Subgroups — C₂×C₂₃×#3486 #19210408

Order 5888.

11254 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂₃×2₋¹⁺⁴	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂⁴×Q₈×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3486	C₂	1	32	yes	maximal, direct product factor
4	C₂×D₄×Q₈×C₂₃	C₂	1	8	yes	maximal, complemented
5	C₂×C₂₃×#548	C₂	1	8	yes	maximal, complemented
6	C₂×C₂₃×#536	C₂	1	6	yes	maximal, complemented
7	C₂×C₂₃×#525	C₂	1	6	yes	maximal, complemented
8	C₂×C₂₃×#519	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₂₃×2₋¹⁺⁴	—	2	16	no	—
10	C₂×C₂₃×2₋¹⁺⁴	C₂²	1	16	yes	complemented
11	C₂²×#41×C₂₃	C₂²	1	6	yes	—
12	C₂²×#41×C₂₃	C₂²	1	4	yes	complemented
13	C₂³×Q₈×C₂₃	—	2	16	no	—
14	C₂³×Q₈×C₂₃	C₂²	1	3	yes	—
15	C₂³×Q₈×C₂₃	C₂²	1	8	yes	—
16	C₂³×Q₈×C₂₃	C₂²	1	1	yes	characteristic
17	C₂³×Q₈×C₂₃	C₂²	1	16	yes	complemented
18	C₂³×Q₈×C₂₃	C₂²	1	4	yes	complemented
19	C₂⁴×C₄×C₂₃	C₂²	1	3	yes	—
20	D₄×Q₈×C₂₃	C₂²	1	128	yes	complemented
21	C₂₃×#548	C₂²	1	128	yes	complemented
22	C₂₃×#536	C₂²	1	96	yes	complemented
23	C₂×C₂₃×C₄⋊Q₈	C₂²	1	12	yes	complemented
24	C₂×C₂₃×#124	C₂²	1	12	yes	complemented
25	C₂₃×#525	C₂²	1	96	yes	complemented
26	C₂×C₂₃×#123	C₂²	1	12	yes	—
27	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	12	yes	—
28	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	24	yes	complemented
29	C₂×C₂₃×#121	C₂²	1	12	yes	—
30	C₂×C₂₃×#120	C₂²	1	4	yes	—
31	C₂₃×#519	C₂²	1	16	yes	complemented
32	C₂×C₄×Q₈×C₂₃	C₂²	1	8	yes	complemented
33	C₂×C₄×D₄×C₂₃	C₂²	1	24	yes	—
34	C₂×C₂₃×#117	C₂²	1	6	yes	—
35	C₂⁵×C₂₃	C₂³	1	1	yes	characteristic
36	C₂₃×2₋¹⁺⁴	—	2	128	no	—
37	C₂×#41×C₂₃	—	2	32	no	—
38	C₂×#41×C₂₃	—	2	48	no	—
39	C₂×#41×C₂₃	C₂³	1	48	yes	—
40	C₂×#41×C₂₃	C₂³	1	32	yes	complemented
41	C₂²×Q₈×C₂₃	—	2	64	no	—
42	C₂²×Q₈×C₂₃	—	2	32	no	—
43	C₂²×Q₈×C₂₃	—	2	128	no	—
44	C₂²×Q₈×C₂₃	—	2	24	no	—
45	C₂²×Q₈×C₂₃	D₄	1	16	yes	complemented
46	C₂²×Q₈×C₂₃	C₂³	1	64	yes	—
47	C₂²×Q₈×C₂₃	C₂³	1	12	yes	—
48	C₂²×Q₈×C₂₃	C₂³	1	24	yes	—
