Subgroups — C₂×C₂₃×#3123 #19208178

Order 5888.

2210 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#541	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂₃×#518	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#3123	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₂₃×#485	C₂	1	2	yes	maximal, complemented
5	C₂×C₂₃×#478	C₂	1	2	yes	maximal, complemented
6	C₂×C₂₃×#461	C₂	1	2	yes	maximal, complemented
7	C₂×C₂₃×#450	C₂	1	2	yes	maximal
8	C₂×C₂₃×#440	C₂	1	2	yes	maximal, complemented
9	C₂×C₄×SD₁₆×C₂₃	C₂	1	2	yes	maximal, complemented
10	C₂×C₂₃×#433	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂³×D₄×C₂₃	C₂²	1	1	yes	characteristic
12	C₂²×SD₁₆×C₂₃	C₂²	1	2	yes	—
13	C₂²×SD₁₆×C₂₃	C₂²	1	2	yes	—
14	C₂₃×#541	—	2	4	no	—
15	C₂₃×#541	C₂²	1	8	yes	complemented
16	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
17	C₂×C₂₃×#126	—	2	2	no	—
18	C₂×C₂₃×#125	—	2	2	no	—
19	C₂×C₂₃×#125	C₂²	1	1	yes	characteristic, complemented
20	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	—
21	C₂₃×#518	—	2	4	no	—
22	C₂₃×#518	C₂²	1	8	yes	—
23	C₂×C₄×Q₈×C₂₃	C₂²	1	2	yes	complemented
24	C₂×C₄×D₄×C₂₃	—	2	2	no	—
25	C₂×C₄×D₄×C₂₃	C₂²	1	2	yes	—
26	C₂×C₄×D₄×C₂₃	C₂²	1	2	yes	—
27	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
28	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
29	C₂²×C₂²⋊C₄×C₂₃	—	2	2	no	—
30	C₂₃×#485	C₂²	1	16	yes	complemented
31	C₂₃×#478	C₂²	1	16	yes	complemented
32	C₂₃×#461	C₂²	1	16	yes	complemented
33	C₂₃×#450	C₂²	1	16	yes	—
34	C₂₃×#440	C₂²	1	16	yes	complemented
35	C₄×SD₁₆×C₂₃	C₂²	1	16	yes	complemented
36	C₂₃×#433	C₂²	1	8	yes	complemented
37	C₂×C₂₃×#107	C₂²	1	2	yes	complemented
38	C₂×C₂₃×#106	C₂²	1	2	yes	complemented
39	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	complemented
40	C₂×C₂₃×#103	C₂²	1	2	yes	complemented
41	C₂×C₂₃×#103	C₂²	1	2	yes	complemented
42	C₂×C₂₃×#102	C₂²	1	2	yes	—
43	C₂×C₂₃×#102	C₂²	1	2	yes	—
44	C₂×C₂₃×C₂²⋊C₈	C₂²	1	2	yes	—
45	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic, complemented
46	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic, complemented
47	C₂⁵×C₂₃	—	2	1	no	—
48	C₂²×Q₈×C₂₃	C₂³	1	2	yes	—
49	C₂²×D₄×C₂₃	—	2	8	no	—
50	C₂²×D₄×C₂₃	—	2	1	no	—
51	C₂²×D₄×C₂₃	—	2	2	no	—
52	C₂²×D₄×C₂₃	C₂³	1	4	yes	—
53	C₂²×D₄×C₂₃	C₂³	1	2	yes	—
54	C₂³×C₄×C₂₃	—	2	2	no	—
55	C₂³×C₄×C₂₃	—	2	2	no	—
56	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
57	C₂×SD₁₆×C₂₃	—	2	8	no	—
58	C₂×SD₁₆×C₂₃	—	2	8	no	—
59	C₂×SD₁₆×C₂₃	C₂³	1	8	yes	—
60	C₂×SD₁₆×C₂₃	C₂³	1	8	yes	—
61	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
62	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
