Subgroups — C₂²×#66103 #13013109

Order 1280.

11462 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#4149	C₂	1	2	yes	maximal, complemented
2	C₂²×C₅×#135	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#4095	C₂	1	2	yes	maximal, complemented
4	C₂×#66103	C₂	1	48	yes	maximal, direct product factor
5	C₂²×#4065	C₂	1	2	yes	maximal, complemented
6	C₂²×#4060	C₂	1	4	yes	maximal, complemented
7	C₂²×D₅×D₈	C₂	1	2	yes	maximal, complemented
8	C₂³×D₄₀	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#4047	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×C₅×#41	C₂²	1	2	yes	complemented
11	C₂×#4149	—	2	24	no	—
12	C₂×#4149	C₂²	1	48	yes	complemented
13	C₂²×#713	C₂²	1	4	yes	complemented
14	C₂²×D₄×D₅	—	2	4	no	—
15	C₂²×D₄×D₅	C₂²	1	4	yes	complemented
16	C₂²×#709	—	2	2	no	—
17	C₂²×#709	C₂²	1	1	yes	characteristic
18	C₂³×D₂₀	—	2	2	no	—
19	C₂³×D₂₀	C₂²	1	2	yes	complemented
20	C₂×C₅×#135	C₂²	1	24	yes	complemented
21	C₂²×C₅×Q₁₆	C₂²	1	1	yes	characteristic, complemented
22	C₂²×C₅×SD₁₆	C₂²	1	2	yes	complemented
23	C₂²×C₅×D₈	C₂²	1	1	yes	characteristic, complemented
24	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic, complemented
25	C₂×#4095	C₂²	1	48	yes	complemented
26	C₂²×#688	C₂²	1	4	yes	complemented
27	C₂²×#686	C₂²	1	4	yes	complemented
28	C₂²×#681	C₂²	1	1	yes	characteristic, complemented
29	#66103	C₂²	1	256	yes	direct product factor
30	C₂×#4065	C₂²	1	48	yes	complemented
31	C₂×#4060	C₂²	1	96	yes	complemented
32	C₂×D₅×D₈	C₂²	1	48	yes	complemented
33	C₂²×D₄₀	C₂²	1	2	yes	complemented
34	C₂²×D₄₀	C₂²	1	2	yes	complemented
35	C₂²×D₄₀	C₂²	1	24	yes	complemented
36	C₂×#4047	C₂²	1	24	yes	complemented
37	C₂²×#676	C₂²	1	2	yes	complemented
38	C₂²×C₈×D₅	C₂²	1	2	yes	complemented
39	C₂²×#576	—	5	1	no	maximal, Sylow
40	C₂⁴×D₅	—	2	2	no	—
41	C₂⁴×D₅	—	2	4	no	—
42	C₂×C₅×#41	—	2	12	no	—
43	C₂×C₅×#41	C₂³	1	24	yes	complemented
44	C₂²×C₅×Q₈	C₂³	1	2	yes	—
45	C₂²×C₅×D₄	—	2	2	no	—
46	C₂²×C₅×D₄	C₂³	1	2	yes	—
47	C₂³×C₄×C₅	—	2	2	no	—
48	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
49	C₂²×C₂²⋊D₅	—	2	4	no	—
50	C₂²×C₂²⋊D₅	D₄	1	2	yes	—
51	#4149	—	2	192	no	—
52	#4149	C₂³	1	128	yes	complemented
53	C₂×#713	—	2	24	no	—
54	C₂×#713	C₂³	1	48	yes	complemented
55	C₂×D₄×D₅	—	2	96	no	—
