Finite Group Databasebeta

← C2×#22608973 #33058219

Subgroups — C2×#22608973 #33058219

Order 18688.

984 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988920C212yesmaximal
2C2×#17988526C211yesmaximal, characteristic
3#22608973C218yesmaximal, direct product factor
4C2×#17988324C212yesmaximal
5C2×#17988313C211yesmaximal, characteristic
6C2×#544×C73C211yesmaximal, characteristic
7C22×C4⋊Dic73C2211yescharacteristic
8#17988920—24no—
9#17988920C2218yes—
10C2×#14556524—22no—
11C2×#14556515—22no—
12C2×C22⋊#10596831C2212yes—
13#17988526—22no—
14#17988526C2214yes—
15C2×#14556498—21no—
16C2×#14556498—21no—
17C2×#14556498C2211yescharacteristic
18#17988324C2218yes—
19#17988313C2214yes—
20#544×C73—24no—
21#544×C73—22no—
22#544×C73C2214yes—
23C2×C4⋊Q8×C73—22no—
24C2×#125×C73—22no—
25C2×C22⋊Q8×C73—21no—
26C2×C22⋊Q8×C73C2212yes—
27C2×#117×C73—21no—
28C22×C4⋊C4×C73C2211yescharacteristic
29C23×Dic73—21no—
30C23×Dic73—21no—
31C22×#10596831—22no—
32C2×C4⋊Dic73—22no—
33C2×C4⋊Dic73—22no—
34C2×C4⋊Dic73—21no—
35C2×C4⋊Dic73C2312yes—
36#14556524—28no—
37#14556515—28no—
38C22⋊#10596831—22no—
39C22⋊#10596831C2314yes—
40#14556498—24no—
41#14556498—24no—
42#14556498—21no—
43#14556498C2312yes—
44C22×Q8×C73—21no—
45C22×Q8×C73D412yes—
46C23×C4×C73D411yescharacteristic
47C23×C4×C73D411yescharacteristic
48C23×C4×C73C2311yescharacteristic
49C4⋊Q8×C73—28no—
50C4⋊Q8×C73—28no—
51#125×C73—28no—
52#125×C73—28no—
53C22⋊Q8×C73—24no—
54C22⋊Q8×C73—24no—
55C22⋊Q8×C73—22no—
56C22⋊Q8×C73—22no—
57C22⋊Q8×C73D414yes—
58C22⋊Q8×C73C2314yes—
59#117×C73—24no—
60#117×C73—24no—
61C2×C4⋊C4×C73—21no—
62C2×C4⋊C4×C73—21no—
63C2×C4⋊C4×C73—21no—
64C2×C4⋊C4×C73—22no—
65C2×C4⋊C4×C73—22no—
66C2×C4⋊C4×C73—22no—
67C2×C4⋊C4×C73—22no—
68C2×C4⋊C4×C73D412yes—
69C2×C4⋊C4×C73D412yes—
70C2×C4⋊C4×C73C2312yes—
71C2×C22⋊C4×C73—22no—
72C2×C42×C73—22no—
73C22×Dic73—22no—
74C22×Dic73—22no—
75C22×Dic73—42no—
76C22×Dic73—42no—
77C22×Dic73—44no—
78C22×Dic73—41no—
79C22×Dic73—41no—
80C22×Dic73—42no—
81C2×#10596831—24no—
82C2×#10596831—44no—
83C4⋊Dic73—24no—
84C4⋊Dic73—42no—
85C24×C73C2×D411yescharacteristic
86C2×Q8×C73—24no—
87C2×Q8×C73—22no—
88C2×Q8×C73—44no—
89C2×Q8×C73—82no—
90C2×Q8×C73C2×D414yes—
91C22×C4×C73—21no—
92C22×C4×C73—21no—
93C22×C4×C73—22no—
94C22×C4×C73—22no—
95C22×C4×C73—22no—
96C22×C4×C73—21no—
97C22×C4×C73—21no—
98C22×C4×C73—42no—
99C22×C4×C73—42no—
100C22×C4×C73Q1614yes—
101C22×C4×C73C2×D412yes—
102C22×C4×C73C2×D412yes—
103C22×C4×C73C2×D412yes—
104C22×C4×C73C2×D411yescharacteristic
105C22×C4×C73C2×D412yes—
106C22×C4×C73C2411yescharacteristic
107C4⋊C4×C73—24no—
108C4⋊C4×C73—24no—
109C4⋊C4×C73—24no—
110C4⋊C4×C73—28no—
111C4⋊C4×C73—24no—
112C4⋊C4×C73—24no—
113C4⋊C4×C73—22no—
114C4⋊C4×C73—24no—
115C4⋊C4×C73—24no—
