Finite Group Databasebeta

← #30766764

Subgroups — #30766764

Order 17152.

412 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 91 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#22560289C211yesmaximal, characteristic
2C2×#16863122C211yesmaximal, characteristic
3#22558239C212yesmaximal, complemented
4#22558215C212yesmaximal, complemented
5C67×#2583C211yesmaximal, characteristic
6#16863360—21no—
7C2×C22⋊(C67⋊C8)—21no—
8#16863165—21no—
9#16863165C2212yes—
10#16863122—21no—
11#16863122C2212yes—
12C2×C22⋊Q8×C67C411yescharacteristic
13C22×C4⋊C4×C67C2211yescharacteristic
14#398×C67C411yescharacteristic
15#398×C67C2211yescharacteristic
16#395×C67C411yescharacteristic
17#395×C67C2211yescharacteristic
18#389×C67—21no—
19#389×C67—21no—
20#385×C67—21no—
21#385×C67—21no—
22C4×C22⋊C4×C67C411yescharacteristic
23C2×#11761612—22no—
24C22×C67⋊C8—41no—
25C22⋊(C67⋊C8)—22no—
26C22⋊(C67⋊C8)—41no—
27C22⋊(C67⋊C8)—41no—
28C22⋊(C67⋊C8)—41no—
29C22×Q8×C67C2×C411yescharacteristic
30C23×C4×C67C2×C411yescharacteristic
31C23×C4×C67D411yescharacteristic
32C23×C4×C67D411yescharacteristic
33C22⋊Q8×C67—41no—
34C22⋊Q8×C67—41no—
35C2×C4⋊C4×C67—21no—
36C2×C4⋊C4×C67—21no—
37C2×C4⋊C4×C67—21no—
38C2×C4⋊C4×C67—21no—
39C2×C4⋊C4×C67—21no—
40C2×C4⋊C4×C67—41no—
41C2×C4⋊C4×C67—41no—
42C2×C4⋊C4×C67C2×C411yescharacteristic
43C2×C4⋊C4×C67C2×C411yescharacteristic
44C2×C4⋊C4×C67D411yescharacteristic
45C2×C4⋊C4×C67D411yescharacteristic
46C2×C4⋊C4×C67C2311yescharacteristic
47C2×C22⋊C4×C67—21no—
48C2×C22⋊C4×C67—21no—
49C2×C42×C67—21no—
50#95×C67—21no—
51#95×C67—21no—
52#95×C67—21no—
53#95×C67—21no—
54#95×C67C2×C411yescharacteristic
55#95×C67C2×C411yescharacteristic
56#11761612—42no—
57#11761612—42no—
58C2×C67⋊C8—42no—
59C2×C67⋊C8—42no—
60C2×C67⋊C8—82no—
61C24×C67C22⋊C411yescharacteristic
62C2×Q8×C67—41no—
63C2×Q8×C67—81no—
64C22×C4×C67—21no—
65C22×C4×C67—21no—
66C22×C4×C67—21no—
67C22×C4×C67—21no—
68C22×C4×C67—21no—
69C22×C4×C67—21no—
70C22×C4×C67—21no—
71C22×C4×C67—21no—
72C22×C4×C67—22no—
73C22×C4×C67—21no—
74C22×C4×C67—21no—
75C22×C4×C67—41no—
76C22×C4×C67—81no—
77C22×C4×C67—81no—
78C22×C4×C67C22⋊C411yescharacteristic
79C22×C4×C67C22⋊C411yescharacteristic
80C22×C4×C67C22⋊C411yescharacteristic
81C22×C4×C67C22×C411yescharacteristic
82C22×C4×C67C2×D411yescharacteristic
83C22×C4×C67C2×D411yescharacteristic
84C4⋊C4×C67—41no—
85C4⋊C4×C67—41no—
86C4⋊C4×C67—41no—
87C4⋊C4×C67—41no—
88C4⋊C4×C67—41no—
89C4⋊C4×C67—42no—
90C4⋊C4×C67—42no—
91C4⋊C4×C67—42no—
92C4⋊C4×C67—81no—
93C4⋊C4×C67—81no—
94C22⋊C4×C67—24no—
95C22⋊C4×C67—41no—
96C22⋊C4×C67—41no—
97C42×C67—81no—
98C23×C67—22no—
99C23×C67—21no—
100C23×C67—21no—
101C23×C67—41no—
102C23×C67#9911yescharacteristic
103C23×C67#9911yescharacteristic
104C23×C67C2×C22⋊C411yescharacteristic
105Q8×C67—82no—
106C2×C4×C67—21no—
107C2×C4×C67—21no—
108C2×C4×C67—22no—
109C2×C4×C67—22no—
110C2×C4×C67—21no—
111C2×C4×C67—21no—
112C2×C4×C67—21no—
113C2×C4×C67—21no—
114C2×C4×C67—21no—
115C2×C4×C67—21no—
116C2×C4×C67—21no—
117C2×C4×C67—42no—
118C2×C4×C67—42no—
119C2×C4×C67—41no—
120C2×C4×C67—41no—
121C2×C4×C67—42no—
122C2×C4×C67—41no—
123C2×C4×C67—41no—
124C2×C4×C67—41no—
125C2×C4×C67—41no—
126C2×C4×C67—41no—
127C2×C4×C67—41no—
128C2×C4×C67—41no—
129C2×C4×C67—41no—
130C2×C4×C67—41no—
131C2×C4×C67—41no—
132C2×C4×C67—41no—
133C2×C4×C67—41no—
134C2×C4×C67—81no—
135C2×C4×C67—81no—
136C2×C4×C67—81no—
137C2×C4×C67#13512yes—
138C67⋊C8—82no—
139C67⋊C8—82no—
140C22×C67—21no—
141C22×C67—21no—
142C22×C67—21no—
143C22×C67—21no—
144C22×C67—21no—
145C22×C67—21no—
146C22×C67—21no—
147C22×C67—42no—
148C22×C67—42no—
149C22×C67#35612yes—
150C22×C67C2×#9911yescharacteristic
151C22×C67#41611yescharacteristic
152C22×C67#42111yescharacteristic
153C4×C67—22no—
154C4×C67—42no—
155C4×C67—42no—
156C4×C67—41no—
157C4×C67—41no—
