Finite Group Databasebeta

← #31902276

Subgroups — #31902276

Order 18176.

330 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 87 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#22588347C211yesmaximal, characteristic
2#22588329C211yesmaximal, characteristic
3#22587115C211yesmaximal, characteristic, complemented
4#22586732C211yesmaximal, characteristic
5#22586235C211yesmaximal, characteristic, complemented
6#22586198C211yesmaximal, characteristic, complemented
7C71×#1286C211yesmaximal, characteristic, complemented
8C2×C4×D284C2211yescharacteristic
9#17986052C2211yescharacteristic
10#17986023C2211yescharacteristic
11C2×C4⋊(C71⋊C8)C2211yescharacteristic
12C2×#14556048—21no—
13C2×#14556044—21no—
14#17985767—21no—
15#17985761—21no—
16#17985752—22no—
17#17985728—22no—
18#431×C71C2211yescharacteristic
19#423×C71C2211yescharacteristic
20#382×C71C2211yescharacteristic
21C22×D284D411yescharacteristic
22C22×C4×D71—21no—
23C2×#11765849—21no—
24C22×C71⋊C8—21no—
25C4×D284—21no—
26C4×D284—21no—
27C4×D284—21no—
28C4×D284—21no—
29C2×C4⋊Dic71D411yescharacteristic
30C2×(C2×D71)⋊C4—21no—
31C4⋊(C71⋊C8)—21no—
32C4⋊(C71⋊C8)—21no—
33C4⋊(C71⋊C8)—21no—
34C4⋊(C71⋊C8)—21no—
35C4⋊(C71⋊C8)—21no—
36#14556048—42no—
37#14556044—42no—
38C22⋊(C71⋊C8)—21no—
39C4×C71⋊C8—21no—
40C2×M16×C71—21no—
41C2×C4⋊C4×C71—21no—
42C2×C42×C71C2311yescharacteristic
43C4⋊C8×C71—21no—
44C4⋊C8×C71—21no—
45C4⋊C8×C71—21no—
46C22⋊C8×C71—21no—
47C4×C8×C71D412yes—
48#95×C71—21no—
49#95×C71—21no—
50C23×D71—21no—
51C2×D284—21no—
52C2×D284—21no—
53C2×D284—21no—
54C2×D284—21no—
55C2×D284—21no—
56C2×D284D812yes—
57C22×Dic71—21no—
58C2×C4×D71—41no—
59C2×C4×D71—41no—
60C2×C4×D71—82no—
61#11765849—41no—
62#11765849—41no—
63C2×C71⋊C8—41no—
64C2×C71⋊C8—41no—
65C2×C71⋊C8—41no—
66C2×C71⋊C8—41no—
67C2×C71⋊C8—41no—
68C4⋊Dic71—21no—
69C4⋊Dic71—21no—
70(C2×D71)⋊C4—41no—
71(C2×D71)⋊C4—41no—
72C22×C4×C71—41no—
73C22×C4×C71C2×D411yescharacteristic
74C22×C4×C71#4111yescharacteristic
75C22×C4×C71#4111yescharacteristic
76M16×C71—41no—
77M16×C71—41no—
78C2×C8×C71—22no—
79C2×C8×C71—41no—
80C4⋊C4×C71—41no—
81C4⋊C4×C71—41no—
82C42×C71C2×D411yescharacteristic
83C42×C71#4111yescharacteristic
84C42×C71#4111yescharacteristic
85C42×C71#4111yescharacteristic
86C23×C71#12311yescharacteristic
87C22×D71—22no—
88C22×D71—41no—
89C22×D71—41no—
90C22×D71—81no—
91C22×D71—81no—
92C2×C4×C71—41no—
93C2×C4×C71—41no—
94C2×C4×C71—81no—
95C2×C4×C71—81no—
96C2×C4×C71—81no—
97C2×C4×C71#12111yescharacteristic
98C2×C4×C71#12311yescharacteristic
99C2×C4×C71#12311yescharacteristic
100C2×C4×C71#12411yescharacteristic
101C2×C4×C71#12611yescharacteristic
102C2×C4×C71#12611yescharacteristic
103C2×C4×C71C2×D811yescharacteristic
104C2×C4×C71#13511yescharacteristic
105C2×C4×C71#13711yescharacteristic
106C2×C4×C71#13711yescharacteristic
107C2×Dic71—41no—
108C2×Dic71—81no—
109D284—22no—
110D284—22no—
111D284—41no—
112D284—41no—
113C4×D71—82no—
114C4×D71—82no—
115C8×C71—42no—
116C8×C71—81no—
117C71⋊C8—81no—
118C71⋊C8—81no—
119C71⋊C8—81no—
120C22×C71—21no—
121C22×C71—21no—
122C22×C71#39711yescharacteristic
123C22×C71#48011yescharacteristic
124C22×C71#48511yescharacteristic
125C2×D71—42no—
126C2×D71—82no—
127C2×D71—81no—
128C2×D71—81no—
129C2×D71—82no—
130C4×C71—21no—
131C4×C71—21no—
132C4×C71—21no—
133C4×C71—21no—
134C4×C71—161no—
135C4×C71#46611yescharacteristic
136C4×C71#47011yescharacteristic
137C4×C71#48011yescharacteristic
138C4×C71#48511yescharacteristic
139Dic71—81no—
140#10637—711nomaximal, Sylow
141C2×C71—21no—
142C2×C71—21no—
143C2×C71#207511yescharacteristic
144C2×C71#208011yescharacteristic
145C2×C71#208111yescharacteristic
146D71—82no—
147#1804—711no—
148#1800—711no—
149#1571—711no—
