Finite Group Databasebeta

← #32995217

Subgroups — #32995217

Order 18688.

418 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988658C211yesmaximal, characteristic
2C2×C22⋊#11924003C211yesmaximal, characteristic, complemented
3#22607870C212yesmaximal, complemented
4#22607854C212yesmaximal, complemented
5C73×#2112C211yesmaximal, characteristic, complemented
6C22×#11924011C411yescharacteristic
7C2×C4×C22⋊D73C411yescharacteristic
8C22×#11924003—21no—
9#17988658—21no—
10#17988658—21no—
11#17988658—21no—
12#17988658—21no—
13C2×#14556520C2211yescharacteristic
14C2×C22⋊#10596831C411yescharacteristic
15C2×#14556503C2211yescharacteristic
16C2×#14556501C411yescharacteristic
17C22⋊#11924003—21no—
18C22⋊#11924003C2212yes—
19C23×C8×C73C2211yescharacteristic
20C2×M32×C73—21no—
21C22⋊C16×C73—21no—
22C22⋊C16×C73C2212yes—
23C22×C22⋊D73C811yescharacteristic
24C22×C4×D73C2×C411yescharacteristic
25C2×#11924011—21no—
26C2×#11924011—21no—
27C2×#11924011—42no—
28C22×#10596831C2×C411yescharacteristic
29C22×#10596831C2×C411yescharacteristic
30C4×C22⋊D73—21no—
31C4×C22⋊D73—21no—
32C4×C22⋊D73—21no—
33C4×C22⋊D73—21no—
34C2×#11924007C811yescharacteristic
35C2×C22⋊Dic73C811yescharacteristic
36C2×#11924005C811yescharacteristic
37C2×C4×Dic73C2×C411yescharacteristic
38C2×#11924003—22no—
39C2×#11924003—42no—
40#14556520—21no—
41#14556520—21no—
42C22⋊#10596831—21no—
43C22⋊#10596831—21no—
44#14556503—21no—
45#14556503—21no—
46#14556501—21no—
47#14556501—21no—
48C23×C4×C73C2×C411yescharacteristic
49C22×C8×C73—42no—
50C22×C8×C73C2×C411yescharacteristic
51C22×C8×C73D411yescharacteristic
52C22×C8×C73D411yescharacteristic
53C22×C8×C73D411yescharacteristic
54C22×C8×C73D411yescharacteristic
55C22×C8×C73C2311yescharacteristic
56M32×C73—42no—
57C2×C16×C73—22no—
58C23×D73C2×C811yescharacteristic
59C2×C22⋊D73—21no—
60C2×C22⋊D73—21no—
61C2×C22⋊D73—42no—
62C22×Dic73C2×C811yescharacteristic
63C22×Dic73C2×C811yescharacteristic
64C22×Dic73C2×C811yescharacteristic
65C2×C4×D73—21no—
66C2×C4×D73—21no—
67C2×C4×D73—41no—
68C2×C4×D73—41no—
69#11924011—42no—
70#11924011—42no—
71C2×#10596831—21no—
72C2×#10596831—21no—
73C2×#10596831—41no—
74C2×#10596831—41no—
75#11924007—21no—
76#11924007—21no—
77C22⋊Dic73—21no—
78C22⋊Dic73—21no—
79#11924005—21no—
80#11924005—21no—
81C4×Dic73—21no—
82C4×Dic73—21no—
83#11924003—42no—
84C24×C73C2×C811yescharacteristic
85C22×C4×C73—41no—
86C22×C4×C73—41no—
87C22×C4×C73C22⋊C411yescharacteristic
88C22×C4×C73C22⋊C411yescharacteristic
89C22×C4×C73C2×C811yescharacteristic
90C22×C4×C73M1611yescharacteristic
91C22×C4×C73M1611yescharacteristic
92C22×C4×C73C22×C411yescharacteristic
93C2×C8×C73—22no—
94C2×C8×C73—22no—
95C2×C8×C73—22no—
96C2×C8×C73—41no—
97C2×C8×C73—41no—
98C2×C8×C73—41no—
99C2×C8×C73C22⋊C411yescharacteristic
100C2×C8×C73C22⋊C411yescharacteristic
101C2×C8×C73C2×D411yescharacteristic
