Subgroups — C₂²×C₂₃×#417 #19207708

Order 5888.

5502 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#41×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂³×M₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂³×C₄⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₂₃×#417	C₂	1	24	yes	maximal, direct product factor
5	C₂²×C₂₃×#103	C₂	1	2	yes	maximal, complemented
6	C₂²×C₂₃×#102	C₂	1	2	yes	maximal, complemented
7	C₂²×#41×C₂₃	—	2	8	no	—
8	C₂²×#41×C₂₃	—	2	6	no	—
9	C₂²×#41×C₂₃	C₄	1	16	yes	complemented
10	C₂²×#41×C₂₃	C₂²	1	12	yes	—
11	C₂³×Q₈×C₂₃	C₂²	1	1	yes	characteristic
12	C₂³×D₄×C₂₃	—	2	1	no	—
13	C₂³×D₄×C₂₃	C₂²	1	1	yes	characteristic
14	C₂⁴×C₄×C₂₃	—	2	1	no	—
15	C₂⁴×C₄×C₂₃	—	2	1	no	—
16	C₂⁴×C₄×C₂₃	C₂²	1	1	yes	characteristic
17	C₂²×M₁₆×C₂₃	—	2	8	no	—
18	C₂²×M₁₆×C₂₃	C₂²	1	12	yes	complemented
19	C₂³×C₈×C₂₃	C₂²	1	2	yes	—
20	C₂²×C₄⋊C₄×C₂₃	—	2	6	no	—
21	C₂²×C₄⋊C₄×C₂₃	—	2	1	no	—
22	C₂²×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
23	C₂²×C₄⋊C₄×C₂₃	C₂²	1	12	yes	complemented
24	C₂₃×#417	C₂²	1	64	yes	direct product factor
25	C₂×C₂₃×#103	C₂²	1	24	yes	complemented
26	C₂×C₂₃×#102	C₂²	1	24	yes	complemented
27	C₂⁵×C₂₃	—	2	1	no	—
28	C₂⁵×C₂₃	D₄	1	1	yes	characteristic
29	C₂×#41×C₂₃	—	2	96	no	—
30	C₂×#41×C₂₃	—	2	48	no	—
31	C₂×#41×C₂₃	—	2	24	no	—
32	C₂×#41×C₂₃	C₂×C₄	1	96	yes	complemented
33	C₂×#41×C₂₃	C₂³	1	16	yes	—
34	C₂²×Q₈×C₂₃	—	2	4	no	—
35	C₂²×Q₈×C₂₃	—	2	3	no	—
36	C₂²×Q₈×C₂₃	C₂×C₄	1	8	yes	complemented
37	C₂²×Q₈×C₂₃	C₂³	1	6	yes	—
38	C₂²×D₄×C₂₃	—	2	16	no	—
39	C₂²×D₄×C₂₃	—	2	12	no	—
40	C₂²×D₄×C₂₃	—	2	3	no	—
41	C₂²×D₄×C₂₃	—	2	4	no	—
42	C₂²×D₄×C₂₃	C₂×C₄	1	8	yes	complemented
43	C₂²×D₄×C₂₃	C₂³	1	6	yes	—
44	C₂³×C₄×C₂₃	—	2	1	no	—
45	C₂³×C₄×C₂₃	—	2	16	no	—
46	C₂³×C₄×C₂₃	—	2	12	no	—
47	C₂³×C₄×C₂₃	—	2	12	no	—
48	C₂³×C₄×C₂₃	—	2	2	no	—
49	C₂³×C₄×C₂₃	—	4	8	no	—
50	C₂³×C₄×C₂₃	C₂×C₄	1	8	yes	—
51	C₂³×C₄×C₂₃	D₄	1	6	yes	—
52	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
53	C₂³×C₄×C₂₃	D₄	1	8	yes	—
54	C₂³×C₄×C₂₃	C₂³	1	6	yes	—