49	C₂²×Q₈×C₂₃	C₂³	1	6	yes	—
50	C₂²×Q₈×C₂₃	C₂³	1	4	yes	—
51	C₂²×Q₈×C₂₃	C₂³	1	8	yes	—
52	C₂²×Q₈×C₂₃	C₂³	1	32	yes	complemented
53	C₂²×D₄×C₂₃	C₂³	1	4	yes	—
54	C₂²×D₄×C₂₃	C₂³	1	6	yes	—
55	C₂²×D₄×C₂₃	C₂³	1	8	yes	—
56	C₂³×C₄×C₂₃	—	2	24	no	—
57	C₂³×C₄×C₂₃	C₂³	1	12	yes	—
58	C₂³×C₄×C₂₃	C₂³	1	3	yes	—
59	C₂³×C₄×C₂₃	C₂³	1	24	yes	—
60	C₂³×C₄×C₂₃	C₂³	1	12	yes	—
61	C₂³×C₄×C₂₃	C₂³	1	3	yes	—
62	C₂₃×C₄⋊Q₈	C₂³	1	96	yes	complemented
63	C₂₃×#124	C₂³	1	96	yes	complemented
64	C₂₃×#123	C₂³	1	96	yes	—
65	C₂₃×C₂²⋊Q₈	C₂³	1	96	yes	—
66	C₂₃×C₂²⋊Q₈	C₂³	1	192	yes	complemented
67	C₂₃×#121	C₂³	1	96	yes	—
68	C₂₃×#120	C₂³	1	32	yes	—
69	C₄×Q₈×C₂₃	C₂³	1	64	yes	complemented
70	C₄×D₄×C₂₃	C₂³	1	192	yes	—
71	C₂₃×#117	C₂³	1	48	yes	—
72	C₂×C₄⋊C₄×C₂₃	C₂³	1	12	yes	—
73	C₂×C₄⋊C₄×C₂₃	C₂³	1	24	yes	—
74	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
75	C₂×C₂²⋊C₄×C₂₃	C₂³	1	8	yes	—
76	C₂×C₂²⋊C₄×C₂₃	C₂³	1	24	yes	—
77	C₂×C₄²×C₂₃	C₂³	1	12	yes	—
78	C₂⁴×C₂₃	—	2	4	no	—
79	C₂⁴×C₂₃	—	2	8	no	—
80	C₂⁴×C₂₃	C₂⁴	1	4	yes	—
81	C₂⁴×C₂₃	C₂⁴	1	4	yes	—
82	C₂⁴×C₂₃	C₂⁴	1	1	yes	characteristic
83	C₂⁴×C₂₃	C₂⁴	1	2	yes	—
84	#41×C₂₃	—	2	128	no	—
85	#41×C₂₃	—	2	192	no	—
86	C₂×Q₈×C₂₃	—	2	256	no	—
87	C₂×Q₈×C₂₃	—	2	128	no	—
88	C₂×Q₈×C₂₃	—	2	48	no	—
89	C₂×Q₈×C₂₃	—	2	96	no	—
90	C₂×Q₈×C₂₃	—	2	24	no	—
91	C₂×Q₈×C₂₃	C₂×D₄	1	32	yes	—
92	C₂×Q₈×C₂₃	C₂×D₄	1	64	yes	complemented
93	C₂×Q₈×C₂₃	C₂⁴	1	24	yes	—
94	C₂×Q₈×C₂₃	C₂⁴	1	48	yes	—
95	C₂×Q₈×C₂₃	C₂⁴	1	16	yes	—
96	C₂×D₄×C₂₃	—	2	16	no	—
97	C₂×D₄×C₂₃	—	2	32	no	—
98	C₂×D₄×C₂₃	—	2	24	no	—
99	C₂×D₄×C₂₃	C₂⁴	1	32	yes	—
100	C₂×D₄×C₂₃	C₂⁴	1	16	yes	—
101	C₂×D₄×C₂₃	C₂⁴	1	24	yes	—
102	C₂²×C₄×C₂₃	—	2	12	no	—
103	C₂²×C₄×C₂₃	—	2	48	no	—
104	C₂²×C₄×C₂₃	—	2	96	no	—
105	C₂²×C₄×C₂₃	—	2	48	no	—
106	C₂²×C₄×C₂₃	C₂×D₄	1	24	yes	—
107	C₂²×C₄×C₂₃	C₂⁴	1	48	yes	—
108	C₂²×C₄×C₂₃	C₂⁴	1	48	yes	—
109	C₂²×C₄×C₂₃	C₂⁴	1	12	yes	—
110	C₂²×C₄×C₂₃	C₂⁴	1	8	yes	—
111	C₂²×C₄×C₂₃	C₂⁴	1	6	yes	—
112	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
