Finite Group Databasebeta

← #31907758

Subgroups — #31907758

Order 18176.

452 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 115 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#22588079C211yesmaximal, characteristic
2#22586654C211yesmaximal, characteristic
3#22586370C212yesmaximal, complemented
4#22586367C212yesmaximal, complemented
5C71×#2396C211yesmaximal, characteristic
6C22×#11765849—21no—
7#17986052—21no—
8#17985768—21no—
9#17985768C2212yes—
10#17985725C2212yes—
11#17985723—21no—
12C2×C4×Q8×C71C411yescharacteristic
13C2×#117×C71C2211yescharacteristic
14#400×C71C2211yescharacteristic
15#399×C71C411yescharacteristic
16#398×C71—21no—
17#396×C71—21no—
18#395×C71C2211yescharacteristic
19#393×C71C411yescharacteristic
20C4×C4⋊C4×C71—21no—
21C4×C4⋊C4×C71C411yescharacteristic
22C4×C22⋊C4×C71—21no—
23C2×#11765849—24no—
24C2×#11765849—22no—
25C2×#11765849—41no—
26C2×#11765849—42no—
27C22×C71⋊C8—41no—
28C4⋊(C71⋊C8)—41no—
29C22⋊(C71⋊C8)—22no—
30C4×C71⋊C8—41no—
31C22×Q8×C71C2×C411yescharacteristic
32C23×C4×C71D411yescharacteristic
33C4×Q8×C71—41no—
34C4×Q8×C71—41no—
35#117×C71—41no—
36#117×C71—41no—
37C2×C4⋊C4×C71—21no—
38C2×C4⋊C4×C71—21no—
39C2×C4⋊C4×C71—21no—
40C2×C4⋊C4×C71—21no—
41C2×C4⋊C4×C71C2×C411yescharacteristic
42C2×C4⋊C4×C71D411yescharacteristic
43C2×C4⋊C4×C71C2311yescharacteristic
44C2×C22⋊C4×C71—21no—
45C2×C22⋊C4×C71—21no—
46C2×C22⋊C4×C71C2×C411yescharacteristic
47C2×C22⋊C4×C71D411yescharacteristic
48C2×C42×C71—21no—
49C2×C42×C71—21no—
50C2×C42×C71C2×C411yescharacteristic
51C2×C42×C71D411yescharacteristic
52#95×C71—21no—
53#95×C71—21no—
54#95×C71—21no—
55#95×C71—21no—
56#95×C71C2×C411yescharacteristic
57#95×C71C2×C411yescharacteristic
58#11765849—24no—
59#11765849—44no—
60#11765849—42no—
61C2×C71⋊C8—42no—
62C2×C71⋊C8—44no—
63C2×C71⋊C8—42no—
64C24×C71C22⋊C411yescharacteristic
65C2×Q8×C71—42no—
66C2×Q8×C71—41no—
67C22×C4×C71—21no—
68C22×C4×C71—21no—
69C22×C4×C71—22no—
70C22×C4×C71—21no—
71C22×C4×C71—21no—
72C22×C4×C71—21no—
73C22×C4×C71—21no—
74C22×C4×C71C22⋊C411yescharacteristic
75C22×C4×C71C22⋊C411yescharacteristic
76C22×C4×C71C22⋊C411yescharacteristic
77C22×C4×C71SD1612yes—
78C22×C4×C71Q1612yes—
79C22×C4×C71C22×C411yescharacteristic
80C22×C4×C71C2×D411yescharacteristic
81C22×C4×C71C2×D411yescharacteristic
82C4⋊C4×C71—41no—
83C4⋊C4×C71—41no—
84C4⋊C4×C71—41no—
85C4⋊C4×C71—42no—
86C4⋊C4×C71—42no—
87C4⋊C4×C71—42no—
88C4⋊C4×C71—42no—
89C22⋊C4×C71—41no—
90C22⋊C4×C71—41no—
91C22⋊C4×C71—42no—
92C22⋊C4×C71—42no—
93C42×C71—21no—
94C42×C71—21no—
95C42×C71—21no—
96C42×C71—21no—
97C42×C71—41no—
98C42×C71—41no—
99C42×C71—42no—
