Subgroups — #3506×C₂₂₃ #43252521

Order 28544.

1140 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#559×C₂₂₃	C₂	1	4	yes	maximal, complemented
2	#556×C₂₂₃	C₂	1	2	yes	maximal, complemented
3	#555×C₂₂₃	C₂	1	2	yes	maximal, complemented
4	#553×C₂₂₃	C₂	1	4	yes	maximal, complemented
5	#550×C₂₂₃	C₂	1	2	yes	maximal, complemented
6	#547×C₂₂₃	C₂	1	2	yes	maximal, complemented
7	#538×C₂₂₃	C₂	1	4	yes	maximal, complemented
8	#535×C₂₂₃	C₂	1	2	yes	maximal, complemented
9	#533×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
10	#532×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
11	#525×C₂₂₃	C₂	1	4	yes	maximal, complemented
12	#520×C₂₂₃	C₂	1	2	yes	maximal, complemented
13	C₂×C₄×D₄×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×#41×C₂₂₃	C₂²	1	2	yes	—
15	C₂×#41×C₂₂₃	C₂²	1	4	yes	complemented
16	C₂²×D₄×C₂₂₃	C₂²	1	1	yes	characteristic
17	C₂³×C₄×C₂₂₃	C₂²	1	2	yes	—
18	C₄⋊Q₈×C₂₂₃	C₂²	1	1	yes	characteristic, complemented
19	C₄⋊Q₈×C₂₂₃	C₂²	1	1	yes	characteristic, complemented
20	C₄⋊Q₈×C₂₂₃	C₂²	1	1	yes	characteristic, complemented
21	#127×C₂₂₃	C₂²	1	4	yes	—
22	#127×C₂₂₃	C₂²	1	1	yes	characteristic, complemented
23	#126×C₂₂₃	C₂²	1	8	yes	complemented
24	#125×C₂₂₃	C₂²	1	2	yes	complemented
25	#124×C₂₂₃	C₂²	1	4	yes	—
26	#124×C₂₂₃	C₂²	1	2	yes	complemented
27	#124×C₂₂₃	C₂²	1	8	yes	complemented
28	#123×C₂₂₃	C₂²	1	8	yes	complemented
29	#123×C₂₂₃	C₂²	1	8	yes	complemented
30	C₂²⋊Q₈×C₂₂₃	C₂²	1	2	yes	—
31	C₂²⋊Q₈×C₂₂₃	C₂²	1	2	yes	—
32	C₂²⋊Q₈×C₂₂₃	C₂²	1	4	yes	complemented
33	#121×C₂₂₃	C₂²	1	2	yes	—
34	#121×C₂₂₃	C₂²	1	2	yes	—
35	#121×C₂₂₃	C₂²	1	4	yes	—
36	#121×C₂₂₃	C₂²	1	4	yes	—
37	#121×C₂₂₃	C₂²	1	4	yes	complemented
38	#121×C₂₂₃	C₂²	1	8	yes	complemented
39	#120×C₂₂₃	C₂²	1	8	yes	—
40	C₄×Q₈×C₂₂₃	C₂²	1	4	yes	complemented
41	C₄×Q₈×C₂₂₃	C₂²	1	2	yes	complemented
42	C₄×Q₈×C₂₂₃	C₂²	1	2	yes	complemented
43	C₄×D₄×C₂₂₃	C₂²	1	8	yes	—
44	C₄×D₄×C₂₂₃	C₂²	1	2	yes	—
45	C₄×D₄×C₂₂₃	C₂²	1	2	yes	—
46	C₄×D₄×C₂₂₃	C₂²	1	2	yes	—
47	C₄×D₄×C₂₂₃	C₂²	1	2	yes	—
48	C₄×D₄×C₂₂₃	C₂²	1	4	yes	complemented
