Finite Group Databasebeta

← C23×#12853 #19160479

Subgroups — C23×#12853 #19160479

Order 5888.

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

normal subgroups only

This group has 656 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×#2297C211yesmaximal, characteristic
2C23×#1785C211yesmaximal, characteristic
3C23×#1782C211yesmaximal, characteristic
4C23×#1390C214yesmaximal, complemented
5C22×#41×C23C2211yescharacteristic
6C2×C23×#117C411yescharacteristic, complemented
7C22×C42×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—
18C4×C22⋊C4×C23—21no—
19C4×C22⋊C4×C23—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×C22⋊C4×C23—22no—
35C2×C22⋊C4×C23—22no—
36C2×C22⋊C4×C23—22no—
37C2×C22⋊C4×C23C2×C412yes—
38C2×C42×C23—21no—
39C2×C42×C23—21no—
40C2×C42×C23—21no—
41C2×C42×C23—42no—
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—44no—
78C22×C4×C23—42no—
79C22×C4×C23—42no—
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—
88C22⋊C4×C23—44no—
89C22⋊C4×C23—44no—
90C22⋊C4×C23—42no—
91C22⋊C4×C23—42no—
92C22⋊C4×C23—42no—
93C42×C23—42no—
94C42×C23—42no—
95C42×C23—42no—
96#12853C2311yesmaximal, 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—42no—
105C23×C23—42no—
106C23×C23—42no—
107C23×C23C2×C22⋊C411yescharacteristic
108C23×C23#11712yes—
109C23×C23#12312yes—
110Q8×C23—44no—
111D4×C23—44no—
112D4×C23—44no—
113D4×C23—44no—
114C2×C4×C23—28no—
115C2×C4×C23—21no—
116C2×C4×C23—21no—
117C2×C4×C23—21no—
118C2×C4×C23—21no—
119C2×C4×C23—21no—
120C2×C4×C23—21no—
121C2×C4×C23—24no—
122C2×C4×C23—21no—
123C2×C4×C23—21no—
124C2×C4×C23—21no—
125C2×C4×C23—44no—
126C2×C4×C23—44no—
127C2×C4×C23—42no—
128C2×C4×C23—44no—
129C2×C4×C23—42no—
130C2×C4×C23—44no—
131C2×C4×C23—42no—
132C2×C4×C23—42no—
133C2×C4×C23—42no—
134C2×C4×C23—82no—
135C2×C4×C23—84no—
136C2×C4×C23#9912yes—
137C2×C4×C23#9912yes—
138C2×C4×C23#12312yes—
139C8×C23—82no—
140#2297C2×C2311yescharacteristic
141#1785C2×C2311yescharacteristic
142#1782C2×C2311yescharacteristic
143#1390C2×C2314yescomplemented
144C22×C23—22no—
145C22×C23—22no—
146C22×C23—22no—
147C22×C23—42no—
148C22×C23—42no—
149C22×C23—42no—
150C22×C23—42no—
151C22×C23—42no—
152C22×C23—42no—
153C22×C23—41no—
154C22×C23—42no—
155C22×C23—81no—
156C22×C23—81no—
157C22×C23#38111yescharacteristic
158C22×C23C2×#9911yescharacteristic
159C22×C23C2×#9911yescharacteristic
160C4×C23—22no—
161C4×C23—22no—
162C4×C23—22no—
163C4×C23—41no—
164C4×C23—44no—
165C4×C23—82no—
166C4×C23—162no—
167C22×#41C22×C2311yescharacteristic
168C2×#117C4×C2311yescharacteristic, complemented
169C22×C42C4×C2311yescharacteristic, complemented
170C2×#101—21no—
171C2×#100C22×C2312yes—
172C2×#99—21no—
173C2×#99C22×C2312yes—
174#407—21no—
175#391—21no—
176#390—21no—
177#386C4×C2312yescomplemented
178#386C22×C2312yes—
179#381—21no—
180C4×C22⋊C4—21no—
181C4×C22⋊C4—21no—
182C2×C23—21no—
183C2×C23—21no—
184C2×C23—42no—
185C2×C23—42no—
186C2×C23—81no—