56	C₂×D₄×D₅	—	2	24	no	—
57	C₂×D₄×D₅	C₂³	1	48	yes	complemented
58	C₂×#709	—	2	48	no	—
59	C₂×#709	D₄	1	12	yes	—
60	C₂×#709	C₂³	1	12	yes	—
61	C₂²×D₂₀	—	2	4	no	—
62	C₂²×D₂₀	—	2	2	no	—
63	C₂²×D₂₀	—	2	12	no	—
64	C₂²×D₂₀	—	2	48	no	—
65	C₂²×D₂₀	D₄	1	1	yes	characteristic
66	C₂²×D₂₀	C₂³	1	4	yes	—
67	C₂²×D₂₀	C₂³	1	24	yes	complemented
68	C₂²×C₄×D₅	—	2	4	no	—
69	C₂²×C₄×D₅	C₂³	1	2	yes	—
70	C₂²×Dic₁₀	D₄	1	1	yes	characteristic
71	C₅×#135	C₂³	1	64	yes	complemented
72	C₂×C₅×Q₁₆	C₂³	1	12	yes	complemented
73	C₂×C₅×SD₁₆	C₂³	1	24	yes	complemented
74	C₂×C₅×D₈	C₂³	1	12	yes	complemented
75	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
76	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
77	C₂²×C₅×C₈	C₂³	1	12	yes	complemented
78	#4095	C₂³	1	128	yes	complemented
79	C₂×#688	C₂³	1	48	yes	complemented
80	C₂×#686	C₂³	1	48	yes	complemented
81	C₂×#681	C₂³	1	12	yes	complemented
82	C₂²×#168	C₂³	1	2	yes	—
83	#4065	C₂³	1	128	yes	complemented
84	#4060	C₂³	1	256	yes	complemented
85	D₅×D₈	C₂³	1	128	yes	complemented
86	C₂×D₄₀	C₂³	1	24	yes	complemented
87	C₂×D₄₀	C₂³	1	24	yes	complemented
88	C₂×D₄₀	C₂³	1	64	yes	complemented
89	#4047	C₂³	1	64	yes	complemented
90	C₂×#676	C₂³	1	24	yes	complemented
91	C₂×C₈×D₅	C₂³	1	24	yes	complemented
92	C₂²×H(5,𝔽₂)	—	5	2	no	—
93	C₂×#576	—	5	48	no	—
94	C₂²×AGL(1,ℤ/8)	—	5	2	no	—
95	C₂²×AGL(1,ℤ/8)	—	5	4	no	—
96	C₂²×#135	—	5	2	no	—
97	C₂²×#135	D₅	1	1	yes	characteristic, complemented
98	C₂³×D₈	—	5	2	no	—
99	C₂³×D₈	—	5	1	no	—
100	C₂²×#131	—	5	1	no	—
101	C₂⁴×C₅	—	2	2	no	—
102	C₂⁴×C₅	C₂×D₄	1	1	yes	characteristic
103	C₂³×D₅	—	2	4	no	—
104	C₂³×D₅	—	2	24	no	—
105	C₂³×D₅	—	2	48	no	—
106	C₂³×D₅	—	8	16	no	—
107	C₂³×D₅	—	8	8	no	—
108	C₂³×D₅	C₂×D₄	1	2	yes	—
109	C₅×#41	—	2	48	no	—
110	C₅×#41	C₂⁴	1	32	yes	complemented
111	C₂×C₅×Q₈	—	2	6	no	—
112	C₂×C₅×Q₈	C₂⁴	1	12	yes	—
113	C₂×C₅×D₄	—	2	24	no	—
114	C₂×C₅×D₄	—	2	6	no	—
115	C₂×C₅×D₄	C₂⁴	1	12	yes	—
116	C₂²×C₄×C₅	—	2	2	no	—
117	C₂²×C₄×C₅	—	2	24	no	—
118	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
119	C₂²×C₄×C₅	C₂×D₄	1	6	yes	—
120	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
121	C₂²×C₄×C₅	C₂⁴	1	6	yes	—
122	C₂×C₂²⋊D₅	—	2	48	no	—
123	C₂×C₂²⋊D₅	C₂×D₄	1	24	yes	—
124	C₂²×Dic₅	C₂×D₄	1	2	yes	—
125	#713	—	2	96	no	—
