Finite Group Databasebeta

← C23×#12094 #19159720

Subgroups — C23×#12094 #19159720

Order 5888.

570 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 570 conjugacy classes of subgroups — too many to draw. Only its 74 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×#2815C211yesmaximal, characteristic, complemented
2C23×#2022C211yesmaximal, characteristic, complemented
3C23×#1595C212yesmaximal, complemented
4C23×#1343C212yesmaximal, complemented
5C23×#1317C211yesmaximal, characteristic, complemented
6C23×#534C2211yescharacteristic
7C2×C23×#123—21no—
8C2×C23×#121—21no—
9C23×#447—22no—
10C2×C23×#99—21no—
11C23×#397—22no—
12C23×#392—22no—
13C23×#392—21no—
14C23×#392C2212yes—
15C23×#381C2211yescharacteristic
16C23×#342C2211yescharacteristic
17C23×#327—22no—
18C23×#323C2212yes—
19C22×D4×C23—21no—
20C22×D4×C23—21no—
21C22×D4×C23—22no—
22C22×D4×C23—21no—
23C22×D4×C23D412yes—
24C23×C4×C23D411yescharacteristic
25C2×D8×C23—42no—
26C23×#123—42no—
27C23×#123—41no—
28C23×#123—42no—
29C23×#121—22no—
30C23×#121—42no—
31C23×#121—42no—
32C23×#120—41no—
33C2×C4⋊C4×C23—21no—
34C2×C4⋊C4×C23—41no—
35C2×C22⋊C4×C23—21no—
36C2×C22⋊C4×C23—22no—
37C2×C22⋊C4×C23—22no—
38C2×C22⋊C4×C23—22no—
39C2×C22⋊C4×C23—41no—
40C2×C22⋊C4×C23D411yescharacteristic
41C2×C22⋊C4×C23C2311yescharacteristic
42C23×#102—42no—
43C23×#99—42no—
44C23×C22⋊C8—22no—
45C23×#95—21no—
46C23×#95—22no—
47C23×#95D412yes—
48C24×C23—21no—
49C24×C23—22no—
50C24×C23C2×D411yescharacteristic
51C2×D4×C23—22no—
52C2×D4×C23—42no—
53C2×D4×C23—42no—
54C2×D4×C23—41no—
55C2×D4×C23—41no—
56C2×D4×C23—41no—
57C2×D4×C23—41no—
58C2×D4×C23—42no—
59C2×D4×C23—41no—
60C2×D4×C23—42no—
61C2×D4×C23—41no—
62C2×D4×C23—82no—
63C2×D4×C23—81no—
64C2×D4×C23—81no—
65C2×D4×C23—82no—
66C2×D4×C23—82no—
67C2×D4×C23—82no—
68C2×D4×C23—81no—
69C22×C4×C23—21no—
70C22×C4×C23—21no—
71C22×C4×C23—21no—
72C22×C4×C23—22no—
73C22×C4×C23—21no—
74C22×C4×C23—41no—
75C22×C4×C23—41no—
76C22×C4×C23—41no—
77C22×C4×C23D812yescomplemented
78C22×C4×C23C2×D412yes—
79D8×C23—84no—
80C2×C8×C23—42no—
81C4⋊C4×C23—22no—
82C4⋊C4×C23—42no—
83C4⋊C4×C23—44no—
84C22⋊C4×C23—22no—
85C22⋊C4×C23—41no—
86C22⋊C4×C23—44no—
87C22⋊C4×C23—42no—
88C22⋊C4×C23—44no—
89C22⋊C4×C23—41no—
90C22⋊C4×C23—41no—
91C22⋊C4×C23—82no—
92C22⋊C4×C23—82no—
93#12094C2311yesmaximal, characteristic, direct product factor, Sylow
94C23×C23—21no—
95C23×C23—21no—
96C23×C23—42no—
97C23×C23—41no—
98C23×C23—42no—
99C23×C23—42no—
100C23×C23—42no—
101C23×C23—42no—
102C23×C23—41no—
103C23×C23—42no—
104C23×C23—42no—
105C23×C23—41no—
106C23×C23—42no—
107C23×C23—82no—
108C23×C23#12011yescharacteristic
109C23×C23C2×D811yescharacteristic
110C23×C23AGL(1,ℤ/8)11yescharacteristic
111D4×C23—44no—
112D4×C23—82no—
113D4×C23—82no—
114D4×C23—84no—
115D4×C23—81no—
116D4×C23—81no—
117D4×C23—84no—
118D4×C23—84no—
119D4×C23—84no—
120D4×C23—81no—
121D4×C23—81no—
122C2×C4×C23—22no—
123C2×C4×C23—22no—
124C2×C4×C23—41no—
125C2×C4×C23—41no—
126C2×C4×C23—42no—
127C2×C4×C23—42no—
128C2×C4×C23—42no—
129C2×C4×C23—42no—
130C2×C4×C23—42no—
131C2×C4×C23—42no—
132C2×C4×C23—41no—
133C2×C4×C23—41no—
134C2×C4×C23—42no—
135C2×C4×C23—41no—
136C2×C4×C23—41no—
137C2×C4×C23—81no—
138C2×C4×C23—82no—
139C8×C23—82no—
140#2815C2×C2311yescharacteristic, complemented
141#2022C2×C2311yescharacteristic, complemented
142#1595C2×C2312yescomplemented
143#1343C2×C2312yescomplemented
144#1317C2×C2311yescharacteristic, complemented
145C22×C23—22no—
146C22×C23—21no—
147C22×C23—21no—
148C22×C23—41no—
149C22×C23—41no—
150C22×C23—42no—
151C22×C23—81no—
152C22×C23—82no—
153C22×C23—82no—
154C22×C23—82no—
155C22×C23—82no—
156C22×C23—82no—
157C22×C23—82no—
158C22×C23—81no—
159C22×C23—82no—
160C22×C23—82no—
161C22×C23—82no—
162C22×C23—161no—
163C22×C23#44711yescharacteristic