49	C₄×D₄×C₂₂₃	C₂²	1	8	yes	complemented
50	C₄×D₄×C₂₂₃	C₂²	1	2	yes	complemented
51	C₄×D₄×C₂₂₃	C₂²	1	2	yes	complemented
52	#117×C₂₂₃	C₂²	1	2	yes	—
53	#117×C₂₂₃	C₂²	1	4	yes	complemented
54	#117×C₂₂₃	C₂²	1	4	yes	complemented
55	C₂×C₄⋊C₄×C₂₂₃	C₂²	1	1	yes	characteristic
56	C₂×C₄⋊C₄×C₂₂₃	C₂²	1	4	yes	complemented
57	C₂×C₂²⋊C₄×C₂₂₃	C₂²	1	2	yes	—
58	C₂×C₄²×C₂₂₃	C₂²	1	1	yes	characteristic
59	C₂⁴×C₂₂₃	C₂³	1	2	yes	—
60	#41×C₂₂₃	—	2	8	no	—
61	#41×C₂₂₃	—	4	8	no	—
62	C₂×Q₈×C₂₂₃	C₂³	1	2	yes	—
63	C₂×Q₈×C₂₂₃	C₂³	1	2	yes	—
64	C₂×Q₈×C₂₂₃	C₂³	1	2	yes	—
65	C₂×D₄×C₂₂₃	—	2	4	no	—
66	C₂×D₄×C₂₂₃	C₂³	1	4	yes	—
67	C₂×D₄×C₂₂₃	C₂³	1	2	yes	—
68	C₂×D₄×C₂₂₃	C₂³	1	2	yes	—
69	C₂×D₄×C₂₂₃	C₂³	1	2	yes	—
70	C₂×D₄×C₂₂₃	C₂³	1	2	yes	—
71	C₂×D₄×C₂₂₃	C₂³	1	8	yes	—
72	C₂×D₄×C₂₂₃	C₂³	1	2	yes	—
73	C₂×D₄×C₂₂₃	C₂³	1	8	yes	—
74	C₂²×C₄×C₂₂₃	—	4	4	no	—
75	C₂²×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
76	C₂²×C₄×C₂₂₃	C₂³	1	4	yes	—
77	C₂²×C₄×C₂₂₃	C₂³	1	4	yes	—
78	C₂²×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
79	C₂²×C₄×C₂₂₃	C₂³	1	2	yes	—
80	C₂²×C₄×C₂₂₃	C₂³	1	8	yes	—
81	C₂²×C₄×C₂₂₃	C₂³	1	4	yes	—
82	C₂²×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
83	C₄⋊C₄×C₂₂₃	C₂³	1	2	yes	—
84	C₄⋊C₄×C₂₂₃	C₂³	1	2	yes	—
85	C₄⋊C₄×C₂₂₃	C₂³	1	2	yes	—
86	C₄⋊C₄×C₂₂₃	C₂³	1	8	yes	—
87	C₄⋊C₄×C₂₂₃	C₂³	1	2	yes	—
88	C₄⋊C₄×C₂₂₃	C₂³	1	2	yes	—
89	C₄⋊C₄×C₂₂₃	C₂³	1	2	yes	—
90	C₄⋊C₄×C₂₂₃	C₂³	1	4	yes	—
91	C₄⋊C₄×C₂₂₃	C₂³	1	2	yes	—
92	C₄⋊C₄×C₂₂₃	C₂³	1	4	yes	—
93	C₄⋊C₄×C₂₂₃	C₂³	1	2	yes	—
94	C₂²⋊C₄×C₂₂₃	C₂³	1	4	yes	—
95	C₂²⋊C₄×C₂₂₃	C₂³	1	4	yes	—
96	C₂²⋊C₄×C₂₂₃	C₂³	1	4	yes	—
97	C₂²⋊C₄×C₂₂₃	C₂³	1	8	yes	—
98	C₂²⋊C₄×C₂₂₃	C₂³	1	8	yes	—
99	C₂²⋊C₄×C₂₂₃	C₂³	1	4	yes	—
100	C₂²⋊C₄×C₂₂₃	C₂³	1	8	yes	—
101	C₂²⋊C₄×C₂₂₃	C₂³	1	4	yes	—
102	C₄²×C₂₂₃	C₂³	1	1	yes	characteristic
103	C₄²×C₂₂₃	C₂³	1	1	yes	characteristic
104	C₄²×C₂₂₃	C₂³	1	1	yes	characteristic
105	C₄²×C₂₂₃	C₂³	1	1	yes	characteristic
106	C₄²×C₂₂₃	C₂³	1	4	yes	—
107	C₄²×C₂₂₃	C₂³	1	8	yes	—
108	C₂³×C₂₂₃	—	2	4	no	—
109	C₂³×C₂₂₃	—	4	2	no	—
110	C₂³×C₂₂₃	—	4	2	no	—