126	#713	C₂⁴	1	64	yes	complemented
127	D₄×D₅	—	2	256	no	—
128	D₄×D₅	—	2	96	no	—
129	D₄×D₅	C₂⁴	1	64	yes	complemented
130	#709	—	2	128	no	—
131	#709	C₂×D₄	1	48	yes	—
132	#709	C₂⁴	1	16	yes	—
133	C₂×D₂₀	—	2	48	no	—
134	C₂×D₂₀	—	2	24	no	—
135	C₂×D₂₀	—	2	12	no	—
136	C₂×D₂₀	—	2	48	no	—
137	C₂×D₂₀	—	2	128	no	—
138	C₂×D₂₀	C₂×D₄	1	12	yes	—
139	C₂×D₂₀	C₂⁴	1	24	yes	—
140	C₂×D₂₀	C₂⁴	1	32	yes	complemented
141	C₂×C₄×D₅	—	2	48	no	—
142	C₂×C₄×D₅	C₂×D₄	1	12	yes	—
143	C₂×C₄×D₅	C₂⁴	1	12	yes	—
144	C₂×Dic₁₀	C₂×D₄	1	12	yes	—
145	C₅×Q₁₆	C₂⁴	1	16	yes	complemented
146	C₅×SD₁₆	C₂⁴	1	32	yes	complemented
147	C₅×D₈	C₂⁴	1	16	yes	complemented
148	C₂×C₅×C₈	C₂⁴	1	6	yes	—
149	C₂×C₅×C₈	C₂⁴	1	6	yes	—
150	C₂×C₅×C₈	C₂⁴	1	16	yes	complemented
151	#688	C₂⁴	1	64	yes	complemented
152	#686	C₂⁴	1	64	yes	complemented
153	#681	C₂⁴	1	16	yes	complemented
154	C₂×#168	C₂⁴	1	12	yes	—
155	D₄₀	C₂⁴	1	32	yes	complemented
156	D₄₀	C₂⁴	1	32	yes	complemented
157	#676	C₂⁴	1	32	yes	complemented
158	C₈×D₅	C₂⁴	1	32	yes	complemented
159	C₂×H(5,𝔽₂)	—	5	48	no	—
160	C₂×H(5,𝔽₂)	—	10	24	no	—
161	C₂²×#41	—	5	1	no	—
162	C₂²×#41	—	5	4	no	—
163	C₂²×#41	—	10	2	no	—
164	C₂²×#41	C₂×D₅	1	2	yes	complemented
165	C₂³×D₄	—	5	4	no	—
166	C₂³×D₄	—	5	2	no	—
167	C₂³×D₄	—	10	2	no	—
168	C₂³×D₄	—	10	4	no	—
169	#576	—	5	256	no	—
170	C₂×AGL(1,ℤ/8)	—	5	48	no	—
171	C₂×AGL(1,ℤ/8)	—	5	96	no	—
172	C₂×#135	—	5	48	no	—
173	C₂×#135	C₂×D₅	1	24	yes	complemented
174	C₂²×Q₁₆	C₂×D₅	1	1	yes	characteristic, complemented
175	C₂²×SD₁₆	—	5	4	no	—
176	C₂²×SD₁₆	C₂×D₅	1	2	yes	complemented
177	C₂²×D₈	—	5	4	no	—
178	C₂²×D₈	—	5	48	no	—
179	C₂²×D₈	—	5	2	no	—
180	C₂²×D₈	—	5	2	no	—
181	C₂²×D₈	—	5	24	no	—
182	C₂²×D₈	C₂×D₅	1	1	yes	characteristic, complemented
183	C₂×#131	—	5	24	no	—
184	C₂²×M₁₆	—	5	1	no	—
185	C₂²×M₁₆	—	5	2	no	—
186	C₂³×C₈	—	5	2	no	—
187	C₂³×C₈	C₂×D₅	1	1	yes	characteristic, complemented
188	C₂³×C₅	—	2	4	no	—
189	C₂³×C₅	—	2	12	no	—
190	C₂³×C₅	—	4	8	no	—
191	C₂³×C₅	C₂²×D₄	1	6	yes	—
192	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
193	C₂²×D₅	—	2	8	no	—
194	C₂²×D₅	—	2	24	no	—
195	C₂²×D₅	—	2	32	no	—
196	C₂²×D₅	—	2	64	no	—
197	C₂²×D₅	—	4	16	no	—
198	C₂²×D₅	—	8	96	no	—
199	C₂²×D₅	—	8	48	no	—
200	C₂²×D₅	C₂²×D₄	1	12	yes	—
201	C₅×Q₈	—	2	12	no	—
202	C₅×Q₈	C₂⁵	1	8	yes	—
