Subgroups — C₂×C₄₁×#3291 #23837303

Order 10496.

2682 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#577	C₂	1	4	yes	maximal, complemented
2	C₄₁×#3291	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₄₁×#492	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₄₁×#471	C₂	1	4	yes	maximal, complemented
5	C₂×C₄₁×#456	C₂	1	2	yes	maximal, complemented
6	C₂×C₄₁×#453	C₂	1	2	yes	maximal, complemented
7	C₂×C₄₁×#443	C₂	1	2	yes	maximal, complemented
8	C₂×2₋¹⁺⁴×C₄₁	C₂²	1	2	yes	—
9	C₂×H(5,𝔽₂)×C₄₁	C₂²	1	2	yes	complemented
10	C₄₁×#577	—	2	16	no	—
11	C₄₁×#577	C₂²	1	32	yes	complemented
12	C₂×#137×C₄₁	—	2	4	no	—
13	C₂×#137×C₄₁	C₂²	1	4	yes	complemented
14	C₂×AGL(1,ℤ/8)×C₄₁	—	2	4	no	—
15	C₂×AGL(1,ℤ/8)×C₄₁	C₂²	1	4	yes	—
16	C₂×#135×C₄₁	—	2	4	no	—
17	C₂×#135×C₄₁	C₂²	1	2	yes	—
18	C₂²×SD₁₆×C₄₁	—	2	4	no	—
19	C₂²×SD₁₆×C₄₁	C₂²	1	4	yes	complemented
20	C₂×#131×C₄₁	C₂²	1	4	yes	complemented
21	C₂×C₄⋊Q₈×C₄₁	C₂²	1	1	yes	characteristic, complemented
22	C₂×#127×C₄₁	C₂²	1	1	yes	characteristic
23	C₄₁×#492	C₂²	1	8	yes	complemented
24	C₄₁×#471	C₂²	1	32	yes	complemented
25	C₄₁×#456	C₂²	1	16	yes	complemented
26	C₄₁×#453	C₂²	1	16	yes	complemented
27	C₄₁×#443	C₂²	1	16	yes	complemented
28	C₂×#108×C₄₁	C₂²	1	2	yes	complemented
29	C₂×#104×C₄₁	C₂²	1	4	yes	complemented
30	C₂×#101×C₄₁	C₂²	1	2	yes	complemented
31	C₂×#100×C₄₁	C₂²	1	2	yes	complemented
32	C₂×C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic, complemented
33	2₋¹⁺⁴×C₄₁	—	2	8	no	—
34	2₋¹⁺⁴×C₄₁	—	4	2	no	—
35	2₋¹⁺⁴×C₄₁	C₂³	1	8	yes	—
36	H(5,𝔽₂)×C₄₁	—	2	8	no	—
37	H(5,𝔽₂)×C₄₁	—	4	2	no	—
38	H(5,𝔽₂)×C₄₁	C₂³	1	8	yes	complemented
39	C₂×#41×C₄₁	—	2	4	no	—
40	C₂×#41×C₄₁	—	2	4	no	—
41	C₂×#41×C₄₁	—	4	2	no	—
42	C₂×#41×C₄₁	C₂³	1	4	yes	—
43	C₂²×Q₈×C₄₁	—	2	4	no	—
44	C₂²×Q₈×C₄₁	C₂³	1	2	yes	—
45	C₂²×D₄×C₄₁	—	2	4	no	—
46	C₂²×D₄×C₄₁	—	4	1	no	—
47	C₂²×D₄×C₄₁	—	4	2	no	—
48	C₂²×D₄×C₄₁	C₂³	1	2	yes	—
49	#137×C₄₁	—	2	8	no	—
50	#137×C₄₁	—	2	32	no	—
51	#137×C₄₁	C₂³	1	16	yes	complemented
52	AGL(1,ℤ/8)×C₄₁	—	2	8	no	—
53	AGL(1,ℤ/8)×C₄₁	—	2	32	no	—
54	AGL(1,ℤ/8)×C₄₁	C₂³	1	16	yes	—
55	#135×C₄₁	—	2	32	no	—
56	#135×C₄₁	D₄	1	8	yes	complemented