102C2×C8×C73C2×D411yescharacteristic
103C16×C73—42no—
104C23×C73—22no—
105C23×C73—22no—
106C23×C73—41no—
107C23×C73C22⋊C811yescharacteristic
108C23×C73C22⋊C811yescharacteristic
109C23×C73C22×C811yescharacteristic
110C22×D73—21no—
111C22×D73—22no—
112C22×D73—41no—
113C22×D73—41no—
114C2×C4×C73—21no—
115C2×C4×C73—21no—
116C2×C4×C73—21no—
117C2×C4×C73—21no—
118C2×C4×C73—21no—
119C2×C4×C73—21no—
120C2×C4×C73—41no—
121C2×C4×C73—41no—
122C2×C4×C73—41no—
123C2×C4×C73C22⋊C811yescharacteristic
124C2×C4×C73C22⋊C811yescharacteristic
125C2×C4×C73C2×C22⋊C411yescharacteristic
126C2×C4×C73C2×M1611yescharacteristic
127C22⋊D73—42no—
128C22⋊D73—42no—
129C2×Dic73—21no—
130C2×Dic73—21no—
131C2×Dic73—21no—
132C2×Dic73—22no—
133C2×Dic73—41no—
134C2×Dic73—41no—
135C4×D73—42no—
136C4×D73—41no—
137C4×D73—41no—
138C8×C73—21no—
139C8×C73—21no—
140C8×C73—21no—
141C8×C73#9912yes—
142#10596831—41no—
143#10596831—41no—
144C22×C73—21no—
145C22×C73—21no—
146C22×C73—21no—
147C22×C73—21no—
148C22×C73—21no—
149C22×C73—21no—
150C22×C73—21no—
151C22×C73—44no—
152C22×C73#34912yes—
153C22×C73C2×C22⋊C811yescharacteristic
154C22×C73#50412yes—
155C2×D73—41no—
156C2×D73—41no—
157C2×D73—41no—
158C2×D73—82no—
159C4×C73—21no—
160C4×C73—21no—
161C4×C73—21no—
162C4×C73C2×#9911yescharacteristic
163C4×C73#41311yescharacteristic
164Dic73—41no—
165Dic73—41no—
166Dic73—41no—
167#11467—731nomaximal, Sylow
168C2×C73—22no—
169C2×C73—22no—
170C2×C73—41no—
171C2×C73#145911yescharacteristic
172C2×C73#211311yescharacteristic
173C2×C73C2×#34911yescharacteristic
174D73—81no—
175C2×#435—731no—
176#2112D7311yescharacteristic, complemented
177C2×C22⋊C16—731no—
178#1322—732no—
179#1314—732no—
180C73#1146711yescharacteristic, complemented, Sylow
181C22×M16—731no—
182C23×C8C2×D7311yescharacteristic
183C2×C4×D4—731no—
184C2×M32—21no—
185C22×C16—1461no—
186#435—1461no—
187#435—1461no—
188#435—1461no—
189#435—1461no—
190C2×C4⋊C8—731no—
191C2×C22⋊C8—731no—
192C2×C22⋊C8—731no—
193C2×#97—731no—
194C22⋊C16—21no—
195C22⋊C16—732no—
196C22⋊C16—1461no—
197C22⋊C16C2×D7312yes—
198C22×D4—731no—
199C23×C4—731no—
200C23×C4C4×D7311yescharacteristic
201C2×M16—2922no—
202C2×M16—1461no—
203C2×M16—1461no—
204C22×C8—42no—
205C22×C8—731no—
206C22×C8—731no—
207C22×C8C4×D7311yescharacteristic
208C22×C8D29211yescharacteristic
209C22×C8D29211yescharacteristic
210C22×C8C22⋊D7311yescharacteristic
211C22×C8C22⋊D7311yescharacteristic
212C22×C8C22×D7311yescharacteristic
213C4×D4—1461no—
214C4×D4—1461no—
215C4×D4—1461no—
216C4×D4—1461no—
217C2×C4⋊C4—731no—
218C2×C22⋊C4—731no—
219C2×C22⋊C4—731no—
220C2×C42—731no—
221M32—42no—
222C2×C16—22no—
223C2×C16—2922no—
224C2×C16—1462no—
225C4⋊C8—1461no—
226C4⋊C8—1461no—
227C22⋊C8—1461no—
228C22⋊C8—1461no—
229C22⋊C8—1461no—
230C22⋊C8—1461no—
231#97—1461no—
232#97—1461no—
233C24—731no—
234C24C8×D7311yescharacteristic
