Finite Group Databasebeta

← #31907353

Subgroups — #31907353

Order 18176.

306 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 89 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#22597314C211yesmaximal, characteristic
2#22590141C211yesmaximal, characteristic, complemented
3#22590094C211yesmaximal, characteristic, complemented
4#22586322C211yesmaximal, characteristic, complemented
5#22586299C211yesmaximal, characteristic, complemented
6#22586297C211yesmaximal, characteristic, complemented
7C71×#1338C211yesmaximal, characteristic, complemented
8#17986536C411yescharacteristic
9C2×#14556067—21no—
10Q8×C71⋊C8C411yescharacteristic
11#17986078C2211yescharacteristic
12#17986072—21no—
13#17986064—21no—
14#17986064C411yescharacteristic
15#17986059—21no—
16#17986055C411yescharacteristic
17#17985956—21no—
18C4×C8×D71C2211yescharacteristic
19C4⋊(C71⋊C16)C2211yescharacteristic
20Dic71⋊C16—21no—
21Dic71⋊C16—21no—
22(C2×D71)⋊C16—21no—
23#17985775C2211yescharacteristic
24#446×C71C2211yescharacteristic
25C4⋊C16×C71C2211yescharacteristic
26#346×C71C2211yescharacteristic
27C2×#11765848—21no—
28C2×C8×D71—21no—
29C2×C8×D71D411yescharacteristic
30C2×C8×D71D411yescharacteristic
31#14556123—21no—
32#14556119C811yescharacteristic
33#14556117C811yescharacteristic
34#14556113—21no—
35#14556104C811yescharacteristic
36#14556103C811yescharacteristic
37C4×D284—21no—
38C4×D284C2×C411yescharacteristic
39#14556085—21no—
40C42×D71C2×C411yescharacteristic
41#14556067—42no—
42C2×C71⋊C16—21no—
43C2×C71⋊C16—21no—
44C4⋊(C71⋊C8)—21no—
45C4⋊(C71⋊C8)C2×C411yescharacteristic
46(C2×D71)⋊C8—21no—
47(C2×D71)⋊C8—21no—
48(C2×D71)⋊C8—21no—
49#14556038—21no—
50C4×C71⋊C8—21no—
51C4×C71⋊C8C2×C411yescharacteristic
52C8×Dic71D411yescharacteristic
53C8×Dic71D411yescharacteristic
54C4×Q8×C71C2×C411yescharacteristic
55C2×C16×C71—21no—
56C2×C16×C71—21no—
57C4⋊C8×C71—21no—
58C4⋊C8×C71C2×C411yescharacteristic
59#97×C71—21no—
60C4×C8×C71C2311yescharacteristic
61C2×D284—21no—
62C2×D284C2×C811yescharacteristic
63C2×C4×D71—21no—
64C2×C4×D71C22⋊C411yescharacteristic
65C2×C4×D71C2×C811yescharacteristic
66C2×C4×D71M1611yescharacteristic
67#11765848—41no—
68#11765848—41no—
69C2×C71⋊C8—21no—
70C2×C71⋊C8C22⋊C411yescharacteristic
71C2×C71⋊C8C22⋊C411yescharacteristic
72C8×D71—21no—
73C8×D71—21no—
74C8×D71—41no—
75C8×D71—42no—
76C4⋊Dic71—21no—
77C4⋊Dic71C2×C811yescharacteristic
78Dic71⋊C4—21no—
79Dic71⋊C4—21no—
80(C2×D71)⋊C4—21no—
81(C2×D71)⋊C4—21no—
82(C2×D71)⋊C4—21no—
83(C2×D71)⋊C4—21no—
84(C2×D71)⋊C4—21no—
85C4×Dic71C22⋊C411yescharacteristic
86C4×Dic71C2×C811yescharacteristic
87C4×Dic71M1611yescharacteristic
88C71⋊C16—41no—
89C71⋊C16—41no—
90C2×Q8×C71C2×C811yescharacteristic
91C2×C8×C71—21no—
92C2×C8×C71C2×D411yescharacteristic
93C2×C8×C71C2×D411yescharacteristic
94C4⋊C4×C71—21no—
95C4⋊C4×C71C2×C811yescharacteristic
96C42×C71—21no—
97C42×C71C22×C411yescharacteristic
98C16×C71—41no—
99C16×C71—41no—
100C22×D71—21no—
101C22×D71C22⋊C811yescharacteristic
102Q8×C71—41no—
103C2×C4×C71—21no—
104C2×C4×C71—21no—
105C2×C4×C71C2×C22⋊C411yescharacteristic
106C2×C4×C71C22×C811yescharacteristic
107C2×C4×C71C2×M1611yescharacteristic
108C2×Dic71—21no—
109C2×Dic71C22⋊C811yescharacteristic
110C2×Dic71C22⋊C811yescharacteristic
111C2×Dic71C22⋊C811yescharacteristic
112D284—41no—
113D284—42no—
114C4×D71—21no—
115C4×D71—21no—
116C4×D71—41no—
117C4×D71—41no—
118C4×D71—41no—
119C4×D71—41no—
120C8×C71—21no—
121C8×C71—41no—
122C8×C71AGL(1,ℤ/8)11yescharacteristic
123C8×C71#13711yescharacteristic
124C71⋊C8—21no—
125C71⋊C8—21no—
126C71⋊C8—41no—
127C22×C71C2×C22⋊C811yescharacteristic
128C2×D71—21no—
129C2×D71—21no—
130C2×D71—21no—
131C2×D71—41no—
132C2×D71—41no—
133C2×D71—41no—
134C4×C71—21no—
135C4×C71—21no—
136C4×C71—41no—
137C4×C71—41no—
138C4×C71#41711yescharacteristic
139C4×C71#42111yescharacteristic
