Finite Group Databasebeta

← #31907306

Subgroups — #31907306

Order 18176.

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

normal subgroups only

This group has 324 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#22597318C211yesmaximal, characteristic
2#22590138C211yesmaximal, characteristic, complemented
3#346×D71C211yesmaximal, characteristic, complemented
4#22586323C211yesmaximal, characteristic, complemented
5#22586321C211yesmaximal, characteristic, complemented
6#22586298C211yesmaximal, characteristic, complemented
7C71×#1330C211yesmaximal, characteristic, complemented
8#17986531C411yescharacteristic
9C2×C16×D71—21no—
10#17986129C2211yescharacteristic
11#17986123C411yescharacteristic
12#17986068—21no—
13D4×C71⋊C8C411yescharacteristic
14(C22⋊D71)⋊C8—21no—
15#17986057—21no—
16#17986055C411yescharacteristic
17#17985957—21no—
18C4×C8×D71C2211yescharacteristic
19C4⋊(C71⋊C16)C2211yescharacteristic
20(C2×D71)⋊C16—21no—
21#17985775C2211yescharacteristic
22#17985774—21no—
23C16×Dic71—21no—
24#436×C71C2211yescharacteristic
25C4⋊C16×C71C2211yescharacteristic
26#346×C71C2211yescharacteristic
27C22×C71⋊C8—21no—
28C2×C8×D71D411yescharacteristic
29C2×C8×D71D411yescharacteristic
30#14556121—21no—
31#14556119C811yescharacteristic
32#14556115C811yescharacteristic
33#14556110—21no—
34#14556108C811yescharacteristic
35#14556099C811yescharacteristic
36C4×Dic142C2×C411yescharacteristic
37C4×C22⋊D71—21no—
38#14556087—21no—
39C42×D71C2×C411yescharacteristic
40C2×C71⋊C16—21no—
41C2×C71⋊C16—21no—
42C16×D71—24no—
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
61C2×Dic142C2×C811yescharacteristic
62C2×C22⋊D71—21no—
63C2×C4×D71C22⋊C411yescharacteristic
64C2×C4×D71C2×C811yescharacteristic
65C2×C4×D71M1611yescharacteristic
66C2×C71⋊C8—21no—
67C2×C71⋊C8—42no—
68C2×C71⋊C8C22⋊C411yescharacteristic
69C2×C71⋊C8C22⋊C411yescharacteristic
70C8×D71—21no—
71C8×D71—21no—
72C8×D71—22no—
73C4⋊Dic71—21no—
74C4⋊Dic71—21no—
75C4⋊Dic71C2×C811yescharacteristic
76Dic71⋊C4—21no—
77Dic71⋊C4—21no—
78C22⋊Dic71—21no—
79C22⋊Dic71—21no—
80C22⋊Dic71—21no—
81(C2×D71)⋊C4—21no—
82C4×Dic71—21no—
83C4×Dic71C22⋊C411yescharacteristic
84C4×Dic71C2×C811yescharacteristic
85C4×Dic71M1611yescharacteristic
86C71⋊C16—22no—
87C71⋊C16—41no—
88C2×D4×C71C2×C811yescharacteristic
89C22×C4×C71—21no—
90M16×C71—41no—
91M16×C71—41no—
92C2×C8×C71—21no—
93C2×C8×C71C2×D411yescharacteristic
94C2×C8×C71C2×D411yescharacteristic
95C4⋊C4×C71C2×C811yescharacteristic
96C22⋊C4×C71—21no—
97C42×C71C22×C411yescharacteristic
98C16×C71—22no—
99C16×C71—41no—
100C23×C71—21no—
101C22×D71C22⋊C811yescharacteristic
102D4×C71—41no—
103C2×C4×C71—21no—
104C2×C4×C71—41no—
105C2×C4×C71—41no—
106C2×C4×C71C2×C22⋊C411yescharacteristic
107C2×C4×C71C22×C811yescharacteristic
108C2×C4×C71C2×M1611yescharacteristic
109C22⋊D71—42no—
110C2×Dic71—21no—
111C2×Dic71—21no—
112C2×Dic71C22⋊C811yescharacteristic
113C2×Dic71C22⋊C811yescharacteristic
114C2×Dic71C22⋊C811yescharacteristic
115C4×D71—22no—
116C4×D71—21no—
117C4×D71—21no—
118C4×D71—41no—
119Dic142—41no—
120C8×C71—21no—
121C8×C71—41no—
122C8×C71#13511yescharacteristic
123C8×C71#13511yescharacteristic
124C71⋊C8—21no—
125C71⋊C8—41no—
126C71⋊C8#10412yes—
127C22×C71—41no—
128C22×C71—41no—
129C22×C71—41no—
130C22×C71C2×C22⋊C811yescharacteristic
131C2×D71—21no—
132C2×D71—21no—
133C2×D71#35012yes—
134C4×C71—21no—
135C4×C71—21no—
136C4×C71—41no—
137C4×C71#41611yescharacteristic