116C4⋊C4×C73C2×D414yes—
117C22⋊C4×C73—24no—
118C22⋊C4×C73—24no—
119C42×C73—28no—
120C23×C73—22no—
121C23×C73—22no—
122C23×C73#12011yescharacteristic
123C23×C73#12011yescharacteristic
124C23×C73C2×Q1612yes—
125C23×C73#13712yes—
126C23×C73C22×D411yescharacteristic
127Q8×C73—44no—
128Q8×C73—82no—
129Q8×C73—84no—
130C2×C4×C73—22no—
131C2×C4×C73—22no—
132C2×C4×C73—24no—
133C2×C4×C73—24no—
134C2×C4×C73—24no—
135C2×C4×C73—24no—
136C2×C4×C73—24no—
137C2×C4×C73—22no—
138C2×C4×C73—42no—
139C2×C4×C73—41no—
140C2×C4×C73—41no—
141C2×C4×C73—44no—
142C2×C4×C73—42no—
143C2×C4×C73—44no—
144C2×C4×C73—42no—
145C2×C4×C73—81no—
146C2×C4×C73#12012yes—
147C2×C4×C73C2×Q1614yes—
148C2×C4×C73C22×D412yes—
149C2×Dic73—42no—
150C2×Dic73—42no—
151C2×Dic73—44no—
152C2×Dic73—42no—
153C2×Dic73—42no—
154C2×Dic73—84no—
155C2×Dic73—82no—
156C2×Dic73—82no—
157#10596831—44no—
158C22×C73—24no—
159C22×C73—24no—
160C22×C73—21no—
161C22×C73—21no—
162C22×C73—21no—
163C22×C73—21no—
164C22×C73#45112yes—
165C22×C73#45112yes—
166C22×C73#45412yes—
167C22×C73#45812yes—
168C22×C73C2×#12011yescharacteristic
169C22×C73C22×Q1611yescharacteristic
170C22×C73C2×#13711yescharacteristic
171C4×C73—24no—
172C4×C73—22no—
173C4×C73—22no—
174C4×C73—44no—
175C4×C73—42no—
176C4×C73—44no—
177Dic73—82no—
178Dic73—82no—
179C2×#1617—731nomaximal, Sylow
180C2×C73—21no—
181C2×C73—21no—
182C2×C73—22no—
183C2×C73#161714yescomplemented
184C2×C73C2×#45111yescharacteristic
185C2×C73C2×#45411yescharacteristic
186C2×C73C2×#45811yescharacteristic
187C2×#544D7311yescharacteristic
188C2×#484—732no—
189C2×#400—731no—
190#1617—738no—
191C2×#328—732no—
192C2×#324—731no—
193C73C2×#161711yescharacteristic, complemented, Sylow
194#544—24no—
195#544—22no—
196#544C2×D7314yes—
197C2×C4⋊Q8—22no—
198C2×#125—22no—
199C2×C22⋊Q8—21no—
200C2×C22⋊Q8C2×D7312yes—
201C2×#117—21no—
202C22×C4⋊C4—731no—
203C22×C4⋊C4C2×D7311yescharacteristic
204#484—738no—
205#484—1464no—
206C2×#107—1462no—
207C2×#103—1462no—
208C2×C22⋊C8—732no—
209#400—734no—
210#400—1462no—
211C2×#95—731no—
212C2×#95—1461no—
213C2×#95—1461no—
214#328—738no—
215#324—734no—
216C22×Q8—21no—
217C22×Q8C22⋊D7312yes—
218C23×C4—1461no—
219C23×C4—1461no—
220C23×C4C22⋊D7311yescharacteristic
221C23×C4C22⋊D7311yescharacteristic
222C23×C4C22×D7311yescharacteristic
223C22×C8—1462no—
224C4⋊Q8—28no—
225C4⋊Q8—28no—
226#125—28no—
227#125—28no—
228C22⋊Q8—24no—
229C22⋊Q8—24no—
230C22⋊Q8—22no—
231C22⋊Q8—22no—
232C22⋊Q8C22⋊D7314yes—
233C22⋊Q8C22×D7314yes—
234#117—24no—
235#117—24no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—732no—
244C2×C4⋊C4—1461no—
245C2×C4⋊C4—1462no—
246C2×C4⋊C4—1462no—
247C2×C4⋊C4C22⋊D7312yes—
248C2×C4⋊C4C22⋊D7312yes—
249C2×C4⋊C4C22×D7312yes—
250C2×C22⋊C4—22no—
251C2×C42—22no—
252#107—1468no—
253#103—1468no—
