Finite Group Databasebeta

← #30751852

Subgroups — #30751852

Order 17152.

524 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 127 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#22568773C211yesmaximal, characteristic
2#22565975C211yesmaximal, characteristic
3#22558880C212yesmaximal, complemented
4#22558068C212yesmaximal, complemented
5C67×#1458C211yesmaximal, characteristic
6C22×#11761612—21no—
7C2×#14524043C2211yescharacteristic
8C2×#14524022—21no—
9#16863423—24no—
10#16863420—21no—
11#16863420C2212yes—
12#16863326—21no—
13#16863326C412yes—
14#16863321—21no—
15#16863321C412yes—
16#16863302—21no—
17#16863262C2211yescharacteristic
18#16863141—24no—
19C2×#117×C67C2211yescharacteristic
20#431×C67C2212yes—
21#324×C67—21no—
22#323×C67—21no—
23C22×Dic134—21no—
24C2×#11761612—22no—
25C2×#11761612—22no—
26C2×#11761612—42no—
27C2×#11761612—42no—
28C22×C67⋊C8—22no—
29#14524043—22no—
30#14524043—22no—
31#14524022—42no—
32#14524022—42no—
33C2×C4⋊Dic67—21no—
34C2×C4⋊Dic67—21no—
35C2×C4⋊Dic67C2×C412yes—
36C2×Dic67⋊C4—22no—
37C2×C22⋊Dic67—22no—
38C4⋊(C67⋊C8)—24no—
39C4⋊(C67⋊C8)—22no—
40C4⋊(C67⋊C8)D414yes—
41C67⋊#95—22no—
42C67⋊#95—22no—
43C67⋊#95—21no—
44C67⋊#95C2×C412yes—
45C23×C4×C67D411yescharacteristic
46#117×C67—22no—
47#117×C67—22no—
48C2×C4⋊C4×C67C2×C411yescharacteristic
49C2×C4⋊C4×C67D411yescharacteristic
50C2×C22⋊C4×C67C2×C411yescharacteristic
51C2×C22⋊C4×C67D411yescharacteristic
52C2×C42×C67D411yescharacteristic
53C2×C42×C67C2311yescharacteristic
54C4⋊C8×C67—22no—
55C22⋊C8×C67—22no—
56C4×C8×C67—22no—
57C2×Dic134—41no—
58C2×Dic134—41no—
59C2×Dic134—41no—
60C2×Dic134—41no—
61C2×Dic134—41no—
62C2×Dic134—41no—
63C22×Dic67—22no—
64C22×Dic67—22no—
65#11761612—44no—
66#11761612—42no—
67#11761612—42no—
68C2×C67⋊C8—24no—
69C2×C67⋊C8—44no—
70C4⋊Dic67—24no—
71C4⋊Dic67—42no—
72C4⋊Dic67—41no—
73C4⋊Dic67—41no—
74Dic67⋊C4—44no—
75C22⋊Dic67—44no—
76C24×C67C22⋊C411yescharacteristic
77C22×C4×C67—21no—
78C22×C4×C67—21no—
79C22×C4×C67—21no—
80C22×C4×C67—21no—
81C22×C4×C67C22⋊C411yescharacteristic
82C22×C4×C67C22⋊C411yescharacteristic
83C22×C4×C67C22⋊C411yescharacteristic
84C22×C4×C67D812yes—
85C22×C4×C67SD1612yes—
86C22×C4×C67C22×C411yescharacteristic
87C22×C4×C67C2×D411yescharacteristic
88C22×C4×C67C2×D411yescharacteristic
89C2×C8×C67—42no—
90C4⋊C4×C67—22no—
91C4⋊C4×C67—22no—
92C22⋊C4×C67—22no—
93C22⋊C4×C67—22no—
94C42×C67—21no—
95C42×C67—21no—
96C42×C67C2×D412yes—
97C42×C67#4112yes—
98C23×C67—21no—
99C23×C67—21no—
100C23×C67—41no—
101C23×C67—41no—
102C23×C67#10212yes—
103C23×C67C2×C22⋊C411yescharacteristic
104C2×C4×C67—21no—
