Finite Group Databasebeta

← C23×#10461 #19158087

Subgroups — C23×#10461 #19158087

Order 5888.

406 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3077C211yesmaximal, characteristic
2C23×#1800C211yesmaximal, characteristic
3C23×#1567C211yesmaximal, characteristic, complemented
4C23×#1550C211yesmaximal, characteristic, complemented
5C23×#1493C211yesmaximal, characteristic, complemented
6C2×C23×D4⋊C8C211yesmaximal, characteristic
7C23×#1292C211yesmaximal, characteristic, complemented
8C23×#533C2211yescharacteristic
9C2×C4×D4×C23C2211yescharacteristic
10C23×#478—22no—
11C23×#476—21no—
12C23×#474—21no—
13C23×#428—21no—
14C23×#423C2211yescharacteristic
15C23×#423C2211yescharacteristic
16C23×#423C2211yescharacteristic
17C2×C23×C4⋊C8C2211yescharacteristic
18C23×#418—21no—
19C2×C23×#100—21no—
20C2×C4×C8×C23C2211yescharacteristic
21C23×#343—21no—
22C23×#335—22no—
23C23×#333—21no—
24C23×#332—21no—
25C23×Q8⋊C8—21no—
26C23×Q8⋊C8—21no—
27C23×D4⋊C8—21no—
28C23×D4⋊C8—21no—
29C22×D4×C23D411yescharacteristic
30C23×C4×C23—21no—
31C2×M16×C23—21no—
32C2×M16×C23—21no—
33C2×M16×C23—21no—
34C22×C8×C23—21no—
35C22×C8×C23—21no—
36C23×C4⋊Q8D411yescharacteristic
37C23×C4⋊Q8D411yescharacteristic
38C23×#125—21no—
39C23×C22⋊Q8—21no—
40C23×C22⋊Q8—21no—
41C4×Q8×C23—21no—
42C4×Q8×C23—21no—
43C4×D4×C23—21no—
44C4×D4×C23—21no—
45C4×D4×C23—21no—
46C4×D4×C23—21no—
47C23×#117—21no—
48C2×C4⋊C4×C23D411yescharacteristic
49C2×C22⋊C4×C23—21no—
50C2×C42×C23C2311yescharacteristic
51C23×#107—22no—
52C23×#106—22no—
53C23×C4⋊C8—21no—
54C23×C4⋊C8—21no—
55C23×C4⋊C8—21no—
56C23×C4⋊C8—21no—
57C23×C4⋊C8—21no—
58C23×C4⋊C8—21no—
59C23×C4⋊C8—21no—
60C23×C4⋊C8D412yes—
61C23×#102—22no—
62C23×#102—22no—
63C23×#100—42no—
64C4×C8×C23—21no—
65C4×C8×C23D411yescharacteristic
66C4×C8×C23D411yescharacteristic
67C24×C23—21no—
68C2×Q8×C23—21no—
69C2×Q8×C23—21no—
70C2×D4×C23—21no—
71C2×D4×C23—21no—
72C2×D4×C23—21no—
73C2×D4×C23—21no—
74C2×D4×C23—21no—
75C2×D4×C23Q1612yes—
76C22×C4×C23—21no—
77C22×C4×C23—41no—
78C22×C4×C23—41no—
79C22×C4×C23—81no—
80C22×C4×C23—81no—
81C22×C4×C23C2×D411yescharacteristic
82C22×C4×C23#4111yescharacteristic
83C22×C4×C23#4111yescharacteristic
84M16×C23—41no—
85M16×C23—41no—
86M16×C23—41no—
87M16×C23—41no—
88M16×C23—41no—
89M16×C23—41no—
90C2×C8×C23—21no—
91C2×C8×C23—22no—
92C2×C8×C23—21no—
93C2×C8×C23—42no—
94C2×C8×C23—41no—
95C2×C8×C23—41no—
96C2×C8×C23—41no—
97C2×C8×C23—81no—
98C4⋊C4×C23—21no—
99C4⋊C4×C23—21no—
100C4⋊C4×C23—21no—
101C4⋊C4×C23—21no—
102C4⋊C4×C23—21no—
103C4⋊C4×C23—21no—
104C4⋊C4×C23—41no—
105C4⋊C4×C23—41no—
106C22⋊C4×C23—41no—
107C22⋊C4×C23—41no—
108C22⋊C4×C23—41no—
109C42×C23—41no—
110C42×C23C2×D411yescharacteristic
111C42×C23C2×D411yescharacteristic
112C42×C23C2×D411yescharacteristic
113C42×C23#4111yescharacteristic
114#10461C2311yesmaximal, characteristic, direct product factor, Sylow
115C23×C23—22no—
116C23×C23—41no—
117C23×C23—41no—
118C23×C23—81no—
119C23×C23—81no—
120C23×C23#12311yescharacteristic
