Finite Group Databasebeta

← #30766574

Subgroups — #30766574

Order 17152.

388 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 87 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#22560292C211yesmaximal, characteristic
2#22558543C211yesmaximal, characteristic
3#22558239C212yesmaximal, complemented
4#22558236C212yesmaximal, complemented
5C67×#2396C211yesmaximal, characteristic
6#16863453—21no—
7#16863360—21no—
8#16863165—21no—
9#16863165C2212yes—
10#16863122C2212yes—
11#16863120—21no—
12C2×C4×Q8×C67C411yescharacteristic
13C2×#117×C67C2211yescharacteristic
14#400×C67C2211yescharacteristic
15#399×C67C411yescharacteristic
16#398×C67—21no—
17#396×C67—21no—
18#395×C67C2211yescharacteristic
19#393×C67C411yescharacteristic
20C4×C4⋊C4×C67—21no—
21C4×C4⋊C4×C67C411yescharacteristic
22C4×C22⋊C4×C67—21no—
23C2×#11761612—22no—
24C4⋊(C67⋊C8)—41no—
25C22⋊(C67⋊C8)—22no—
26C22⋊(C67⋊C8)—41no—
27C22⋊(C67⋊C8)—41no—
28#14523954—41no—
29C22×Q8×C67C2×C411yescharacteristic
30C23×C4×C67D411yescharacteristic
31C4×Q8×C67—41no—
32C4×Q8×C67—41no—
33#117×C67—41no—
34#117×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×C67C2×C411yescharacteristic
40C2×C4⋊C4×C67D411yescharacteristic
41C2×C4⋊C4×C67C2311yescharacteristic
42C2×C22⋊C4×C67—21no—
43C2×C22⋊C4×C67—21no—
44C2×C22⋊C4×C67C2×C411yescharacteristic
45C2×C22⋊C4×C67D411yescharacteristic
46C2×C42×C67—21no—
47C2×C42×C67—21no—
48C2×C42×C67C2×C411yescharacteristic
49C2×C42×C67D411yescharacteristic
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—
60C24×C67C22⋊C411yescharacteristic
61C2×Q8×C67—42no—
62C2×Q8×C67—41no—
63C22×C4×C67—21no—
64C22×C4×C67—21no—
65C22×C4×C67—21no—
66C22×C4×C67—21no—
67C22×C4×C67—22no—
68C22×C4×C67—21no—
69C22×C4×C67—21no—
70C22×C4×C67—41no—
71C22×C4×C67C22⋊C411yescharacteristic
72C22×C4×C67C22⋊C411yescharacteristic
73C22×C4×C67C22⋊C411yescharacteristic
74C22×C4×C67C22×C411yescharacteristic
75C22×C4×C67C2×D411yescharacteristic
76C22×C4×C67C2×D411yescharacteristic
77C4⋊C4×C67—41no—
78C4⋊C4×C67—41no—
79C4⋊C4×C67—41no—
80C4⋊C4×C67—42no—
81C4⋊C4×C67—42no—
82C4⋊C4×C67—42no—
83C4⋊C4×C67—42no—
84C22⋊C4×C67—41no—
85C22⋊C4×C67—41no—
86C22⋊C4×C67—42no—
87C22⋊C4×C67—42no—
88C42×C67—21no—
89C42×C67—21no—
90C42×C67—21no—
91C42×C67—21no—
92C42×C67—42no—
93C42×C67—41no—
94C42×C67—41no—
95C23×C67—21no—
96C23×C67—21no—
97C23×C67—21no—
98C23×C67—41no—
99C23×C67—41no—
100C23×C67C2×C22⋊C411yescharacteristic
101Q8×C67—42no—
102Q8×C67—42no—
103C2×C4×C67—21no—
104C2×C4×C67—21no—
105C2×C4×C67—21no—
106C2×C4×C67—21no—
107C2×C4×C67—22no—
108C2×C4×C67—22no—
109C2×C4×C67—21no—
110C2×C4×C67—21no—
111C2×C4×C67—21no—
112C2×C4×C67—21no—
113C2×C4×C67—42no—
114C2×C4×C67—42no—
115C2×C4×C67—41no—
116C2×C4×C67—41no—
117C2×C4×C67—41no—
118C2×C4×C67—41no—
119C2×C4×C67—41no—
120C2×C4×C67—41no—
121C2×C4×C67—41no—
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#9911yescharacteristic
130C2×C4×C67#9911yescharacteristic
131C2×C4×C67#13712yes—
132C67⋊C8—82no—
133C67⋊C8—82no—
134C22×C67—21no—
135C22×C67—21no—
136C22×C67—42no—
137C22×C67—41no—
138C22×C67—41no—
139C22×C67—41no—
140C22×C67—81no—
141C22×C67C2×#9911yescharacteristic
142C22×C67#41911yescharacteristic
143C22×C67#42111yescharacteristic
144C4×C67—22no—
145C4×C67—21no—
146C4×C67—21no—
147C4×C67—42no—
148C4×C67—41no—
149C4×C67—42no—
150C4×C67—41no—
151C4×C67—41no—
152C4×C67—41no—
153C4×C67—81no—
154C4×C67—81no—
155#11816—671nomaximal, Sylow
156C2×C67—21no—
157C2×C67—21no—
158C2×C67—81no—