203	C₅×D₄	—	2	32	no	—
204	C₅×D₄	—	2	12	no	—
205	C₅×D₄	C₂⁵	1	8	yes	—
206	C₂×C₄×C₅	—	2	12	no	—
207	C₂×C₄×C₅	—	2	32	no	—
208	C₂×C₄×C₅	C₂²×D₄	1	6	yes	—
209	C₂×C₄×C₅	C₂²×D₄	1	3	yes	—
210	C₂×C₄×C₅	C₂²×D₄	1	12	yes	—
211	C₂×C₄×C₅	C₂⁵	1	3	yes	—
212	C₂×C₄×C₅	C₂⁵	1	4	yes	—
213	C₂²⋊D₅	—	2	64	no	—
214	C₂²⋊D₅	C₂²×D₄	1	32	yes	—
215	C₂×Dic₅	C₂²×D₄	1	12	yes	—
216	D₂₀	—	2	64	no	—
217	D₂₀	—	2	32	no	—
218	D₂₀	—	2	24	no	—
219	D₂₀	C₂²×D₄	1	16	yes	—
220	D₂₀	C₂⁵	1	16	yes	—
221	C₄×D₅	—	2	64	no	—
222	C₄×D₅	C₂²×D₄	1	24	yes	—
223	C₄×D₅	C₂⁵	1	8	yes	—
224	Dic₁₀	C₂²×D₄	1	16	yes	—
225	C₅×C₈	C₂⁵	1	4	yes	—
226	C₅×C₈	C₂⁵	1	4	yes	—
227	#168	C₂⁵	1	8	yes	—
228	C₂⁵	—	10	4	no	—
229	C₂⁵	—	10	2	no	—
230	H(5,𝔽₂)	—	5	128	no	—
231	H(5,𝔽₂)	—	10	192	no	—
232	C₂×#41	—	2	12	no	—
233	C₂×#41	—	5	12	no	—
234	C₂×#41	—	5	12	no	—
235	C₂×#41	—	5	48	no	—
236	C₂×#41	—	10	48	no	—
237	C₂×#41	—	10	24	no	—
238	C₂×#41	C₂²×D₅	1	24	yes	complemented
239	C₂²×Q₈	—	5	1	no	—
240	C₂²×Q₈	C₂²×D₅	1	2	yes	—
241	C₂²×D₄	—	2	2	no	—
242	C₂²×D₄	—	5	2	no	—
243	C₂²×D₄	—	5	48	no	—
244	C₂²×D₄	—	5	1	no	—
245	C₂²×D₄	—	5	4	no	—
246	C₂²×D₄	—	5	24	no	—
247	C₂²×D₄	—	10	48	no	—
248	C₂²×D₄	—	10	4	no	—
249	C₂²×D₄	—	10	24	no	—
250	C₂²×D₄	—	10	4	no	—
251	C₂²×D₄	—	10	96	no	—
252	C₂²×D₄	—	10	2	no	—
253	C₂²×D₄	—	10	12	no	—
254	C₂²×D₄	C₂²×D₅	1	2	yes	—
255	C₂³×C₄	—	2	2	no	—
256	C₂³×C₄	—	5	2	no	—
257	C₂³×C₄	—	10	4	no	—
258	C₂³×C₄	C₂²×D₅	1	1	yes	characteristic
259	AGL(1,ℤ/8)	—	5	128	no	—
260	AGL(1,ℤ/8)	—	5	256	no	—
261	#135	—	5	128	no	—
262	#135	C₂²×D₅	1	64	yes	complemented
263	C₂×Q₁₆	C₂²×D₅	1	12	yes	complemented
264	C₂×SD₁₆	—	5	48	no	—
265	C₂×SD₁₆	C₂²×D₅	1	24	yes	complemented
266	C₂×D₈	—	5	48	no	—
267	C₂×D₈	—	5	128	no	—
268	C₂×D₈	—	5	24	no	—
269	C₂×D₈	—	5	24	no	—
270	C₂×D₈	—	5	64	no	—
271	C₂×D₈	C₂²×D₅	1	12	yes	complemented
272	#131	—	5	64	no	—
273	C₂×M₁₆	—	5	12	no	—
274	C₂×M₁₆	—	5	24	no	—
275	C₂²×C₈	—	5	2	no	—
276	C₂²×C₈	—	5	24	no	—
277	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
278	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
279	C₂²×C₈	C₂²×D₅	1	12	yes	complemented
280	C₂²×C₅	—	2	12	no	—
281	C₂²×C₅	—	2	8	no	—
282	C₂²×C₅	—	4	24	no	—
283	C₂²×C₅	#576	1	4	yes	complemented
284	C₂²×C₅	C₂³×D₄	1	4	yes	—
285	C₂²×C₅	C₂³×D₄	1	3	yes	—