55	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
56	C₂×M₁₆×C₂₃	—	2	48	no	—
57	C₂×M₁₆×C₂₃	C₂³	1	16	yes	complemented
58	C₂²×C₈×C₂₃	—	2	8	no	—
59	C₂²×C₈×C₂₃	C₂³	1	12	yes	—
60	C₂×C₄⋊C₄×C₂₃	—	2	6	no	—
61	C₂×C₄⋊C₄×C₂₃	—	2	24	no	—
62	C₂×C₄⋊C₄×C₂₃	—	2	12	no	—
63	C₂×C₄⋊C₄×C₂₃	C₂³	1	12	yes	—
64	C₂×C₄⋊C₄×C₂₃	C₂³	1	16	yes	complemented
65	C₂₃×#103	C₂³	1	32	yes	complemented
66	C₂₃×#102	C₂³	1	32	yes	complemented
67	C₂⁴×C₂₃	—	2	4	no	—
68	C₂⁴×C₂₃	—	2	4	no	—
69	C₂⁴×C₂₃	—	2	8	no	—
70	C₂⁴×C₂₃	—	2	6	no	—
71	C₂⁴×C₂₃	—	4	4	no	—
72	C₂⁴×C₂₃	—	4	4	no	—
73	C₂⁴×C₂₃	C₂²⋊C₄	1	8	yes	—
74	C₂⁴×C₂₃	C₂×D₄	1	6	yes	—
75	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
76	#41×C₂₃	—	2	192	no	—
77	#41×C₂₃	—	2	32	no	—
78	#41×C₂₃	C₂²×C₄	1	64	yes	complemented
79	C₂×Q₈×C₂₃	—	2	24	no	—
80	C₂×Q₈×C₂₃	—	2	12	no	—
81	C₂×Q₈×C₂₃	—	2	6	no	—
82	C₂×Q₈×C₂₃	C₂²×C₄	1	24	yes	complemented
83	C₂×Q₈×C₂₃	C₂⁴	1	4	yes	—
84	C₂×D₄×C₂₃	—	2	96	no	—
85	C₂×D₄×C₂₃	—	2	16	no	—
86	C₂×D₄×C₂₃	—	2	24	no	—
87	C₂×D₄×C₂₃	—	2	6	no	—
88	C₂×D₄×C₂₃	—	2	12	no	—
89	C₂×D₄×C₂₃	C₂²×C₄	1	24	yes	complemented
90	C₂×D₄×C₂₃	C₂⁴	1	4	yes	—
91	C₂²×C₄×C₂₃	—	2	8	no	—
92	C₂²×C₄×C₂₃	—	2	6	no	—
93	C₂²×C₄×C₂₃	—	2	96	no	—
94	C₂²×C₄×C₂₃	—	2	16	no	—
95	C₂²×C₄×C₂₃	—	2	12	no	—
96	C₂²×C₄×C₂₃	—	2	16	no	—
97	C₂²×C₄×C₂₃	—	4	48	no	—
98	C₂²×C₄×C₂₃	—	4	8	no	—
99	C₂²×C₄×C₂₃	C₂²⋊C₄	1	8	yes	—
100	C₂²×C₄×C₂₃	C₂²⋊C₄	1	48	yes	—
101	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
102	C₂²×C₄×C₂₃	C₂²×C₄	1	24	yes	—
103	C₂²×C₄×C₂₃	C₂×D₄	1	6	yes	—
104	C₂²×C₄×C₂₃	C₂×D₄	1	3	yes	—
105	C₂²×C₄×C₂₃	C₂×D₄	1	12	yes	—
106	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
107	C₂²×C₄×C₂₃	C₂×D₄	1	24	yes	—
108	C₂²×C₄×C₂₃	C₂⁴	1	3	yes	—
109	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
110	M₁₆×C₂₃	—	2	32	no	—
111	C₂×C₈×C₂₃	—	2	24	no	—
112	C₂×C₈×C₂₃	C₂⁴	1	8	yes	—
113	C₄⋊C₄×C₂₃	—	2	12	no	—
114	C₄⋊C₄×C₂₃	—	2	16	no	—
115	C₄⋊C₄×C₂₃	C₂⁴	1	8	yes	—
116	C₂²×#417	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
117	C₂³×C₂₃	—	2	12	no	—
