Finite Group Databasebeta

← #34030145

Subgroups — #34030145

Order 18688.

600 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22626297C214yesmaximal, complemented
2#22626293C212yesmaximal, complemented
3#22626289C212yesmaximal, complemented
4#22626268C212yesmaximal, complemented
5#22626158C211yesmaximal, characteristic
6C2×C22⋊#11924031C211yesmaximal, characteristic
7#22626142C212yesmaximal, complemented
8C2×#17988545C211yesmaximal, characteristic
9C2×#14556665C2212yes—
10C22×#11924031C2212yes—
11#17989508C2214yescomplemented
12#17989508C2214yescomplemented
13C22⋊#11924031C2211yescharacteristic
14C22⋊#11924031C2211yescharacteristic
15C22⋊#11924031C2212yes—
16C22⋊#11924031C2212yes—
17C22⋊#11924031C2212yes—
18#17989476C2214yescomplemented
19#17989473C2214yescomplemented
20C23×#10596831C2211yescharacteristic
21C2×#14556550C411yescharacteristic
22#17988545C411yescharacteristic
23#17988545C412yes—
24#17988545C411yescharacteristic
25#17988545C2211yescharacteristic
26#17988545C2211yescharacteristic
27#17988545C2212yes—
28C2×#14556501C412yes—
29C2×C8×Dic73C2212yes—
30C22×M16×C73C411yescharacteristic
31#14556665—24no—
32C2×#11924031—24no—
33C2×#11924031C2312yes—
34C2×#11924031C2312yes—
35C2×#11924031C2314yes—
36C22×#10596831—24no—
37C22×#10596831C2×C411yescharacteristic
38C22×#10596831C2×C411yescharacteristic
39C22×#10596831C2×C411yescharacteristic
40C22×#10596831C2311yescharacteristic
41C22×#10596831C2311yescharacteristic
42C22×#10596831C2311yescharacteristic
43#14556550C812yes—
44#14556550C811yescharacteristic
45#14556550C811yescharacteristic
46#14556550C2×C412yes—
47#14556550C2×C411yescharacteristic
48#14556550C2×C411yescharacteristic
49C2×C4⋊Dic73C811yescharacteristic
50C2×C4⋊Dic73C811yescharacteristic
51C2×C22⋊Dic73C812yes—
52C2×C4×Dic73C2×C412yes—
53#14556501C2×C412yes—
54#14556501C2×C412yes—
55#14556501C2×C412yes—
56#14556501C2×C412yes—
57C8×Dic73C2×C412yes—
58C8×Dic73C2×C412yes—
59C8×Dic73C2312yes—
60C8×Dic73C2312yes—
61C23×C4×C73C2×C411yescharacteristic
62C2×M16×C73—24no—
63C2×M16×C73C2×C411yescharacteristic
64C2×M16×C73C2×C412yes—
65C2×M16×C73C2×C411yescharacteristic
66C22×C8×C73C2×C412yes—
67#11924031—22no—
68#11924031—22no—
69#11924031—24no—
70C22×Dic73C2×C812yes—
71C22×Dic73C2×C812yes—
72C2×#10596831—22no—
73C2×#10596831—22no—
74C2×#10596831—22no—
75C2×#10596831—22no—
76C2×#10596831—22no—
77C2×#10596831C22×C411yescharacteristic
78C2×#10596831C22×C411yescharacteristic
79C2×#10596831C22×C411yescharacteristic
80C2×#10596831#4114yes—
81C2×#10596831C2411yescharacteristic
82C4⋊Dic73C2×C812yes—
83C4⋊Dic73C2×C812yes—
84C4⋊Dic73C2×C811yescharacteristic
85C4⋊Dic73C2×C811yescharacteristic
86C4⋊Dic73C2×C811yescharacteristic
87C4⋊Dic73C2×C811yescharacteristic
88C22⋊Dic73C2×C812yes—
89C22⋊Dic73C2×C812yes—
90C22⋊Dic73C2×C812yes—
91C22⋊Dic73C2×C812yes—
92C4×Dic73C2×C812yes—
93C4×Dic73C2×C812yes—
94C4×Dic73C22×C412yes—
95C4×Dic73C22×C412yes—