57	#135×C₄₁	C₂³	1	8	yes	—
58	C₂×Q₁₆×C₄₁	—	2	4	no	—
59	C₂×Q₁₆×C₄₁	D₄	1	2	yes	complemented
60	C₂×SD₁₆×C₄₁	—	2	4	no	—
61	C₂×SD₁₆×C₄₁	—	2	4	no	—
62	C₂×SD₁₆×C₄₁	—	2	32	no	—
63	C₂×SD₁₆×C₄₁	—	2	4	no	—
64	C₂×SD₁₆×C₄₁	—	2	4	no	—
65	C₂×SD₁₆×C₄₁	—	2	8	no	—
66	C₂×SD₁₆×C₄₁	D₄	1	4	yes	complemented
67	C₂×SD₁₆×C₄₁	C₂³	1	16	yes	complemented
68	C₂×D₈×C₄₁	—	2	4	no	—
69	C₂×D₈×C₄₁	D₄	1	2	yes	complemented
70	#131×C₄₁	—	2	8	no	—
71	#131×C₄₁	C₂³	1	16	yes	complemented
72	C₂×M₁₆×C₄₁	—	2	4	no	—
73	C₂×M₁₆×C₄₁	C₂³	1	4	yes	—
74	C₂²×C₈×C₄₁	—	2	4	no	—
75	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
76	C₄⋊Q₈×C₄₁	—	4	1	no	—
77	C₄⋊Q₈×C₄₁	C₂³	1	4	yes	complemented
78	#127×C₄₁	—	4	1	no	—
79	#127×C₄₁	C₂³	1	4	yes	—
80	C₂×C₄⋊C₄×C₄₁	—	4	1	no	—
81	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
82	#108×C₄₁	C₂³	1	8	yes	complemented
83	#104×C₄₁	C₂³	1	16	yes	complemented
84	#101×C₄₁	C₂³	1	8	yes	complemented
85	#100×C₄₁	C₂³	1	8	yes	complemented
86	C₄×C₈×C₄₁	C₂³	1	4	yes	complemented
87	C₂⁴×C₄₁	—	2	2	no	—
88	C₂⁴×C₄₁	—	4	2	no	—
89	#41×C₄₁	—	2	4	no	—
90	#41×C₄₁	—	2	4	no	—
91	#41×C₄₁	—	2	16	no	—
92	#41×C₄₁	—	2	16	no	—
93	#41×C₄₁	—	4	8	no	—
94	#41×C₄₁	—	4	8	no	—
95	#41×C₄₁	—	4	8	no	—
96	#41×C₄₁	—	4	8	no	—
97	#41×C₄₁	C₂×D₄	1	8	yes	—
98	#41×C₄₁	C₂⁴	1	8	yes	—
99	C₂×Q₈×C₄₁	—	2	2	no	—
100	C₂×Q₈×C₄₁	—	2	16	no	—
101	C₂×Q₈×C₄₁	—	2	4	no	—
102	C₂×Q₈×C₄₁	—	2	4	no	—
103	C₂×Q₈×C₄₁	—	4	2	no	—
104	C₂×Q₈×C₄₁	—	4	8	no	—
105	C₂×Q₈×C₄₁	C₂×D₄	1	4	yes	—
106	C₂×Q₈×C₄₁	C₂⁴	1	4	yes	—
107	C₂×D₄×C₄₁	—	2	4	no	—
108	C₂×D₄×C₄₁	—	2	16	no	—
109	C₂×D₄×C₄₁	—	2	4	no	—
110	C₂×D₄×C₄₁	—	2	4	no	—
111	C₂×D₄×C₄₁	—	2	2	no	—
112	C₂×D₄×C₄₁	—	4	8	no	—
113	C₂×D₄×C₄₁	—	4	4	no	—
114	C₂×D₄×C₄₁	—	4	4	no	—
115	C₂×D₄×C₄₁	—	4	8	no	—
116	C₂×D₄×C₄₁	—	4	8	no	—
117	C₂×D₄×C₄₁	—	4	2	no	—
118	C₂×D₄×C₄₁	—	8	4	no	—
119	C₂×D₄×C₄₁	C₂×D₄	1	4	yes	—
120	C₂×D₄×C₄₁	C₂⁴	1	4	yes	—
121	C₂²×C₄×C₄₁	—	2	1	no	—
122	C₂²×C₄×C₄₁	—	2	4	no	—
123	C₂²×C₄×C₄₁	—	2	2	no	—
124	C₂²×C₄×C₄₁	—	4	2	no	—
125	C₂²×C₄×C₄₁	—	4	4	no	—
126	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
127	Q₁₆×C₄₁	—	2	16	no	—
128	Q₁₆×C₄₁	C₂×D₄	1	8	yes	complemented
129	SD₁₆×C₄₁	—	2	16	no	—