118	C₂³×C₂₃	—	2	12	no	—
119	C₂³×C₂₃	—	2	32	no	—
120	C₂³×C₂₃	—	2	24	no	—
121	C₂³×C₂₃	—	2	4	no	—
122	C₂³×C₂₃	—	4	12	no	—
123	C₂³×C₂₃	—	4	12	no	—
124	C₂³×C₂₃	—	4	32	no	—
125	C₂³×C₂₃	C₂×C₂²⋊C₄	1	24	yes	—
126	C₂³×C₂₃	C₂×C₂²⋊C₄	1	4	yes	—
127	C₂³×C₂₃	AGL(1,ℤ/8)	1	4	yes	—
128	C₂³×C₂₃	#137	1	4	yes	—
129	C₂³×C₂₃	C₂²×D₄	1	4	yes	—
130	C₂³×C₂₃	C₂²×D₄	1	3	yes	—
131	Q₈×C₂₃	—	2	24	no	—
132	Q₈×C₂₃	—	2	4	no	—
133	Q₈×C₂₃	C₂³×C₄	1	8	yes	complemented
134	D₄×C₂₃	—	2	64	no	—
135	D₄×C₂₃	—	2	24	no	—
136	D₄×C₂₃	—	2	4	no	—
137	D₄×C₂₃	C₂³×C₄	1	8	yes	complemented
138	C₂×C₄×C₂₃	—	2	24	no	—
139	C₂×C₄×C₂₃	—	2	4	no	—
140	C₂×C₄×C₂₃	—	2	64	no	—
141	C₂×C₄×C₂₃	—	2	8	no	—
142	C₂×C₄×C₂₃	—	4	32	no	—
143	C₂×C₄×C₂₃	—	4	24	no	—
144	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	24	yes	—
145	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	12	yes	—
146	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	48	yes	—
147	C₂×C₄×C₂₃	C₂³×C₄	1	6	yes	—
148	C₂×C₄×C₂₃	C₂³×C₄	1	8	yes	—
149	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
150	C₂×C₄×C₂₃	C₂²×D₄	1	3	yes	—
151	C₂×C₄×C₂₃	C₂²×D₄	1	6	yes	—
152	C₂×C₄×C₂₃	C₂²×D₄	1	8	yes	—
153	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
154	C₈×C₂₃	—	2	8	no	—
155	C₂³×#41	C₂×C₂₃	1	1	yes	characteristic
156	C₂³×M₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
157	C₂³×C₄⋊C₄	C₂×C₂₃	1	1	yes	characteristic, complemented
158	C₂×#417	C₂×C₂₃	1	24	yes	direct product factor
159	C₂²×#103	C₂×C₂₃	1	2	yes	complemented
160	C₂²×#102	C₂×C₂₃	1	2	yes	complemented
161	C₂²×C₂₃	—	2	48	no	—
162	C₂²×C₂₃	—	2	4	no	—
163	C₂²×C₂₃	—	2	4	no	—
164	C₂²×C₂₃	—	2	8	no	—
165	C₂²×C₂₃	—	4	48	no	—
166	C₂²×C₂₃	—	4	4	no	—
167	C₂²×C₂₃	—	4	4	no	—
168	C₂²×C₂₃	#417	1	16	yes	direct product factor
169	C₂²×C₂₃	C₂²×C₂²⋊C₄	1	8	yes	—
170	C₂²×C₂₃	C₂²×C₂²⋊C₄	1	6	yes	—
171	C₂²×C₂₃	C₂×AGL(1,ℤ/8)	1	6	yes	—
172	C₂²×C₂₃	C₂×#137	1	6	yes	—
173	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
174	C₄×C₂₃	—	2	8	no	—
175	C₄×C₂₃	—	4	8	no	—