63	C₂₃×C₄⋊Q₈	—	2	2	no	—
64	C₂₃×C₄⋊Q₈	C₂³	1	4	yes	complemented
65	C₂₃×#126	—	2	16	no	—
66	C₂₃×#125	—	2	16	no	—
67	C₂₃×#125	—	2	2	no	—
68	C₂₃×#125	C₂³	1	4	yes	complemented
69	C₂₃×C₂²⋊Q₈	—	2	4	no	—
70	C₂₃×C₂²⋊Q₈	C₂³	1	8	yes	—
71	C₄×Q₈×C₂₃	—	2	4	no	—
72	C₄×Q₈×C₂₃	C₂³	1	8	yes	complemented
73	C₄×D₄×C₂₃	—	2	16	no	—
74	C₄×D₄×C₂₃	—	2	4	no	—
75	C₄×D₄×C₂₃	—	2	4	no	—
76	C₄×D₄×C₂₃	C₂³	1	8	yes	—
77	C₄×D₄×C₂₃	C₂³	1	8	yes	—
78	C₂₃×#117	—	2	2	no	—
79	C₂₃×#117	D₄	1	4	yes	—
80	C₂₃×#117	C₂³	1	4	yes	—
81	C₂₃×#117	C₂³	1	4	yes	—
82	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
83	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
84	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
85	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
86	C₂×C₄⋊C₄×C₂₃	D₄	1	2	yes	—
87	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
88	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
89	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
90	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
91	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
92	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
93	C₂×C₂²⋊C₄×C₂₃	—	2	16	no	—
94	C₂×C₂²⋊C₄×C₂₃	D₄	1	2	yes	—
95	C₂×C₄²×C₂₃	—	2	1	no	—
96	C₂×C₄²×C₂₃	—	2	2	no	—
97	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
98	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
99	C₂₃×#107	C₂³	1	8	yes	complemented
100	C₂₃×#106	C₂³	1	8	yes	complemented
101	C₂₃×C₄⋊C₈	C₂³	1	8	yes	complemented
102	C₂₃×#103	C₂³	1	8	yes	complemented
103	C₂₃×#103	C₂³	1	8	yes	complemented
104	C₂₃×#102	C₂³	1	8	yes	—
105	C₂₃×#102	C₂³	1	8	yes	—
106	C₂₃×C₂²⋊C₈	C₂³	1	8	yes	—
107	C₂₃×#97	C₂³	1	4	yes	complemented
108	C₄×C₈×C₂₃	C₂³	1	4	yes	complemented
109	C₂⁴×C₂₃	—	2	8	no	—
110	C₂⁴×C₂₃	—	2	2	no	—
111	C₂⁴×C₂₃	—	2	4	no	—
112	C₂⁴×C₂₃	—	8	4	no	—
113	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
114	C₂×Q₈×C₂₃	—	2	2	no	—
115	C₂×Q₈×C₂₃	—	2	4	no	—
116	C₂×Q₈×C₂₃	—	2	4	no	—
117	C₂×Q₈×C₂₃	C₂⁴	1	4	yes	—
118	C₂×D₄×C₂₃	—	2	8	no	—
119	C₂×D₄×C₂₃	—	2	16	no	—
120	C₂×D₄×C₂₃	—	2	16	no	—
121	C₂×D₄×C₂₃	—	2	4	no	—
122	C₂×D₄×C₂₃	—	2	2	no	—
123	C₂×D₄×C₂₃	—	2	4	no	—
124	C₂×D₄×C₂₃	#41	1	8	yes	—
125	C₂×D₄×C₂₃	C₂⁴	1	4	yes	—
126	C₂²×C₄×C₂₃	—	2	4	no	—
127	C₂²×C₄×C₂₃	—	2	2	no	—
128	C₂²×C₄×C₂₃	—	2	2	no	—
129	C₂²×C₄×C₂₃	—	2	2	no	—
130	C₂²×C₄×C₂₃	—	2	8	no	—
131	C₂²×C₄×C₂₃	—	2	8	no	—
132	C₂²×C₄×C₂₃	—	8	4	no	—
133	C₂²×C₄×C₂₃	—	8	4	no	—
134	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
135	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
136	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
