Subgroups — C₂²×C₂₃×#398 #19182164

Order 5888.

4158 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₄⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂³×C₄⋊C₄×C₂₃	C₂	1	2	yes	maximal
4	C₂×C₂₃×#398	C₂	1	24	yes	maximal, direct product factor
5	C₂²×C₂₃×#95	C₂	1	2	yes	maximal
6	C₂²×C₂₃×#95	C₂	1	1	yes	maximal, characteristic
7	C₂⁴×C₄×C₂₃	C₂²	1	1	yes	characteristic
8	C₂⁴×C₄×C₂₃	C₂²	1	2	yes	—
9	C₂⁴×C₄×C₂₃	C₂²	1	1	yes	characteristic
10	C₂⁴×C₄×C₂₃	C₂²	1	1	yes	characteristic
11	C₂⁴×C₄×C₂₃	C₂²	1	2	yes	—
12	C₂²×C₄⋊C₄×C₂₃	—	2	8	no	—
13	C₂²×C₄⋊C₄×C₂₃	—	2	16	no	—
14	C₂²×C₄⋊C₄×C₂₃	—	2	8	no	—
15	C₂²×C₄⋊C₄×C₂₃	C₂²	1	12	yes	—
16	C₂²×C₄⋊C₄×C₂₃	C₂²	1	12	yes	—
17	C₂²×C₄⋊C₄×C₂₃	C₂²	1	24	yes	—
18	C₂₃×#398	C₂²	1	64	yes	direct product factor
19	C₂×C₂₃×#95	C₂²	1	24	yes	—
20	C₂×C₂₃×#95	C₂²	1	12	yes	—
21	C₂⁵×C₂₃	C₂³	1	1	yes	characteristic
22	C₂³×C₄×C₂₃	—	2	4	no	—
23	C₂³×C₄×C₂₃	—	2	8	no	—
24	C₂³×C₄×C₂₃	—	2	4	no	—
25	C₂³×C₄×C₂₃	—	2	8	no	—
26	C₂³×C₄×C₂₃	—	2	8	no	—
27	C₂³×C₄×C₂₃	—	2	8	no	—
28	C₂³×C₄×C₂₃	—	2	8	no	—
29	C₂³×C₄×C₂₃	—	2	4	no	—
30	C₂³×C₄×C₂₃	—	2	8	no	—
31	C₂³×C₄×C₂₃	—	2	8	no	—
32	C₂³×C₄×C₂₃	D₄	1	8	yes	—
33	C₂³×C₄×C₂₃	D₄	1	8	yes	—
34	C₂³×C₄×C₂₃	Q₈	1	16	yes	—
35	C₂³×C₄×C₂₃	C₂³	1	6	yes	—
36	C₂³×C₄×C₂₃	C₂³	1	12	yes	—
37	C₂³×C₄×C₂₃	C₂³	1	6	yes	—
38	C₂³×C₄×C₂₃	C₂³	1	6	yes	—
39	C₂³×C₄×C₂₃	C₂³	1	12	yes	—
40	C₂×C₄⋊C₄×C₂₃	—	2	48	no	—
41	C₂×C₄⋊C₄×C₂₃	—	2	96	no	—
42	C₂×C₄⋊C₄×C₂₃	—	2	48	no	—
43	C₂×C₄⋊C₄×C₂₃	C₂³	1	16	yes	—
44	C₂×C₄⋊C₄×C₂₃	C₂³	1	32	yes	—
45	C₂×C₄⋊C₄×C₂₃	C₂³	1	16	yes	—
46	C₂₃×#95	C₂³	1	32	yes	—
47	C₂₃×#95	C₂³	1	16	yes	—
48	C₂⁴×C₂₃	C₂×D₄	1	4	yes	—
49	C₂⁴×C₂₃	C₂×D₄	1	4	yes	—
50	C₂⁴×C₂₃	C₂×Q₈	1	8	yes	—
51	C₂⁴×C₂₃	#41	1	4	yes	—
52	C₂⁴×C₂₃	#41	1	8	yes	—
53	C₂⁴×C₂₃	C₂⁴	1	3	yes	—
54	C₂²×C₄×C₂₃	—	2	32	no	—
55	C₂²×C₄×C₂₃	—	2	64	no	—
56	C₂²×C₄×C₂₃	—	2	32	no	—
57	C₂²×C₄×C₂₃	—	2	12	no	—
58	C₂²×C₄×C₂₃	—	2	24	no	—
59	C₂²×C₄×C₂₃	—	2	12	no	—
60	C₂²×C₄×C₂₃	—	2	24	no	—
61	C₂²×C₄×C₂₃	—	2	24	no	—
62	C₂²×C₄×C₂₃	—	2	24	no	—
63	C₂²×C₄×C₂₃	—	2	24	no	—
64	C₂²×C₄×C₂₃	—	2	12	no	—
65	C₂²×C₄×C₂₃	—	2	24	no	—
66	C₂²×C₄×C₂₃	—	2	24	no	—
67	C₂²×C₄×C₂₃	—	4	32	no	—
68	C₂²×C₄×C₂₃	—	4	16	no	—
69	C₂²×C₄×C₂₃	C₂×D₄	1	24	yes	—
70	C₂²×C₄×C₂₃	C₂×D₄	1	24	yes	—
71	C₂²×C₄×C₂₃	C₂×Q₈	1	48	yes	—