150#1478—711no—
151#1301—711no—
152#1286—711no—
153#1286D7111yescharacteristic, complemented
154C71#1063711yescharacteristic, complemented, Sylow
155C2×C4×D4—711no—
156#431—711no—
157#431C2×D7111yescharacteristic
158#423—711no—
159#423C2×D7111yescharacteristic
160C2×C4⋊C8—711no—
161C2×#102—1421no—
162C2×#100—1421no—
163#382C2×D7111yescharacteristic
164#343—1421no—
165#340—1421no—
166#335—1422no—
167D4⋊C8—1422no—
168C22×D4—711no—
169C23×C4—1421no—
170C2×M16—21no—
171C2×M16—1421no—
172C22×C8—1421no—
173C4×D4—1421no—
174C4×D4—1421no—
175C4×D4—1421no—
176C4×D4—1421no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—711no—
179C2×C22⋊C4—1421no—
180C2×C42C22×D7111yescharacteristic
181C4⋊C8—21no—
182C4⋊C8—21no—
183C4⋊C8—21no—
184C4⋊C8—1421no—
185C4⋊C8—1421no—
186C4⋊C8—1421no—
187C4⋊C8—1421no—
188C4⋊C8—1421no—
189#102—2842no—
190#100—2842no—
191C22⋊C8—21no—
192C22⋊C8—1421no—
193C4×C8—1421no—
194C4×C8D28412yes—
195#95—21no—
196#95—21no—
197C24—1421no—
198C2×D4—712no—
199C2×D4—1421no—
200C2×D4—1421no—
201C2×D4—1421no—
202C2×D4—1421no—
203C2×D4—1421no—
204C22×C4—41no—
205C22×C4—2841no—
206C22×C4—2841no—
207C22×C4—5682no—
208C22×C4—1421no—
209C22×C4D4×D7111yescharacteristic
210C22×C4#1176586411yescharacteristic
211C22×C4#1176586511yescharacteristic
212M16—41no—
213M16—41no—
214M16—2841no—
215M16—2841no—
216C2×C8—22no—
217C2×C8—41no—
218C2×C8—2841no—
219C2×C8—2841no—
220C2×C8—2841no—
221C2×C8—2841no—
222C2×C8—2841no—
223C4⋊C4—41no—
224C4⋊C4—41no—
225C4⋊C4—1421no—
226C4⋊C4—1421no—
227C22⋊C4—2841no—
228C22⋊C4—2841no—
229C42C2×D28411yescharacteristic
230C42#1176586411yescharacteristic
231C42#1176586511yescharacteristic
232C42#1176586611yescharacteristic
233C23—2841no—
234C23—2841no—
235C23—5681no—
236C23—5681no—
237C23—1422no—
238C23#1455611011yescharacteristic
239D4—2841no—
240D4—2841no—
241D4—1422no—
242D4—1422no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—81no—
246C2×C4—81no—
247C2×C4—81no—
248C2×C4—2841no—
249C2×C4—5682no—
250C2×C4—5682no—
251C2×C4—5681no—
252C2×C4#1455610311yescharacteristic
253C2×C4#1455611011yescharacteristic
254C2×C4#1455611211yescharacteristic
255C2×C4#1455611611yescharacteristic
256C2×C4#1455612311yescharacteristic
257C2×C4#1455612311yescharacteristic
258C2×C4D8×D7111yescharacteristic
259C2×C4#1455615111yescharacteristic
260C2×C4#1455615711yescharacteristic
261C2×C4#1455615711yescharacteristic
262C8—42no—
263C8—81no—
264C8—5681no—
265C8—5681no—
266C8—5681no—
267C22—21no—
268C22—21no—
269C22—2842no—
270C22—5682no—
271C22—5682no—
272C22—5681no—
273C22—5681no—
274C22#1798592611yescharacteristic
275C22#1798631111yescharacteristic
276C22#1798633911yescharacteristic
277C4—21no—
278C4—21no—
279C4—21no—
280C4—21no—
281C4—161no—
282C4—5681no—
283C4#1798623611yescharacteristic
284C4#1798625811yescharacteristic
285C4#1798631111yescharacteristic
286C4#1798633911yescharacteristic
287C2—21no—
288C2—21no—
289C2—5682no—
290C2#2258976511yescharacteristic
291C2#2258979211yescharacteristic
292C2#2258979911yescharacteristic
subgroupG/H
centerC22#17985926
commutator, cosocleC2×C42×C71C23
FrattiniC2×C42C22×D71
FittingC71×#1286C2
socleC22×C71#397
prime radicalC22×D284D4
hypercenter#1286D71
hypocenterC71#10637
LCSG ▷ C2×C42×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1286

Complements

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

NH ≅ G/Ntype
3. #22587115C2semidirect
5. #22586235C2semidirect
6. #22586198C2semidirect
7. C71×#1286C2semidirect
153. #1286D71semidirect
154. C71#10637semidirect

Maximal subgroups

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

subgroupindex
#225883472
#225883292
#225871152
#225867322
#225862352
#225861982
C71×#12862
#1063771