113	C₂²×C₄×C₂₃	C₂⁴	1	12	yes	—
114	C₂²×C₄×C₂₃	C₂⁴	1	6	yes	—
115	C₄⋊C₄×C₂₃	C₂⁴	1	48	yes	—
116	C₄⋊C₄×C₂₃	C₂⁴	1	96	yes	—
117	C₂²⋊C₄×C₂₃	C₂⁴	1	16	yes	—
118	C₂²⋊C₄×C₂₃	C₂⁴	1	32	yes	—
119	C₂²⋊C₄×C₂₃	C₂⁴	1	96	yes	—
120	C₄²×C₂₃	C₂⁴	1	48	yes	—
121	C₂×#3486	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
122	C₂³×C₂₃	—	2	32	no	—
123	C₂³×C₂₃	—	2	8	no	—
124	C₂³×C₂₃	—	2	4	no	—
125	C₂³×C₂₃	—	2	4	no	—
126	C₂³×C₂₃	—	2	4	no	—
127	C₂³×C₂₃	—	2	16	no	—
128	C₂³×C₂₃	—	2	8	no	—
129	C₂³×C₂₃	—	2	16	no	—
130	C₂³×C₂₃	C₂²×D₄	1	4	yes	—
131	C₂³×C₂₃	2₋¹⁺⁴	1	8	yes	complemented
132	C₂³×C₂₃	C₂⁵	1	8	yes	—
133	C₂³×C₂₃	C₂⁵	1	2	yes	—
134	C₂³×C₂₃	C₂⁵	1	4	yes	—
135	C₂³×C₂₃	C₂⁵	1	1	yes	characteristic
136	Q₈×C₂₃	—	2	96	no	—
137	Q₈×C₂₃	—	2	48	no	—
138	Q₈×C₂₃	C₂²×D₄	1	64	yes	—
139	D₄×C₂₃	—	2	32	no	—
140	D₄×C₂₃	—	2	64	no	—
141	D₄×C₂₃	—	2	48	no	—
142	C₂×C₄×C₂₃	—	2	8	no	—
143	C₂×C₄×C₂₃	—	2	24	no	—
144	C₂×C₄×C₂₃	—	2	12	no	—
145	C₂×C₄×C₂₃	—	2	16	no	—
146	C₂×C₄×C₂₃	—	2	96	no	—
147	C₂×C₄×C₂₃	—	2	96	no	—
148	C₂×C₄×C₂₃	C₂²×D₄	1	24	yes	—
149	C₂×C₄×C₂₃	C₂²×D₄	1	48	yes	—
150	C₂×C₄×C₂₃	C₂⁵	1	16	yes	—
151	C₂×C₄×C₂₃	C₂⁵	1	12	yes	—
152	C₂×C₄×C₂₃	C₂⁵	1	8	yes	—
153	C₂×C₄×C₂₃	C₂⁵	1	12	yes	—
154	C₂²×2₋¹⁺⁴	C₂×C₂₃	1	1	yes	characteristic, complemented
155	C₂⁴×Q₈	C₂×C₂₃	1	1	yes	characteristic, complemented
156	#3486	C₂×C₂₃	1	32	yes	direct product factor
157	C₂×D₄×Q₈	C₂×C₂₃	1	8	yes	complemented
158	C₂×#548	C₂×C₂₃	1	8	yes	complemented
159	C₂×#536	C₂×C₂₃	1	6	yes	complemented
160	C₂×#525	C₂×C₂₃	1	6	yes	complemented
161	C₂×#519	C₂×C₂₃	1	1	yes	characteristic, complemented
162	C₂²×C₂₃	—	2	32	no	—
163	C₂²×C₂₃	—	2	4	no	—
164	C₂²×C₂₃	—	2	4	no	—
165	C₂²×C₂₃	—	2	4	no	—
166	C₂²×C₂₃	—	2	8	no	—
167	C₂²×C₂₃	—	2	16	no	—
168	C₂²×C₂₃	—	2	8	no	—
169	C₂²×C₂₃	—	2	16	no	—
170	C₂²×C₂₃	—	4	16	no	—
171	C₂²×C₂₃	C₂³×D₄	1	4	yes	—
172	C₂²×C₂₃	C₂³×D₄	1	2	yes	—
173	C₂²×C₂₃	C₂×2₋¹⁺⁴	1	4	yes	—
174	C₂²×C₂₃	C₂×2₋¹⁺⁴	1	8	yes	complemented
175	C₂²×C₂₃	C₂⁶	1	1	yes	characteristic