96C24×C73C2×C811yescharacteristic
97C22×C4×C73—22no—
98C22×C4×C73—22no—
99C22×C4×C73C2×C811yescharacteristic
100C22×C4×C73C2×C811yescharacteristic
101C22×C4×C73C2×C811yescharacteristic
102C22×C4×C73C22×C411yescharacteristic
103C22×C4×C73C22×C411yescharacteristic
104C22×C4×C73C22×C411yescharacteristic
105M16×C73—22no—
106M16×C73—24no—
107M16×C73M1614yes—
108C2×C8×C73—24no—
109C2×C8×C73C22×C412yes—
110C2×C8×C73C22×C412yes—
111C23×C73—22no—
112C23×C73—22no—
113C23×C73—41no—
114C23×C73C22×C811yescharacteristic
115C23×C73C22×C811yescharacteristic
116C23×C73C22×C811yescharacteristic
117C2×C4×C73—22no—
118C2×C4×C73—21no—
119C2×C4×C73—21no—
120C2×C4×C73—21no—
121C2×C4×C73—21no—
122C2×C4×C73—22no—
123C2×C4×C73—22no—
124C2×C4×C73#11712yes—
125C2×C4×C73C22×C811yescharacteristic
126C2×C4×C73C22×C811yescharacteristic
127C2×C4×C73C22×C811yescharacteristic
128C2×C4×C73C2×M1612yes—
129C2×C4×C73C23×C411yescharacteristic
130C2×Dic73—24no—
131C2×Dic73—42no—
132C2×Dic73C22×C812yes—
133C2×Dic73C22×C812yes—
134C2×Dic73C22×C812yes—
135C2×Dic73C22×C812yes—
136C8×C73—22no—
137C8×C73C2×M1614yes—
138#10596831—21no—
139#10596831—21no—
140#10596831#11712yes—
141#10596831C2×#4112yes—
142C22×C73—21no—
143C22×C73—21no—
144C22×C73—21no—
145C22×C73—21no—
146C22×C73—21no—
147C22×C73—21no—
148C22×C73—21no—
149C22×C73—44no—
150C22×C73#43112yes—
151C22×C73#50412yes—
152C22×C73C23×C811yescharacteristic
153C4×C73—21no—
154C4×C73—21no—
155C4×C73#43112yes—
156C4×C73C2×#11711yescharacteristic
157C4×C73C22×M1611yescharacteristic
158Dic73—22no—
159Dic73—22no—
160Dic73—22no—
161Dic73—22no—
162#33422—731nomaximal, Sylow
163C2×C73—22no—
164C2×C73—22no—
165C2×C73—41no—
166C2×C73C2×#43111yescharacteristic
167C2×C73C2×#50411yescharacteristic
168C2×C73#340711yescharacteristic
169C2×#405—731no—
170#2164—734no—
171#2163—732no—
172#2162—732no—
173#2151—732no—
174#2112—731no—
175C2×C22⋊C16—731no—
176#2108—732no—
177C73#3342211yescharacteristic, complemented, Sylow
178C22×M16#6440811yescharacteristic
179C23×C8—731no—
180C2×#117—731no—
181C2×M32—732no—
182C22×C16—732no—
183#405—731no—
184#405—731no—
185#405—732no—
186#405—731no—
187#405—732no—
188#405—731no—
189C2×#97—732no—
190C2×C4×C8—732no—
191C4⋊C16—734no—
192C4⋊C16—734no—
193C22⋊C16—731no—
194C22⋊C16—731no—
195C22⋊C16—732no—
196C22⋊C16—732no—
197C22⋊C16—732no—
198#346—734no—
199C4×C16—734no—
200C23×C4C2×#6440811yescharacteristic
201C2×M16—24no—
202C2×M16C2×#6440811yescharacteristic
203C2×M16C2×#6440812yes—
204C2×M16C2×#6440811yescharacteristic
205C22×C8—731no—
206C22×C8—731no—
207C22×C8—731no—
208C22×C8—731no—
209C22×C8—731no—
210C22×C8—731no—
211C22×C8—1464no—
212C22×C8C2×#6440812yes—
213#117—731no—
214#117—731no—
215#117—732no—
216#117—732no—
217#117—731no—
218#117—731no—
219C2×C4⋊C4—731no—
220C2×C4⋊C4—731no—
221C2×C22⋊C4—732no—
222C2×C42—732no—
223M32—1464no—
224C2×C16—732no—
225C2×C16—734no—
226C2×C16—732no—
227C2×C16—1464no—