164C22×C23#45311yescharacteristic
165C22×C23B(4,𝔽2)11yescharacteristic
166C4×C23—42no—
167C4×C23—42no—
168C4×C23—82no—
169C4×C23—81no—
170C4×C23—82no—
171C4×C23—81no—
172#534C22×C2311yescharacteristic
173C2×#123—21no—
174C2×#121—21no—
175#447—22no—
176C2×#99—21no—
177#397—22no—
178#392—22no—
179#392—21no—
180#392C22×C2312yes—
181#381C22×C2311yescharacteristic
182#342C22×C2311yescharacteristic
183#327—22no—
184#323C22×C2312yes—
185C2×C23—21no—
186C2×C23—21no—
187C2×C23—41no—
188C2×C23—41no—
189C2×C23—162no—
190C2×C23—161no—
191C2×C23#159511yescharacteristic
192C2×C23#220012yes—
193C22×D4—21no—
194C22×D4—21no—
195C22×D4—22no—
196C22×D4—21no—
197C22×D4D4×C2312yes—
198C23×C4D4×C2311yescharacteristic
199C2×D8—42no—
200#123—42no—
201#123—41no—
202#123—42no—
203#121—22no—
204#121—42no—
205#121—42no—
206#120—41no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—41no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—41no—
214C2×C22⋊C4D4×C2311yescharacteristic
215C2×C22⋊C4C23×C2311yescharacteristic
216#102—42no—
217#99—42no—
218C22⋊C8—22no—
219#95—21no—
220#95—22no—
221#95D4×C2312yes—
222C23#1209411yescharacteristic, direct product factor, Sylow
223C24—21no—
224C24—22no—
225C24C2×D4×C2311yescharacteristic
226C2×D4—22no—
227C2×D4—42no—
228C2×D4—42no—
229C2×D4—41no—
230C2×D4—41no—
231C2×D4—41no—
232C2×D4—41no—
233C2×D4—42no—
234C2×D4—41no—
235C2×D4—42no—
236C2×D4—41no—
237C2×D4—82no—
238C2×D4—81no—
239C2×D4—81no—
240C2×D4—82no—
241C2×D4—82no—
242C2×D4—82no—
243C2×D4—81no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—22no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4D8×C2312yescomplemented
253C22×C4C2×D4×C2312yes—
254D8—84no—
255C2×C8—42no—
256C4⋊C4—22no—
257C4⋊C4—42no—
258C4⋊C4—44no—
259C22⋊C4—22no—
260C22⋊C4—41no—
261C22⋊C4—44no—
262C22⋊C4—42no—
263C22⋊C4—44no—
264C22⋊C4—41no—
265C22⋊C4—41no—
266C22⋊C4—82no—
267C22⋊C4—82no—
268C23—21no—
269C23—21no—
270C23—42no—
271C23—41no—
272C23—42no—
273C23—42no—
274C23—42no—
275C23—42no—
276C23—41no—
277C23—42no—
278C23—42no—
279C23—41no—
280C23—42no—
281C23—82no—
282C23C23×#12011yescharacteristic
283C23C2×D8×C2311yescharacteristic
284C23C23×AGL(1,ℤ/8)11yescharacteristic
285D4—44no—
286D4—82no—
287D4—82no—
288D4—84no—
289D4—81no—
290D4—81no—
291D4—84no—
292D4—84no—
293D4—84no—
294D4—81no—
295D4—81no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—42no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—81no—
312C2×C4—82no—
313C8—82no—
314C22—22no—
315C22—21no—
316C22—21no—
317C22—41no—
318C22—41no—
319C22—42no—
320C22—81no—
321C22—82no—
322C22—82no—
323C22—82no—
324C22—82no—
325C22—82no—
326C22—82no—
327C22—82no—
328C22—82no—
329C22—82no—
330C22—81no—
331C22—161no—
332C22C23×#44711yescharacteristic
333C22C23×#45311yescharacteristic
334C22C23×B(4,𝔽2)11yescharacteristic
335C4—42no—
336C4—42no—
337C4—82no—
338C4—81no—
339C4—82no—
340C4—81no—
341C2—21no—
342C2—21no—
343C2—41no—
344C2—41no—
345C2—162no—
346C2—161no—
347C2C23×#159511yescharacteristic
348C2C23×#220012yes—
subgroupG/H
center, socleC22×C23B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23×C23
LCSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C23×C23 ◁ C2×C22⋊C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
1. C23×#2815C2semidirect
2. C23×#2022C2semidirect
3. C23×#1595C2semidirect
4. C23×#1343C2semidirect
5. C23×#1317C2semidirect
77. C22×C4×C23D8semidirect
93. #12094C23direct (N × H)
140. #2815C2×C23semidirect
141. #2022C2×C23semidirect
142. #1595C2×C23semidirect
143. #1343C2×C23semidirect
144. #1317C2×C23semidirect
222. C23#12094direct (N × H)
252. C22×C4D8×C23semidirect

Maximal subgroups

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

subgroupindex
C23×#28152
C23×#20222
C23×#15952
C23×#13432
C23×#13172
#1209423