Finite Group Databasebeta

← #30859143

Subgroups — #30859143

Order 17152.

1002 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16863422C211yesmaximal, characteristic
2C2×#16863422C211yesmaximal, characteristic
3C2×#16863420C211yesmaximal, characteristic
4C2×#16863351C211yesmaximal, characteristic
5#22562692C211yesmaximal, characteristic
6#22560931C214yesmaximal, complemented
7#22560926C214yesmaximal, complemented
8#22559905C211yesmaximal, characteristic
9C2×#336×C67C211yesmaximal, characteristic
10C22×#11761612C2211yescharacteristic
11C22×#11761612C2211yescharacteristic
12C2×#14524002C2211yescharacteristic
13C2×#14524002C2211yescharacteristic
14#16863422—24no—
15#16863422—24no—
16#16863420—24no—
17C2×C4⋊(C67⋊C8)C2214yes—
18C2×Dic67⋊C8C2214yes—
19C2×Dic67⋊C8C2214yes—
20#16863351—24no—
21C2×#14523953C2212yes—
22C2×C8×Dic67C2212yes—
23#16863294C412yes—
24#16863294C412yes—
25#16863269C414yes—
26C4×Dic67⋊C4C2214yes—
27#16863150C2212yes—
28#16863150C2212yes—
29C23×C8×C67C2211yescharacteristic
30C23×C8×C67C2211yescharacteristic
31C22×C42×C67C2211yescharacteristic
32#336×C67C2212yes—
33#336×C67C2212yes—
34C2×#11761612—22no—
35C2×#11761612—22no—
36C2×#11761612—22no—
37C2×#11761612—22no—
38C2×#11761612—22no—
39C2×#11761612D414yes—
40C22×C67⋊C8C2312yes—
41C22×C67⋊C8C2312yes—
42#14524002—22no—
43#14524002—22no—
44#14524002D414yes—
45#14524002Q814yes—
46C2×C4⋊Dic67C2×C412yes—
47C2×C4⋊Dic67C2×C412yes—
48C2×Dic67⋊C4C2×C414yes—
49C2×Dic67⋊C4C2×C414yes—
50C2×C22⋊Dic67C2×C412yes—
51C2×C22⋊Dic67C2×C412yes—
52C2×C4×Dic67C2312yes—
53C2×C4×Dic67C2312yes—
54C4⋊(C67⋊C8)—28no—
55Dic67⋊C8—28no—
56Dic67⋊C8—28no—
57#14523953—24no—
58C8×Dic67—24no—
59C67⋊#95C2×C414yes—
60C67⋊#95C2×C414yes—
61C23×C4×C67C2×C412yes—
62C23×C4×C67C2311yescharacteristic
63C22×C8×C67—22no—
64C22×C8×C67—22no—
65C22×C8×C67—22no—
66C22×C8×C67—22no—
67C22×C8×C67—22no—
68C22×C8×C67Q814yes—
69C22×C8×C67C2312yes—
70C22×C8×C67C2311yescharacteristic
71C22×C8×C67C2311yescharacteristic
72C2×C42×C67—22no—
73C2×C42×C67—22no—
74C2×C42×C67C2×C411yescharacteristic
75C2×C42×C67C2×C411yescharacteristic
76C2×C42×C67C2312yes—
77C22×Dic67C22×C414yes—
78C22×Dic67C22×C414yes—
79#11761612—28no—
80#11761612—44no—
81C2×C67⋊C8—22no—
82C2×C67⋊C8—22no—
83C2×C67⋊C8—22no—
84C2×C67⋊C8—22no—
85C2×C67⋊C8—22no—
86C2×C67⋊C8C2×D414yes—
87C4⋊Dic67—24no—
88C4⋊Dic67C4⋊C418yes—
89Dic67⋊C4—28no—
90Dic67⋊C4—44no—
91C22⋊Dic67—24no—
92C22⋊Dic67C4⋊C418yes—
93C4×Dic67—22no—
94C4×Dic67—22no—
95C4×Dic67C2×D414yes—
96C4×Dic67C2×Q814yes—
97C24×C67C22×C411yescharacteristic
98C22×C4×C67—24no—
99C22×C4×C67—24no—
100C22×C4×C67—42no—
101C22×C4×C67C22×C412yes—
102C22×C4×C67C22×C412yes—
103C22×C4×C67C22×C411yescharacteristic
104C22×C4×C67C2×D412yes—
105C22×C4×C67C2×D412yes—
106C22×C4×C67C2×Q812yes—
107C22×C4×C67C2×Q812yes—
108C22×C4×C67#4112yes—
109C22×C4×C67#4112yes—
110C22×C4×C67C2411yescharacteristic
111C2×C8×C67—28no—
112C2×C8×C67—21no—
113C2×C8×C67—21no—
114C2×C8×C67—21no—
115C2×C8×C67—21no—
116C2×C8×C67—22no—
117C2×C8×C67—22no—
118C2×C8×C67—22no—
119C2×C8×C67—44no—
120C2×C8×C67C2×Q812yes—