111	C₂³×C₂₂₃	C₂⁴	1	4	yes	—
112	C₂³×C₂₂₃	C₂⁴	1	4	yes	—
113	C₂³×C₂₂₃	C₂⁴	1	1	yes	characteristic
114	Q₈×C₂₂₃	—	2	4	no	—
115	Q₈×C₂₂₃	—	2	4	no	—
116	Q₈×C₂₂₃	—	4	2	no	—
117	D₄×C₂₂₃	—	2	8	no	—
118	D₄×C₂₂₃	—	2	4	no	—
119	D₄×C₂₂₃	—	2	4	no	—
120	D₄×C₂₂₃	—	2	4	no	—
121	D₄×C₂₂₃	—	2	4	no	—
122	D₄×C₂₂₃	—	4	2	no	—
123	D₄×C₂₂₃	—	4	8	no	—
124	D₄×C₂₂₃	—	4	8	no	—
125	C₂×C₄×C₂₂₃	—	2	8	no	—
126	C₂×C₄×C₂₂₃	—	2	2	no	—
127	C₂×C₄×C₂₂₃	—	2	8	no	—
128	C₂×C₄×C₂₂₃	—	4	2	no	—
129	C₂×C₄×C₂₂₃	—	4	4	no	—
130	C₂×C₄×C₂₂₃	—	4	1	no	—
131	C₂×C₄×C₂₂₃	—	4	8	no	—
132	C₂×C₄×C₂₂₃	#41	1	4	yes	complemented
133	C₂×C₄×C₂₂₃	C₂⁴	1	4	yes	—
134	C₂×C₄×C₂₂₃	C₂⁴	1	1	yes	characteristic
135	C₂×C₄×C₂₂₃	C₂⁴	1	4	yes	—
136	C₂×C₄×C₂₂₃	C₂⁴	1	1	yes	characteristic
137	C₂×C₄×C₂₂₃	C₂⁴	1	1	yes	characteristic
138	C₂×C₄×C₂₂₃	C₂⁴	1	4	yes	—
139	C₂×C₄×C₂₂₃	C₂⁴	1	4	yes	—
140	C₂×C₄×C₂₂₃	C₂⁴	1	1	yes	characteristic
141	C₂×C₄×C₂₂₃	C₂⁴	1	1	yes	characteristic
142	C₂×C₄×C₂₂₃	C₂⁴	1	1	yes	characteristic
143	C₂²×C₂₂₃	—	2	1	no	—
144	C₂²×C₂₂₃	—	2	1	no	—
145	C₂²×C₂₂₃	—	2	4	no	—
146	C₂²×C₂₂₃	—	2	4	no	—
147	C₂²×C₂₂₃	—	2	4	no	—
148	C₂²×C₂₂₃	—	2	4	no	—
149	C₂²×C₂₂₃	—	2	4	no	—
150	C₂²×C₂₂₃	—	2	4	no	—
151	C₂²×C₂₂₃	—	4	8	no	—
152	C₂²×C₂₂₃	C₂×#41	1	2	yes	—
153	C₂²×C₂₂₃	C₂⁵	1	1	yes	characteristic
154	C₄×C₂₂₃	—	2	4	no	—
155	C₄×C₂₂₃	—	2	1	no	—
156	C₄×C₂₂₃	—	2	4	no	—
157	C₄×C₂₂₃	—	2	4	no	—
158	C₄×C₂₂₃	—	2	1	no	—
159	C₄×C₂₂₃	—	2	1	no	—
160	C₄×C₂₂₃	—	2	4	no	—
161	C₄×C₂₂₃	C₂×#41	1	2	yes	—
162	C₄×C₂₂₃	C₂×#41	1	2	yes	—
163	C₄×C₂₂₃	H(5,𝔽₂)	1	2	yes	—
164	C₂×C₂₂₃	—	2	2	no	—
165	C₂×C₂₂₃	—	4	4	no	—
166	C₂×C₂₂₃	—	4	4	no	—
167	C₂×C₂₂₃	C₂²×#41	1	1	yes	characteristic
168	C₂×C₂₂₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
169	C₂×C₂₂₃	#585	1	1	yes	characteristic
170	C₂₂₃	#3506	1	1	yes	characteristic, direct product factor, Sylow
171	#3506	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
172	#559	C₂×C₂₂₃	1	4	yes	complemented
173	#556	C₂×C₂₂₃	1	2	yes	complemented
174	#555	C₂×C₂₂₃	1	2	yes	complemented
175	#553	C₂×C₂₂₃	1	4	yes	complemented