176	C₄×C₂₃	C₂²×C₂²⋊C₄	1	8	yes	—
177	C₄×C₂₃	C₂²×C₂²⋊C₄	1	6	yes	—
178	C₄×C₂₃	C₂⁴×C₄	1	1	yes	characteristic
179	C₄×C₂₃	C₂³×D₄	1	1	yes	characteristic
180	C₂²×#41	—	2	8	no	—
181	C₂²×#41	—	2	6	no	—
182	C₂²×#41	C₄×C₂₃	1	16	yes	complemented
183	C₂²×#41	C₂²×C₂₃	1	12	yes	—
184	C₂³×Q₈	C₂²×C₂₃	1	1	yes	characteristic
185	C₂³×D₄	—	2	1	no	—
186	C₂³×D₄	C₂²×C₂₃	1	1	yes	characteristic
187	C₂⁴×C₄	—	2	1	no	—
188	C₂⁴×C₄	—	2	1	no	—
189	C₂⁴×C₄	C₂²×C₂₃	1	1	yes	characteristic
190	C₂²×M₁₆	—	2	8	no	—
191	C₂²×M₁₆	C₂²×C₂₃	1	12	yes	complemented
192	C₂³×C₈	C₂²×C₂₃	1	2	yes	—
193	C₂²×C₄⋊C₄	—	2	6	no	—
194	C₂²×C₄⋊C₄	—	2	1	no	—
195	C₂²×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
196	C₂²×C₄⋊C₄	C₂²×C₂₃	1	12	yes	complemented
197	#417	C₂²×C₂₃	1	64	yes	direct product factor
198	C₂×#103	C₂²×C₂₃	1	24	yes	complemented
199	C₂×#102	C₂²×C₂₃	1	24	yes	complemented
200	C₂×C₂₃	—	2	8	no	—
201	C₂×C₂₃	—	4	8	no	—
202	C₂×C₂₃	C₂×#417	1	12	yes	direct product factor
203	C₂×C₂₃	C₂³×C₂²⋊C₄	1	1	yes	characteristic
204	C₂×C₂₃	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
205	C₂×C₂₃	C₂²×#137	1	1	yes	characteristic
206	C₂⁵	—	2	1	no	—
207	C₂⁵	D₄×C₂₃	1	1	yes	characteristic
208	C₂×#41	—	2	96	no	—
209	C₂×#41	—	2	48	no	—
210	C₂×#41	—	2	24	no	—
211	C₂×#41	C₂×C₄×C₂₃	1	96	yes	complemented
212	C₂×#41	C₂³×C₂₃	1	16	yes	—
213	C₂²×Q₈	—	2	4	no	—
214	C₂²×Q₈	—	2	3	no	—
215	C₂²×Q₈	C₂×C₄×C₂₃	1	8	yes	complemented
216	C₂²×Q₈	C₂³×C₂₃	1	6	yes	—
217	C₂²×D₄	—	2	16	no	—
218	C₂²×D₄	—	2	12	no	—
219	C₂²×D₄	—	2	3	no	—
220	C₂²×D₄	—	2	4	no	—
221	C₂²×D₄	C₂×C₄×C₂₃	1	8	yes	complemented
222	C₂²×D₄	C₂³×C₂₃	1	6	yes	—
223	C₂³×C₄	—	2	1	no	—
224	C₂³×C₄	—	2	16	no	—
225	C₂³×C₄	—	2	12	no	—
226	C₂³×C₄	—	2	2	no	—
227	C₂³×C₄	—	2	12	no	—
228	C₂³×C₄	—	4	8	no	—
229	C₂³×C₄	C₂×C₄×C₂₃	1	8	yes	—
230	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
231	C₂³×C₄	D₄×C₂₃	1	6	yes	—
232	C₂³×C₄	D₄×C₂₃	1	8	yes	—
233	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
234	C₂³×C₄	C₂³×C₂₃	1	6	yes	—
235	C₂×M₁₆	—	2	48	no	—