137	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
138	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
139	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
140	SD₁₆×C₂₃	—	2	16	no	—
141	SD₁₆×C₂₃	—	2	16	no	—
142	C₂×C₈×C₂₃	—	2	4	no	—
143	C₂×C₈×C₂₃	—	2	4	no	—
144	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
145	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
146	C₄⋊C₄×C₂₃	—	2	8	no	—
147	C₄⋊C₄×C₂₃	—	2	8	no	—
148	C₄⋊C₄×C₂₃	—	2	8	no	—
149	C₄⋊C₄×C₂₃	—	2	8	no	—
150	C₄⋊C₄×C₂₃	—	2	2	no	—
151	C₄⋊C₄×C₂₃	—	2	2	no	—
152	C₄⋊C₄×C₂₃	C₂×D₄	1	8	yes	—
153	C₄⋊C₄×C₂₃	C₂⁴	1	4	yes	—
154	C₄⋊C₄×C₂₃	C₂⁴	1	4	yes	—
155	C₂²⋊C₄×C₂₃	—	2	4	no	—
156	C₂²⋊C₄×C₂₃	—	2	8	no	—
157	C₂²⋊C₄×C₂₃	—	2	8	no	—
158	C₂²⋊C₄×C₂₃	—	2	8	no	—
159	C₂²⋊C₄×C₂₃	C₂×D₄	1	8	yes	—
160	C₄²×C₂₃	—	2	4	no	—
161	C₄²×C₂₃	—	2	8	no	—
162	C₄²×C₂₃	C₂×D₄	1	2	yes	—
163	C₄²×C₂₃	C₂×D₄	1	2	yes	—
164	C₄²×C₂₃	C₂⁴	1	2	yes	—
165	C₄²×C₂₃	C₂⁴	1	2	yes	—
166	C₂×#3123	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
167	C₂³×C₂₃	—	2	16	no	—
168	C₂³×C₂₃	—	2	2	no	—
169	C₂³×C₂₃	—	2	8	no	—
170	C₂³×C₂₃	—	2	4	no	—
171	C₂³×C₂₃	—	2	4	no	—
172	C₂³×C₂₃	—	4	4	no	—
173	C₂³×C₂₃	—	4	4	no	—
174	C₂³×C₂₃	—	8	8	no	—
175	C₂³×C₂₃	—	8	16	no	—
176	C₂³×C₂₃	C₂²×D₄	1	2	yes	—
177	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
178	Q₈×C₂₃	—	2	4	no	—
179	Q₈×C₂₃	—	2	4	no	—
180	Q₈×C₂₃	—	4	4	no	—
181	D₄×C₂₃	—	2	16	no	—
182	D₄×C₂₃	—	2	8	no	—
183	D₄×C₂₃	—	2	4	no	—
184	D₄×C₂₃	C₂×#41	1	8	yes	—
185	C₂×C₄×C₂₃	—	2	2	no	—
186	C₂×C₄×C₂₃	—	2	4	no	—
187	C₂×C₄×C₂₃	—	2	4	no	—
188	C₂×C₄×C₂₃	—	2	4	no	—
189	C₂×C₄×C₂₃	—	2	4	no	—
190	C₂×C₄×C₂₃	—	2	4	no	—
191	C₂×C₄×C₂₃	—	2	4	no	—
192	C₂×C₄×C₂₃	—	2	4	no	—
193	C₂×C₄×C₂₃	—	4	4	no	—
194	C₂×C₄×C₂₃	—	4	4	no	—
195	C₂×C₄×C₂₃	—	4	4	no	—
196	C₂×C₄×C₂₃	—	8	8	no	—
197	C₂×C₄×C₂₃	—	8	8	no	—
198	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
199	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
200	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
201	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
202	C₂×C₄×C₂₃	C₂×#41	1	4	yes	—
203	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
204	C₈×C₂₃	—	2	4	no	—
205	C₈×C₂₃	—	2	4	no	—
206	C₂×#541	C₂×C₂₃	1	1	yes	characteristic, complemented
207	C₂×#518	C₂×C₂₃	1	1	yes	characteristic
208	#3123	C₂×C₂₃	1	16	yes	direct product factor
209	C₂×#485	C₂×C₂₃	1	2	yes	complemented
210	C₂×#478	C₂×C₂₃	1	2	yes	complemented
211	C₂×#461	C₂×C₂₃	1	2	yes	complemented
212	C₂×#450	C₂×C₂₃	1	2	yes	—
213	C₂×#440	C₂×C₂₃	1	2	yes	complemented
214	C₂×C₄×SD₁₆	C₂×C₂₃	1	2	yes	complemented