121Q8×C23—41no—
122Q8×C23—41no—
123Q8×C23—81no—
124D4×C23—22no—
125D4×C23—22no—
126D4×C23—41no—
127D4×C23—41no—
128C2×C4×C23—21no—
129C2×C4×C23—21no—
130C2×C4×C23—41no—
131C2×C4×C23—41no—
132C2×C4×C23—41no—
133C2×C4×C23—41no—
134C2×C4×C23—82no—
135C2×C4×C23—81no—
136C2×C4×C23—81no—
137C2×C4×C23—81no—
138C2×C4×C23#12011yescharacteristic
139C2×C4×C23#12111yescharacteristic
140C2×C4×C23#12111yescharacteristic
141C2×C4×C23#12111yescharacteristic
142C2×C4×C23#12311yescharacteristic
143C2×C4×C23#12411yescharacteristic
144C2×C4×C23C2×Q1611yescharacteristic
145C2×C4×C23#13511yescharacteristic
146C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
147C2×C4×C23#13711yescharacteristic
148C8×C23—22no—
149C8×C23—41no—
150C8×C23—42no—
151C8×C23—81no—
152C8×C23—81no—
153C8×C23—81no—
154#3077C2×C2311yescharacteristic
155#1800C2×C2311yescharacteristic
156#1567C2×C2311yescharacteristic, complemented
157#1550C2×C2311yescharacteristic, complemented
158#1493C2×C2311yescharacteristic, complemented
159C2×D4⋊C8C2×C2311yescharacteristic
160#1292C2×C2311yescharacteristic, complemented
161C22×C23—21no—
162C22×C23—21no—
163C22×C23—42no—
164C22×C23—82no—
165C22×C23—82no—
166C22×C23—81no—
167C22×C23—81no—
168C22×C23#39411yescharacteristic
169C22×C23#48411yescharacteristic
170C22×C23#48511yescharacteristic
171C4×C23—21no—
172C4×C23—21no—
173C4×C23—21no—
174C4×C23—81no—
175C4×C23—81no—
176C4×C23—81no—
177C4×C23#44911yescharacteristic
178C4×C23#45111yescharacteristic
179C4×C23#46711yescharacteristic
180C4×C23#46811yescharacteristic
181C4×C23#47111yescharacteristic
182C4×C23#47211yescharacteristic
183#533C22×C2311yescharacteristic
184C2×C4×D4C22×C2311yescharacteristic
185#478—22no—
186#476—21no—
187#474—21no—
188#428—21no—
189#423C22×C2311yescharacteristic
190#423C22×C2311yescharacteristic
191#423C22×C2311yescharacteristic
192C2×C4⋊C8C22×C2311yescharacteristic
193#418—21no—
194C2×#100—21no—
195C2×C4×C8C22×C2311yescharacteristic
196#343—21no—
197#335—22no—
198#333—21no—
199#332—21no—
200Q8⋊C8—21no—
201Q8⋊C8—21no—
202D4⋊C8—21no—
203D4⋊C8—21no—
204C2×C23—21no—
205C2×C23—21no—
206C2×C23—82no—
207C2×C23#203611yescharacteristic
208C2×C23#205011yescharacteristic
209C2×C23#205111yescharacteristic
210C22×D4D4×C2311yescharacteristic
211C23×C4—21no—
212C2×M16—21no—
213C2×M16—21no—
214C2×M16—21no—
215C22×C8—21no—
216C22×C8—21no—
217C4⋊Q8D4×C2311yescharacteristic
218C4⋊Q8D4×C2311yescharacteristic
219#125—21no—
220C22⋊Q8—21no—
221C22⋊Q8—21no—
222C4×Q8—21no—
223C4×Q8—21no—
224C4×D4—21no—
225C4×D4—21no—
226C4×D4—21no—
227C4×D4—21no—
228#117—21no—
229C2×C4⋊C4D4×C2311yescharacteristic
230C2×C22⋊C4—21no—
231C2×C42C23×C2311yescharacteristic
232#107—22no—
233#106—22no—
234C4⋊C8—21no—
235C4⋊C8—21no—
236C4⋊C8—21no—
237C4⋊C8—21no—
238C4⋊C8—21no—
239C4⋊C8—21no—
240C4⋊C8—21no—
241C4⋊C8D4×C2312yes—
242#102—22no—
243#102—22no—
244#100—42no—
245C4×C8—21no—
246C4×C8D4×C2311yescharacteristic
247C4×C8D4×C2311yescharacteristic