286	C₂×D₅	—	2	24	no	—
287	C₂×D₅	—	2	16	no	—
288	C₂×D₅	—	4	48	no	—
289	C₂×D₅	—	8	64	no	—
290	C₂×D₅	—	8	32	no	—
291	C₂×D₅	C₂³×D₄	1	8	yes	—
292	C₄×C₅	—	2	8	no	—
293	C₄×C₅	C₂³×D₄	1	4	yes	—
294	C₄×C₅	C₂³×D₄	1	3	yes	—
295	C₄×C₅	C₂⁶	1	1	yes	characteristic
296	Dic₅	C₂³×D₄	1	8	yes	—
297	C₂⁴	—	2	2	no	—
298	C₂⁴	—	5	2	no	—
299	C₂⁴	—	10	48	no	—
300	C₂⁴	—	10	4	no	—
301	C₂⁴	—	10	24	no	—
302	C₂⁴	—	40	8	no	—
303	C₂⁴	—	40	16	no	—
304	C₂⁴	D₄×D₅	1	1	yes	characteristic
305	#41	—	2	48	no	—
306	#41	—	5	48	no	—
307	#41	—	5	16	no	—
308	#41	—	5	64	no	—
309	#41	—	10	128	no	—
310	#41	—	10	96	no	—
311	#41	C₂³×D₅	1	32	yes	complemented
312	C₂×Q₈	—	2	6	no	—
313	C₂×Q₈	—	5	12	no	—
314	C₂×Q₈	C₂³×D₅	1	12	yes	—
315	C₂×D₄	—	2	24	no	—
316	C₂×D₄	—	2	6	no	—
317	C₂×D₄	—	5	24	no	—
318	C₂×D₄	—	5	64	no	—
319	C₂×D₄	—	5	12	no	—
320	C₂×D₄	—	5	24	no	—
321	C₂×D₄	—	5	32	no	—
322	C₂×D₄	—	10	128	no	—
323	C₂×D₄	—	10	48	no	—
324	C₂×D₄	—	10	96	no	—
325	C₂×D₄	—	10	48	no	—
326	C₂×D₄	—	10	256	no	—
327	C₂×D₄	—	10	24	no	—
328	C₂×D₄	—	10	12	no	—
329	C₂×D₄	—	10	48	no	—
330	C₂×D₄	C₂³×D₅	1	12	yes	—
331	C₂²×C₄	—	2	2	no	—
332	C₂²×C₄	—	2	24	no	—
333	C₂²×C₄	—	5	2	no	—
334	C₂²×C₄	—	5	12	no	—
335	C₂²×C₄	—	5	12	no	—
336	C₂²×C₄	—	10	48	no	—
337	C₂²×C₄	D₄×D₅	1	6	yes	—
338	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
339	C₂²×C₄	C₂³×D₅	1	6	yes	—
340	C₂²×C₄	C₂³×D₅	1	1	yes	characteristic
341	Q₁₆	C₂³×D₅	1	16	yes	complemented
342	SD₁₆	—	5	64	no	—
343	SD₁₆	C₂³×D₅	1	32	yes	complemented
344	D₈	—	5	64	no	—
345	D₈	—	5	32	no	—
346	D₈	—	5	32	no	—
347	D₈	C₂³×D₅	1	16	yes	complemented
348	M₁₆	—	5	16	no	—
349	M₁₆	—	5	32	no	—
350	C₂×C₈	—	5	12	no	—
351	C₂×C₈	—	5	32	no	—
352	C₂×C₈	C₂³×D₅	1	6	yes	—
353	C₂×C₈	C₂³×D₅	1	6	yes	—
354	C₂×C₈	C₂³×D₅	1	16	yes	complemented
355	C₂×C₅	—	2	4	no	—
356	C₂×C₅	—	4	8	no	—
357	C₂×C₅	C₂×#576	1	6	yes	complemented
358	C₂×C₅	C₂⁴×D₄	1	1	yes	characteristic
359	D₅	—	2	8	no	—
360	D₅	—	4	16	no	—
361	C₂³	—	2	4	no	—
362	C₂³	—	2	12	no	—
363	C₂³	—	4	8	no	—
364	C₂³	—	5	12	no	—
365	C₂³	—	10	64	no	—
366	C₂³	—	10	8	no	—
367	C₂³	—	10	24	no	—
368	C₂³	—	10	32	no	—
369	C₂³	—	20	16	no	—
370	C₂³	—	40	48	no	—
371	C₂³	—	40	96	no	—
372	C₂³	C₂×D₄×D₅	1	6	yes	—
373	C₂³	C₂×D₄×D₅	1	1	yes	characteristic
374	Q₈	—	2	12	no	—
375	Q₈	—	5	16	no	—