158C4×C67—41no—
159C4×C67—41no—
160C4×C67—41no—
161C4×C67—81no—
162C4×C67—81no—
163#11854—671nomaximal, Sylow
164C2×C67—21no—
165C2×C67—21no—
166C2×C67—22no—
167C2×C67—41no—
168C2×C67#151611yescharacteristic
169C2×C67#212111yescharacteristic
170C2×C67C2×#35611yescharacteristic
171#2583D6711yescharacteristic
172#1821—671no—
173C2×#324—671no—
174#1354—672no—
175#1346—672no—
176C67#1185411yescharacteristic, complemented, Sylow
177C2×C22⋊Q8Dic6711yescharacteristic
178C22×C4⋊C4C2×D6711yescharacteristic
179#407—1341no—
180C2×C22⋊C8—1341no—
181#398Dic6711yescharacteristic
182#398C2×D6711yescharacteristic
183#395Dic6711yescharacteristic
184#395C2×D6711yescharacteristic
185#389—21no—
186#389—21no—
187#385—21no—
188#385—21no—
189C4×C22⋊C4Dic6711yescharacteristic
190#343—672no—
191#343—1341no—
192#324—672no—
193#324—1341no—
194C22×Q8C2×Dic6711yescharacteristic
195C23×C4C2×Dic6711yescharacteristic
196C23×C4C22⋊D6711yescharacteristic
197C23×C4C22⋊D6711yescharacteristic
198C2×M16—1342no—
199C22×C8—2681no—
200C22⋊Q8—41no—
201C22⋊Q8—41no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—41no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4C2×Dic6711yescharacteristic
210C2×C4⋊C4C2×Dic6711yescharacteristic
211C2×C4⋊C4C22⋊D6711yescharacteristic
212C2×C4⋊C4C22⋊D6711yescharacteristic
213C2×C4⋊C4C22×D6711yescharacteristic
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C42—21no—
217C22⋊C8—2681no—
218C22⋊C8—2681no—
219C22⋊C8—2681no—
220C22⋊C8—1342no—
221#95—21no—
222#95—21no—
223#95—21no—
224#95—21no—
225#95C2×Dic6711yescharacteristic
226#95C2×Dic6711yescharacteristic
227C24C22⋊Dic6711yescharacteristic
228C2×Q8—41no—
229C2×Q8—81no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—22no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—41no—
242C22×C4—81no—
243C22×C4—81no—
244C22×C4C22⋊Dic6711yescharacteristic
245C22×C4C22⋊Dic6711yescharacteristic
246C22×C4C22⋊Dic6711yescharacteristic
247C22×C4C22×Dic6711yescharacteristic
248C22×C4C2×C22⋊D6711yescharacteristic
249C22×C4C2×C22⋊D6711yescharacteristic
250M16—2682no—
251M16—2682no—
252C2×C8—2682no—
253C2×C8—2682no—
254C2×C8—5362no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—42no—
261C4⋊C4—42no—
262C4⋊C4—42no—
263C4⋊C4—81no—
264C4⋊C4—81no—
265C22⋊C4—24no—
266C22⋊C4—41no—
267C22⋊C4—41no—
268C42—81no—
269C23—21no—
270C23—21no—
271C23—22no—
272C23—41no—
273C23#1452395811yescharacteristic
274C23#1452395811yescharacteristic
275C23C2×C22⋊Dic6711yescharacteristic
276Q8—82no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—42no—
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—81no—
306C2×C4—81no—
307C2×C4—81no—
308C2×C4#1452406312yes—
309C8—5362no—
310C8—5362no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—42no—
319C22—42no—
320C22#1686319512yes—
321C22C2×#1452395811yescharacteristic
322C22#1686339211yescharacteristic
323C22#1686341311yescharacteristic
324C4—22no—
325C4—42no—
326C4—42no—
327C4—41no—
328C4—41no—
329C4—41no—
330C4—41no—
331C4—41no—
332C4—81no—
333C4—81no—
334C2—21no—
335C2—21no—
336C2—22no—
337C2—41no—
338C2#2255876511yescharacteristic
339C2#2256185511yescharacteristic
340C2C2×#1686319511yescharacteristic
subgroupG/H
centerC22C2×#14523958
commutatorC22×C4×C67C22×C4
FrattiniC2×C4⋊C4C22×D67
FittingC67×#2583C2
socleC22×C67C2×#99
prime radicalC24×C67C22⋊C4
cosocleC2×C4⋊C4×C67C23
hypercenter#2583D67
hypocenterC67#11854
LCSG ▷ C22×C4×C67 ▷ C23×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ #2583

Complements

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

NH ≅ G/Ntype
3. #22558239C2semidirect
4. #22558215C2semidirect
176. C67#11854semidirect

Maximal subgroups

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

subgroupindex
#225602892
C2×#168631222
#225582392
#225582152
C67×#25832
#1185467