176	C₄×C₂₃	—	2	8	no	—
177	C₄×C₂₃	—	2	12	no	—
178	C₄×C₂₃	—	2	16	no	—
179	C₄×C₂₃	C₂³×D₄	1	24	yes	—
180	C₂×2₋¹⁺⁴	—	2	16	no	—
181	C₂×2₋¹⁺⁴	C₂²×C₂₃	1	16	yes	complemented
182	C₂²×#41	C₂²×C₂₃	1	6	yes	—
183	C₂²×#41	C₂²×C₂₃	1	4	yes	complemented
184	C₂³×Q₈	—	2	16	no	—
185	C₂³×Q₈	C₂²×C₂₃	1	3	yes	—
186	C₂³×Q₈	C₂²×C₂₃	1	8	yes	—
187	C₂³×Q₈	C₂²×C₂₃	1	1	yes	characteristic
188	C₂³×Q₈	C₂²×C₂₃	1	16	yes	complemented
189	C₂³×Q₈	C₂²×C₂₃	1	4	yes	complemented
190	C₂⁴×C₄	C₂²×C₂₃	1	3	yes	—
191	D₄×Q₈	C₂²×C₂₃	1	128	yes	complemented
192	#548	C₂²×C₂₃	1	128	yes	complemented
193	#536	C₂²×C₂₃	1	96	yes	complemented
194	C₂×C₄⋊Q₈	C₂²×C₂₃	1	12	yes	complemented
195	C₂×#124	C₂²×C₂₃	1	12	yes	complemented
196	#525	C₂²×C₂₃	1	96	yes	complemented
197	C₂×#123	C₂²×C₂₃	1	12	yes	—
198	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	12	yes	—
199	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	24	yes	complemented
200	C₂×#121	C₂²×C₂₃	1	12	yes	—
201	C₂×#120	C₂²×C₂₃	1	4	yes	—
202	#519	C₂²×C₂₃	1	16	yes	complemented
203	C₂×C₄×Q₈	C₂²×C₂₃	1	8	yes	complemented
204	C₂×C₄×D₄	C₂²×C₂₃	1	24	yes	—
205	C₂×#117	C₂²×C₂₃	1	6	yes	—
206	C₂×C₂₃	—	2	4	no	—
207	C₂×C₂₃	—	2	8	no	—
208	C₂×C₂₃	—	4	8	no	—
209	C₂×C₂₃	#3486	1	4	yes	direct product factor
210	C₂×C₂₃	C₂⁴×D₄	1	1	yes	characteristic
211	C₂×C₂₃	C₂²×2₋¹⁺⁴	1	2	yes	—
212	C₂⁵	C₂³×C₂₃	1	1	yes	characteristic
213	2₋¹⁺⁴	—	2	128	no	—
214	C₂×#41	—	2	32	no	—
215	C₂×#41	—	2	48	no	—
216	C₂×#41	C₂³×C₂₃	1	48	yes	—
217	C₂×#41	C₂³×C₂₃	1	32	yes	complemented
218	C₂²×Q₈	—	2	64	no	—
219	C₂²×Q₈	—	2	32	no	—
220	C₂²×Q₈	—	2	128	no	—
221	C₂²×Q₈	—	2	24	no	—
222	C₂²×Q₈	D₄×C₂₃	1	16	yes	complemented
223	C₂²×Q₈	C₂³×C₂₃	1	6	yes	—
224	C₂²×Q₈	C₂³×C₂₃	1	12	yes	—
225	C₂²×Q₈	C₂³×C₂₃	1	24	yes	—
226	C₂²×Q₈	C₂³×C₂₃	1	64	yes	—
227	C₂²×Q₈	C₂³×C₂₃	1	4	yes	—
228	C₂²×Q₈	C₂³×C₂₃	1	8	yes	—
229	C₂²×Q₈	C₂³×C₂₃	1	32	yes	complemented
230	C₂²×D₄	C₂³×C₂₃	1	8	yes	—
231	C₂²×D₄	C₂³×C₂₃	1	4	yes	—
232	C₂²×D₄	C₂³×C₂₃	1	6	yes	—
233	C₂³×C₄	—	2	24	no	—