72	C₂²×C₄×C₂₃	C₂⁴	1	8	yes	—
73	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
74	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
75	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
76	C₂²×C₄×C₂₃	C₂⁴	1	8	yes	—
77	C₄⋊C₄×C₂₃	—	2	32	no	—
78	C₄⋊C₄×C₂₃	—	2	64	no	—
79	C₄⋊C₄×C₂₃	—	2	32	no	—
80	C₂²×#398	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
81	C₂³×C₂₃	#121	1	16	yes	—
82	C₂³×C₂₃	C₂²⋊Q₈	1	32	yes	—
83	C₂³×C₂₃	#123	1	16	yes	—
84	C₂³×C₂₃	#125	1	32	yes	—
85	C₂³×C₂₃	C₄⋊Q₈	1	16	yes	—
86	C₂³×C₂₃	C₂²×D₄	1	6	yes	—
87	C₂³×C₂₃	C₂²×D₄	1	6	yes	—
88	C₂³×C₂₃	C₂²×Q₈	1	12	yes	—
89	C₂³×C₂₃	C₂×#41	1	6	yes	—
90	C₂³×C₂₃	C₂×#41	1	12	yes	—
91	C₂³×C₂₃	C₂⁵	1	1	yes	characteristic
92	C₂×C₄×C₂₃	—	2	48	no	—
93	C₂×C₄×C₂₃	—	2	96	no	—
94	C₂×C₄×C₂₃	—	2	48	no	—
95	C₂×C₄×C₂₃	—	2	4	no	—
96	C₂×C₄×C₂₃	—	2	8	no	—
97	C₂×C₄×C₂₃	—	2	4	no	—
98	C₂×C₄×C₂₃	—	2	8	no	—
99	C₂×C₄×C₂₃	—	2	8	no	—
100	C₂×C₄×C₂₃	—	2	8	no	—
101	C₂×C₄×C₂₃	—	2	8	no	—
102	C₂×C₄×C₂₃	—	2	4	no	—
103	C₂×C₄×C₂₃	—	2	8	no	—
104	C₂×C₄×C₂₃	—	2	8	no	—
105	C₂×C₄×C₂₃	—	4	48	no	—
106	C₂×C₄×C₂₃	—	4	24	no	—
107	C₂×C₄×C₂₃	C₂²×D₄	1	8	yes	—
108	C₂×C₄×C₂₃	C₂²×D₄	1	8	yes	—
109	C₂×C₄×C₂₃	C₂²×Q₈	1	16	yes	—
110	C₂³×C₄⋊C₄	C₂×C₂₃	1	1	yes	characteristic
111	C₂³×C₄⋊C₄	C₂×C₂₃	1	2	yes	—
112	C₂³×C₄⋊C₄	C₂×C₂₃	1	1	yes	characteristic
113	C₂×#398	C₂×C₂₃	1	24	yes	direct product factor
114	C₂²×#95	C₂×C₂₃	1	2	yes	—
115	C₂²×#95	C₂×C₂₃	1	1	yes	characteristic
116	C₂²×C₂₃	#398	1	64	yes	direct product factor
117	C₂²×C₂₃	C₂×#121	1	12	yes	—
118	C₂²×C₂₃	C₂×C₂²⋊Q₈	1	24	yes	—
119	C₂²×C₂₃	C₂×#123	1	12	yes	—
120	C₂²×C₂₃	C₂×#125	1	24	yes	—
121	C₂²×C₂₃	C₂×C₄⋊Q₈	1	12	yes	—
122	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
123	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
124	C₂²×C₂₃	C₂³×Q₈	1	2	yes	—
125	C₂²×C₂₃	C₂²×#41	1	1	yes	characteristic
126	C₂²×C₂₃	C₂²×#41	1	2	yes	—
127	C₄×C₂₃	—	2	8	no	—
128	C₄×C₂₃	—	2	16	no	—
129	C₄×C₂₃	—	2	8	no	—
130	C₄×C₂₃	—	4	8	no	—
131	C₄×C₂₃	—	4	4	no	—
132	C₂⁴×C₄	C₂²×C₂₃	1	2	yes	—
133	C₂⁴×C₄	C₂²×C₂₃	1	1	yes	characteristic
134	C₂⁴×C₄	C₂²×C₂₃	1	1	yes	characteristic
135	C₂⁴×C₄	C₂²×C₂₃	1	1	yes	characteristic
136	C₂⁴×C₄	C₂²×C₂₃	1	2	yes	—
137	C₂²×C₄⋊C₄	—	2	8	no	—
138	C₂²×C₄⋊C₄	—	2	16	no	—