215	C₂×#433	C₂×C₂₃	1	1	yes	characteristic, complemented
216	C₂²×C₂₃	—	2	2	no	—
217	C₂²×C₂₃	—	2	8	no	—
218	C₂²×C₂₃	—	2	4	no	—
219	C₂²×C₂₃	—	4	4	no	—
220	C₂²×C₂₃	—	4	16	no	—
221	C₂²×C₂₃	—	4	4	no	—
222	C₂²×C₂₃	—	4	4	no	—
223	C₂²×C₂₃	—	8	16	no	—
224	C₂²×C₂₃	#525	1	2	yes	—
225	C₂²×C₂₃	#577	1	4	yes	—
226	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
227	C₄×C₂₃	—	2	2	no	—
228	C₄×C₂₃	—	2	4	no	—
229	C₄×C₂₃	—	2	4	no	—
230	C₄×C₂₃	—	4	4	no	—
231	C₄×C₂₃	—	4	4	no	—
232	C₄×C₂₃	—	4	4	no	—
233	C₄×C₂₃	#525	1	2	yes	—
234	C₄×C₂₃	C₂²×#41	1	2	yes	—
235	C₂³×D₄	C₂²×C₂₃	1	1	yes	characteristic
236	C₂²×SD₁₆	C₂²×C₂₃	1	2	yes	—
237	C₂²×SD₁₆	C₂²×C₂₃	1	2	yes	—
238	#541	—	2	4	no	—
239	#541	C₂²×C₂₃	1	8	yes	complemented
240	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
241	C₂×#126	—	2	2	no	—
242	C₂×#125	—	2	2	no	—
243	C₂×#125	C₂²×C₂₃	1	1	yes	characteristic, complemented
244	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	—
245	#518	—	2	4	no	—
246	#518	C₂²×C₂₃	1	8	yes	—
247	C₂×C₄×Q₈	C₂²×C₂₃	1	2	yes	complemented
248	C₂×C₄×D₄	—	2	2	no	—
249	C₂×C₄×D₄	C₂²×C₂₃	1	2	yes	—
250	C₂×C₄×D₄	C₂²×C₂₃	1	2	yes	—
251	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
252	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
253	C₂²×C₂²⋊C₄	—	2	2	no	—
254	#485	C₂²×C₂₃	1	16	yes	complemented
255	#478	C₂²×C₂₃	1	16	yes	complemented
256	#461	C₂²×C₂₃	1	16	yes	complemented
257	#450	C₂²×C₂₃	1	16	yes	—
258	#440	C₂²×C₂₃	1	16	yes	complemented
259	C₄×SD₁₆	C₂²×C₂₃	1	16	yes	complemented
260	#433	C₂²×C₂₃	1	8	yes	complemented
261	C₂×#107	C₂²×C₂₃	1	2	yes	complemented
262	C₂×#106	C₂²×C₂₃	1	2	yes	complemented
263	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	complemented
264	C₂×#103	C₂²×C₂₃	1	2	yes	complemented
265	C₂×#103	C₂²×C₂₃	1	2	yes	complemented
266	C₂×#102	C₂²×C₂₃	1	2	yes	—
267	C₂×#102	C₂²×C₂₃	1	2	yes	—
268	C₂×C₂²⋊C₈	C₂²×C₂₃	1	2	yes	—
269	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic, complemented
270	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
271	C₂×C₂₃	—	4	2	no	—
272	C₂×C₂₃	—	4	8	no	—
273	C₂×C₂₃	#3123	1	4	yes	direct product factor
274	C₂×C₂₃	C₂×#525	1	1	yes	characteristic
275	C₂×C₂₃	C₂×#577	1	2	yes	—
276	C₂⁵	—	2	1	no	—
277	C₂²×Q₈	C₂³×C₂₃	1	2	yes	—
278	C₂²×D₄	—	2	8	no	—
279	C₂²×D₄	—	2	1	no	—
280	C₂²×D₄	—	2	2	no	—
281	C₂²×D₄	C₂³×C₂₃	1	4	yes	—
282	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
283	C₂³×C₄	—	2	2	no	—
284	C₂³×C₄	—	2	2	no	—
285	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
286	C₂×SD₁₆	—	2	8	no	—
287	C₂×SD₁₆	—	2	8	no	—
288	C₂×SD₁₆	C₂³×C₂₃	1	8	yes	—
289	C₂×SD₁₆	C₂³×C₂₃	1	8	yes	—