100C23×C71—21no—
101C23×C71—21no—
102C23×C71—41no—
103C23×C71—41no—
104C23×C71#10312yes—
105C23×C71C2×C22⋊C411yescharacteristic
106Q8×C71—42no—
107Q8×C71—42no—
108C2×C4×C71—21no—
109C2×C4×C71—21no—
110C2×C4×C71—22no—
111C2×C4×C71—22no—
112C2×C4×C71—22no—
113C2×C4×C71—22no—
114C2×C4×C71—22no—
115C2×C4×C71—22no—
116C2×C4×C71—21no—
117C2×C4×C71—21no—
118C2×C4×C71—21no—
119C2×C4×C71—21no—
120C2×C4×C71—41no—
121C2×C4×C71—41no—
122C2×C4×C71—41no—
123C2×C4×C71—41no—
124C2×C4×C71—41no—
125C2×C4×C71—41no—
126C2×C4×C71—41no—
127C2×C4×C71—41no—
128C2×C4×C71—41no—
129C2×C4×C71—41no—
130C2×C4×C71—41no—
131C2×C4×C71—41no—
132C2×C4×C71—41no—
133C2×C4×C71—41no—
134C2×C4×C71#9911yescharacteristic
135C2×C4×C71#9911yescharacteristic
136C2×C4×C71#10312yes—
137C2×C4×C71#10412yes—
138C2×C4×C71C2×SD1611yescharacteristic
139C2×C4×C71C2×Q1611yescharacteristic
140C71⋊C8—44no—
141C71⋊C8—82no—
142C22×C71—21no—
143C22×C71—21no—
144C22×C71—22no—
145C22×C71—42no—
146C22×C71—41no—
147C22×C71—41no—
148C22×C71—81no—
149C22×C71C2×#9911yescharacteristic
150C22×C71C2×#10311yescharacteristic
151C22×C71C2×#10411yescharacteristic
152C4×C71—21no—
153C4×C71—21no—
154C4×C71—42no—
155C4×C71—41no—
156C4×C71—42no—
157C4×C71—41no—
158C4×C71—41no—
159C4×C71—41no—
160C4×C71—81no—
161C4×C71—81no—
162C4×C71#32812yes—
163C4×C71#32812yes—
164#11721—711nomaximal, Sylow
165C2×C71—21no—
166C2×C71—21no—
167C2×C71—42no—
168C2×C71C2×#32811yescharacteristic
169C2×C71#212011yescharacteristic
170C2×C71#213211yescharacteristic
171#2396D7111yescharacteristic
172#1742—711no—
173#1458—711no—
174#1354—712no—
175#1353—712no—
176C71#1172111yescharacteristic, complemented, Sylow
177C22×M16—1421no—
178C2×C4×Q8Dic7111yescharacteristic
179C2×#117C2×D7111yescharacteristic
180#431—1421no—
181#400C2×D7111yescharacteristic
182#399Dic7111yescharacteristic
183#398—21no—
184#396—21no—
185#395C2×D7111yescharacteristic
186#393Dic7111yescharacteristic
187C4×C4⋊C4—21no—
188C4×C4⋊C4Dic7111yescharacteristic
189C4×C22⋊C4—21no—
190#343—712no—
191#343—1421no—
192#324—712no—
193#323—1421no—
194C22×Q8C2×Dic7111yescharacteristic
195C23×C4C22⋊D7111yescharacteristic
196C2×M16—2841no—
197C2×M16—2842no—
198C2×M16—1424no—
199C2×M16—1422no—
200C22×C8—2841no—
201C4×Q8—41no—
202C4×Q8—41no—
203#117—41no—
204#117—41no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4C2×Dic7111yescharacteristic
210C2×C4⋊C4C22⋊D7111yescharacteristic
211C2×C4⋊C4C22×D7111yescharacteristic
212C2×C22⋊C4—21no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4C2×Dic7111yescharacteristic
215C2×C22⋊C4C22⋊D7111yescharacteristic
216C2×C42—21no—
217C2×C42—21no—
