Finite Group Databasebeta

← C23×#12855 #19160481

Subgroups — C23×#12855 #19160481

Order 5888.

628 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 628 conjugacy classes of subgroups — too many to draw. Only its 118 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
1C23×#2311C211yesmaximal, characteristic
2C23×#1785C211yesmaximal, characteristic
3C23×#1782C211yesmaximal, characteristic
4C23×#1390C214yesmaximal, complemented
5C22×#41×C23C2211yescharacteristic
6C2×C23×#117C411yescharacteristic, complemented
7C22×C4⋊C4×C23C411yescharacteristic, complemented
8C2×C23×#101—21no—
9C2×C23×#100C2212yes—
10C2×C23×#99—21no—
11C2×C23×#99C2212yes—
12C23×#407—21no—
13C23×#391—21no—
14C23×#390—21no—
15C23×#386C412yescomplemented
16C23×#386C2212yes—
17C23×#381—21no—
18C23×#380—21no—
19C23×#380—21no—
20C2×#41×C23—24no—
21C2×#41×C23—42no—
22C2×#41×C23—42no—
23C22×Q8×C23D411yescharacteristic
24C22×D4×C23C2×C411yescharacteristic
25C22×D4×C23D411yescharacteristic
26C22×D4×C23C2311yescharacteristic
27C23×C4×C23C2×C411yescharacteristic
28C23×C4×C23C2×C412yes—
29C23×C4×C23D411yescharacteristic
30C23×C4×C23D411yescharacteristic
31C2×M16×C23—22no—
32C23×#117—42no—
33C2×C4⋊C4×C23—21no—
34C2×C4⋊C4×C23—21no—
35C2×C4⋊C4×C23—21no—
36C2×C4⋊C4×C23—42no—
37C2×C22⋊C4×C23—22no—
38C2×C22⋊C4×C23—22no—
39C2×C22⋊C4×C23—22no—
40C2×C22⋊C4×C23C2×C412yes—
41C2×C42×C23—21no—
42C23×#101—42no—
43C23×#100—24no—
44C23×#99—24no—
45C23×#99—42no—
46C23×C22⋊C8—42no—
47C23×#95—22no—
48C23×#95—22no—
49C23×#95—42no—
50C24×C23C22⋊C411yescharacteristic
51C24×C23C22×C411yescharacteristic
52C24×C23C2×D411yescharacteristic
53#41×C23—416no—
54C2×Q8×C23—22no—
55C2×Q8×C23—22no—
56C2×Q8×C23—41no—
57C2×D4×C23—22no—
58C2×D4×C23—22no—
59C2×D4×C23—22no—
60C2×D4×C23—41no—
61C2×D4×C23—41no—
62C2×D4×C23—41no—
63C2×D4×C23—41no—
64C2×D4×C23—41no—
65C2×D4×C23#4114yes—
66C22×C4×C23—22no—
67C22×C4×C23—22no—
68C22×C4×C23—22no—
69C22×C4×C23—22no—
70C22×C4×C23—22no—
71C22×C4×C23—22no—
72C22×C4×C23—22no—
73C22×C4×C23—22no—
74C22×C4×C23—22no—
75C22×C4×C23—41no—
76C22×C4×C23—41no—
77C22×C4×C23—42no—
78C22×C4×C23—42no—
79C22×C4×C23—82no—
80C22×C4×C23C22⋊C411yescharacteristic
81C22×C4×C23C22⋊C411yescharacteristic
82C22×C4×C23C22⋊C411yescharacteristic
83C22×C4×C23C2×D411yescharacteristic
84M16×C23—42no—
85M16×C23—42no—
86C2×C8×C23—42no—
87C4⋊C4×C23—42no—
88C4⋊C4×C23—42no—
89C4⋊C4×C23—42no—
90C22⋊C4×C23—42no—
91C22⋊C4×C23—42no—
92C22⋊C4×C23—42no—
93C22⋊C4×C23—82no—
94C22⋊C4×C23—82no—
95C42×C23—42no—
96#12855C2311yesmaximal, characteristic, direct product factor, Sylow
97C23×C23—21no—
98C23×C23—21no—
99C23×C23—21no—
100C23×C23—21no—
101C23×C23—21no—
102C23×C23—21no—
103C23×C23—21no—
104C23×C23—41no—
105C23×C23—41no—
106C23×C23—41no—
107C23×C23—41no—
108C23×C23—41no—
109C23×C23—41no—
110C23×C23C2×C22⋊C411yescharacteristic
111C23×C23#11712yes—
112C23×C23#12312yes—
113Q8×C23—44no—
114D4×C23—44no—
115D4×C23—44no—
116D4×C23—44no—
117C2×C4×C23—28no—
118C2×C4×C23—21no—
119C2×C4×C23—21no—
120C2×C4×C23—21no—
121C2×C4×C23—21no—
122C2×C4×C23—21no—
123C2×C4×C23—21no—
124C2×C4×C23—21no—
125C2×C4×C23—21no—
126C2×C4×C23—21no—
127C2×C4×C23—44no—
128C2×C4×C23—44no—
129C2×C4×C23—42no—
130C2×C4×C23—42no—
131C2×C4×C23—42no—
132C2×C4×C23—42no—
133C2×C4×C23—42no—
134C2×C4×C23—42no—
135C2×C4×C23—84no—
136C2×C4×C23—82no—
137C2×C4×C23—82no—
138C2×C4×C23—82no—
139C2×C4×C23#9912yes—
140C2×C4×C23#9912yes—
141C2×C4×C23#12312yes—
142C8×C23—82no—
143#2311C2×C2311yescharacteristic
144#1785C2×C2311yescharacteristic
145#1782C2×C2311yescharacteristic
146#1390C2×C2314yescomplemented
147C22×C23—21no—
148C22×C23—21no—
149C22×C23—22no—
150C22×C23—22no—
151C22×C23—42no—
152C22×C23—42no—
153C22×C23—42no—
154C22×C23—42no—
155C22×C23—41no—
156C22×C23—41no—
157C22×C23—41no—
158C22×C23—41no—
159C22×C23—41no—
160C22×C23—41no—
161C22×C23—41no—
