Finite Group Databasebeta

← #30602030

Subgroups — #30602030

Order 15616.

536 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 86 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#21461698C211yesmaximal, characteristic, complemented
2#21461555C211yesmaximal, characteristic, complemented
3#21461216C212yesmaximal, complemented
4#21461096C212yesmaximal, complemented
5C2×#16839999C211yesmaximal, characteristic
6C2×#14519847—21no—
7#16840733C2212yes—
8D61⋊(C22⋊C8)C2211yescharacteristic
9C2×#14519831—21no—
10C4×C22⋊#6923C2211yescharacteristic
11#16840638—21no—
12(C2×#6923)⋊C8—21no—
13#16840614—24no—
14#16840606—24no—
15C22×D488C411yescharacteristic, complemented
16C22×C8×D61C2211yescharacteristic
17C2×#14519752C2212yes—
18#16839999—21no—
19#16839999—21no—
20#16839999—21no—
21#16839999—21no—
22C2×#14519697C412yescomplemented
23C2×#107×C61C411yescharacteristic, complemented
24C2×#11758213—22no—
25C2×D61⋊C8—21no—
26C2×C22⋊#6923—21no—
27C2×C4×#6923—21no—
28#14519847—42no—
29#14519834—21no—
30#14519834—22no—
31C22⋊#92420—22no—
32C22⋊#92420D412yes—
33#14519831—42no—
34(C2×#6923)⋊C4D412yes—
35C22×D244—22no—
36C22×D244C2×C412yes—
37C22×C4×D61C2311yescharacteristic
38C2×D488—21no—
39C2×D488—21no—
40C2×D488—42no—
41C22×#92418C2×C411yescharacteristic
42C2×C8×D61—42no—
43C2×C8×D61D411yescharacteristic
44C2×C8×D61D411yescharacteristic
45C2×C8×D61D411yescharacteristic
46C2×C8×D61D411yescharacteristic
47#14519752—22no—
48#14519752—22no—
49#14519752—22no—
50#14519752—22no—
51C2×#11758198—22no—
52#14519697—22no—
53#14519697—22no—
54C22×C8×C61C2×C411yescharacteristic
55C2×C4⋊C4×C61C2×C412yes—
56#107×C61—21no—
57#107×C61—21no—
58C23×D61—22no—
59C23×D61#4111yescharacteristic
60C22×#6923—22no—
61C22×C4×C61—22no—
62C22×C4×C61C22×C411yescharacteristic
63C22×Dic61C2×D411yescharacteristic
64C2×D244—22no—
65C2×D244—22no—
66C2×D244—42no—
67C2×D244—42no—
68C2×D244—42no—
69C2×D244—42no—
70C2×D244—84no—
71C2×C4×D61—41no—
72C2×C4×D61—41no—
73C2×C4×D61C2×D411yescharacteristic
74C2×C4×D61C2×D411yescharacteristic
75C2×C4×D61C2×D411yescharacteristic
76C2×C4×D61#4111yescharacteristic
77C22⋊#6923—42no—
78C2×#92420—22no—
79C2×#92420—42no—
80C4×#6923—41no—
81C4×#6923—41no—
82#11758213—42no—
83#11758213—42no—
84D61⋊C8—41no—
85D61⋊C8—41no—
86C2×C8×C61—22no—
87C2×C8×C61C22⋊C411yescharacteristic
88C2×C8×C61C22⋊C411yescharacteristic
89C4⋊C4×C61—22no—
90C4⋊C4×C61—22no—
91#11758198—42no—
92#11758198—42no—
93C2×#92418—41no—
94C2×#92418C22⋊C411yescharacteristic
95C2×#92418C22⋊C411yescharacteristic
96D488—42no—
97D488—42no—
98C8×D61—22no—
99C8×D61—22no—
100C8×D61—41no—
101C8×D61—41no—
102C23×C61C4×D411yescharacteristic
103C22×D61—22no—
104C22×D61—22no—
105C22×D61—41no—
106C22×D61—42no—
107C22×D61—44no—
108C22×D61—82no—
109C22×D61—82no—
110C22×D61#12111yescharacteristic
111C22×D61#12411yescharacteristic
112C2×#6923—42no—
113C2×#6923—42no—
114C2×#6923—42no—
115C2×C4×C61—21no—
116C2×C4×C61—21no—
117C2×C4×C61—42no—
118C2×C4×C61—82no—
119C2×C4×C61C2×C22⋊C411yescharacteristic
120C2×C4×C61C4×D411yescharacteristic
121C2×Dic61—41no—
122C2×Dic61#12111yescharacteristic
123C2×Dic61#12711yescharacteristic
124D244—42no—
125D244—42no—
126D244—82no—
127D244—84no—
128D244—84no—
129C4×D61—21no—
130C4×D61—21no—
131C4×D61—21no—
132C4×D61—21no—
133C4×D61—41no—
134C4×D61—41no—
135#92420—42no—
136#92420—82no—
137C8×C61—21no—
138C8×C61#9912yes—
139#92418—21no—
140#92418—21no—
141#9818—611nomaximal, Sylow
142C22×C61—21no—
143C22×C61—21no—
144C22×C61#39011yescharacteristic
145C22×C61#44412yes—
146C2×D61—21no—
147C2×D61—21no—
148C2×D61—21no—
149C2×D61—21no—
150C2×D61—21no—
151C2×D61—44no—
152C2×D61—82no—
153C2×D61—82no—
154C2×D61—82no—
155C2×D61—164no—
156C2×D61#47212yes—
157#6923—82no—
158C4×C61—21no—
159C4×C61—82no—
160C4×C61C2×#9911yescharacteristic
161C4×C61#41011yescharacteristic
162Dic61—21no—
163Dic61—21no—
164C2×#466—611no—
165#1810—611no—
166#1752—611no—
167#1507—612no—
168#1471—612no—
