Finite Group Databasebeta

← #31886002

Subgroups — #31886002

Order 18176.

470 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 107 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#22596679C211yesmaximal, characteristic
2#22588320C211yesmaximal, characteristic
3#22586165C212yesmaximal, complemented
4#22586162C212yesmaximal, complemented
5C71×#1749C211yesmaximal, characteristic
6C2×#14556107C411yescharacteristic
7C2×C4⋊(C71⋊C8)—21no—
8C2×C22⋊(C71⋊C8)—21no—
9#17985905C411yescharacteristic
10#17985905C2211yescharacteristic
11#17985897C411yescharacteristic
12#17985897C2211yescharacteristic
13#17985883—21no—
14#17985883—21no—
15#17985872—21no—
16#17985862—21no—
17#17985859C411yescharacteristic
18#17985754C2212yes—
19C22×C42×C71C2211yescharacteristic
20C2×C22⋊C8×C71—21no—
21C2×#97×C71—21no—
22#336×C71C2212yes—
23C22×Dic142C2×C411yescharacteristic
24C22×C71⋊C8—22no—
25#14556107—42no—
26C2×C4⋊Dic71C2×C411yescharacteristic
27C2×C4⋊Dic71C2×C411yescharacteristic
28C2×C4⋊Dic71C2×C411yescharacteristic
29C2×Dic71⋊C4—21no—
30C2×Dic71⋊C4—21no—
31C2×Dic71⋊C4—21no—
32C2×C22⋊Dic71—21no—
33C2×C22⋊Dic71—21no—
34C4⋊(C71⋊C8)—42no—
35C22⋊(C71⋊C8)—42no—
36C71⋊#95—21no—
37C71⋊#95—21no—
38C71⋊#95—21no—
39C71⋊#95—21no—
40C71⋊#95—21no—
41C71⋊#95—21no—
42C71⋊#95—21no—
43C23×C4×C71C2×C411yescharacteristic
44C23×C4×C71D411yescharacteristic
45C23×C4×C71D411yescharacteristic
46C22×C8×C71—22no—
47C2×C42×C71—42no—
48C2×C42×C71C2×C411yescharacteristic
49C2×C42×C71D411yescharacteristic
50C2×C42×C71D411yescharacteristic
51C2×C42×C71C2311yescharacteristic
52C22⋊C8×C71—24no—
53#97×C71—42no—
54C2×Dic142—22no—
55C2×Dic142—41no—
56C2×Dic142—41no—
57C22×Dic71—21no—
58C22×Dic71—21no—
59C22×Dic71—21no—
60C22×Dic71—21no—
61C2×C71⋊C8—42no—
62C2×C71⋊C8—42no—
63C2×C71⋊C8—42no—
64C4⋊Dic71—22no—
65C4⋊Dic71—41no—
66C4⋊Dic71—41no—
67Dic71⋊C4—42no—
68Dic71⋊C4—42no—
69Dic71⋊C4—41no—
70Dic71⋊C4—41no—
71C22⋊Dic71—42no—
72C22⋊Dic71—42no—
73C24×C71C22⋊C411yescharacteristic
74C22×C4×C71—22no—
75C22×C4×C71—22no—
76C22×C4×C71—22no—
77C22×C4×C71—21no—
78C22×C4×C71—21no—
79C22×C4×C71—22no—
80C22×C4×C71—41no—
81C22×C4×C71—41no—
82C22×C4×C71—41no—
83C22×C4×C71—41no—
84C22×C4×C71C22⋊C411yescharacteristic
85C22×C4×C71C22⋊C411yescharacteristic
86C22×C4×C71C22⋊C411yescharacteristic
87C22×C4×C71C22×C411yescharacteristic
88C22×C4×C71C2×D411yescharacteristic
89C22×C4×C71C2×D411yescharacteristic
90C2×C8×C71—24no—
91C2×C8×C71—42no—
92C2×C8×C71—42no—
93C42×C71—22no—
94C42×C71—22no—
95C42×C71—41no—
96C42×C71—41no—
97C23×C71—21no—
98C23×C71—21no—
99C23×C71—21no—
100C23×C71—21no—
101C23×C71—21no—
102C23×C71—21no—
103C23×C71#10412yes—
104C23×C71C2×C22⋊C411yescharacteristic
105C2×C4×C71—21no—
106C2×C4×C71—21no—
107C2×C4×C71—21no—
