—711no—
246C42—1421no—
247C42—1421no—
248C42—1421no—
249C42C2×C4×D7111yescharacteristic
250C16—41no—
251C16—41no—
252C16—2841no—
253C16—2841no—
254C23—21no—
255C23—711no—
256Q8—2841no—
257D4—41no—
258D4—2842no—
259C2×C4—21no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—2841no—
263C2×C4—2842no—
264C2×C4—2842no—
265C2×C4—711no—
266C2×C4—711no—
267C2×C4—711no—
268C2×C4—1422no—
269C2×C4—1421no—
270C2×C4—1421no—
271C2×C4—1421no—
272C2×C4—1421no—
273C2×C4C22⋊C4×D7111yescharacteristic
274C2×C4C2×C8×D7111yescharacteristic
275C2×C4M16×D7111yescharacteristic
276C8—21no—
277C8—41no—
278C8—2841no—
279C8—1421no—
280C8—1421no—
281C8#1455615211yescharacteristic
282C8#1455615711yescharacteristic
283C22—41no—
284C22—41no—
285C22—41no—
286C22—1421no—
287C22—1421no—
288C22—1421no—
289C22C22⋊C8×D7111yescharacteristic
290C4—21no—
291C4—21no—
292C4—41no—
293C4—2841no—
294C4—2841no—
295C4—712no—
296C4—1421no—
297C4—1421no—
298C4#1798599711yescharacteristic
299C4#1798601411yescharacteristic
300C2—81no—
301C2—2841no—
302C2#2258672411yescharacteristic
303C2#349×D7111yescharacteristic
304C2#2258996611yescharacteristic
subgroupG/H
centerC2×C4C22⋊C4×D71
commutatorC2×C4×C71C22×C8
FrattiniC4×C8C22×D71
FittingC71×#1330C2
socleC22×C71C2×C22⋊C8
prime radicalC2×#11765864C8
cosocleC4×C8×C71C23
hypercenter#1330D71
hypocenterC71#11628
LCSG ▷ C2×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1330

Complements

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

NH ≅ G/Ntype
2. #22590141C2semidirect
3. #22590094C2semidirect
4. #22586323C2semidirect
5. #22586321C2semidirect
6. #22586298C2semidirect
7. C71×#1330C2semidirect
159. #1330D71semidirect
160. C71#11628semidirect

Maximal subgroups

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

subgroupindex
#225973032
#225901412
#225900942
#225863232
#225863212
#225862982
C71×#13302
#1162871