169C2×C61—22no—
170C2×C61#196411yescharacteristic
171C2×C61#197511yescharacteristic
172C2×C61#197811yescharacteristic
173D61—22no—
174D61—41no—
175D61—162no—
176C22×D8—611no—
177C23×C8—611no—
178C2×#121—612no—
179#466—1221no—
180#466—1221no—
181#466—1221no—
182#466—1221no—
183C2×#108—1221no—
184C2×#107#692311yescharacteristic, complemented
185C2×#102—612no—
186#407—612no—
187C2×C22⋊C8—611no—
188C2×#97—1221no—
189C4×C22⋊C4—611no—
190#341—1221no—
191#336—1221no—
192#327—1224no—
193#323—1224no—
194C61#981811yescharacteristic, complemented, Sylow
195C22×D4—612no—
196C22×D4—1222no—
197C23×C4—611no—
198C2×D8—1221no—
199C2×D8—1221no—
200C2×D8—2442no—
201C2×M16—1222no—
202C22×C8—611no—
203C22×C8—611no—
204C22×C8—611no—
205C22×C8—611no—
206C22×C8—611no—
207C22×C8—1221no—
208C22×C8—2442no—
209C22×C8C2×#692311yescharacteristic
210#121—1222no—
211#121—1222no—
212#121—1222no—
213#121—1222no—
214C2×C4⋊C4C2×#692312yes—
215C2×C22⋊C4—1222no—
216C2×C22⋊C4—1221no—
217C2×C42—1221no—
218#108—2442no—
219#107—21no—
220#107—21no—
221#102—1222no—
222#102—1222no—
223C22⋊C8—612no—
224C22⋊C8—1221no—
225C22⋊C8—1222no—
226C22⋊C8—1222no—
227#97—2442no—
228#95—612no—
229C24—611no—
230C24—1222no—
231C2×D4—1222no—
232C2×D4—1222no—
233C2×D4—4884no—
234C2×D4—2442no—
235C2×D4—2442no—
236C2×D4—2442no—
237C2×D4—2442no—
238C22×C4—22no—
239C22×C4—611no—
240C22×C4—611no—
241C22×C4—611no—
242C22×C4—611no—
243C22×C4—611no—
244C22×C4—1222no—
245C22×C4—2441no—
246C22×C4—2441no—
247C22×C4C22×#692311yescharacteristic
248D8—2442no—
249D8—2442no—
250M16—2442no—
251M16—2442no—
252C2×C8—22no—
253C2×C8—611no—
254C2×C8—611no—
255C2×C8—1222no—
256C2×C8—1222no—
257C2×C8—1222no—
258C2×C8—2441no—
259C2×C8—2441no—
260C2×C8—2441no—
261C2×C8—2442no—
262C2×C8—2441no—
263C2×C8—2441no—
264C2×C8C22⋊#692311yescharacteristic
265C2×C8C22⋊#692311yescharacteristic
266C4⋊C4—22no—
267C4⋊C4—22no—
268C22⋊C4—2442no—
269C22⋊C4—2442no—
270C22⋊C4—2442no—
271C42—2441no—
272C42—2441no—
273C23—611no—
274C23—611no—
275C23—1222no—
276C23—1222no—
277C23—4882no—
278C23—4882no—
279C23—2441no—
280C23—2444no—
281C23—2442no—
282C23D4×#692311yescharacteristic
283D4—4882no—
284D4—4884no—
285D4—4884no—
286D4—2442no—
287D4—2442no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—42no—
291C2×C4—82no—
292C2×C4—611no—
293C2×C4—611no—
294C2×C4—1221no—
295C2×C4—1221no—
296C2×C4—1221no—
297C2×C4—1221no—
298C2×C4—2441no—
299C2×C4—2441no—
300C2×C4—2441no—
301C2×C4—2442no—
302C2×C4—2442no—
303C2×C4—2442no—
304C2×C4C2×C22⋊#692311yescharacteristic
305C2×C4D4×#692311yescharacteristic
306C8—21no—
307C8—1221no—
308C8—1221no—
309C8—4882no—
310C8—2442no—
311C8#1451983612yes—
312C22—21no—
313C22—21no—
314C22—612no—
315C22—1221no—
316C22—1221no—
317C22—1221no—
318C22—1221no—
319C22—1221no—
320C22—9764no—
321C22—4882no—
322C22—4882no—
323C22—4882no—
324C22—2444no—
325C22#1684071411yescharacteristic
326C22#1684080812yes—
327C4—21no—
328C4—82no—
329C4—1221no—
330C4—1221no—
331C4—4882no—
332C4C2×#1451983611yescharacteristic
333C4#1684074311yescharacteristic
334C2—22no—
335C2—1222no—
336C2—9762no—
337C2—2441no—
338C2#2146195911yescharacteristic
339C2#2146197311yescharacteristic
340C2#2146198011yescharacteristic
subgroupG/H
centerC22#16840714
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×#107×C61C4
socleC22×C61#390
prime radicalC2×#16839999C2
cosocleC22×C4×D61C23
hypercenterC2×#107#6923
hypocenterC61#9818
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#107

Complements

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

NH ≅ G/Ntype
1. #21461698C2semidirect
2. #21461555C2semidirect
3. #21461216C2semidirect
4. #21461096C2semidirect
15. C22×D488C4semidirect
22. C2×#14519697C4semidirect
23. C2×#107×C61C4semidirect
184. C2×#107#6923semidirect
194. C61#9818semidirect

Maximal subgroups

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

subgroupindex
#214616982
#214615552
#214612162
#214610962
C2×#168399992
#981861