176	#550	C₂×C₂₂₃	1	2	yes	complemented
177	#547	C₂×C₂₂₃	1	2	yes	complemented
178	#538	C₂×C₂₂₃	1	4	yes	complemented
179	#535	C₂×C₂₂₃	1	2	yes	complemented
180	#533	C₂×C₂₂₃	1	1	yes	characteristic, complemented
181	#532	C₂×C₂₂₃	1	1	yes	characteristic, complemented
182	#525	C₂×C₂₂₃	1	4	yes	complemented
183	#520	C₂×C₂₂₃	1	2	yes	complemented
184	C₂×C₄×D₄	C₂×C₂₂₃	1	1	yes	characteristic, complemented
185	C₂×#41	C₂²×C₂₂₃	1	2	yes	—
186	C₂×#41	C₂²×C₂₂₃	1	4	yes	complemented
187	C₂²×D₄	C₂²×C₂₂₃	1	1	yes	characteristic
188	C₂³×C₄	C₂²×C₂₂₃	1	2	yes	—
189	C₄⋊Q₈	C₂²×C₂₂₃	1	1	yes	characteristic, complemented
190	C₄⋊Q₈	C₂²×C₂₂₃	1	1	yes	characteristic, complemented
191	C₄⋊Q₈	C₂²×C₂₂₃	1	1	yes	characteristic, complemented
192	#127	C₂²×C₂₂₃	1	4	yes	—
193	#127	C₂²×C₂₂₃	1	1	yes	characteristic, complemented
194	#126	C₂²×C₂₂₃	1	8	yes	complemented
195	#125	C₂²×C₂₂₃	1	2	yes	complemented
196	#124	C₂²×C₂₂₃	1	4	yes	—
197	#124	C₂²×C₂₂₃	1	2	yes	complemented
198	#124	C₂²×C₂₂₃	1	8	yes	complemented
199	#123	C₂²×C₂₂₃	1	8	yes	complemented
200	#123	C₂²×C₂₂₃	1	8	yes	complemented
201	C₂²⋊Q₈	C₂²×C₂₂₃	1	2	yes	—
202	C₂²⋊Q₈	C₂²×C₂₂₃	1	2	yes	—
203	C₂²⋊Q₈	C₂²×C₂₂₃	1	4	yes	complemented
204	#121	C₂²×C₂₂₃	1	2	yes	—
205	#121	C₂²×C₂₂₃	1	2	yes	—
206	#121	C₂²×C₂₂₃	1	4	yes	—
207	#121	C₂²×C₂₂₃	1	4	yes	—
208	#121	C₂²×C₂₂₃	1	4	yes	complemented
209	#121	C₂²×C₂₂₃	1	8	yes	complemented
210	#120	C₂²×C₂₂₃	1	8	yes	—
211	C₄×Q₈	C₂²×C₂₂₃	1	4	yes	complemented
212	C₄×Q₈	C₂²×C₂₂₃	1	2	yes	complemented
213	C₄×Q₈	C₂²×C₂₂₃	1	2	yes	complemented
214	C₄×D₄	C₂²×C₂₂₃	1	2	yes	—
215	C₄×D₄	C₂²×C₂₂₃	1	2	yes	—
216	C₄×D₄	C₂²×C₂₂₃	1	2	yes	—
217	C₄×D₄	C₂²×C₂₂₃	1	2	yes	—
218	C₄×D₄	C₂²×C₂₂₃	1	8	yes	—
219	C₄×D₄	C₂²×C₂₂₃	1	4	yes	complemented
220	C₄×D₄	C₂²×C₂₂₃	1	8	yes	complemented
221	C₄×D₄	C₂²×C₂₂₃	1	2	yes	complemented
222	C₄×D₄	C₂²×C₂₂₃	1	2	yes	complemented
223	#117	C₂²×C₂₂₃	1	2	yes	—
224	#117	C₂²×C₂₂₃	1	4	yes	complemented
225	#117	C₂²×C₂₂₃	1	4	yes	complemented
226	C₂×C₄⋊C₄	C₂²×C₂₂₃	1	1	yes	characteristic