236	C₂×M₁₆	C₂³×C₂₃	1	16	yes	complemented
237	C₂²×C₈	—	2	8	no	—
238	C₂²×C₈	C₂³×C₂₃	1	12	yes	—
239	C₂×C₄⋊C₄	—	2	6	no	—
240	C₂×C₄⋊C₄	—	2	24	no	—
241	C₂×C₄⋊C₄	—	2	12	no	—
242	C₂×C₄⋊C₄	C₂³×C₂₃	1	12	yes	—
243	C₂×C₄⋊C₄	C₂³×C₂₃	1	16	yes	complemented
244	#103	C₂³×C₂₃	1	32	yes	complemented
245	#102	C₂³×C₂₃	1	32	yes	complemented
246	C₂₃	C₂²×#417	1	1	yes	characteristic, direct product factor, Sylow
247	C₂⁴	—	2	4	no	—
248	C₂⁴	—	2	4	no	—
249	C₂⁴	—	2	8	no	—
250	C₂⁴	—	2	6	no	—
251	C₂⁴	—	4	4	no	—
252	C₂⁴	—	4	4	no	—
253	C₂⁴	C₂²⋊C₄×C₂₃	1	8	yes	—
254	C₂⁴	C₂×D₄×C₂₃	1	6	yes	—
255	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
256	#41	—	2	192	no	—
257	#41	—	2	32	no	—
258	#41	C₂²×C₄×C₂₃	1	64	yes	complemented
259	C₂×Q₈	—	2	24	no	—
260	C₂×Q₈	—	2	6	no	—
261	C₂×Q₈	—	2	12	no	—
262	C₂×Q₈	C₂²×C₄×C₂₃	1	24	yes	complemented
263	C₂×Q₈	C₂⁴×C₂₃	1	4	yes	—
264	C₂×D₄	—	2	96	no	—
265	C₂×D₄	—	2	16	no	—
266	C₂×D₄	—	2	24	no	—
267	C₂×D₄	—	2	12	no	—
268	C₂×D₄	—	2	6	no	—
269	C₂×D₄	C₂²×C₄×C₂₃	1	24	yes	complemented
270	C₂×D₄	C₂⁴×C₂₃	1	4	yes	—
271	C₂²×C₄	—	2	8	no	—
272	C₂²×C₄	—	2	6	no	—
273	C₂²×C₄	—	2	96	no	—
274	C₂²×C₄	—	2	16	no	—
275	C₂²×C₄	—	2	12	no	—
276	C₂²×C₄	—	2	16	no	—
277	C₂²×C₄	—	4	48	no	—
278	C₂²×C₄	—	4	8	no	—
279	C₂²×C₄	C₂²⋊C₄×C₂₃	1	8	yes	—
280	C₂²×C₄	C₂²⋊C₄×C₂₃	1	48	yes	—
281	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
282	C₂²×C₄	C₂²×C₄×C₂₃	1	24	yes	—
283	C₂²×C₄	C₂×D₄×C₂₃	1	6	yes	—
284	C₂²×C₄	C₂×D₄×C₂₃	1	3	yes	—
285	C₂²×C₄	C₂×D₄×C₂₃	1	12	yes	—
286	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
287	C₂²×C₄	C₂×D₄×C₂₃	1	24	yes	—
288	C₂²×C₄	C₂⁴×C₂₃	1	3	yes	—
289	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
290	M₁₆	—	2	32	no	—
291	C₂×C₈	—	2	24	no	—
292	C₂×C₈	C₂⁴×C₂₃	1	8	yes	—
293	C₄⋊C₄	—	2	12	no	—
294	C₄⋊C₄	—	2	16	no	—
295	C₄⋊C₄	C₂⁴×C₂₃	1	8	yes	—
296	C₂³	—	2	12	no	—
297	C₂³	—	2	12	no	—
298	C₂³	—	2	32	no	—
299	C₂³	—	2	4	no	—
300	C₂³	—	2	24	no	—
301	C₂³	—	4	12	no	—
302	C₂³	—	4	12	no	—
303	C₂³	—	4	32	no	—
304	C₂³	C₂×C₂²⋊C₄×C₂₃	1	24	yes	—