121C2×C8×C67C2×Q812yes—
122C42×C67—22no—
123C42×C67—22no—
124C42×C67—22no—
125C42×C67—22no—
126C23×C67—22no—
127C23×C67—22no—
128C23×C67C2×C4⋊C412yes—
129C23×C67C4×D412yes—
130C23×C67C4×Q812yes—
131C23×C67C23×C411yescharacteristic
132C2×C4×C67—22no—
133C2×C4×C67—22no—
134C2×C4×C67—22no—
135C2×C4×C67—22no—
136C2×C4×C67—22no—
137C2×C4×C67—48no—
138C2×C4×C67C2×C4⋊C412yes—
139C2×C4×C67C4×D412yes—
140C2×C4×C67C4×Q812yes—
141C2×C4×C67#12114yes—
142C2×C4×C67C22⋊Q814yes—
143C2×C4×C67#12514yes—
144C2×C4×C67C4⋊Q814yes—
145C2×C4×C67#13114yes—
146C2×C4×C67C22×D411yescharacteristic
147C2×C4×C67C22×D411yescharacteristic
148C2×C4×C67C22×Q811yescharacteristic
149C2×C4×C67C22×Q811yescharacteristic
150C2×C4×C67C2×#4111yescharacteristic
151C2×C4×C67C2×#4111yescharacteristic
152C2×Dic67—24no—
153C2×Dic67—24no—
154C2×Dic67—24no—
155C2×Dic67—24no—
156C2×Dic67—24no—
157C2×Dic67C2×C4⋊C418yes—
158C8×C67—22no—
159C8×C67—22no—
160C8×C67—42no—
161C67⋊C8—24no—
162C67⋊C8—42no—
163C22×C67—24no—
164C22×C67—24no—
165C22×C67—22no—
166C22×C67—22no—
167C22×C67#38914yes—
168C22×C67C22×C4⋊C411yescharacteristic
169C22×C67C2×C4×D411yescharacteristic
170C22×C67C2×C4×Q811yescharacteristic
171C22×C67C2×#13112yes—
172C22×C67#56812yes—
173C4×C67—24no—
174C4×C67—42no—
175C4×C67#38914yes—
176C4×C67C2×#12111yescharacteristic
177C4×C67C2×C22⋊Q811yescharacteristic
178C4×C67C2×#12511yescharacteristic
179C4×C67C2×C4⋊Q811yescharacteristic
180Dic67—28no—
181Dic67—44no—
182#21148—671nomaximal, Sylow
183C2×C67—24no—
184C2×C67C2×#38911yescharacteristic
185C2×C67#290612yes—
186C2×C67#293312yes—
187C2×C67#296011yescharacteristic
188C2×C67#296111yescharacteristic
189C2×#424—671no—
190C2×#424—671no—
191C2×#423—671no—
192C2×#405—671no—
193#2296—671no—
194#1940—674no—
195#1939—674no—
196#1729—671no—
197C2×#336D6711yescharacteristic
198C67#2114811yescharacteristic, complemented, Sylow
199C22×M16—671no—
200C22×M16—671no—
201C23×C8C2×D6711yescharacteristic
202C23×C8C2×D6711yescharacteristic
203C2×#117—671no—
204C2×#117—671no—
205C22×C42C2×D6711yescharacteristic
206#424—1344no—
207#424—1344no—
208#423—1344no—
209C2×C4⋊C8—674no—
210C2×C4⋊C8—674no—
211C2×C4⋊C8—674no—
212#405—1344no—
213C2×#97—672no—
214C2×C4×C8—672no—
215#389—672no—
216#389—672no—
217#385—674no—
218C4×C4⋊C4—674no—
219#336—672no—
220#336—672no—
221#336C2×D6712yes—
222#336C2×D6712yes—
223C23×C4C4×D6712yes—
224C23×C4C22×D6711yescharacteristic
225C2×M16—674no—
226C2×M16—1342no—
227C2×M16—1342no—
228C2×M16—1342no—
229C2×M16—1342no—
230C2×M16—1342no—
231C22×C8—22no—
232C22×C8—22no—
233C22×C8—22no—
234C22×C8—22no—
235C22×C8—22no—
236C22×C8—672no—
237C22×C8—672no—
238C22×C8Dic13414yes—
239C22×C8C22×D6712yes—
240C22×C8C22×D6711yescharacteristic
241C22×C8C22×D6711yescharacteristic
242#117—674no—
243#117—674no—
244#117—1342no—
245#117—1342no—
246C2×C4⋊C4—674no—
247C2×C4⋊C4—672no—
248C2×C4⋊C4—672no—
249C2×C4⋊C4—674no—
250C2×C22⋊C4—672no—
251C2×C22⋊C4—672no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—672no—
255C2×C42—672no—
256C2×C42C4×D6711yescharacteristic
257C2×C42C4×D6711yescharacteristic