290	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
291	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
292	C₄⋊Q₈	—	2	2	no	—
293	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	complemented
294	#126	—	2	16	no	—
295	#125	—	2	16	no	—
296	#125	—	2	2	no	—
297	#125	C₂³×C₂₃	1	4	yes	complemented
298	C₂²⋊Q₈	—	2	4	no	—
299	C₂²⋊Q₈	C₂³×C₂₃	1	8	yes	—
300	C₄×Q₈	—	2	4	no	—
301	C₄×Q₈	C₂³×C₂₃	1	8	yes	complemented
302	C₄×D₄	—	2	16	no	—
303	C₄×D₄	—	2	4	no	—
304	C₄×D₄	—	2	4	no	—
305	C₄×D₄	C₂³×C₂₃	1	8	yes	—
306	C₄×D₄	C₂³×C₂₃	1	8	yes	—
307	#117	—	2	2	no	—
308	#117	D₄×C₂₃	1	4	yes	—
309	#117	C₂³×C₂₃	1	4	yes	—
310	#117	C₂³×C₂₃	1	4	yes	—
311	C₂×C₄⋊C₄	—	2	2	no	—
312	C₂×C₄⋊C₄	—	2	2	no	—
313	C₂×C₄⋊C₄	—	2	2	no	—
314	C₂×C₄⋊C₄	—	2	2	no	—
315	C₂×C₄⋊C₄	D₄×C₂₃	1	2	yes	—
316	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
317	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
318	C₂×C₂²⋊C₄	—	2	1	no	—
319	C₂×C₂²⋊C₄	—	2	2	no	—
320	C₂×C₂²⋊C₄	—	2	2	no	—
321	C₂×C₂²⋊C₄	—	2	2	no	—
322	C₂×C₂²⋊C₄	—	2	16	no	—
323	C₂×C₂²⋊C₄	D₄×C₂₃	1	2	yes	—
324	C₂×C₄²	—	2	1	no	—
325	C₂×C₄²	—	2	2	no	—
326	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
327	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
328	#107	C₂³×C₂₃	1	8	yes	complemented
329	#106	C₂³×C₂₃	1	8	yes	complemented
330	C₄⋊C₈	C₂³×C₂₃	1	8	yes	complemented
331	#103	C₂³×C₂₃	1	8	yes	complemented
332	#103	C₂³×C₂₃	1	8	yes	complemented
333	#102	C₂³×C₂₃	1	8	yes	—
334	#102	C₂³×C₂₃	1	8	yes	—
335	C₂²⋊C₈	C₂³×C₂₃	1	8	yes	—
336	#97	C₂³×C₂₃	1	4	yes	complemented
337	C₄×C₈	C₂³×C₂₃	1	4	yes	complemented
338	C₂₃	C₂×#3123	1	1	yes	characteristic, direct product factor, Sylow
339	C₂⁴	—	2	8	no	—
340	C₂⁴	—	2	2	no	—
341	C₂⁴	—	2	4	no	—
342	C₂⁴	—	8	4	no	—
343	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
344	C₂×Q₈	—	2	2	no	—
345	C₂×Q₈	—	2	4	no	—
346	C₂×Q₈	—	2	4	no	—
347	C₂×Q₈	C₂⁴×C₂₃	1	4	yes	—
348	C₂×D₄	—	2	8	no	—
349	C₂×D₄	—	2	16	no	—
350	C₂×D₄	—	2	16	no	—
351	C₂×D₄	—	2	4	no	—
352	C₂×D₄	—	2	2	no	—
353	C₂×D₄	—	2	4	no	—
354	C₂×D₄	#41×C₂₃	1	8	yes	—
355	C₂×D₄	C₂⁴×C₂₃	1	4	yes	—
356	C₂²×C₄	—	2	4	no	—
357	C₂²×C₄	—	2	2	no	—
358	C₂²×C₄	—	2	2	no	—
359	C₂²×C₄	—	2	2	no	—
360	C₂²×C₄	—	2	8	no	—
361	C₂²×C₄	—	2	8	no	—
362	C₂²×C₄	—	8	4	no	—
363	C₂²×C₄	—	8	4	no	—
364	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
365	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
366	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
367	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
368	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
369	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