105C2×C4×C67—21no—
106C2×C4×C67—22no—
107C2×C4×C67—22no—
108C2×C4×C67—22no—
109C2×C4×C67—22no—
110C2×C4×C67—21no—
111C2×C4×C67—21no—
112C2×C4×C67—41no—
113C2×C4×C67—41no—
114C2×C4×C67—41no—
115C2×C4×C67—41no—
116C2×C4×C67#10212yes—
117C2×C4×C67C4×D412yes—
118C2×C4×C67#12011yescharacteristic
119C2×C4×C67#12112yes—
120C2×C4×C67#12311yescharacteristic
121C2×C4×C67C2×D811yescharacteristic
122C2×C4×C67C2×SD1611yescharacteristic
123C2×C4×C67AGL(1,ℤ/8)11yescharacteristic
124C2×C4×C67#13711yescharacteristic
125C2×Dic67—42no—
126C2×Dic67—42no—
127C2×Dic67—42no—
128C2×Dic67—42no—
129C2×Dic67—42no—
130C2×Dic67—42no—
131Dic134—81no—
132Dic134—81no—
133Dic134—81no—
134Dic134—81no—
135C8×C67—82no—
136C67⋊C8—44no—
137C22×C67—21no—
138C22×C67—21no—
139C22×C67—22no—
140C22×C67—42no—
141C22×C67—41no—
142C22×C67—41no—
143C22×C67—81no—
144C22×C67#38611yescharacteristic
145C22×C67C2×#10211yescharacteristic
146C22×C67#41711yescharacteristic
147C4×C67—22no—
148C4×C67—22no—
149C4×C67—22no—
150C4×C67—41no—
151C4×C67—41no—
152C4×C67#44711yescharacteristic
153C4×C67#44811yescharacteristic
154C4×C67#48011yescharacteristic
155C4×C67#48111yescharacteristic
156Dic67—82no—
157Dic67—82no—
158#9195—671nomaximal, Sylow
159C2×C67—21no—
160C2×C67—21no—
161C2×C67—42no—
162C2×C67#187411yescharacteristic
163C2×C67#190911yescharacteristic
164C2×C67#191511yescharacteristic
165#2905—671no—
166#2621—671no—
167#1546—672no—
168#1458D6711yescharacteristic
169#1286—672no—
170C67#919511yescharacteristic, complemented, Sylow
171C22×M16—1341no—
172C2×C4⋊Q8—671no—
173C2×C22⋊Q8—1341no—
174C2×#117C2×D6711yescharacteristic
175#431C2×D6712yes—
176#424—1344no—
177#423—672no—
178#423—1341no—
179#398—672no—
180#398—1341no—
181#397—672no—
182#397—1341no—
183#391—1341no—
184#382—671no—
185#332—1344no—
186#324—21no—
187#323—21no—
188C22×Q8—1341no—
189C23×C4D26811yescharacteristic
190C2×M16—2682no—
191C2×M16—2682no—
192C2×M16—1342no—
193C2×M16—1342no—
194C22×C8—1342no—
195C4⋊Q8—1342no—
196C4⋊Q8—1342no—
197C22⋊Q8—2682no—
198C22⋊Q8—2682no—
199#117—22no—
200#117—22no—
201C2×C4⋊C4—672no—
202C2×C4⋊C4—1342no—
203C2×C4⋊C4—1341no—
204C2×C4⋊C4—1341no—
205C2×C4⋊C4C4×D6711yescharacteristic
206C2×C4⋊C4C22⋊D6711yescharacteristic
207C2×C22⋊C4—1342no—
208C2×C22⋊C4C4×D6711yescharacteristic
209C2×C22⋊C4C22⋊D6711yescharacteristic
210C2×C42D26811yescharacteristic
211C2×C42C22×D6711yescharacteristic
212C4⋊C8—22no—
213C4⋊C8—674no—
214C4⋊C8—1344no—
215C4⋊C8—1342no—
216C22⋊C8—22no—
217C4×C8—22no—
218#95—672no—
219#95—1342no—
220#95—1342no—
221#95—1341no—
222C24(C2×D67)⋊C411yescharacteristic