140Dic71—21no—
141Dic71—21no—
142Dic71—21no—
143Dic71—41no—
144#11627—711nomaximal, Sylow
145C2×C71#147711yescharacteristic
146C2×C71#211511yescharacteristic
147C2×C71#211711yescharacteristic
148D71—41no—
149D71—81no—
150#2966—711no—
151#2163—711no—
152#2150—711no—
153#1338—711no—
154#1338D7111yescharacteristic, complemented
155#1330—711no—
156#1330—711no—
157C71#1162711yescharacteristic, complemented, Sylow
158#529—711no—
159C2×M32—1421no—
160#446C2×D7111yescharacteristic
161C8×Q8—711no—
162#436—711no—
163#435—1421no—
164C8×D4—711no—
165C8×D4—1421no—
166#433—1421no—
167#432—711no—
168#405—1421no—
169C2×C4×C8—711no—
170C4⋊C16—711no—
171C4⋊C16—1421no—
172C4⋊C16—1421no—
173C4⋊C16C2×D7111yescharacteristic
174C22⋊C16—1421no—
175#346—711no—
176#346C2×D7111yescharacteristic
177C2×M16—1421no—
178C22×C8—711no—
179C22×C8—711no—
180C22×C8—1421no—
181#126—1421no—
182#125—711no—
183#124—711no—
184#123—1421no—
185C22⋊Q8—711no—
186#121—711no—
187C4×Q8C4×D7111yescharacteristic
188C4×D4—711no—
189C4×D4—1421no—
190#117—1421no—
191C2×C42—711no—
192M32—2842no—
193C2×C16—21no—
194C2×C16—21no—
195C2×C16—1421no—
196C2×C16—1421no—
197C4⋊C8—21no—
198C4⋊C8—711no—
199C4⋊C8—1421no—
200C4⋊C8C4×D7111yescharacteristic
201C22⋊C8—1421no—
202C22⋊C8—1421no—
203C22⋊C8—1421no—
204#97—21no—
205#97—1421no—
206C4×C8—711no—
207C4×C8—711no—
208C4×C8—711no—
209C4×C8—1421no—
210C4×C8C22×D7111yescharacteristic
211C2×Q8C8×D7111yescharacteristic
212C2×D4—711no—
213C2×D4—1421no—
214C22×C4—711no—
215C22×C4—711no—
216C22×C4—711no—
217C22×C4—1421no—
218M16—2841no—
219M16—2841no—
220C2×C8—21no—
221C2×C8—2841no—
222C2×C8—2842no—
223C2×C8—711no—
224C2×C8—711no—
225C2×C8—1421no—
226C2×C8—1421no—
227C2×C8—1421no—
228C2×C8D4×D7111yescharacteristic
229C2×C8D4×D7111yescharacteristic
230C4⋊C4—21no—
231C4⋊C4—711no—
232C4⋊C4—1421no—
233C4⋊C4—1421no—
234C4⋊C4—1421no—
235C4⋊C4C8×D7111yescharacteristic
236C22⋊C4—1421no—
237C22⋊C4—1421no—
238C22⋊C4—1421no—
239C22⋊C4—1421no—
240C22⋊C4—1421no—
241C42—21no—
242C42—711no—
243C42—711no—
244C42—711no—
245C42C2×C4×D7111yescharacteristic
246C16—41no—
247C16—41no—
248C16—2841no—
249C16—2841no—
250C23—711no—
251C23—1421no—
252Q8—41no—
253D4—2841no—
254D4—2842no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—2841no—
258C2×C4—2841no—
259C2×C4—2841no—
260C2×C4—2841no—
261C2×C4—711no—
262C2×C4—711no—
263C2×C4—711no—
264C2×C4—1421no—
265C2×C4—1421no—
266C2×C4—1421no—
267C2×C4C22⋊C4×D7111yescharacteristic
268C2×C4C2×C8×D7111yescharacteristic
269C2×C4M16×D7111yescharacteristic
270C8—21no—
271C8—41no—
272C8—2841no—
273C8—1421no—
274C8—1421no—
275C8#1455615511yescharacteristic
276C8#1455616011yescharacteristic
277C22—2841no—
278C22—2841no—
279C22—2841no—
280C22—1421no—
281C22—1421no—
282C22—1421no—
283C22C22⋊C8×D7111yescharacteristic
284C4—21no—
285C4—21no—
286C4—41no—
287C4—41no—
288C4—2841no—
289C4—1421no—
290C4—1421no—
291C4—1421no—
292C4#1798599911yescharacteristic
293C4#1798601411yescharacteristic
294C2—2841no—
295C2—5681no—
296C2#2258672611yescharacteristic
297C2#2258995911yescharacteristic
298C2#2258996611yescharacteristic
subgroupG/H
centerC2×C4C22⋊C4×D71
commutatorC2×C4×C71C22×C8
FrattiniC4×C8C22×D71
FittingC71×#1338C2
socleC22×C71C2×C22⋊C8
prime radical#14556103C8
cosocleC4×C8×C71C23
hypercenter#1338D71
hypocenterC71#11627
LCSG ▷ C2×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1338

Complements

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

NH ≅ G/Ntype
2. #22590141C2semidirect
3. #22590094C2semidirect
4. #22586322C2semidirect
5. #22586299C2semidirect
6. #22586297C2semidirect
7. C71×#1338C2semidirect
154. #1338D71semidirect
157. C71#11627semidirect

Maximal subgroups

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

subgroupindex
#225973142
#225901412
#225900942
#225863222
#225862992
#225862972
C71×#13382
#1162771