138C4×C71C2×#10411yescharacteristic
139Dic71—21no—
140Dic71—21no—
141Dic71—21no—
142Dic71—41no—
143Dic71—41no—
144#11618—711nomaximal, Sylow
145C2×C71—81no—
146C2×C71#147611yescharacteristic
147C2×C71#211511yescharacteristic
148C2×C71C2×#35011yescharacteristic
149D71—22no—
150#2966—711no—
151#2162—711no—
152C2×#346—711no—
153#1338—711no—
154#1338—711no—
155#1330—711no—
156#1330D7111yescharacteristic, complemented
157C71#1161811yescharacteristic, complemented, Sylow
158#529—711no—
159C22×C16—1421no—
160#446—711no—
161C8×Q8—711no—
162#436C2×D7111yescharacteristic
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⋊C16C2×D7111yescharacteristic
172C22⋊C16—1421no—
173#346—711no—
174#346—1421no—
175#346C2×D7111yescharacteristic
176C4×C16—1421no—
177C2×M16—21no—
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×Q8—711no—
188C4×D4—1421no—
189C4×D4C4×D7111yescharacteristic
190#117—1421no—
191C2×C42—711no—
192C2×C16—21no—
193C2×C16—21no—
194C2×C16—1424no—
195C2×C16—1421no—
196C2×C16—1421no—
197C4⋊C8—711no—
198C4⋊C8—1421no—
199C4⋊C8—1421no—
200C4⋊C8C4×D7111yescharacteristic
201C22⋊C8—21no—
202C22⋊C8—1421no—
203C22⋊C8—1421no—
204#97—1421no—
205#97—1421no—
206C4×C8—711no—
207C4×C8—711no—
208C4×C8—711no—
209C4×C8—1421no—
210C4×C8C22×D7111yescharacteristic
211C2×Q8—711no—
212C2×D4—1421no—
213C2×D4C8×D7111yescharacteristic
214C22×C4—21no—
215C22×C4—711no—
216C22×C4—711no—
217C22×C4—711no—
218M16—41no—
219M16—41no—
220C2×C8—21no—
221C2×C8—2842no—
222C2×C8—711no—
223C2×C8—711no—
224C2×C8—1422no—
225C2×C8—1421no—
226C2×C8—1421no—
227C2×C8—1421no—
228C2×C8D4×D7111yescharacteristic
229C2×C8D4×D7111yescharacteristic
230C4⋊C4—711no—
231C4⋊C4—1421no—
232C4⋊C4—1421no—
233C4⋊C4—1421no—
234C4⋊C4—1421no—
235C4⋊C4C8×D7111yescharacteristic
236C22⋊C4—21no—
237C22⋊C4—1421no—
238C22⋊C4—1421no—
239C22⋊C4—1421no—
240C22⋊C4—1421no—
241C42—711no—
242C42—711no—
243C42—711no—
244C42—1421no—
245C42C2×C4×D7111yescharacteristic
246C16—22no—
247C16—41no—
248C16—2841no—
249C16—1422no—
250C23—21no—
251C23—711no—
252Q8—2841no—
253D4—41no—
254D4—2842no—
255C2×C4—21no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—2841no—
259C2×C4—711no—
260C2×C4—711no—
261C2×C4—711no—
262C2×C4—1422no—
263C2×C4—1421no—
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—712no—
274C8—1421no—
275C8#1455614711yescharacteristic
276C8#1455614911yescharacteristic
277C22—41no—
278C22—41no—
279C22—41no—
280C22—712no—
281C22—1421no—
282C22—1421no—
283C22C22⋊C8×D7111yescharacteristic
284C4—21no—
285C4—21no—
286C4—41no—
287C4—2841no—
288C4—2841no—
289C4—1421no—
290C4—1421no—
291C4—1421no—
292C4#1798599211yescharacteristic
293C4#104×D7111yescharacteristic
294C2—81no—
295C2—1422no—
296C2#2258671911yescharacteristic
297C2#2258995911yescharacteristic
298C2#350×D7111yescharacteristic
subgroupG/H
centerC2×C4C22⋊C4×D71
commutatorC2×C4×C71C22×C8
FrattiniC4×C8C22×D71
FittingC71×#1330C2
socleC22×C71C2×C22⋊C8
prime radical#14556099C8
cosocleC4×C8×C71C23
hypercenter#1330D71
hypocenterC71#11618
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. #22590138C2semidirect
3. #346×D71C2semidirect
4. #22586323C2semidirect
5. #22586321C2semidirect
6. #22586298C2semidirect
7. C71×#1330C2semidirect
156. #1330D71semidirect
157. C71#11618semidirect

Maximal subgroups

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

subgroupindex
#225973182
#225901382
#346×D712
#225863232
#225863212
#225862982
C71×#13302
#1161871