370	SD₁₆	—	2	16	no	—
371	SD₁₆	—	2	16	no	—
372	C₂×C₈	—	2	4	no	—
373	C₂×C₈	—	2	4	no	—
374	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
375	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
376	C₄⋊C₄	—	2	8	no	—
377	C₄⋊C₄	—	2	8	no	—
378	C₄⋊C₄	—	2	8	no	—
379	C₄⋊C₄	—	2	8	no	—
380	C₄⋊C₄	—	2	2	no	—
381	C₄⋊C₄	—	2	2	no	—
382	C₄⋊C₄	C₂×D₄×C₂₃	1	8	yes	—
383	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
384	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
385	C₂²⋊C₄	—	2	4	no	—
386	C₂²⋊C₄	—	2	8	no	—
387	C₂²⋊C₄	—	2	8	no	—
388	C₂²⋊C₄	—	2	8	no	—
389	C₂²⋊C₄	C₂×D₄×C₂₃	1	8	yes	—
390	C₄²	—	2	4	no	—
391	C₄²	—	2	8	no	—
392	C₄²	C₂×D₄×C₂₃	1	2	yes	—
393	C₄²	C₂×D₄×C₂₃	1	2	yes	—
394	C₄²	C₂⁴×C₂₃	1	2	yes	—
395	C₄²	C₂⁴×C₂₃	1	2	yes	—
396	C₂³	—	2	16	no	—
397	C₂³	—	2	4	no	—
398	C₂³	—	2	4	no	—
399	C₂³	—	2	2	no	—
400	C₂³	—	2	8	no	—
401	C₂³	—	4	4	no	—
402	C₂³	—	4	4	no	—
403	C₂³	—	8	8	no	—
404	C₂³	—	8	16	no	—
405	C₂³	C₂²×D₄×C₂₃	1	2	yes	—
406	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
407	Q₈	—	2	4	no	—
408	Q₈	—	2	4	no	—
409	Q₈	—	4	4	no	—
410	D₄	—	2	16	no	—
411	D₄	—	2	4	no	—
412	D₄	—	2	8	no	—
413	D₄	C₂×#41×C₂₃	1	8	yes	—
414	C₂×C₄	—	2	4	no	—
415	C₂×C₄	—	2	4	no	—
416	C₂×C₄	—	2	2	no	—
417	C₂×C₄	—	2	4	no	—
418	C₂×C₄	—	2	4	no	—
419	C₂×C₄	—	2	4	no	—
420	C₂×C₄	—	2	4	no	—
421	C₂×C₄	—	2	4	no	—
422	C₂×C₄	—	4	4	no	—
423	C₂×C₄	—	4	4	no	—
424	C₂×C₄	—	4	4	no	—
425	C₂×C₄	—	8	8	no	—
426	C₂×C₄	—	8	8	no	—
427	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
428	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
429	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
430	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
431	C₂×C₄	C₂×#41×C₂₃	1	4	yes	—
432	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
433	C₈	—	2	4	no	—
434	C₈	—	2	4	no	—
435	C₂²	—	2	2	no	—
436	C₂²	—	2	8	no	—
437	C₂²	—	2	4	no	—
438	C₂²	—	4	4	no	—
439	C₂²	—	4	4	no	—
440	C₂²	—	4	4	no	—
441	C₂²	—	4	16	no	—
442	C₂²	—	8	16	no	—
443	C₂²	C₂₃×#525	1	2	yes	—
444	C₂²	C₂₃×#577	1	4	yes	—
445	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
446	C₄	—	2	2	no	—
447	C₄	—	2	4	no	—
448	C₄	—	2	4	no	—
449	C₄	—	4	4	no	—
450	C₄	—	4	4	no	—
451	C₄	—	4	4	no	—
452	C₄	C₂₃×#525	1	2	yes	—
453	C₄	C₂²×#41×C₂₃	1	2	yes	—
454	C₂	—	4	2	no	—
455	C₂	—	4	8	no	—
456	C₂	C₂₃×#3123	1	4	yes	direct product factor
457	C₂	C₂×C₂₃×#525	1	1	yes	characteristic
458	C₂	C₂×C₂₃×#577	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
prime radical	C₂³×D₄×C₂₃	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂×C₂₃×#117 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₂₃×#541	C₂	semidirect