227	C₂×C₄⋊C₄	C₂²×C₂₂₃	1	4	yes	complemented
228	C₂×C₂²⋊C₄	C₂²×C₂₂₃	1	2	yes	—
229	C₂×C₄²	C₂²×C₂₂₃	1	1	yes	characteristic
230	C₂⁴	C₂³×C₂₂₃	1	2	yes	—
231	#41	—	2	8	no	—
232	#41	—	4	8	no	—
233	C₂×Q₈	C₂³×C₂₂₃	1	2	yes	—
234	C₂×Q₈	C₂³×C₂₂₃	1	2	yes	—
235	C₂×Q₈	C₂³×C₂₂₃	1	2	yes	—
236	C₂×D₄	—	2	4	no	—
237	C₂×D₄	C₂³×C₂₂₃	1	4	yes	—
238	C₂×D₄	C₂³×C₂₂₃	1	2	yes	—
239	C₂×D₄	C₂³×C₂₂₃	1	2	yes	—
240	C₂×D₄	C₂³×C₂₂₃	1	2	yes	—
241	C₂×D₄	C₂³×C₂₂₃	1	2	yes	—
242	C₂×D₄	C₂³×C₂₂₃	1	8	yes	—
243	C₂×D₄	C₂³×C₂₂₃	1	2	yes	—
244	C₂×D₄	C₂³×C₂₂₃	1	8	yes	—
245	C₂²×C₄	—	4	4	no	—
246	C₂²×C₄	C₂³×C₂₂₃	1	1	yes	characteristic
247	C₂²×C₄	C₂³×C₂₂₃	1	4	yes	—
248	C₂²×C₄	C₂³×C₂₂₃	1	4	yes	—
249	C₂²×C₄	C₂³×C₂₂₃	1	1	yes	characteristic
250	C₂²×C₄	C₂³×C₂₂₃	1	2	yes	—
251	C₂²×C₄	C₂³×C₂₂₃	1	8	yes	—
252	C₂²×C₄	C₂³×C₂₂₃	1	4	yes	—
253	C₂²×C₄	C₂³×C₂₂₃	1	1	yes	characteristic
254	C₄⋊C₄	C₂³×C₂₂₃	1	2	yes	—
255	C₄⋊C₄	C₂³×C₂₂₃	1	2	yes	—
256	C₄⋊C₄	C₂³×C₂₂₃	1	2	yes	—
257	C₄⋊C₄	C₂³×C₂₂₃	1	8	yes	—
258	C₄⋊C₄	C₂³×C₂₂₃	1	2	yes	—
259	C₄⋊C₄	C₂³×C₂₂₃	1	2	yes	—
260	C₄⋊C₄	C₂³×C₂₂₃	1	2	yes	—
261	C₄⋊C₄	C₂³×C₂₂₃	1	4	yes	—
262	C₄⋊C₄	C₂³×C₂₂₃	1	2	yes	—
263	C₄⋊C₄	C₂³×C₂₂₃	1	4	yes	—
264	C₄⋊C₄	C₂³×C₂₂₃	1	2	yes	—
265	C₂²⋊C₄	C₂³×C₂₂₃	1	4	yes	—
266	C₂²⋊C₄	C₂³×C₂₂₃	1	4	yes	—
267	C₂²⋊C₄	C₂³×C₂₂₃	1	4	yes	—
268	C₂²⋊C₄	C₂³×C₂₂₃	1	8	yes	—
269	C₂²⋊C₄	C₂³×C₂₂₃	1	8	yes	—
270	C₂²⋊C₄	C₂³×C₂₂₃	1	4	yes	—
271	C₂²⋊C₄	C₂³×C₂₂₃	1	8	yes	—
272	C₂²⋊C₄	C₂³×C₂₂₃	1	4	yes	—
273	C₄²	C₂³×C₂₂₃	1	1	yes	characteristic
274	C₄²	C₂³×C₂₂₃	1	1	yes	characteristic
275	C₄²	C₂³×C₂₂₃	1	1	yes	characteristic
276	C₄²	C₂³×C₂₂₃	1	1	yes	characteristic
277	C₄²	C₂³×C₂₂₃	1	4	yes	—
278	C₄²	C₂³×C₂₂₃	1	8	yes	—
279	C₂³	—	2	4	no	—
280	C₂³	—	4	2	no	—
281	C₂³	—	4	2	no	—
282	C₂³	C₂⁴×C₂₂₃	1	4	yes	—
283	C₂³	C₂⁴×C₂₂₃	1	4	yes	—
284	C₂³	C₂⁴×C₂₂₃	1	1	yes	characteristic
285	Q₈	—	2	4	no	—
286	Q₈	—	2	4	no	—
287	Q₈	—	4	2	no	—
288	D₄	—	2	8	no	—
289	D₄	—	2	4	no	—
290	D₄	—	2	4	no	—
291	D₄	—	2	4	no	—
292	D₄	—	2	4	no	—
293	D₄	—	4	8	no	—
294	D₄	—	4	2	no	—
295	D₄	—	4	8	no	—