162C22×C23—81no—
163C22×C23—81no—
164C22×C23#38111yescharacteristic
165C22×C23C2×#9911yescharacteristic
166C22×C23C2×#9911yescharacteristic
167C4×C23—22no—
168C4×C23—22no—
169C4×C23—22no—
170C4×C23—41no—
171C4×C23—82no—
172C4×C23—82no—
173C4×C23—162no—
174C22×#41C22×C2311yescharacteristic
175C2×#117C4×C2311yescharacteristic, complemented
176C22×C4⋊C4C4×C2311yescharacteristic, complemented
177C2×#101—21no—
178C2×#100C22×C2312yes—
179C2×#99—21no—
180C2×#99C22×C2312yes—
181#407—21no—
182#391—21no—
183#390—21no—
184#386C4×C2312yescomplemented
185#386C22×C2312yes—
186#381—21no—
187#380—21no—
188#380—21no—
189C2×C23—21no—
190C2×C23—21no—
191C2×C23—42no—
192C2×C23—42no—
193C2×C23—81no—
194C2×C23#181911yescharacteristic
195C2×C23#212211yescharacteristic
196C2×C23#212311yescharacteristic
197C2×#41—24no—
198C2×#41—42no—
199C2×#41—42no—
200C22×Q8D4×C2311yescharacteristic
201C22×D4C2×C4×C2311yescharacteristic
202C22×D4D4×C2311yescharacteristic
203C22×D4C23×C2311yescharacteristic
204C23×C4C2×C4×C2311yescharacteristic
205C23×C4C2×C4×C2312yes—
206C23×C4D4×C2311yescharacteristic
207C23×C4D4×C2311yescharacteristic
208C2×M16—22no—
209#117—42no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—42no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4C2×C4×C2312yes—
218C2×C42—21no—
219#101—42no—
220#100—24no—
221#99—24no—
222#99—42no—
223C22⋊C8—42no—
224#95—22no—
225#95—22no—
226#95—42no—
227C23#1285511yescharacteristic, direct product factor, Sylow
228C24C22⋊C4×C2311yescharacteristic
229C24C22×C4×C2311yescharacteristic
230C24C2×D4×C2311yescharacteristic
231#41—416no—
232C2×Q8—22no—
233C2×Q8—22no—
234C2×Q8—41no—
235C2×D4—22no—
236C2×D4—22no—
237C2×D4—22no—
238C2×D4—41no—
239C2×D4—41no—
240C2×D4—41no—
241C2×D4—41no—
242C2×D4—41no—
243C2×D4#41×C2314yes—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—82no—
258C22×C4C22⋊C4×C2311yescharacteristic
259C22×C4C22⋊C4×C2311yescharacteristic
260C22×C4C22⋊C4×C2311yescharacteristic
261C22×C4C2×D4×C2311yescharacteristic
262M16—42no—
263M16—42no—
264C2×C8—42no—
265C4⋊C4—42no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C22⋊C4—42no—
269C22⋊C4—42no—
270C22⋊C4—42no—
271C22⋊C4—82no—
272C22⋊C4—82no—
273C42—42no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23C2×C22⋊C4×C2311yescharacteristic
288C23C23×#11712yes—
289C23C23×#12312yes—
290Q8—44no—
291D4—44no—
292D4—44no—
293D4—44no—
294C2×C4—28no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—44no—
305C2×C4—44no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—84no—
313C2×C4—82no—
314C2×C4—82no—
315C2×C4—82no—
316C2×C4C23×#9912yes—
317C2×C4C23×#9912yes—
318C2×C4C23×#12312yes—
319C8—82no—
320C22—21no—
321C22—21no—
322C22—22no—
323C22—22no—
324C22—42no—
325C22—42no—
326C22—42no—
327C22—42no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—81no—
336C22—81no—
337C22C23×#38111yescharacteristic
338C22C2×C23×#9911yescharacteristic
339C22C2×C23×#9911yescharacteristic
340C4—22no—
341C4—22no—
342C4—22no—
343C4—41no—
344C4—82no—
345C4—82no—
346C4—162no—
347C2—21no—
348C2—21no—
349C2—42no—
350C2—42no—
351C2—81no—
352C2C23×#181911yescharacteristic
353C2C23×#212211yescharacteristic
354C2C23×#212311yescharacteristic
subgroupG/H
center, socleC22×C23C2×#99
commutatorC24C22×C4×C23
Frattini, cosocleC22×D4C23×C23
prime radicalC22×#41×C23C22
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C22×C4×C23 ◁ C22×#41×C23 ◁ G

Complements

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

NH ≅ G/Ntype
4. C23×#1390C2semidirect
6. C2×C23×#117C4semidirect
7. C22×C4⋊C4×C23C4semidirect
15. C23×#386C4semidirect
96. #12855C23direct (N × H)
146. #1390C2×C23semidirect
175. C2×#117C4×C23semidirect
176. C22×C4⋊C4C4×C23semidirect
184. #386C4×C23semidirect
227. C23#12855direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#23112
C23×#17852
C23×#17822
C23×#13902
#1285523