305	C₂³	C₂×C₂²⋊C₄×C₂₃	1	4	yes	—
306	C₂³	C₂₃×AGL(1,ℤ/8)	1	4	yes	—
307	C₂³	C₂₃×#137	1	4	yes	—
308	C₂³	C₂²×D₄×C₂₃	1	4	yes	—
309	C₂³	C₂²×D₄×C₂₃	1	3	yes	—
310	Q₈	—	2	4	no	—
311	Q₈	—	2	24	no	—
312	Q₈	C₂³×C₄×C₂₃	1	8	yes	complemented
313	D₄	—	2	64	no	—
314	D₄	—	2	4	no	—
315	D₄	—	2	24	no	—
316	D₄	C₂³×C₄×C₂₃	1	8	yes	complemented
317	C₂×C₄	—	2	4	no	—
318	C₂×C₄	—	2	24	no	—
319	C₂×C₄	—	2	64	no	—
320	C₂×C₄	—	2	8	no	—
321	C₂×C₄	—	4	32	no	—
322	C₂×C₄	—	4	24	no	—
323	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	24	yes	—
324	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	12	yes	—
325	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	48	yes	—
326	C₂×C₄	C₂³×C₄×C₂₃	1	6	yes	—
327	C₂×C₄	C₂³×C₄×C₂₃	1	8	yes	—
328	C₂×C₄	C₂²×D₄×C₂₃	1	6	yes	—
329	C₂×C₄	C₂²×D₄×C₂₃	1	8	yes	—
330	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
331	C₂×C₄	C₂²×D₄×C₂₃	1	3	yes	—
332	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
333	C₈	—	2	8	no	—
334	C₂²	—	2	4	no	—
335	C₂²	—	2	4	no	—
336	C₂²	—	2	48	no	—
337	C₂²	—	2	8	no	—
338	C₂²	—	4	4	no	—
339	C₂²	—	4	4	no	—
340	C₂²	—	4	48	no	—
341	C₂²	C₂₃×#417	1	16	yes	direct product factor
342	C₂²	C₂²×C₂²⋊C₄×C₂₃	1	8	yes	—
343	C₂²	C₂²×C₂²⋊C₄×C₂₃	1	6	yes	—
344	C₂²	C₂×C₂₃×AGL(1,ℤ/8)	1	6	yes	—
345	C₂²	C₂×C₂₃×#137	1	6	yes	—
346	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
347	C₄	—	2	8	no	—
348	C₄	—	4	8	no	—
349	C₄	C₂²×C₂²⋊C₄×C₂₃	1	8	yes	—
350	C₄	C₂²×C₂²⋊C₄×C₂₃	1	6	yes	—
351	C₄	C₂⁴×C₄×C₂₃	1	1	yes	characteristic
352	C₄	C₂³×D₄×C₂₃	1	1	yes	characteristic
353	C₂	—	2	8	no	—
354	C₂	—	4	8	no	—
355	C₂	C₂×C₂₃×#417	1	12	yes	direct product factor
356	C₂	C₂³×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
357	C₂	C₂²×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
358	C₂	C₂²×C₂₃×#137	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₂₃	C₂×D₄
commutator	C₄	C₂⁴×C₄×C₂₃
Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