296	C₂×C₄	—	2	8	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	8	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	4	no	—
301	C₂×C₄	—	4	8	no	—
302	C₂×C₄	—	4	1	no	—
303	C₂×C₄	#41×C₂₂₃	1	4	yes	complemented
304	C₂×C₄	C₂⁴×C₂₂₃	1	4	yes	—
305	C₂×C₄	C₂⁴×C₂₂₃	1	1	yes	characteristic
306	C₂×C₄	C₂⁴×C₂₂₃	1	4	yes	—
307	C₂×C₄	C₂⁴×C₂₂₃	1	1	yes	characteristic
308	C₂×C₄	C₂⁴×C₂₂₃	1	1	yes	characteristic
309	C₂×C₄	C₂⁴×C₂₂₃	1	4	yes	—
310	C₂×C₄	C₂⁴×C₂₂₃	1	4	yes	—
311	C₂×C₄	C₂⁴×C₂₂₃	1	1	yes	characteristic
312	C₂×C₄	C₂⁴×C₂₂₃	1	1	yes	characteristic
313	C₂×C₄	C₂⁴×C₂₂₃	1	1	yes	characteristic
314	C₂²	—	2	1	no	—
315	C₂²	—	2	1	no	—
316	C₂²	—	2	4	no	—
317	C₂²	—	2	4	no	—
318	C₂²	—	2	4	no	—
319	C₂²	—	2	4	no	—
320	C₂²	—	2	4	no	—
321	C₂²	—	2	4	no	—
322	C₂²	—	4	8	no	—
323	C₂²	C₂×#41×C₂₂₃	1	2	yes	—
324	C₂²	C₂⁵×C₂₂₃	1	1	yes	characteristic
325	C₄	—	2	4	no	—
326	C₄	—	2	1	no	—
327	C₄	—	2	4	no	—
328	C₄	—	2	4	no	—
329	C₄	—	2	1	no	—
330	C₄	—	2	1	no	—
331	C₄	—	2	4	no	—
332	C₄	C₂×#41×C₂₂₃	1	2	yes	—
333	C₄	C₂×#41×C₂₂₃	1	2	yes	—
334	C₄	H(5,𝔽₂)×C₂₂₃	1	2	yes	—
335	C₂	—	2	2	no	—
336	C₂	—	4	4	no	—
337	C₂	—	4	4	no	—
338	C₂	C₂²×#41×C₂₂₃	1	1	yes	characteristic
339	C₂	C₂×H(5,𝔽₂)×C₂₂₃	1	1	yes	characteristic
340	C₂	#585×C₂₂₃	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₂₃	C₂⁵
commutator, Frattini, cosocle	C₂²	C₂⁵×C₂₂₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. #559×C₂₂₃	C₂	semidirect
2. #556×C₂₂₃	C₂	semidirect
3. #555×C₂₂₃	C₂	semidirect
4. #553×C₂₂₃	C₂	semidirect
5. #550×C₂₂₃	C₂	semidirect
6. #547×C₂₂₃	C₂	semidirect
7. #538×C₂₂₃	C₂	semidirect
8. #535×C₂₂₃	C₂	semidirect
9. #533×C₂₂₃	C₂	semidirect
10. #532×C₂₂₃	C₂	semidirect
11. #525×C₂₂₃	C₂	semidirect
12. #520×C₂₂₃	C₂	semidirect
13. C₂×C₄×D₄×C₂₂₃	C₂	semidirect
15. C₂×#41×C₂₂₃	C₂²	semidirect
18. C₄⋊Q₈×C₂₂₃	C₂²	semidirect
19. C₄⋊Q₈×C₂₂₃	C₂²	semidirect
20. C₄⋊Q₈×C₂₂₃	C₂²	semidirect
22. #127×C₂₂₃	C₂²	semidirect
23. #126×C₂₂₃	C₂²	semidirect
24. #125×C₂₂₃	C₂²	semidirect
26. #124×C₂₂₃	C₂²	semidirect
27. #124×C₂₂₃	C₂²	semidirect
28. #123×C₂₂₃	C₂²	semidirect