108C2×C4×C71—21no—
109C2×C4×C71—21no—
110C2×C4×C71—21no—
111C2×C4×C71—21no—
112C2×C4×C71—21no—
113C2×C4×C71—21no—
114C2×C4×C71—21no—
115C2×C4×C71—21no—
116C2×C4×C71—21no—
117C2×C4×C71—21no—
118C2×C4×C71—21no—
119C2×C4×C71—44no—
120C2×C4×C71—44no—
121C2×C4×C71—42no—
122C2×C4×C71—42no—
123C2×C4×C71#9912yes—
124C2×C4×C71#13112yes—
125C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
126C2×C4×C71#13711yescharacteristic
127C2×C4×C71#13712yes—
128C2×Dic71—41no—
129C2×Dic71—41no—
130C2×Dic71—41no—
131C2×Dic71—41no—
132C2×Dic71—41no—
133C2×Dic71—41no—
134C2×Dic71—41no—
135C2×Dic71—41no—
136C2×Dic71—41no—
137C2×Dic71—41no—
138C2×Dic71—41no—
139C2×Dic71—41no—
140Dic142—82no—
141C8×C71—44no—
142C71⋊C8—82no—
143C22×C71—22no—
144C22×C71—22no—
145C22×C71—22no—
146C22×C71—41no—
147C22×C71—41no—
148C22×C71—41no—
149C22×C71—41no—
150C22×C71#40811yescharacteristic
151C22×C71C2×#9911yescharacteristic
152C22×C71#41311yescharacteristic
153C22×C71#41711yescharacteristic
154C22×C71#41911yescharacteristic
155C22×C71C2×#10411yescharacteristic
156C22×C71#42111yescharacteristic
157C4×C71—22no—
158C4×C71—22no—
159C4×C71—22no—
160C4×C71—21no—
161C4×C71—21no—
162C4×C71—41no—
163C4×C71—41no—
164Dic71—81no—
165Dic71—81no—
166Dic71—81no—
167Dic71—81no—
168#7701—711nomaximal, Sylow
169C2×C71—42no—
170C2×C71#146811yescharacteristic
171C2×C71#148311yescharacteristic
172C2×C71#149711yescharacteristic
173C2×C71#150811yescharacteristic
174C2×C71#151311yescharacteristic
175C2×C71#153211yescharacteristic
176C2×C71#155911yescharacteristic
177#2855—711no—
178#1797—711no—
179#1749D7111yescharacteristic
180#1272—712no—
181#1271—712no—
182C71#770111yescharacteristic, complemented, Sylow
183C2×C22⋊Q8—711no—
184C22×C42C2×D7111yescharacteristic
185C2×C4⋊C8—1421no—
186C2×C22⋊C8—21no—
187C2×C22⋊C8—1421no—
188C2×#97—21no—
189#391—711no—
190#391—711no—
191#389—711no—
192#389—711no—
193#387—1421no—
194#387—1421no—
195#385—1421no—
196#381—1421no—
197#380—711no—
198#336—712no—
199#336C2×D7112yes—
200C22×Q8—711no—
201C23×C4C4×D7111yescharacteristic
202C23×C4D28411yescharacteristic
203C23×C4C22⋊D7111yescharacteristic
204C22×C8—22no—
205C22×C8—1422no—
206C22⋊Q8—2842no—
207C2×C4⋊C4—711no—
208C2×C4⋊C4—711no—
209C2×C4⋊C4—711no—
210C2×C4⋊C4—1421no—
211C2×C4⋊C4—1421no—
212C2×C4⋊C4—1421no—
213C2×C22⋊C4—1421no—
214C2×C22⋊C4—1421no—
215C2×C42—42no—
216C2×C42C4×D7111yescharacteristic
217C2×C42D28411yescharacteristic
218C2×C42C22⋊D7111yescharacteristic
219C2×C42C22×D7111yescharacteristic
220C4⋊C8—2842no—
221C22⋊C8—24no—
222C22⋊C8—2842no—
223#97—42no—
224#95—1421no—
225#95—1421no—
226#95—1421no—
227#95—1421no—
228#95—1421no—
229#95—1421no—
230#95—1421no—
231C24(C2×D71)⋊C411yescharacteristic
232C2×Q8—2841no—
233C2×Q8—2841no—