376	Q₈	C₂⁴×D₅	1	8	yes	—
377	D₄	—	2	32	no	—
378	D₄	—	2	12	no	—
379	D₄	—	5	32	no	—
380	D₄	—	5	16	no	—
381	D₄	—	5	16	no	—
382	D₄	—	10	64	no	—
383	D₄	—	10	64	no	—
384	D₄	—	10	32	no	—
385	D₄	—	10	24	no	—
386	D₄	C₂⁴×D₅	1	8	yes	—
387	C₂×C₄	—	2	12	no	—
388	C₂×C₄	—	2	32	no	—
389	C₂×C₄	—	5	12	no	—
390	C₂×C₄	—	5	24	no	—
391	C₂×C₄	—	5	8	no	—
392	C₂×C₄	—	10	64	no	—
393	C₂×C₄	C₂×D₄×D₅	1	6	yes	—
394	C₂×C₄	C₂×D₄×D₅	1	3	yes	—
395	C₂×C₄	C₂×D₄×D₅	1	12	yes	—
396	C₂×C₄	C₂⁴×D₅	1	3	yes	—
397	C₂×C₄	C₂⁴×D₅	1	4	yes	—
398	C₈	—	5	8	no	—
399	C₈	C₂⁴×D₅	1	4	yes	—
400	C₈	C₂⁴×D₅	1	4	yes	—
401	C₅	C₂²×#576	1	1	yes	characteristic, complemented, Sylow
402	C₂²	—	2	12	no	—
403	C₂²	—	2	8	no	—
404	C₂²	—	4	24	no	—
405	C₂²	—	5	8	no	—
406	C₂²	—	10	24	no	—
407	C₂²	—	10	16	no	—
408	C₂²	—	20	48	no	—
409	C₂²	—	40	32	no	—
410	C₂²	—	40	64	no	—
411	C₂²	#66103	1	4	yes	direct product factor
412	C₂²	C₂²×D₄×D₅	1	4	yes	—
413	C₂²	C₂²×D₄×D₅	1	3	yes	—
414	C₄	—	2	8	no	—
415	C₄	—	5	8	no	—
416	C₄	C₂²×D₄×D₅	1	4	yes	—
417	C₄	C₂²×D₄×D₅	1	3	yes	—
418	C₄	C₂⁵×D₅	1	1	yes	characteristic
419	C₂	—	2	4	no	—
420	C₂	—	4	8	no	—
421	C₂	—	10	8	no	—
422	C₂	—	20	16	no	—
423	C₂	C₂×#66103	1	6	yes	direct product factor
424	C₂	C₂³×D₄×D₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₄×D₅
commutator, cosocle	C₄×C₅	C₂⁶
Frattini	C₄	C₂⁵×D₅
Fitting	C₂²×C₅×#135	C₂
socle	C₂³×C₅	C₂²×D₄
hypercenter	C₂²×#135	D₅
hypocenter	C₅	C₂²×#576
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×#135

Complements

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

N	H ≅ G/N	type
1. C₂²×#4149	C₂	semidirect
2. C₂²×C₅×#135	C₂	semidirect
3. C₂²×#4095	C₂	semidirect
4. C₂×#66103	C₂	direct (N × H)
5. C₂²×#4065	C₂	semidirect
6. C₂²×#4060	C₂	semidirect
7. C₂²×D₅×D₈	C₂	semidirect
8. C₂³×D₄₀	C₂	semidirect
9. C₂²×#4047	C₂	semidirect
10. C₂²×C₅×#41	C₂²	semidirect
12. C₂×#4149	C₂²	semidirect
13. C₂²×#713	C₂²	semidirect
15. C₂²×D₄×D₅	C₂²	semidirect
19. C₂³×D₂₀	C₂²	semidirect
20. C₂×C₅×#135	C₂²	semidirect
21. C₂²×C₅×Q₁₆	C₂²	semidirect
22. C₂²×C₅×SD₁₆	C₂²	semidirect
23. C₂²×C₅×D₈	C₂²	semidirect
24. C₂³×C₅×C₈	C₂²	semidirect
25. C₂×#4095	C₂²	semidirect
26. C₂²×#688	C₂²	semidirect
27. C₂²×#686	C₂²	semidirect
28. C₂²×#681	C₂²	semidirect
29. #66103	C₂²	direct (N × H)