29. #123×C₂₂₃	C₂²	semidirect
32. C₂²⋊Q₈×C₂₂₃	C₂²	semidirect
37. #121×C₂₂₃	C₂²	semidirect
38. #121×C₂₂₃	C₂²	semidirect
40. C₄×Q₈×C₂₂₃	C₂²	semidirect
41. C₄×Q₈×C₂₂₃	C₂²	semidirect
42. C₄×Q₈×C₂₂₃	C₂²	semidirect
48. C₄×D₄×C₂₂₃	C₂²	semidirect
49. C₄×D₄×C₂₂₃	C₂²	semidirect
50. C₄×D₄×C₂₂₃	C₂²	semidirect
51. C₄×D₄×C₂₂₃	C₂²	semidirect
53. #117×C₂₂₃	C₂²	semidirect
54. #117×C₂₂₃	C₂²	semidirect
56. C₂×C₄⋊C₄×C₂₂₃	C₂²	semidirect
132. C₂×C₄×C₂₂₃	#41	semidirect
170. C₂₂₃	#3506	direct (N × H)
171. #3506	C₂₂₃	direct (N × H)
172. #559	C₂×C₂₂₃	semidirect
173. #556	C₂×C₂₂₃	semidirect
174. #555	C₂×C₂₂₃	semidirect
175. #553	C₂×C₂₂₃	semidirect
176. #550	C₂×C₂₂₃	semidirect
177. #547	C₂×C₂₂₃	semidirect
178. #538	C₂×C₂₂₃	semidirect
179. #535	C₂×C₂₂₃	semidirect
180. #533	C₂×C₂₂₃	semidirect
181. #532	C₂×C₂₂₃	semidirect
182. #525	C₂×C₂₂₃	semidirect
183. #520	C₂×C₂₂₃	semidirect
184. C₂×C₄×D₄	C₂×C₂₂₃	semidirect
186. C₂×#41	C₂²×C₂₂₃	semidirect
189. C₄⋊Q₈	C₂²×C₂₂₃	semidirect
190. C₄⋊Q₈	C₂²×C₂₂₃	semidirect
191. C₄⋊Q₈	C₂²×C₂₂₃	semidirect
193. #127	C₂²×C₂₂₃	semidirect
194. #126	C₂²×C₂₂₃	semidirect
195. #125	C₂²×C₂₂₃	semidirect
197. #124	C₂²×C₂₂₃	semidirect
198. #124	C₂²×C₂₂₃	semidirect
199. #123	C₂²×C₂₂₃	semidirect
200. #123	C₂²×C₂₂₃	semidirect
203. C₂²⋊Q₈	C₂²×C₂₂₃	semidirect
208. #121	C₂²×C₂₂₃	semidirect
209. #121	C₂²×C₂₂₃	semidirect
211. C₄×Q₈	C₂²×C₂₂₃	semidirect
212. C₄×Q₈	C₂²×C₂₂₃	semidirect
213. C₄×Q₈	C₂²×C₂₂₃	semidirect
219. C₄×D₄	C₂²×C₂₂₃	semidirect
220. C₄×D₄	C₂²×C₂₂₃	semidirect
221. C₄×D₄	C₂²×C₂₂₃	semidirect
222. C₄×D₄	C₂²×C₂₂₃	semidirect
224. #117	C₂²×C₂₂₃	semidirect
225. #117	C₂²×C₂₂₃	semidirect
227. C₂×C₄⋊C₄	C₂²×C₂₂₃	semidirect
303. C₂×C₄	#41×C₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
#559×C₂₂₃	2
#556×C₂₂₃	2
#555×C₂₂₃	2
#553×C₂₂₃	2
#550×C₂₂₃	2
#547×C₂₂₃	2
#538×C₂₂₃	2
#535×C₂₂₃	2
#533×C₂₂₃	2
#532×C₂₂₃	2
#525×C₂₂₃	2
#520×C₂₂₃	2
C₂×C₄×D₄×C₂₂₃	2
#3506	223

Subgroups — #3506×C223 #43252521

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #3506×C₂₂₃ #43252521