228#97—732no—
229#97—732no—
230#97—732no—
231#97—732no—
232C4×C8—732no—
233C4×C8—732no—
234C4×C8—732no—
235C4×C8—732no—
236C24#1192403611yescharacteristic
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—732no—
240C22×C4—732no—
241C22×C4#1192403611yescharacteristic
242C22×C4#1192403611yescharacteristic
243C22×C4#1192403611yescharacteristic
244C22×C4C22×#6440811yescharacteristic
245C22×C4C22×#6440811yescharacteristic
246C22×C4C22×#6440811yescharacteristic
247M16—22no—
248M16—24no—
249M16#1192403714yes—
250C2×C8—24no—
251C2×C8—731no—
252C2×C8—731no—
253C2×C8—734no—
254C2×C8—731no—
255C2×C8—731no—
256C2×C8—1462no—
257C2×C8—1462no—
258C2×C8—1462no—
259C2×C8—1462no—
260C2×C8—1462no—
261C2×C8C22×#6440812yes—
262C2×C8C22×#6440812yes—
263C4⋊C4—732no—
264C4⋊C4—732no—
265C4⋊C4—731no—
266C4⋊C4—731no—
267C4⋊C4—731no—
268C4⋊C4—731no—
269C22⋊C4—732no—
270C22⋊C4—732no—
271C22⋊C4—732no—
272C22⋊C4—732no—
273C42—732no—
274C42—732no—
275C42—732no—
276C42—732no—
277C16—1462no—
278C16—1462no—
279C16—1464no—
280C23—22no—
281C23—22no—
282C23—41no—
283C23C2×#1192403611yescharacteristic
284C23C2×#1192403611yescharacteristic
285C23C2×#1192403611yescharacteristic
286C2×C4—22no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—2922no—
294C2×C4—732no—
295C2×C4—732no—
296C2×C4—732no—
297C2×C4—732no—
298C2×C4—1464no—
299C2×C4#1455667012yes—
300C2×C4C2×#1192403611yescharacteristic
301C2×C4C2×#1192403611yescharacteristic
302C2×C4C2×#1192403611yescharacteristic
303C2×C4C2×#1192403712yes—
304C2×C4C23×#6440811yescharacteristic
305C8—22no—
306C8—732no—
307C8—732no—
308C8—1461no—
309C8—1461no—
310C8C2×#1192403714yes—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—44no—
319C22#1798960412yes—
320C22#1798964612yes—
321C22C22×#1192403611yescharacteristic
322C4—21no—
323C4—21no—
324C4—1462no—
325C4—1462no—
326C4—1462no—
327C4—1462no—
328C4#1798960412yes—
329C4C2×#1455667011yescharacteristic
330C4C22×#1192403711yescharacteristic
331C2—22no—
332C2—22no—
333C2—41no—
334C2C2×#1798960411yescharacteristic
335C2C2×#1798964611yescharacteristic
336C2#2262690011yescharacteristic
subgroupG/H
center, FrattiniC2×C4C23×#64408
commutator, socleC22×C73C23×C8
FittingC22×M16×C73C4
prime radicalC24×C73C2×C8
cosocleC2×#10596831C24
hypercenterC22×M16#64408
hypocenterC73#33422
LCSG ▷ C22×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C22×M16

Complements

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

NH ≅ G/Ntype
1. #22626297C2semidirect
2. #22626293C2semidirect
3. #22626289C2semidirect
4. #22626268C2semidirect
7. #22626142C2semidirect
11. #17989508C22semidirect
12. #17989508C22semidirect
18. #17989476C22semidirect
19. #17989473C22semidirect
177. C73#33422semidirect

Maximal subgroups

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

subgroupindex
#226262972
#226262932
#226262892
#226262682
#226261582
C2×C22⋊#119240312
#226261422
C2×#179885452
#3342273