Finite Group Databasebeta

← C23×#12728 #19160354

Subgroups — C23×#12728 #19160354

Order 5888.

714 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 714 conjugacy classes of subgroups — too many to draw. Only its 138 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×#3510C211yesmaximal, characteristic
2C23×#2127C212yesmaximal
3C2×C23×#354C212yesmaximal, complemented
4C23×#1828C212yesmaximal, complemented
5C23×#559—22no—
6C23×#556—22no—
7C23×#551—24no—
8C23×#542—22no—
9C23×#536C411yescharacteristic, complemented
10C23×#535C2211yescharacteristic
11C23×#533C411yescharacteristic, complemented
12C23×#529—22no—
13C2×C23×#124C2211yescharacteristic
14C23×#519C411yescharacteristic, complemented
15C23×#518C2211yescharacteristic
16C4×#41×C23C411yescharacteristic, complemented
17C2×C23×#99C2214yes—
18C23×#354—22no—
19C23×#354—22no—
20C23×#354—22no—
21C23×#353—22no—
22C23×#342—24no—
23C2×#41×C23C2×C411yescharacteristic
24C22×Q8×C23C2×C411yescharacteristic
25C22×D4×C23C2311yescharacteristic
26C23×C4⋊Q8—21no—
27C23×C4⋊Q8—21no—
28C23×#127D412yes—
29C23×#126—24no—
30C23×#126—24no—
31C23×#125—21no—
32C23×#124—24no—
33C23×#124—21no—
34C23×#124—22no—
35C23×#124D412yes—
36C23×#124D412yes—
37C23×#124D412yes—
38C23×#123—24no—
39C23×#123—24no—
40C23×#123—22no—
41C23×C22⋊Q8—24no—
42C23×C22⋊Q8—22no—
43C23×C22⋊Q8—22no—
44C23×#121—22no—
45C23×#120—22no—
46C4×Q8×C23—21no—
47C4×Q8×C23—22no—
48C4×Q8×C23—22no—
49C4×Q8×C23—22no—
50C4×D4×C23—24no—
51C4×D4×C23—21no—
52C4×D4×C23—22no—
53C4×D4×C23—22no—
54C4×D4×C23—22no—
55C23×#117—24no—
56C23×#117—22no—
57C23×#117—21no—
58C23×#117C2×C411yescharacteristic
59C23×#117C2×C411yescharacteristic
60C23×#117C2×C411yescharacteristic
61C2×C22⋊C4×C23—22no—
62C2×C22⋊C4×C23—24no—
63C2×C22⋊C4×C23D414yes—
64C2×C42×C23—21no—
65C2×C42×C23C2×C411yescharacteristic
66C23×#99—24no—
67C23×#99—24no—
68C24×C23C2×D412yes—
69#41×C23—42no—
70C2×Q8×C23—22no—
71C2×Q8×C23—21no—
72C2×Q8×C23—42no—
73C2×Q8×C23C22⋊C412yes—
74C2×Q8×C23C22⋊C411yescharacteristic
75C2×D4×C23—21no—
76C2×D4×C23—24no—
77C2×D4×C23—42no—
78C2×D4×C23C22⋊C411yescharacteristic
79C2×D4×C23C2×D412yes—
80C2×D4×C23C2×D412yes—
81C22×C4×C23—21no—
82C22×C4×C23—22no—
83C22×C4×C23—21no—
84C22×C4×C23—24no—
85C22×C4×C23—21no—
86C22×C4×C23—44no—
87C22×C4×C23C22×C411yescharacteristic
88C4⋊C4×C23—22no—
89C4⋊C4×C23—22no—
90C4⋊C4×C23—24no—
91C4⋊C4×C23—24no—
92C4⋊C4×C23—22no—
93C4⋊C4×C23—22no—
94C4⋊C4×C23—21no—
95C4⋊C4×C23—21no—
96C4⋊C4×C23—21no—
97C22⋊C4×C23—22no—
98C22⋊C4×C23—24no—
99C22⋊C4×C23—21no—
100C22⋊C4×C23—24no—
101C22⋊C4×C23—24no—
102C22⋊C4×C23—24no—
103C22⋊C4×C23—24no—
104C22⋊C4×C23—44no—
105C22⋊C4×C23—44no—
106C22⋊C4×C23C22⋊C412yes—
107C22⋊C4×C23C22⋊C412yes—