130	SD₁₆×C₄₁	—	2	16	no	—
131	SD₁₆×C₄₁	—	2	16	no	—
132	SD₁₆×C₄₁	—	2	16	no	—
133	SD₁₆×C₄₁	C₂×D₄	1	16	yes	complemented
134	D₈×C₄₁	—	2	16	no	—
135	D₈×C₄₁	C₂×D₄	1	8	yes	complemented
136	M₁₆×C₄₁	—	2	4	no	—
137	M₁₆×C₄₁	—	2	16	no	—
138	M₁₆×C₄₁	C₂⁴	1	8	yes	—
139	C₂×C₈×C₄₁	—	2	4	no	—
140	C₂×C₈×C₄₁	—	2	16	no	—
141	C₂×C₈×C₄₁	C₂×D₄	1	2	yes	—
142	C₂×C₈×C₄₁	C₂×D₄	1	2	yes	—
143	C₂×C₈×C₄₁	C₂×D₄	1	4	yes	—
144	C₂×C₈×C₄₁	C₂⁴	1	4	yes	—
145	C₄⋊C₄×C₄₁	—	4	4	no	—
146	C₄²×C₄₁	—	2	1	no	—
147	C₄²×C₄₁	C₂⁴	1	2	yes	—
148	C₂³×C₄₁	—	2	4	no	—
149	C₂³×C₄₁	—	2	4	no	—
150	C₂³×C₄₁	—	4	8	no	—
151	C₂³×C₄₁	—	4	2	no	—
152	C₂³×C₄₁	—	4	2	no	—
153	C₂³×C₄₁	—	8	4	no	—
154	C₂³×C₄₁	—	16	4	no	—
155	C₂³×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
156	Q₈×C₄₁	—	2	4	no	—
157	Q₈×C₄₁	—	2	8	no	—
158	Q₈×C₄₁	—	4	8	no	—
159	Q₈×C₄₁	—	4	4	no	—
160	Q₈×C₄₁	C₂²×D₄	1	8	yes	—
161	D₄×C₄₁	—	2	8	no	—
162	D₄×C₄₁	—	2	8	no	—
163	D₄×C₄₁	—	2	4	no	—
164	D₄×C₄₁	—	2	8	no	—
165	D₄×C₄₁	—	4	16	no	—
166	D₄×C₄₁	—	4	8	no	—
167	D₄×C₄₁	—	4	4	no	—
168	D₄×C₄₁	—	8	8	no	—
169	D₄×C₄₁	C₂²×D₄	1	8	yes	—
170	C₂×C₄×C₄₁	—	2	4	no	—
171	C₂×C₄×C₄₁	—	2	4	no	—
172	C₂×C₄×C₄₁	—	2	2	no	—
173	C₂×C₄×C₄₁	—	2	8	no	—
174	C₂×C₄×C₄₁	—	2	8	no	—
175	C₂×C₄×C₄₁	—	2	4	no	—
176	C₂×C₄×C₄₁	—	4	8	no	—
177	C₂×C₄×C₄₁	—	4	16	no	—
178	C₂×C₄×C₄₁	—	4	2	no	—
179	C₂×C₄×C₄₁	—	4	2	no	—
180	C₂×C₄×C₄₁	C₂²×D₄	1	2	yes	—
181	C₂×C₄×C₄₁	C₂²×D₄	1	2	yes	—
182	C₂×C₄×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
183	C₂×C₄×C₄₁	C₂⁵	1	1	yes	characteristic
184	C₈×C₄₁	—	2	8	no	—
185	C₈×C₄₁	C₂²×D₄	1	4	yes	—
186	C₈×C₄₁	C₂²×D₄	1	4	yes	—
187	C₂×#3291	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
188	C₂²×C₄₁	—	2	2	no	—
189	C₂²×C₄₁	—	2	8	no	—
190	C₂²×C₄₁	—	4	8	no	—
191	C₂²×C₄₁	—	4	4	no	—
192	C₂²×C₄₁	—	8	4	no	—
193	C₂²×C₄₁	—	8	4	no	—
194	C₂²×C₄₁	—	16	2	no	—
195	C₂²×C₄₁	—	16	8	no	—
196	C₂²×C₄₁	D₄²	1	1	yes	characteristic
197	C₂²×C₄₁	D₄²	1	1	yes	characteristic
198	C₂²×C₄₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
199	C₄×C₄₁	—	2	4	no	—
200	C₄×C₄₁	—	2	8	no	—
201	C₄×C₄₁	—	4	4	no	—
202	C₄×C₄₁	D₄²	1	2	yes	—
203	C₄×C₄₁	C₂³×D₄	1	2	yes	—