234C2×Q8—1422no—
235C22×C4—22no—
236C22×C4—21no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—21no—
240C22×C4—22no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—1421no—
246C22×C4—1421no—
247C22×C4—1421no—
248C22×C4—1421no—
249C22×C4(C2×D71)⋊C411yescharacteristic
250C22×C4(C2×D71)⋊C411yescharacteristic
251C22×C4(C2×D71)⋊C411yescharacteristic
252C22×C4C2×C4×D7111yescharacteristic
253C22×C4C2×C22⋊D7111yescharacteristic
254C22×C4C2×D28411yescharacteristic
255C2×C8—24no—
256C2×C8—42no—
257C2×C8—42no—
258C2×C8—2842no—
259C2×C8—2842no—
260C2×C8—2842no—
261C4⋊C4—2841no—
262C4⋊C4—2841no—
263C4⋊C4—2842no—
264C4⋊C4—2841no—
265C4⋊C4—2842no—
266C4⋊C4—2841no—
267C4⋊C4—1422no—
268C22⋊C4—2842no—
269C22⋊C4—2842no—
270C42—22no—
271C42—22no—
272C42—41no—
273C42—41no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23#1455605612yes—
281C23C2×(C2×D71)⋊C411yescharacteristic
282Q8—5682no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—44no—
298C2×C4—44no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—2841no—
302C2×C4—2841no—
303C2×C4—2841no—
304C2×C4—2841no—
305C2×C4—2841no—
306C2×C4—2841no—
307C2×C4—2841no—
308C2×C4—2841no—
309C2×C4—2841no—
310C2×C4—2841no—
311C2×C4—2841no—
312C2×C4—2841no—
313C2×C4#1455604212yes—
314C2×C4#1455613612yes—
315C2×C4#1455615611yescharacteristic
316C2×C4#1455615812yes—
317C2×C4#1455616111yescharacteristic
318C8—44no—
319C8—5682no—
320C22—22no—
321C22—22no—
322C22—22no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22#1798596711yescharacteristic
328C22C2×#1455604211yescharacteristic
329C22#1798598111yescharacteristic
330C22#1798600111yescharacteristic
331C22#1798600711yescharacteristic
332C22C2×#1455605611yescharacteristic
333C22#1798601511yescharacteristic
334C4—21no—
335C4—21no—
336C4—22no—
337C4—22no—
338C4—22no—
339C4—41no—
340C4—41no—
341C4—5681no—
342C4—5681no—
343C4—5681no—
344C4—5681no—
345C2—42no—
346C2#2258668911yescharacteristic
347C2#2258675411yescharacteristic
348C2#2258681211yescharacteristic
349C2#2258685911yescharacteristic
350C2#2258688411yescharacteristic
351C2#2258695811yescharacteristic
352C2#2258707111yescharacteristic
subgroupG/H
centerC23C2×(C2×D71)⋊C4
commutatorC22×C4×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#1749C2
socleC23×C71C2×C22⋊C4
prime radicalC24×C71C22⋊C4
cosocleC2×C42×C71C23
hypercenter#1749D71
hypocenterC71#7701
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1749

Complements

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

NH ≅ G/Ntype
3. #22586165C2semidirect
4. #22586162C2semidirect
182. C71#7701semidirect

Maximal subgroups

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

subgroupindex
#225966792
#225883202
#225861652
#225861622
C71×#17492
#770171