30. C₂×#4065	C₂²	semidirect
31. C₂×#4060	C₂²	semidirect
32. C₂×D₅×D₈	C₂²	semidirect
33. C₂²×D₄₀	C₂²	semidirect
34. C₂²×D₄₀	C₂²	semidirect
35. C₂²×D₄₀	C₂²	semidirect
36. C₂×#4047	C₂²	semidirect
37. C₂²×#676	C₂²	semidirect
38. C₂²×C₈×D₅	C₂²	semidirect
43. C₂×C₅×#41	C₂³	semidirect
52. #4149	C₂³	semidirect
54. C₂×#713	C₂³	semidirect
57. C₂×D₄×D₅	C₂³	semidirect
67. C₂²×D₂₀	C₂³	semidirect
71. C₅×#135	C₂³	semidirect
72. C₂×C₅×Q₁₆	C₂³	semidirect
73. C₂×C₅×SD₁₆	C₂³	semidirect
74. C₂×C₅×D₈	C₂³	semidirect
77. C₂²×C₅×C₈	C₂³	semidirect
78. #4095	C₂³	semidirect
79. C₂×#688	C₂³	semidirect
80. C₂×#686	C₂³	semidirect
81. C₂×#681	C₂³	semidirect
83. #4065	C₂³	semidirect
84. #4060	C₂³	semidirect
85. D₅×D₈	C₂³	semidirect
86. C₂×D₄₀	C₂³	semidirect
87. C₂×D₄₀	C₂³	semidirect
88. C₂×D₄₀	C₂³	semidirect
89. #4047	C₂³	semidirect
90. C₂×#676	C₂³	semidirect
91. C₂×C₈×D₅	C₂³	semidirect
97. C₂²×#135	D₅	semidirect
110. C₅×#41	C₂⁴	semidirect
126. #713	C₂⁴	semidirect
129. D₄×D₅	C₂⁴	semidirect
140. C₂×D₂₀	C₂⁴	semidirect
145. C₅×Q₁₆	C₂⁴	semidirect
146. C₅×SD₁₆	C₂⁴	semidirect
147. C₅×D₈	C₂⁴	semidirect
150. C₂×C₅×C₈	C₂⁴	semidirect
151. #688	C₂⁴	semidirect
152. #686	C₂⁴	semidirect
153. #681	C₂⁴	semidirect
155. D₄₀	C₂⁴	semidirect
156. D₄₀	C₂⁴	semidirect
157. #676	C₂⁴	semidirect
158. C₈×D₅	C₂⁴	semidirect
164. C₂²×#41	C₂×D₅	semidirect
173. C₂×#135	C₂×D₅	semidirect
174. C₂²×Q₁₆	C₂×D₅	semidirect
176. C₂²×SD₁₆	C₂×D₅	semidirect
182. C₂²×D₈	C₂×D₅	semidirect
187. C₂³×C₈	C₂×D₅	semidirect
238. C₂×#41	C₂²×D₅	semidirect
262. #135	C₂²×D₅	semidirect
263. C₂×Q₁₆	C₂²×D₅	semidirect
265. C₂×SD₁₆	C₂²×D₅	semidirect
271. C₂×D₈	C₂²×D₅	semidirect
279. C₂²×C₈	C₂²×D₅	semidirect
283. C₂²×C₅	#576	semidirect
311. #41	C₂³×D₅	semidirect
341. Q₁₆	C₂³×D₅	semidirect
343. SD₁₆	C₂³×D₅	semidirect
347. D₈	C₂³×D₅	semidirect
354. C₂×C₈	C₂³×D₅	semidirect
357. C₂×C₅	C₂×#576	semidirect
401. C₅	C₂²×#576	semidirect
411. C₂²	#66103	direct (N × H)
423. C₂	C₂×#66103	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×#4149	2
C₂²×C₅×#135	2
C₂²×#4095	2
C₂×#66103	2
C₂²×#4065	2
C₂²×#4060	2
C₂²×D₅×D₈	2
C₂³×D₄₀	2
C₂²×#4047	2
C₂²×#576	5

Subgroups — C22×#66103 #13013109

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#66103 #13013109