258C2×C42C22×D6712yes—
259C4⋊C8—1348no—
260C4⋊C8—1348no—
261C4⋊C8—1348no—
262#97—1344no—
263C4×C8—1344no—
264#95—674no—
265#95—674no—
266C24C2×C4×D6711yescharacteristic
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—42no—
270C22×C4—674no—
271C22×C4—674no—
272C22×C4C2×C4×D6712yes—
273C22×C4C2×C4×D6712yes—
274C22×C4C2×C4×D6711yescharacteristic
275C22×C4D4×D6712yes—
276C22×C4D4×D6712yes—
277C22×C4Q8×D6712yes—
278C22×C4C2×Dic13412yes—
279C22×C4#1176162712yes—
280C22×C4#1176162812yes—
281C22×C4C23×D6711yescharacteristic
282M16—2684no—
283M16—1348no—
284C2×C8—28no—
285C2×C8—21no—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8—21no—
289C2×C8—22no—
290C2×C8—22no—
291C2×C8—22no—
292C2×C8—44no—
293C2×C8—674no—
294C2×C8—1342no—
295C2×C8—1342no—
296C2×C8—1342no—
297C2×C8—1342no—
298C2×C8—1342no—
299C2×C8C2×Dic13412yes—
300C2×C8C2×Dic13412yes—
301C4⋊C4—2684no—
302C4⋊C4—678no—
303C4⋊C4—1344no—
304C4⋊C4—1348no—
305C22⋊C4—678no—
306C22⋊C4—1344no—
307C42—22no—
308C42—22no—
309C42—22no—
310C42—22no—
311C42—674no—
312C42—674no—
313C42—1342no—
314C42—1342no—
315C23—22no—
316C23—22no—
317C23C4⋊C4×D6712yes—
318C23(C22⋊D67)⋊C412yes—
319C23C4×Dic13412yes—
320C23C22×C4×D6711yescharacteristic
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—48no—
327C2×C4—678no—
328C2×C4—1344no—
329C2×C4—1344no—
330C2×C4—1344no—
331C2×C4—1344no—
332C2×C4—1344no—
333C2×C4C4⋊C4×D6712yes—
334C2×C4(C22⋊D67)⋊C412yes—
335C2×C4C4×Dic13412yes—
336C2×C4#1452401614yes—
337C2×C4#1452402014yes—
338C2×C4#1452403314yes—
339C2×C4#1452404214yes—
340C2×C4#1452404914yes—
341C2×C4C2×D4×D6711yescharacteristic
342C2×C4C2×D4×D6711yescharacteristic
343C2×C4C2×Q8×D6711yescharacteristic
344C2×C4C22×Dic13411yescharacteristic
345C2×C4C2×#1176162711yescharacteristic
346C2×C4C2×#1176162811yescharacteristic
347C8—22no—
348C8—22no—
349C8—42no—
350C8—2682no—
351C8—1344no—
352C22—24no—
353C22—22no—
354C22—22no—
355C22—24no—
356C22#1686329314yes—
357C22C2×C4⋊C4×D6711yescharacteristic
358C22C2×(C22⋊D67)⋊C411yescharacteristic
359C22C2×C4×Dic13411yescharacteristic
360C22C2×#1452404912yes—
361C22#1686416212yes—
362C4—24no—
363C4—42no—
364C4—2684no—
365C4—1348no—
366C4#1686329314yes—
367C4C2×#1452401611yescharacteristic
368C4C2×#1452402011yescharacteristic
369C4C2×#1452403311yescharacteristic
370C4C2×#1452404211yescharacteristic
371C2—24no—
372C2C2×#1686329311yescharacteristic
373C2#2256877412yes—
374C2#2256893012yes—
375C2#2256914311yescharacteristic
376C2#2256914911yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D67
commutator, socleC23×C67C23×C4
FittingC2×#336×C67C2
prime radicalC24×C67C22×C4
cosocleC22×C4×C67C24
hypercenterC2×#336D67
hypocenterC67#21148
LCSG ▷ C23×C67 ▷ C67
UCSC1 ◁ C22×C4 ◁ C2×#336

Complements

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

NH ≅ G/Ntype
6. #22560931C2semidirect
7. #22560926C2semidirect
198. C67#21148semidirect

Maximal subgroups

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

subgroupindex
C2×#168634222
C2×#168634202
C2×#168633512
#225626922
#225609312
#225609262
#225599052
C2×#336×C672
#2114867