139	C₂²×C₄⋊C₄	—	2	8	no	—
140	C₂²×C₄⋊C₄	C₂²×C₂₃	1	12	yes	—
141	C₂²×C₄⋊C₄	C₂²×C₂₃	1	24	yes	—
142	C₂²×C₄⋊C₄	C₂²×C₂₃	1	12	yes	—
143	#398	C₂²×C₂₃	1	64	yes	direct product factor
144	C₂×#95	C₂²×C₂₃	1	24	yes	—
145	C₂×#95	C₂²×C₂₃	1	12	yes	—
146	C₂×C₂₃	C₂×#398	1	24	yes	direct product factor
147	C₂×C₂₃	C₂²×#121	1	1	yes	characteristic
148	C₂×C₂₃	C₂²×C₂²⋊Q₈	1	2	yes	—
149	C₂×C₂₃	C₂²×#123	1	1	yes	characteristic
150	C₂×C₂₃	C₂²×#125	1	2	yes	—
151	C₂×C₂₃	C₂²×C₄⋊Q₈	1	1	yes	characteristic
152	C₂⁵	C₂³×C₂₃	1	1	yes	characteristic
153	C₂³×C₄	—	2	4	no	—
154	C₂³×C₄	—	2	8	no	—
155	C₂³×C₄	—	2	4	no	—
156	C₂³×C₄	—	2	8	no	—
157	C₂³×C₄	—	2	8	no	—
158	C₂³×C₄	—	2	8	no	—
159	C₂³×C₄	—	2	8	no	—
160	C₂³×C₄	—	2	4	no	—
161	C₂³×C₄	—	2	8	no	—
162	C₂³×C₄	—	2	8	no	—
163	C₂³×C₄	D₄×C₂₃	1	8	yes	—
164	C₂³×C₄	D₄×C₂₃	1	8	yes	—
165	C₂³×C₄	Q₈×C₂₃	1	16	yes	—
166	C₂³×C₄	C₂³×C₂₃	1	12	yes	—
167	C₂³×C₄	C₂³×C₂₃	1	6	yes	—
168	C₂³×C₄	C₂³×C₂₃	1	6	yes	—
169	C₂³×C₄	C₂³×C₂₃	1	6	yes	—
170	C₂³×C₄	C₂³×C₂₃	1	12	yes	—
171	C₂×C₄⋊C₄	—	2	48	no	—
172	C₂×C₄⋊C₄	—	2	96	no	—
173	C₂×C₄⋊C₄	—	2	48	no	—
174	C₂×C₄⋊C₄	C₂³×C₂₃	1	16	yes	—
175	C₂×C₄⋊C₄	C₂³×C₂₃	1	16	yes	—
176	C₂×C₄⋊C₄	C₂³×C₂₃	1	32	yes	—
177	#95	C₂³×C₂₃	1	32	yes	—
178	#95	C₂³×C₂₃	1	16	yes	—
179	C₂₃	C₂²×#398	1	1	yes	characteristic, direct product factor, Sylow
180	C₂⁴	C₂×D₄×C₂₃	1	4	yes	—
181	C₂⁴	C₂×D₄×C₂₃	1	4	yes	—
182	C₂⁴	C₂×Q₈×C₂₃	1	8	yes	—
183	C₂⁴	#41×C₂₃	1	4	yes	—
184	C₂⁴	#41×C₂₃	1	8	yes	—
185	C₂⁴	C₂⁴×C₂₃	1	3	yes	—
186	C₂²×C₄	—	2	32	no	—
187	C₂²×C₄	—	2	64	no	—
188	C₂²×C₄	—	2	32	no	—
189	C₂²×C₄	—	2	12	no	—
190	C₂²×C₄	—	2	24	no	—
191	C₂²×C₄	—	2	12	no	—
192	C₂²×C₄	—	2	24	no	—
193	C₂²×C₄	—	2	24	no	—
194	C₂²×C₄	—	2	24	no	—
195	C₂²×C₄	—	2	24	no	—
196	C₂²×C₄	—	2	12	no	—
197	C₂²×C₄	—	2	24	no	—
198	C₂²×C₄	—	2	24	no	—
199	C₂²×C₄	—	4	32	no	—
200	C₂²×C₄	—	4	16	no	—
201	C₂²×C₄	C₂×D₄×C₂₃	1	24	yes	—
202	C₂²×C₄	C₂×D₄×C₂₃	1	24	yes	—
203	C₂²×C₄	C₂×Q₈×C₂₃	1	48	yes	—
204	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
205	C₂²×C₄	C₂⁴×C₂₃	1	8	yes	—
206	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
207	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
208	C₂²×C₄	C₂⁴×C₂₃	1	8	yes	—
209	C₄⋊C₄	—	2	32	no	—
210	C₄⋊C₄	—	2	64	no	—
211	C₄⋊C₄	—	2	32	no	—