234	C₂³×C₄	C₂³×C₂₃	1	12	yes	—
235	C₂³×C₄	C₂³×C₂₃	1	12	yes	—
236	C₂³×C₄	C₂³×C₂₃	1	3	yes	—
237	C₂³×C₄	C₂³×C₂₃	1	24	yes	—
238	C₂³×C₄	C₂³×C₂₃	1	3	yes	—
239	C₄⋊Q₈	C₂³×C₂₃	1	96	yes	complemented
240	#124	C₂³×C₂₃	1	96	yes	complemented
241	#123	C₂³×C₂₃	1	96	yes	—
242	C₂²⋊Q₈	C₂³×C₂₃	1	96	yes	—
243	C₂²⋊Q₈	C₂³×C₂₃	1	192	yes	complemented
244	#121	C₂³×C₂₃	1	96	yes	—
245	#120	C₂³×C₂₃	1	32	yes	—
246	C₄×Q₈	C₂³×C₂₃	1	64	yes	complemented
247	C₄×D₄	C₂³×C₂₃	1	192	yes	—
248	#117	C₂³×C₂₃	1	48	yes	—
249	C₂×C₄⋊C₄	C₂³×C₂₃	1	12	yes	—
250	C₂×C₄⋊C₄	C₂³×C₂₃	1	24	yes	—
251	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
252	C₂×C₂²⋊C₄	C₂³×C₂₃	1	8	yes	—
253	C₂×C₂²⋊C₄	C₂³×C₂₃	1	24	yes	—
254	C₂×C₄²	C₂³×C₂₃	1	12	yes	—
255	C₂₃	C₂×#3486	1	1	yes	characteristic, direct product factor, Sylow
256	C₂⁴	—	2	4	no	—
257	C₂⁴	—	2	8	no	—
258	C₂⁴	C₂⁴×C₂₃	1	4	yes	—
259	C₂⁴	C₂⁴×C₂₃	1	4	yes	—
260	C₂⁴	C₂⁴×C₂₃	1	1	yes	characteristic
261	C₂⁴	C₂⁴×C₂₃	1	2	yes	—
262	#41	—	2	128	no	—
263	#41	—	2	192	no	—
264	C₂×Q₈	—	2	256	no	—
265	C₂×Q₈	—	2	128	no	—
266	C₂×Q₈	—	2	48	no	—
267	C₂×Q₈	—	2	96	no	—
268	C₂×Q₈	—	2	24	no	—
269	C₂×Q₈	C₂×D₄×C₂₃	1	32	yes	—
270	C₂×Q₈	C₂×D₄×C₂₃	1	64	yes	complemented
271	C₂×Q₈	C₂⁴×C₂₃	1	24	yes	—
272	C₂×Q₈	C₂⁴×C₂₃	1	48	yes	—
273	C₂×Q₈	C₂⁴×C₂₃	1	16	yes	—
274	C₂×D₄	—	2	16	no	—
275	C₂×D₄	—	2	32	no	—
276	C₂×D₄	—	2	24	no	—
277	C₂×D₄	C₂⁴×C₂₃	1	32	yes	—
278	C₂×D₄	C₂⁴×C₂₃	1	16	yes	—
279	C₂×D₄	C₂⁴×C₂₃	1	24	yes	—
280	C₂²×C₄	—	2	12	no	—
281	C₂²×C₄	—	2	48	no	—
282	C₂²×C₄	—	2	96	no	—
283	C₂²×C₄	—	2	48	no	—
284	C₂²×C₄	C₂×D₄×C₂₃	1	24	yes	—
285	C₂²×C₄	C₂⁴×C₂₃	1	8	yes	—
286	C₂²×C₄	C₂⁴×C₂₃	1	6	yes	—
287	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
288	C₂²×C₄	C₂⁴×C₂₃	1	48	yes	—
289	C₂²×C₄	C₂⁴×C₂₃	1	48	yes	—
290	C₂²×C₄	C₂⁴×C₂₃	1	12	yes	—
291	C₂²×C₄	C₂⁴×C₂₃	1	6	yes	—
292	C₂²×C₄	C₂⁴×C₂₃	1	12	yes	—
293	C₄⋊C₄	C₂⁴×C₂₃	1	48	yes	—
294	C₄⋊C₄	C₂⁴×C₂₃	1	96	yes	—
295	C₂²⋊C₄	C₂⁴×C₂₃	1	16	yes	—
296	C₂²⋊C₄	C₂⁴×C₂₃	1	32	yes	—
297	C₂²⋊C₄	C₂⁴×C₂₃	1	96	yes	—