prime radical	C₂³×#41×C₂₃	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₂₃ ◁ C₂⁴×C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂³×M₁₆×C₂₃	C₂	semidirect
3. C₂³×C₄⋊C₄×C₂₃	C₂	semidirect
4. C₂×C₂₃×#417	C₂	direct (N × H)
5. C₂²×C₂₃×#103	C₂	semidirect
6. C₂²×C₂₃×#102	C₂	semidirect
9. C₂²×#41×C₂₃	C₄	semidirect
18. C₂²×M₁₆×C₂₃	C₂²	semidirect
23. C₂²×C₄⋊C₄×C₂₃	C₂²	semidirect
24. C₂₃×#417	C₂²	direct (N × H)
25. C₂×C₂₃×#103	C₂²	semidirect
26. C₂×C₂₃×#102	C₂²	semidirect
32. C₂×#41×C₂₃	C₂×C₄	semidirect
36. C₂²×Q₈×C₂₃	C₂×C₄	semidirect
42. C₂²×D₄×C₂₃	C₂×C₄	semidirect
57. C₂×M₁₆×C₂₃	C₂³	semidirect
64. C₂×C₄⋊C₄×C₂₃	C₂³	semidirect
65. C₂₃×#103	C₂³	semidirect
66. C₂₃×#102	C₂³	semidirect
78. #41×C₂₃	C₂²×C₄	semidirect
82. C₂×Q₈×C₂₃	C₂²×C₄	semidirect
89. C₂×D₄×C₂₃	C₂²×C₄	semidirect
116. C₂²×#417	C₂₃	direct (N × H)
133. Q₈×C₂₃	C₂³×C₄	semidirect
137. D₄×C₂₃	C₂³×C₄	semidirect
156. C₂³×M₁₆	C₂×C₂₃	semidirect
157. C₂³×C₄⋊C₄	C₂×C₂₃	semidirect
158. C₂×#417	C₂×C₂₃	direct (N × H)
159. C₂²×#103	C₂×C₂₃	semidirect
160. C₂²×#102	C₂×C₂₃	semidirect
168. C₂²×C₂₃	#417	direct (N × H)
182. C₂²×#41	C₄×C₂₃	semidirect
191. C₂²×M₁₆	C₂²×C₂₃	semidirect
196. C₂²×C₄⋊C₄	C₂²×C₂₃	semidirect
197. #417	C₂²×C₂₃	direct (N × H)
198. C₂×#103	C₂²×C₂₃	semidirect
199. C₂×#102	C₂²×C₂₃	semidirect
202. C₂×C₂₃	C₂×#417	direct (N × H)
211. C₂×#41	C₂×C₄×C₂₃	semidirect
215. C₂²×Q₈	C₂×C₄×C₂₃	semidirect
221. C₂²×D₄	C₂×C₄×C₂₃	semidirect
236. C₂×M₁₆	C₂³×C₂₃	semidirect
243. C₂×C₄⋊C₄	C₂³×C₂₃	semidirect
244. #103	C₂³×C₂₃	semidirect
245. #102	C₂³×C₂₃	semidirect
246. C₂₃	C₂²×#417	direct (N × H)
258. #41	C₂²×C₄×C₂₃	semidirect
262. C₂×Q₈	C₂²×C₄×C₂₃	semidirect
269. C₂×D₄	C₂²×C₄×C₂₃	semidirect
312. Q₈	C₂³×C₄×C₂₃	semidirect
316. D₄	C₂³×C₄×C₂₃	semidirect
341. C₂²	C₂₃×#417	direct (N × H)
355. C₂	C₂×C₂₃×#417	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×#41×C₂₃	2
C₂³×M₁₆×C₂₃	2
C₂³×C₄⋊C₄×C₂₃	2
C₂×C₂₃×#417	2
C₂²×C₂₃×#103	2
C₂²×C₂₃×#102	2
C₂²×#417	23

Subgroups — C22×C23×#417 #19207708

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₂₃×#417 #19207708