212	C₂³	C₂₃×#121	1	16	yes	—
213	C₂³	C₂₃×C₂²⋊Q₈	1	32	yes	—
214	C₂³	C₂₃×#123	1	16	yes	—
215	C₂³	C₂₃×#125	1	32	yes	—
216	C₂³	C₂₃×C₄⋊Q₈	1	16	yes	—
217	C₂³	C₂²×D₄×C₂₃	1	6	yes	—
218	C₂³	C₂²×D₄×C₂₃	1	6	yes	—
219	C₂³	C₂²×Q₈×C₂₃	1	12	yes	—
220	C₂³	C₂×#41×C₂₃	1	6	yes	—
221	C₂³	C₂×#41×C₂₃	1	12	yes	—
222	C₂³	C₂⁵×C₂₃	1	1	yes	characteristic
223	C₂×C₄	—	2	4	no	—
224	C₂×C₄	—	2	8	no	—
225	C₂×C₄	—	2	4	no	—
226	C₂×C₄	—	2	8	no	—
227	C₂×C₄	—	2	8	no	—
228	C₂×C₄	—	2	48	no	—
229	C₂×C₄	—	2	96	no	—
230	C₂×C₄	—	2	48	no	—
231	C₂×C₄	—	2	8	no	—
232	C₂×C₄	—	2	8	no	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	2	8	no	—
235	C₂×C₄	—	2	8	no	—
236	C₂×C₄	—	4	48	no	—
237	C₂×C₄	—	4	24	no	—
238	C₂×C₄	C₂²×D₄×C₂₃	1	8	yes	—
239	C₂×C₄	C₂²×D₄×C₂₃	1	8	yes	—
240	C₂×C₄	C₂²×Q₈×C₂₃	1	16	yes	—
241	C₂²	C₂₃×#398	1	64	yes	direct product factor
242	C₂²	C₂×C₂₃×#121	1	12	yes	—
243	C₂²	C₂×C₂₃×C₂²⋊Q₈	1	24	yes	—
244	C₂²	C₂×C₂₃×#123	1	12	yes	—
245	C₂²	C₂×C₂₃×#125	1	24	yes	—
246	C₂²	C₂×C₂₃×C₄⋊Q₈	1	12	yes	—
247	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
248	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
249	C₂²	C₂³×Q₈×C₂₃	1	2	yes	—
250	C₂²	C₂²×#41×C₂₃	1	1	yes	characteristic
251	C₂²	C₂²×#41×C₂₃	1	2	yes	—
252	C₄	—	2	8	no	—
253	C₄	—	2	16	no	—
254	C₄	—	2	8	no	—
255	C₄	—	4	8	no	—
256	C₄	—	4	4	no	—
257	C₂	C₂×C₂₃×#398	1	24	yes	direct product factor
258	C₂	C₂²×C₂₃×#121	1	1	yes	characteristic
259	C₂	C₂²×C₂₃×C₂²⋊Q₈	1	2	yes	—
260	C₂	C₂²×C₂₃×#123	1	1	yes	characteristic
261	C₂	C₂²×C₂₃×#125	1	2	yes	—
262	C₂	C₂²×C₂₃×C₄⋊Q₈	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂⁵×C₂₃	C₂³
commutator, Frattini, cosocle	C₂³	C₂⁵×C₂₃
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂×C₂₃×#398	C₂	direct (N × H)
18. C₂₃×#398	C₂²	direct (N × H)
80. C₂²×#398	C₂₃	direct (N × H)
113. C₂×#398	C₂×C₂₃	direct (N × H)
116. C₂²×C₂₃	#398	direct (N × H)
143. #398	C₂²×C₂₃	direct (N × H)
146. C₂×C₂₃	C₂×#398	direct (N × H)
179. C₂₃	C₂²×#398	direct (N × H)
241. C₂²	C₂₃×#398	direct (N × H)
257. C₂	C₂×C₂₃×#398	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₄⋊C₄×C₂₃	2
C₂×C₂₃×#398	2
C₂²×C₂₃×#95	2
C₂²×#398	23

Subgroups — C22×C23×#398 #19182164

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₂₃×#398 #19182164