298	C₄²	C₂⁴×C₂₃	1	48	yes	—
299	C₂³	—	2	32	no	—
300	C₂³	—	2	8	no	—
301	C₂³	—	2	4	no	—
302	C₂³	—	2	4	no	—
303	C₂³	—	2	4	no	—
304	C₂³	—	2	16	no	—
305	C₂³	—	2	8	no	—
306	C₂³	—	2	16	no	—
307	C₂³	C₂²×D₄×C₂₃	1	4	yes	—
308	C₂³	C₂₃×2₋¹⁺⁴	1	8	yes	complemented
309	C₂³	C₂⁵×C₂₃	1	8	yes	—
310	C₂³	C₂⁵×C₂₃	1	2	yes	—
311	C₂³	C₂⁵×C₂₃	1	4	yes	—
312	C₂³	C₂⁵×C₂₃	1	1	yes	characteristic
313	Q₈	—	2	96	no	—
314	Q₈	—	2	48	no	—
315	Q₈	C₂²×D₄×C₂₃	1	64	yes	—
316	D₄	—	2	32	no	—
317	D₄	—	2	64	no	—
318	D₄	—	2	48	no	—
319	C₂×C₄	—	2	8	no	—
320	C₂×C₄	—	2	24	no	—
321	C₂×C₄	—	2	12	no	—
322	C₂×C₄	—	2	16	no	—
323	C₂×C₄	—	2	96	no	—
324	C₂×C₄	—	2	96	no	—
325	C₂×C₄	C₂²×D₄×C₂₃	1	24	yes	—
326	C₂×C₄	C₂²×D₄×C₂₃	1	48	yes	—
327	C₂×C₄	C₂⁵×C₂₃	1	16	yes	—
328	C₂×C₄	C₂⁵×C₂₃	1	12	yes	—
329	C₂×C₄	C₂⁵×C₂₃	1	8	yes	—
330	C₂×C₄	C₂⁵×C₂₃	1	12	yes	—
331	C₂²	—	2	32	no	—
332	C₂²	—	2	8	no	—
333	C₂²	—	2	16	no	—
334	C₂²	—	2	4	no	—
335	C₂²	—	2	4	no	—
336	C₂²	—	2	4	no	—
337	C₂²	—	2	8	no	—
338	C₂²	—	2	16	no	—
339	C₂²	—	4	16	no	—
340	C₂²	C₂³×D₄×C₂₃	1	4	yes	—
341	C₂²	C₂³×D₄×C₂₃	1	2	yes	—
342	C₂²	C₂×C₂₃×2₋¹⁺⁴	1	4	yes	—
343	C₂²	C₂×C₂₃×2₋¹⁺⁴	1	8	yes	complemented
344	C₂²	C₂⁶×C₂₃	1	1	yes	characteristic
345	C₄	—	2	8	no	—
346	C₄	—	2	12	no	—
347	C₄	—	2	16	no	—
348	C₄	C₂³×D₄×C₂₃	1	24	yes	—
349	C₂	—	2	4	no	—
350	C₂	—	2	8	no	—
351	C₂	—	4	8	no	—
352	C₂	C₂₃×#3486	1	4	yes	direct product factor
353	C₂	C₂⁴×D₄×C₂₃	1	1	yes	characteristic
354	C₂	C₂²×C₂₃×2₋¹⁺⁴	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂⁵
commutator, Frattini, cosocle	C₂²	C₂⁶×C₂₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₂₃×2₋¹⁺⁴	C₂	semidirect
2. C₂⁴×Q₈×C₂₃	C₂	semidirect
3. C₂₃×#3486	C₂	direct (N × H)
4. C₂×D₄×Q₈×C₂₃	C₂	semidirect
5. C₂×C₂₃×#548	C₂	semidirect
6. C₂×C₂₃×#536	C₂	semidirect
7. C₂×C₂₃×#525	C₂	semidirect
8. C₂×C₂₃×#519	C₂	semidirect
10. C₂×C₂₃×2₋¹⁺⁴	C₂²	semidirect
12. C₂²×#41×C₂₃	C₂²	semidirect
17. C₂³×Q₈×C₂₃	C₂²	semidirect
18. C₂³×Q₈×C₂₃	C₂²	semidirect