159C2×C67#151111yescharacteristic
160C2×C67#212011yescharacteristic
161C2×C67#213211yescharacteristic
162#2396D6711yescharacteristic
163#1822—671no—
164#1464—671no—
165#1354—672no—
166#1353—672no—
167C67#1181611yescharacteristic, complemented, Sylow
168C2×C4×Q8Dic6711yescharacteristic
169C2×#117C2×D6711yescharacteristic
170#432—1341no—
171#407—1341no—
172#400C2×D6711yescharacteristic
173#399Dic6711yescharacteristic
174#398—21no—
175#396—21no—
176#395C2×D6711yescharacteristic
177#393Dic6711yescharacteristic
178C4×C4⋊C4—21no—
179C4×C4⋊C4Dic6711yescharacteristic
180C4×C22⋊C4—21no—
181#343—672no—
182#343—1341no—
183#324—672no—
184#323—1341no—
185C22×Q8C2×Dic6711yescharacteristic
186C23×C4C22⋊D6711yescharacteristic
187C2×M16—1342no—
188C4×Q8—41no—
189C4×Q8—41no—
190#117—41no—
191#117—41no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4C2×Dic6711yescharacteristic
197C2×C4⋊C4C22⋊D6711yescharacteristic
198C2×C4⋊C4C22×D6711yescharacteristic
199C2×C22⋊C4—21no—
200C2×C22⋊C4—21no—
201C2×C22⋊C4C2×Dic6711yescharacteristic
202C2×C22⋊C4C22⋊D6711yescharacteristic
203C2×C42—21no—
204C2×C42—21no—
205C2×C42C2×Dic6711yescharacteristic
206C2×C42C22⋊D6711yescharacteristic
207C4⋊C8—2681no—
208C22⋊C8—2681no—
209C22⋊C8—2681no—
210C22⋊C8—1342no—
211#97—2681no—
212#95—21no—
213#95—21no—
214#95—21no—
215#95—21no—
216#95C2×Dic6711yescharacteristic
217#95C2×Dic6711yescharacteristic
218C24C22⋊Dic6711yescharacteristic
219C2×Q8—42no—
220C2×Q8—41no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—22no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—41no—
229C22×C4C22⋊Dic6711yescharacteristic
230C22×C4C22⋊Dic6711yescharacteristic
231C22×C4C22⋊Dic6711yescharacteristic
232C22×C4C22×Dic6711yescharacteristic
233C22×C4C2×C22⋊D6711yescharacteristic
234C22×C4C2×C22⋊D6711yescharacteristic
235M16—2682no—
236M16—2682no—
237C2×C8—2682no—
238C2×C8—2682no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C22⋊C4—41no—
247C22⋊C4—41no—
248C22⋊C4—42no—
249C22⋊C4—42no—
250C42—21no—
251C42—21no—
252C42—21no—
253C42—21no—
254C42—42no—
255C42—41no—
256C42—41no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23—41no—
261C23—41no—
262C23C2×C22⋊Dic6711yescharacteristic
263Q8—42no—
264Q8—42no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4#1452395811yescharacteristic
292C2×C4#1452395811yescharacteristic
293C2×C4#1452407412yes—
294C8—5362no—
295C8—5362no—
296C22—21no—
297C22—21no—
298C22—42no—
299C22—41no—
300C22—41no—
301C22—41no—
302C22—81no—
303C22C2×#1452395811yescharacteristic
304C22#1686340511yescharacteristic
305C22#1686341311yescharacteristic
306C4—22no—
307C4—21no—
308C4—21no—
309C4—42no—
310C4—41no—
311C4—42no—
312C4—41no—
313C4—41no—
314C4—41no—
315C4—81no—
316C4—81no—
317C2—21no—
318C2—21no—
319C2—81no—
320C2#2255874211yescharacteristic
321C2#2256185011yescharacteristic
322C2#2256189411yescharacteristic
subgroupG/H
centerC22C2×#14523958
commutatorC22×C4×C67C22×C4
FrattiniC2×C4⋊C4C22×D67
FittingC67×#2396C2
socleC22×C67C2×#99
prime radicalC24×C67C22⋊C4
cosocleC2×C4⋊C4×C67C23
hypercenter#2396D67
hypocenterC67#11816
LCSG ▷ C22×C4×C67 ▷ C23×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2396

Complements

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

NH ≅ G/Ntype
3. #22558239C2semidirect
4. #22558236C2semidirect
167. C67#11816semidirect

Maximal subgroups

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

subgroupindex
#225602922
#225585432
#225582392
#225582362
C67×#23962
#1181667