254C22⋊C8—734no—
255C22⋊C8—1462no—
256#95—732no—
257#95—1461no—
258#95—1464no—
259#95—1464no—
260C24C2×C22⋊D7311yescharacteristic
261C2×Q8—24no—
262C2×Q8—22no—
263C2×Q8—44no—
264C2×Q8—82no—
265C2×Q8C2×C22⋊D7314yes—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—2922no—
276C22×C4—2922no—
277C22×C4—2924no—
278C22×C4—2921no—
279C22×C4—2921no—
280C22×C4—2922no—
281C22×C4—1462no—
282C22×C4—1462no—
283C22×C4#1192401814yes—
284C22×C4C2×C22⋊D7312yes—
285C22×C4C2×C22⋊D7312yes—
286C22×C4C2×C22⋊D7312yes—
287C22×C4C2×C22⋊D7311yescharacteristic
288C22×C4C2×C22⋊D7312yes—
289C22×C4C23×D7311yescharacteristic
290C2×C8—2924no—
291C2×C8—1464no—
292C4⋊C4—24no—
293C4⋊C4—24no—
294C4⋊C4—24no—
295C4⋊C4—28no—
296C4⋊C4—24no—
297C4⋊C4—24no—
298C4⋊C4—22no—
299C4⋊C4—24no—
300C4⋊C4—24no—
301C4⋊C4—2922no—
302C4⋊C4—1464no—
303C4⋊C4C2×C22⋊D7314yes—
304C22⋊C4—24no—
305C22⋊C4—24no—
306C42—28no—
307C23—22no—
308C23—22no—
309C23#1455656011yescharacteristic
310C23#1455656011yescharacteristic
311C23C2×#1192401812yes—
312C23#1455662212yes—
313C23C22×C22⋊D7311yescharacteristic
314Q8—44no—
315Q8—82no—
316Q8—84no—
317C2×C4—22no—
318C2×C4—24no—
319C2×C4—24no—
320C2×C4—24no—
321C2×C4—22no—
322C2×C4—24no—
323C2×C4—24no—
324C2×C4—22no—
325C2×C4—42no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—44no—
329C2×C4—42no—
330C2×C4—44no—
331C2×C4—42no—
332C2×C4—81no—
333C2×C4—2922no—
334C2×C4—2922no—
335C2×C4—2924no—
336C2×C4—2922no—
337C2×C4—2922no—
338C2×C4—5844no—
339C2×C4—5842no—
340C2×C4—5842no—
341C2×C4#1455656012yes—
342C2×C4C2×#1192401814yes—
343C2×C4C22×C22⋊D7312yes—
344C8—2924no—
345C22—24no—
346C22—24no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22#1798874312yes—
352C22#1798874312yes—
353C22#1798876012yes—
354C22#1798877712yes—
355C22C2×#1455656011yescharacteristic
356C22C22×#1192401811yescharacteristic
357C22C2×#1455662211yescharacteristic
358C4—22no—
359C4—22no—
360C4—24no—
361C4—44no—
362C4—42no—
363C4—44no—
364C4—5842no—
365C4—5842no—
366C2—21no—
367C2—22no—
368C2—21no—
369C2#2260897314yesdirect product factor
370C2C2×#1798874311yescharacteristic
371C2C2×#1798876011yescharacteristic
372C2C2×#1798877711yescharacteristic
subgroupG/H
centerC23#14556560
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC2×#544×C73C2
socleC23×C73#120
prime radicalC24×C73C2×D4
hypercenterC2×#544D73
hypocenterC73C2×#1617
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#544

Complements

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

NH ≅ G/Ntype
3. #22608973C2direct (N × H)
183. C2×C73#1617semidirect
193. C73C2×#1617semidirect
369. C2#22608973direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#179889202
C2×#179885262
#226089732
C2×#179883242
C2×#179883132
C2×#544×C732
C2×#161773