108C42×C23—22no—
109C42×C23—24no—
110C42×C23—21no—
111C42×C23—21no—
112C42×C23—21no—
113C42×C23C22⋊C411yescharacteristic
114C42×C23C22⋊C411yescharacteristic
115C42×C23C22⋊C412yes—
116#12728C2311yesmaximal, characteristic, direct product factor, Sylow
117C23×C23—24no—
118C23×C23—21no—
119C23×C23—82no—
120C23×C23C2×C22⋊C411yescharacteristic
121C23×C23#12014yes—
122Q8×C23—42no—
123Q8×C23—41no—
124Q8×C23—42no—
125Q8×C23—82no—
126D4×C23—42no—
127D4×C23—42no—
128D4×C23—41no—
129D4×C23—42no—
130D4×C23—84no—
131C2×C4×C23—22no—
132C2×C4×C23—22no—
133C2×C4×C23—21no—
134C2×C4×C23—22no—
135C2×C4×C23—22no—
136C2×C4×C23—21no—
137C2×C4×C23—21no—
138C2×C4×C23—22no—
139C2×C4×C23—42no—
140C2×C4×C23—42no—
141C2×C4×C23—82no—
142C2×C4×C23—84no—
143C2×C4×C23—81no—
144C2×C4×C23—84no—
145C2×C4×C23—164no—
146C2×C4×C23C2×C22⋊C411yescharacteristic
147C2×C4×C23C2×C22⋊C411yescharacteristic
148#3510C2×C2311yescharacteristic
149#2127C2×C2312yes—
150C2×#354C2×C2312yescomplemented
151#1828C2×C2312yescomplemented
152C22×C23—24no—
153C22×C23—41no—
154C22×C23—44no—
155C22×C23—42no—
156C22×C23—41no—
157C22×C23—84no—
158C22×C23#37911yescharacteristic
159C22×C23#41012yes—
160C4×C23—21no—
161C4×C23—21no—
162C4×C23—42no—
163C4×C23—41no—
164C4×C23—42no—
165C4×C23—41no—
166C4×C23—42no—
167C4×C23—41no—
168C4×C23—42no—
169C4×C23—164no—
170#559—22no—
171#556—22no—
172#551—24no—
173#542—22no—
174#536C4×C2311yescharacteristic, complemented
175#535C22×C2311yescharacteristic
176#533C4×C2311yescharacteristic, complemented
177#529—22no—
178C2×#124C22×C2311yescharacteristic
179#519C4×C2311yescharacteristic, complemented
180#518C22×C2311yescharacteristic
181C4×#41C4×C2311yescharacteristic, complemented
182C2×#99C22×C2314yes—
183#354—22no—
184#354—22no—
185#354—22no—
186#353—22no—
187#342—24no—
188C2×C23—21no—
189C2×C23—22no—
190C2×C23—44no—
191C2×C23—81no—
192C2×C23#178211yescharacteristic
193C2×#41C2×C4×C2311yescharacteristic
194C22×Q8C2×C4×C2311yescharacteristic
195C22×D4C23×C2311yescharacteristic
196C4⋊Q8—21no—
197C4⋊Q8—21no—
198#127D4×C2312yes—
199#126—24no—
200#126—24no—
201#125—21no—
202#124—24no—
203#124—21no—
204#124—22no—
205#124D4×C2312yes—
206#124D4×C2312yes—
207#124D4×C2312yes—
208#123—24no—
209#123—24no—
210#123—22no—
211C22⋊Q8—24no—
212C22⋊Q8—22no—
213C22⋊Q8—22no—
214#121—22no—
215#120—22no—
216C4×Q8—21no—
217C4×Q8—22no—
218C4×Q8—22no—
219C4×Q8—22no—
220C4×D4—24no—
221C4×D4—21no—
222C4×D4—22no—
223C4×D4—22no—
224C4×D4—22no—
225#117—24no—
226#117—22no—
227#117—21no—
228#117C2×C4×C2311yescharacteristic
229#117C2×C4×C2311yescharacteristic
230#117C2×C4×C2311yescharacteristic
231C2×C22⋊C4—22no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4D4×C2314yes—