248C23#1046111yescharacteristic, direct product factor, Sylow
249C24—21no—
250C2×Q8—21no—
251C2×Q8—21no—
252C2×D4—21no—
253C2×D4—21no—
254C2×D4—21no—
255C2×D4—21no—
256C2×D4—21no—
257C2×D4Q16×C2312yes—
258C22×C4—21no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—81no—
262C22×C4—81no—
263C22×C4C2×D4×C2311yescharacteristic
264C22×C4#41×C2311yescharacteristic
265C22×C4#41×C2311yescharacteristic
266M16—41no—
267M16—41no—
268M16—41no—
269M16—41no—
270M16—41no—
271M16—41no—
272C2×C8—21no—
273C2×C8—22no—
274C2×C8—21no—
275C2×C8—42no—
276C2×C8—41no—
277C2×C8—41no—
278C2×C8—41no—
279C2×C8—81no—
280C4⋊C4—21no—
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—21no—
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C22⋊C4—41no—
289C22⋊C4—41no—
290C22⋊C4—41no—
291C42—41no—
292C42C2×D4×C2311yescharacteristic
293C42C2×D4×C2311yescharacteristic
294C42C2×D4×C2311yescharacteristic
295C42#41×C2311yescharacteristic
296C23—22no—
297C23—41no—
298C23—41no—
299C23—81no—
300C23—81no—
301C23C23×#12311yescharacteristic
302Q8—41no—
303Q8—41no—
304Q8—81no—
305D4—22no—
306D4—22no—
307D4—41no—
308D4—41no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—82no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—81no—
319C2×C4C23×#12011yescharacteristic
320C2×C4C23×#12111yescharacteristic
321C2×C4C23×#12111yescharacteristic
322C2×C4C23×#12111yescharacteristic
323C2×C4C23×#12311yescharacteristic
324C2×C4C23×#12411yescharacteristic
325C2×C4C2×Q16×C2311yescharacteristic
326C2×C4C23×#13511yescharacteristic
327C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
328C2×C4C23×#13711yescharacteristic
329C8—22no—
330C8—41no—
331C8—42no—
332C8—81no—
333C8—81no—
334C8—81no—
335C22—21no—
336C22—21no—
337C22—42no—
338C22—82no—
339C22—82no—
340C22—81no—
341C22—81no—
342C22C23×#39411yescharacteristic
343C22C23×#48411yescharacteristic
344C22C23×#48511yescharacteristic
345C4—21no—
346C4—21no—
347C4—21no—
348C4—81no—
349C4—81no—
350C4—81no—
351C4C23×#44911yescharacteristic
352C4C23×#45111yescharacteristic
353C4C23×#46711yescharacteristic
354C4C23×#46811yescharacteristic
355C4C23×#47111yescharacteristic
356C4C23×#47211yescharacteristic
357C2—21no—
358C2—21no—
359C2—82no—
360C2C23×#203611yescharacteristic
361C2C23×#205011yescharacteristic
362C2C23×#205111yescharacteristic
subgroupG/H
center, socleC22×C23#394
commutator, Frattini, cosocleC2×C42C23×C23
prime radicalC22×D4×C23D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C2×C42×C23 ◁ G

Complements

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

NH ≅ G/Ntype
3. C23×#1567C2semidirect
4. C23×#1550C2semidirect
5. C23×#1493C2semidirect
7. C23×#1292C2semidirect
114. #10461C23direct (N × H)
156. #1567C2×C23semidirect
157. #1550C2×C23semidirect
158. #1493C2×C23semidirect
160. #1292C2×C23semidirect
248. C23#10461direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#30772
C23×#18002
C23×#15672
C23×#15502
C23×#14932
C2×C23×D4⋊C82
C23×#12922
#1046123