218C2×C42C2×Dic7111yescharacteristic
219C2×C42C22⋊D7111yescharacteristic
220C4⋊C8—2841no—
221C22⋊C8—1422no—
222C4×C8—2841no—
223#95—21no—
224#95—21no—
225#95—21no—
226#95—21no—
227#95C2×Dic7111yescharacteristic
228#95C2×Dic7111yescharacteristic
229C24C22⋊Dic7111yescharacteristic
230C2×Q8—42no—
231C2×Q8—41no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—22no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4C22⋊Dic7111yescharacteristic
240C22×C4C22⋊Dic7111yescharacteristic
241C22×C4C22⋊Dic7111yescharacteristic
242C22×C4#1176585212yes—
243C22×C4#1176585512yes—
244C22×C4C22×Dic7111yescharacteristic
245C22×C4C2×C22⋊D7111yescharacteristic
246C22×C4C2×C22⋊D7111yescharacteristic
247M16—2844no—
248M16—2842no—
249M16—1424no—
250C2×C8—2842no—
251C2×C8—2844no—
252C2×C8—2842no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—42no—
257C4⋊C4—42no—
258C4⋊C4—42no—
259C4⋊C4—42no—
260C22⋊C4—41no—
261C22⋊C4—41no—
262C22⋊C4—42no—
263C22⋊C4—42no—
264C42—21no—
265C42—21no—
266C42—21no—
267C42—21no—
268C42—41no—
269C42—41no—
270C42—42no—
271C23—21no—
272C23—21no—
273C23—41no—
274C23—41no—
275C23#1455605212yes—
276C23C2×C22⋊Dic7111yescharacteristic
277Q8—42no—
278Q8—42no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4#1455604311yescharacteristic
306C2×C4#1455604311yescharacteristic
307C2×C4#1455605212yes—
308C2×C4#1455605512yes—
309C2×C4C2×#1176585211yescharacteristic
310C2×C4C2×#1176585511yescharacteristic
311C8—2844no—
312C8—5682no—
313C22—21no—
314C22—21no—
315C22—22no—
316C22—42no—
317C22—41no—
318C22—41no—
319C22—81no—
320C22C2×#1455604311yescharacteristic
321C22C2×#1455605211yescharacteristic
322C22C2×#1455605511yescharacteristic
323C4—21no—
324C4—21no—
325C4—42no—
326C4—41no—
327C4—42no—
328C4—41no—
329C4—41no—
330C4—41no—
331C4—81no—
332C4—81no—
333C4#1798573612yes—
334C4#1798573612yes—
335C2—21no—
336C2—21no—
337C2—42no—
338C2C2×#1798573611yescharacteristic
339C2#2258998111yescharacteristic
340C2#2259002511yescharacteristic
subgroupG/H
centerC2×C4#14556043
commutatorC22×C4×C71C22×C4
FrattiniC2×C4⋊C4C22×D71
FittingC71×#2396C2
socleC22×C71C2×#99
prime radicalC24×C71C22⋊C4
cosocleC2×C4⋊C4×C71C23
hypercenter#2396D71
hypocenterC71#11721
LCSG ▷ C22×C4×C71 ▷ C23×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#117 ◁ #2396

Complements

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

NH ≅ G/Ntype
3. #22586370C2semidirect
4. #22586367C2semidirect
176. C71#11721semidirect

Maximal subgroups

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

subgroupindex
#225880792
#225866542
#225863702
#225863672
C71×#23962
#1172171