234C2×C42—21no—
235C2×C42C2×C4×C2311yescharacteristic
236#99—24no—
237#99—24no—
238C23#1272811yescharacteristic, direct product factor, Sylow
239C24C2×D4×C2312yes—
240#41—42no—
241C2×Q8—22no—
242C2×Q8—21no—
243C2×Q8—42no—
244C2×Q8C22⋊C4×C2312yes—
245C2×Q8C22⋊C4×C2311yescharacteristic
246C2×D4—21no—
247C2×D4—24no—
248C2×D4—42no—
249C2×D4C22⋊C4×C2311yescharacteristic
250C2×D4C2×D4×C2312yes—
251C2×D4C2×D4×C2312yes—
252C22×C4—21no—
253C22×C4—22no—
254C22×C4—21no—
255C22×C4—24no—
256C22×C4—21no—
257C22×C4—44no—
258C22×C4C22×C4×C2311yescharacteristic
259C4⋊C4—22no—
260C4⋊C4—22no—
261C4⋊C4—24no—
262C4⋊C4—24no—
263C4⋊C4—22no—
264C4⋊C4—22no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C22⋊C4—22no—
269C22⋊C4—24no—
270C22⋊C4—21no—
271C22⋊C4—24no—
272C22⋊C4—24no—
273C22⋊C4—24no—
274C22⋊C4—24no—
275C22⋊C4—44no—
276C22⋊C4—44no—
277C22⋊C4C22⋊C4×C2312yes—
278C22⋊C4C22⋊C4×C2312yes—
279C42—22no—
280C42—24no—
281C42—21no—
282C42—21no—
283C42—21no—
284C42C22⋊C4×C2311yescharacteristic
285C42C22⋊C4×C2311yescharacteristic
286C42C22⋊C4×C2312yes—
287C23—24no—
288C23—21no—
289C23—82no—
290C23C2×C22⋊C4×C2311yescharacteristic
291C23C23×#12014yes—
292Q8—42no—
293Q8—41no—
294Q8—42no—
295Q8—82no—
296D4—42no—
297D4—42no—
298D4—41no—
299D4—42no—
300D4—84no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—21no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—22no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—82no—
312C2×C4—84no—
313C2×C4—81no—
314C2×C4—84no—
315C2×C4—164no—
316C2×C4C2×C22⋊C4×C2311yescharacteristic
317C2×C4C2×C22⋊C4×C2311yescharacteristic
318C22—24no—
319C22—41no—
320C22—44no—
321C22—42no—
322C22—41no—
323C22—84no—
324C22C23×#37911yescharacteristic
325C22C23×#41012yes—
326C4—21no—
327C4—21no—
328C4—42no—
329C4—41no—
330C4—42no—
331C4—41no—
332C4—42no—
333C4—41no—
334C4—42no—
335C4—164no—
336C2—21no—
337C2—22no—
338C2—44no—
339C2—81no—
340C2C23×#178211yescharacteristic
subgroupG/H
center, socleC2×C23#1782
commutatorC22×C4C22×C4×C23
Frattini, cosocleC22×D4C23×C23
prime radicalC23×#535C22
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C23 ◁ C23×C23 ◁ C23×#518 ◁ G

Complements

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

NH ≅ G/Ntype
3. C2×C23×#354C2semidirect
4. C23×#1828C2semidirect
9. C23×#536C4semidirect
11. C23×#533C4semidirect
14. C23×#519C4semidirect
16. C4×#41×C23C4semidirect
116. #12728C23direct (N × H)
150. C2×#354C2×C23semidirect
151. #1828C2×C23semidirect
174. #536C4×C23semidirect
176. #533C4×C23semidirect
179. #519C4×C23semidirect
181. C4×#41C4×C23semidirect
238. C23#12728direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#35102
C23×#21272
C2×C23×#3542
C23×#18282
#1272823