204	C₂×#577	C₂×C₄₁	1	4	yes	complemented
205	#3291	C₂×C₄₁	1	16	yes	direct product factor
206	C₂×#492	C₂×C₄₁	1	1	yes	characteristic, complemented
207	C₂×#471	C₂×C₄₁	1	4	yes	complemented
208	C₂×#456	C₂×C₄₁	1	2	yes	complemented
209	C₂×#453	C₂×C₄₁	1	2	yes	complemented
210	C₂×#443	C₂×C₄₁	1	2	yes	complemented
211	C₂×C₄₁	—	2	1	no	—
212	C₂×C₄₁	—	2	1	no	—
213	C₂×C₄₁	—	4	8	no	—
214	C₂×C₄₁	—	8	4	no	—
215	C₂×C₄₁	#3291	1	2	yes	direct product factor
216	C₂×C₄₁	C₂×D₄²	1	1	yes	characteristic
217	C₂×2₋¹⁺⁴	C₂²×C₄₁	1	2	yes	—
218	C₂×H(5,𝔽₂)	C₂²×C₄₁	1	2	yes	complemented
219	#577	—	2	16	no	—
220	#577	C₂²×C₄₁	1	32	yes	complemented
221	C₂×#137	—	2	4	no	—
222	C₂×#137	C₂²×C₄₁	1	4	yes	complemented
223	C₂×AGL(1,ℤ/8)	—	2	4	no	—
224	C₂×AGL(1,ℤ/8)	C₂²×C₄₁	1	4	yes	—
225	C₂×#135	—	2	4	no	—
226	C₂×#135	C₂²×C₄₁	1	2	yes	—
227	C₂²×SD₁₆	—	2	4	no	—
228	C₂²×SD₁₆	C₂²×C₄₁	1	4	yes	complemented
229	C₂×#131	C₂²×C₄₁	1	4	yes	complemented
230	C₂×C₄⋊Q₈	C₂²×C₄₁	1	1	yes	characteristic, complemented
231	C₂×#127	C₂²×C₄₁	1	1	yes	characteristic
232	#492	C₂²×C₄₁	1	8	yes	complemented
233	#471	C₂²×C₄₁	1	32	yes	complemented
234	#456	C₂²×C₄₁	1	16	yes	complemented
235	#453	C₂²×C₄₁	1	16	yes	complemented
236	#443	C₂²×C₄₁	1	16	yes	complemented
237	C₂×#108	C₂²×C₄₁	1	2	yes	complemented
238	C₂×#104	C₂²×C₄₁	1	4	yes	complemented
239	C₂×#101	C₂²×C₄₁	1	2	yes	complemented
240	C₂×#100	C₂²×C₄₁	1	2	yes	complemented
241	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic, complemented
242	C₄₁	C₂×#3291	1	1	yes	characteristic, direct product factor, Sylow
243	2₋¹⁺⁴	—	2	8	no	—
244	2₋¹⁺⁴	—	4	2	no	—
245	2₋¹⁺⁴	C₂³×C₄₁	1	8	yes	—
246	H(5,𝔽₂)	—	2	8	no	—
247	H(5,𝔽₂)	—	4	2	no	—
248	H(5,𝔽₂)	C₂³×C₄₁	1	8	yes	complemented
249	C₂×#41	—	2	4	no	—
250	C₂×#41	—	2	4	no	—
251	C₂×#41	—	4	2	no	—
252	C₂×#41	C₂³×C₄₁	1	4	yes	—
253	C₂²×Q₈	—	2	4	no	—
254	C₂²×Q₈	C₂³×C₄₁	1	2	yes	—
255	C₂²×D₄	—	2	4	no	—
256	C₂²×D₄	—	4	1	no	—
257	C₂²×D₄	—	4	2	no	—
258	C₂²×D₄	C₂³×C₄₁	1	2	yes	—
259	#137	—	2	8	no	—
260	#137	—	2	32	no	—
261	#137	C₂³×C₄₁	1	16	yes	complemented
262	AGL(1,ℤ/8)	—	2	8	no	—
263	AGL(1,ℤ/8)	—	2	32	no	—
264	AGL(1,ℤ/8)	C₂³×C₄₁	1	16	yes	—
265	#135	—	2	32	no	—