187C2×C23#181911yescharacteristic
188C2×C23#212012yes—
189C2×#41—24no—
190C2×#41—42no—
191C2×#41—42no—
192C22×Q8D4×C2311yescharacteristic
193C22×D4C2×C4×C2311yescharacteristic
194C22×D4D4×C2311yescharacteristic
195C22×D4C23×C2311yescharacteristic
196C23×C4C2×C4×C2311yescharacteristic
197C23×C4C2×C4×C2312yes—
198C23×C4D4×C2311yescharacteristic
199C23×C4D4×C2311yescharacteristic
200C2×M16—22no—
201#117—42no—
202C2×C4⋊C4—21no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4C2×C4×C2312yes—
207C2×C42—21no—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—42no—
211#101—42no—
212#100—24no—
213#99—24no—
214#99—42no—
215C22⋊C8—42no—
216#95—22no—
217#95—22no—
218#95—42no—
219C23#1285311yescharacteristic, direct product factor, Sylow
220C24C22⋊C4×C2311yescharacteristic
221C24C22×C4×C2311yescharacteristic
222C24C2×D4×C2311yescharacteristic
223#41—416no—
224C2×Q8—22no—
225C2×Q8—22no—
226C2×Q8—41no—
227C2×D4—22no—
228C2×D4—22no—
229C2×D4—22no—
230C2×D4—41no—
231C2×D4—41no—
232C2×D4—41no—
233C2×D4—41no—
234C2×D4—41no—
235C2×D4#41×C2314yes—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—44no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4C22⋊C4×C2311yescharacteristic
251C22×C4C22⋊C4×C2311yescharacteristic
252C22×C4C22⋊C4×C2311yescharacteristic
253C22×C4C2×D4×C2311yescharacteristic
254M16—42no—
255M16—42no—
256C2×C8—42no—
257C4⋊C4—42no—
258C22⋊C4—44no—
259C22⋊C4—44no—
260C22⋊C4—42no—
261C22⋊C4—42no—
262C22⋊C4—42no—
263C42—42no—
264C42—42no—
265C42—42no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23—42no—
274C23—42no—
275C23—42no—
276C23C2×C22⋊C4×C2311yescharacteristic
277C23C23×#11712yes—
278C23C23×#12312yes—
279Q8—44no—
280D4—44no—
281D4—44no—
282D4—44no—
283C2×C4—28no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—24no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—44no—
295C2×C4—44no—
296C2×C4—42no—
297C2×C4—44no—
298C2×C4—42no—
299C2×C4—44no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—82no—
304C2×C4—84no—
305C2×C4C23×#9912yes—
306C2×C4C23×#9912yes—
307C2×C4C23×#12312yes—
308C8—82no—
309C22—22no—
310C22—22no—
311C22—22no—
312C22—42no—
313C22—42no—
314C22—42no—
315C22—42no—
316C22—42no—
317C22—42no—
318C22—41no—
319C22—42no—
320C22—81no—
321C22—81no—
322C22C23×#38111yescharacteristic
323C22C2×C23×#9911yescharacteristic
324C22C2×C23×#9911yescharacteristic
325C4—22no—
326C4—22no—
327C4—22no—
328C4—41no—
329C4—44no—
330C4—82no—
331C4—162no—
332C2—21no—
333C2—21no—
334C2—42no—
335C2—42no—
336C2—81no—
337C2C23×#181911yescharacteristic
338C2C23×#212012yes—
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×C42×C23C4semidirect
15. C23×#386C4semidirect
96. #12853C23direct (N × H)
143. #1390C2×C23semidirect
168. C2×#117C4×C23semidirect
169. C22×C42C4×C23semidirect
177. #386C4×C23semidirect
219. C23#12853direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#22972
C23×#17852
C23×#17822
C23×#13902
#1285323