3. C₂₃×#3123	C₂	direct (N × H)
4. C₂×C₂₃×#485	C₂	semidirect
5. C₂×C₂₃×#478	C₂	semidirect
6. C₂×C₂₃×#461	C₂	semidirect
8. C₂×C₂₃×#440	C₂	semidirect
9. C₂×C₄×SD₁₆×C₂₃	C₂	semidirect
10. C₂×C₂₃×#433	C₂	semidirect
15. C₂₃×#541	C₂²	semidirect
16. C₂×C₂₃×C₄⋊Q₈	C₂²	semidirect
19. C₂×C₂₃×#125	C₂²	semidirect
23. C₂×C₄×Q₈×C₂₃	C₂²	semidirect
30. C₂₃×#485	C₂²	semidirect
31. C₂₃×#478	C₂²	semidirect
32. C₂₃×#461	C₂²	semidirect
34. C₂₃×#440	C₂²	semidirect
35. C₄×SD₁₆×C₂₃	C₂²	semidirect
36. C₂₃×#433	C₂²	semidirect
37. C₂×C₂₃×#107	C₂²	semidirect
38. C₂×C₂₃×#106	C₂²	semidirect
39. C₂×C₂₃×C₄⋊C₈	C₂²	semidirect
40. C₂×C₂₃×#103	C₂²	semidirect
41. C₂×C₂₃×#103	C₂²	semidirect
45. C₂×C₂₃×#97	C₂²	semidirect
46. C₂×C₄×C₈×C₂₃	C₂²	semidirect
64. C₂₃×C₄⋊Q₈	C₂³	semidirect
68. C₂₃×#125	C₂³	semidirect
72. C₄×Q₈×C₂₃	C₂³	semidirect
99. C₂₃×#107	C₂³	semidirect
100. C₂₃×#106	C₂³	semidirect
101. C₂₃×C₄⋊C₈	C₂³	semidirect
102. C₂₃×#103	C₂³	semidirect
103. C₂₃×#103	C₂³	semidirect
107. C₂₃×#97	C₂³	semidirect
108. C₄×C₈×C₂₃	C₂³	semidirect
166. C₂×#3123	C₂₃	direct (N × H)
206. C₂×#541	C₂×C₂₃	semidirect
208. #3123	C₂×C₂₃	direct (N × H)
209. C₂×#485	C₂×C₂₃	semidirect
210. C₂×#478	C₂×C₂₃	semidirect
211. C₂×#461	C₂×C₂₃	semidirect
213. C₂×#440	C₂×C₂₃	semidirect
214. C₂×C₄×SD₁₆	C₂×C₂₃	semidirect
215. C₂×#433	C₂×C₂₃	semidirect
239. #541	C₂²×C₂₃	semidirect
240. C₂×C₄⋊Q₈	C₂²×C₂₃	semidirect
243. C₂×#125	C₂²×C₂₃	semidirect
247. C₂×C₄×Q₈	C₂²×C₂₃	semidirect
254. #485	C₂²×C₂₃	semidirect
255. #478	C₂²×C₂₃	semidirect
256. #461	C₂²×C₂₃	semidirect
258. #440	C₂²×C₂₃	semidirect
259. C₄×SD₁₆	C₂²×C₂₃	semidirect
260. #433	C₂²×C₂₃	semidirect
261. C₂×#107	C₂²×C₂₃	semidirect
262. C₂×#106	C₂²×C₂₃	semidirect
263. C₂×C₄⋊C₈	C₂²×C₂₃	semidirect
264. C₂×#103	C₂²×C₂₃	semidirect
265. C₂×#103	C₂²×C₂₃	semidirect
269. C₂×#97	C₂²×C₂₃	semidirect
270. C₂×C₄×C₈	C₂²×C₂₃	semidirect
273. C₂×C₂₃	#3123	direct (N × H)
293. C₄⋊Q₈	C₂³×C₂₃	semidirect
297. #125	C₂³×C₂₃	semidirect
301. C₄×Q₈	C₂³×C₂₃	semidirect
328. #107	C₂³×C₂₃	semidirect
329. #106	C₂³×C₂₃	semidirect
330. C₄⋊C₈	C₂³×C₂₃	semidirect
331. #103	C₂³×C₂₃	semidirect
332. #103	C₂³×C₂₃	semidirect
336. #97	C₂³×C₂₃	semidirect
337. C₄×C₈	C₂³×C₂₃	semidirect
338. C₂₃	C₂×#3123	direct (N × H)
456. C₂	C₂₃×#3123	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#541	2
C₂×C₂₃×#518	2
C₂₃×#3123	2
C₂×C₂₃×#485	2
C₂×C₂₃×#478	2
C₂×C₂₃×#461	2
C₂×C₂₃×#450	2
C₂×C₂₃×#440	2
C₂×C₄×SD₁₆×C₂₃	2
C₂×C₂₃×#433	2
C₂×#3123	23

Subgroups — C2×C23×#3123 #19208178

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#3123 #19208178