235C2×D4—2922no—
236C2×D4—1461no—
237C2×D4—1461no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—2921no—
241C22×C4—2921no—
242C22×C4—731no—
243C22×C4—731no—
244C22×C4—731no—
245C22×C4—1461no—
246C22×C4—1461no—
247C22×C4#1192400511yescharacteristic
248C22×C4#1192400511yescharacteristic
249C22×C4C8×D7311yescharacteristic
250C22×C4#1192401111yescharacteristic
251C22×C4#1192401111yescharacteristic
252C22×C4C2×C4×D7311yescharacteristic
253M16—2922no—
254M16—2922no—
255C2×C8—22no—
256C2×C8—22no—
257C2×C8—22no—
258C2×C8—41no—
259C2×C8—41no—
260C2×C8—41no—
261C2×C8—2921no—
262C2×C8—2921no—
263C2×C8—1461no—
264C2×C8—1461no—
265C2×C8#1192400511yescharacteristic
266C2×C8#1192400511yescharacteristic
267C2×C8C2×C22⋊D7311yescharacteristic
268C2×C8C2×D29211yescharacteristic
269C4⋊C4—1461no—
270C4⋊C4—1461no—
271C22⋊C4—1461no—
272C22⋊C4—1461no—
273C22⋊C4—1461no—
274C22⋊C4—1461no—
275C42—1461no—
276C42—1461no—
277C16—42no—
278C16—2922no—
279C23—22no—
280C23—22no—
281C23—41no—
282C23—2921no—
283C23—2921no—
284C23—1461no—
285C23—1462no—
286C23#1455650311yescharacteristic
287C23#1455650311yescharacteristic
288C23C2×C8×D7311yescharacteristic
289D4—2922no—
290D4—2922no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—2922no—
301C2×C4—2921no—
302C2×C4—2921no—
303C2×C4—2921no—
304C2×C4—2921no—
305C2×C4—1462no—
306C2×C4—1461no—
307C2×C4—1461no—
308C2×C4—1461no—
309C2×C4#1455650311yescharacteristic
310C2×C4#1455650311yescharacteristic
311C2×C4C2×#1192400511yescharacteristic
312C2×C4C2×#1192401111yescharacteristic
313C8—21no—
314C8—21no—
315C8—21no—
316C8—2921no—
317C8—2921no—
318C8#1455650512yes—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—44no—
327C22—2921no—
328C22—2921no—
329C22—2921no—
330C22—5842no—
331C22#1798836812yes—
332C22C2×#1455650311yescharacteristic
333C22#1798900512yes—
334C4—21no—
335C4—21no—
336C4—21no—
337C4—2921no—
338C4—2921no—
339C4—2921no—
340C4C2×#1455650511yescharacteristic
341C4#1798856911yescharacteristic
342C2—22no—
343C2—22no—
344C2—41no—
345C2—5841no—
346C2#2260824711yescharacteristic
347C2#2261154611yescharacteristic
348C2C2×#1798836811yescharacteristic
subgroupG/H
centerC2×C4C2×#11924005
commutatorC23×C73C22×C8
FrattiniC22×C8C22×D73
FittingC73×#2112C2
socleC22×C73C2×C22⋊C8
prime radicalC22×C22⋊D73C8
cosocleC22×C8×C73C23
hypercenter#2112D73
hypocenterC73#11467
LCSG ▷ C23×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C23×C8 ◁ #2112

Complements

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

NH ≅ G/Ntype
2. C2×C22⋊#11924003C2semidirect
3. #22607870C2semidirect
4. #22607854C2semidirect
5. C73×#2112C2semidirect
176. #2112D73semidirect
180. C73#11467semidirect

Maximal subgroups

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

subgroupindex
C2×#179886582
C2×C22⋊#119240032
#226078702
#226078542
C73×#21122
#1146773