20. D₄×Q₈×C₂₃	C₂²	semidirect
21. C₂₃×#548	C₂²	semidirect
22. C₂₃×#536	C₂²	semidirect
23. C₂×C₂₃×C₄⋊Q₈	C₂²	semidirect
24. C₂×C₂₃×#124	C₂²	semidirect
25. C₂₃×#525	C₂²	semidirect
28. C₂×C₂₃×C₂²⋊Q₈	C₂²	semidirect
31. C₂₃×#519	C₂²	semidirect
32. C₂×C₄×Q₈×C₂₃	C₂²	semidirect
40. C₂×#41×C₂₃	C₂³	semidirect
45. C₂²×Q₈×C₂₃	D₄	semidirect
52. C₂²×Q₈×C₂₃	C₂³	semidirect
62. C₂₃×C₄⋊Q₈	C₂³	semidirect
63. C₂₃×#124	C₂³	semidirect
66. C₂₃×C₂²⋊Q₈	C₂³	semidirect
69. C₄×Q₈×C₂₃	C₂³	semidirect
92. C₂×Q₈×C₂₃	C₂×D₄	semidirect
121. C₂×#3486	C₂₃	direct (N × H)
131. C₂³×C₂₃	2₋¹⁺⁴	semidirect
154. C₂²×2₋¹⁺⁴	C₂×C₂₃	semidirect
155. C₂⁴×Q₈	C₂×C₂₃	semidirect
156. #3486	C₂×C₂₃	direct (N × H)
157. C₂×D₄×Q₈	C₂×C₂₃	semidirect
158. C₂×#548	C₂×C₂₃	semidirect
159. C₂×#536	C₂×C₂₃	semidirect
160. C₂×#525	C₂×C₂₃	semidirect
161. C₂×#519	C₂×C₂₃	semidirect
174. C₂²×C₂₃	C₂×2₋¹⁺⁴	semidirect
181. C₂×2₋¹⁺⁴	C₂²×C₂₃	semidirect
183. C₂²×#41	C₂²×C₂₃	semidirect
188. C₂³×Q₈	C₂²×C₂₃	semidirect
189. C₂³×Q₈	C₂²×C₂₃	semidirect
191. D₄×Q₈	C₂²×C₂₃	semidirect
192. #548	C₂²×C₂₃	semidirect
193. #536	C₂²×C₂₃	semidirect
194. C₂×C₄⋊Q₈	C₂²×C₂₃	semidirect
195. C₂×#124	C₂²×C₂₃	semidirect
196. #525	C₂²×C₂₃	semidirect
199. C₂×C₂²⋊Q₈	C₂²×C₂₃	semidirect
202. #519	C₂²×C₂₃	semidirect
203. C₂×C₄×Q₈	C₂²×C₂₃	semidirect
209. C₂×C₂₃	#3486	direct (N × H)
217. C₂×#41	C₂³×C₂₃	semidirect
222. C₂²×Q₈	D₄×C₂₃	semidirect
229. C₂²×Q₈	C₂³×C₂₃	semidirect
239. C₄⋊Q₈	C₂³×C₂₃	semidirect
240. #124	C₂³×C₂₃	semidirect
243. C₂²⋊Q₈	C₂³×C₂₃	semidirect
246. C₄×Q₈	C₂³×C₂₃	semidirect
255. C₂₃	C₂×#3486	direct (N × H)
270. C₂×Q₈	C₂×D₄×C₂₃	semidirect
308. C₂³	C₂₃×2₋¹⁺⁴	semidirect
343. C₂²	C₂×C₂₃×2₋¹⁺⁴	semidirect
352. C₂	C₂₃×#3486	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₂₃×2₋¹⁺⁴	2
C₂⁴×Q₈×C₂₃	2
C₂₃×#3486	2
C₂×D₄×Q₈×C₂₃	2
C₂×C₂₃×#548	2
C₂×C₂₃×#536	2
C₂×C₂₃×#525	2
C₂×C₂₃×#519	2
C₂×#3486	23

Subgroups — C2×C23×#3486 #19210408

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#3486 #19210408