223C2×Q8—2681no—
224C2×Q8—2681no—
225C2×Q8—2681no—
226C2×Q8—2681no—
227C2×Q8—2681no—
228C2×Q8—2681no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—1342no—
234C22×C4—1342no—
235C22×C4(C2×D67)⋊C411yescharacteristic
236C22×C4(C2×D67)⋊C411yescharacteristic
237C22×C4(C2×D67)⋊C411yescharacteristic
238C22×C4D53612yes—
239C22×C4#1176161712yes—
240C22×C4C2×C4×D6711yescharacteristic
241C22×C4C2×C22⋊D6711yescharacteristic
242C22×C4C2×D26811yescharacteristic
243M16—2684no—
244M16—2682no—
245M16—2682no—
246C2×C8—42no—
247C2×C8—2684no—
248C2×C8—1344no—
249C4⋊C4—22no—
250C4⋊C4—22no—
251C4⋊C4—2684no—
252C4⋊C4—2682no—
253C4⋊C4—2681no—
254C4⋊C4—2681no—
255C4⋊C4—1344no—
256C22⋊C4—22no—
257C22⋊C4—22no—
258C22⋊C4—2684no—
259C42—21no—
260C42—21no—
261C42D4×D6712yes—
262C42#1176162712yes—
263C23—21no—
264C23—21no—
265C23—41no—
266C23—41no—
267C23#1452396512yes—
268C23C2×(C2×D67)⋊C411yescharacteristic
269Q8—5361no—
270Q8—5361no—
271Q8—5361no—
272Q8—5361no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—2682no—
286C2×C4—2682no—
287C2×C4—2682no—
288C2×C4—2682no—
289C2×C4—2682no—
290C2×C4—2682no—
291C2×C4#1452396512yes—
292C2×C4(C22⋊D67)⋊C412yes—
293C2×C4#1452401311yescharacteristic
294C2×C4#1452401512yes—
295C2×C4#1452402711yescharacteristic
296C2×C4C2×D53611yescharacteristic
297C2×C4C2×#1176161711yescharacteristic
298C2×C4#1452407111yescharacteristic
299C2×C4#1452407611yescharacteristic
300C8—82no—
301C8—2684no—
302C22—21no—
303C22—21no—
304C22—22no—
305C22—42no—
306C22—41no—
307C22—41no—
308C22—81no—
309C22#1686327311yescharacteristic
310C22C2×#1452396511yescharacteristic
311C22#1686339811yescharacteristic
312C4—22no—
313C4—22no—
314C4—22no—
315C4—41no—
316C4—41no—
317C4—5362no—
318C4—5362no—
319C4#1686353211yescharacteristic
320C4#1686353711yescharacteristic
321C4#1686371011yescharacteristic
322C4#1686371511yescharacteristic
323C2—21no—
324C2—21no—
325C2—42no—
326C2#2256052411yescharacteristic
327C2#2256072911yescharacteristic
328C2#2256075911yescharacteristic
subgroupG/H
centerC22#16863273
commutatorC22×C4×C67C22×C4
FrattiniC2×C42C22×D67
FittingC67×#1458C2
socleC22×C67#386
prime radicalC24×C67C22⋊C4
cosocleC2×C42×C67C23
hypercenter#1458D67
hypocenterC67#9195
LCSG ▷ C22×C4×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1458

Complements

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

NH ≅ G/Ntype
3. #22558880C2semidirect
4. #22558068C2semidirect
170. C67#9195semidirect

Maximal subgroups

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

subgroupindex
#225687732
#225659752
#225588802
#225580682
C67×#14582
#919567