266	#135	D₄×C₄₁	1	8	yes	complemented
267	#135	C₂³×C₄₁	1	8	yes	—
268	C₂×Q₁₆	—	2	4	no	—
269	C₂×Q₁₆	D₄×C₄₁	1	2	yes	complemented
270	C₂×SD₁₆	—	2	4	no	—
271	C₂×SD₁₆	—	2	4	no	—
272	C₂×SD₁₆	—	2	32	no	—
273	C₂×SD₁₆	—	2	4	no	—
274	C₂×SD₁₆	—	2	4	no	—
275	C₂×SD₁₆	—	2	8	no	—
276	C₂×SD₁₆	D₄×C₄₁	1	4	yes	complemented
277	C₂×SD₁₆	C₂³×C₄₁	1	16	yes	complemented
278	C₂×D₈	—	2	4	no	—
279	C₂×D₈	D₄×C₄₁	1	2	yes	complemented
280	#131	—	2	8	no	—
281	#131	C₂³×C₄₁	1	16	yes	complemented
282	C₂×M₁₆	—	2	4	no	—
283	C₂×M₁₆	C₂³×C₄₁	1	4	yes	—
284	C₂²×C₈	—	2	4	no	—
285	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
286	C₄⋊Q₈	—	4	1	no	—
287	C₄⋊Q₈	C₂³×C₄₁	1	4	yes	complemented
288	#127	—	4	1	no	—
289	#127	C₂³×C₄₁	1	4	yes	—
290	C₂×C₄⋊C₄	—	4	1	no	—
291	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
292	#108	C₂³×C₄₁	1	8	yes	complemented
293	#104	C₂³×C₄₁	1	16	yes	complemented
294	#101	C₂³×C₄₁	1	8	yes	complemented
295	#100	C₂³×C₄₁	1	8	yes	complemented
296	C₄×C₈	C₂³×C₄₁	1	4	yes	complemented
297	C₂⁴	—	2	2	no	—
298	C₂⁴	—	4	2	no	—
299	#41	—	2	4	no	—
300	#41	—	2	4	no	—
301	#41	—	2	16	no	—
302	#41	—	2	16	no	—
303	#41	—	4	8	no	—
304	#41	—	4	8	no	—
305	#41	—	4	8	no	—
306	#41	—	4	8	no	—
307	#41	C₂×D₄×C₄₁	1	8	yes	—
308	#41	C₂⁴×C₄₁	1	8	yes	—
309	C₂×Q₈	—	2	2	no	—
310	C₂×Q₈	—	2	16	no	—
311	C₂×Q₈	—	2	4	no	—
312	C₂×Q₈	—	2	4	no	—
313	C₂×Q₈	—	4	2	no	—
314	C₂×Q₈	—	4	8	no	—
315	C₂×Q₈	C₂×D₄×C₄₁	1	4	yes	—
316	C₂×Q₈	C₂⁴×C₄₁	1	4	yes	—
317	C₂×D₄	—	2	4	no	—
318	C₂×D₄	—	2	16	no	—
319	C₂×D₄	—	2	4	no	—
320	C₂×D₄	—	2	4	no	—
321	C₂×D₄	—	2	2	no	—
322	C₂×D₄	—	4	8	no	—
323	C₂×D₄	—	4	4	no	—
324	C₂×D₄	—	4	4	no	—
325	C₂×D₄	—	4	8	no	—
326	C₂×D₄	—	4	8	no	—
327	C₂×D₄	—	4	2	no	—
328	C₂×D₄	—	8	4	no	—
329	C₂×D₄	C₂×D₄×C₄₁	1	4	yes	—
330	C₂×D₄	C₂⁴×C₄₁	1	4	yes	—
331	C₂²×C₄	—	2	1	no	—
332	C₂²×C₄	—	2	4	no	—
333	C₂²×C₄	—	2	2	no	—
334	C₂²×C₄	—	4	2	no	—
335	C₂²×C₄	—	4	4	no	—
336	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
337	Q₁₆	—	2	16	no	—
338	Q₁₆	C₂×D₄×C₄₁	1	8	yes	complemented
339	SD₁₆	—	2	16	no	—
340	SD₁₆	—	2	16	no	—
341	SD₁₆	—	2	16	no	—
342	SD₁₆	—	2	16	no	—
343	SD₁₆	C₂×D₄×C₄₁	1	16	yes	complemented
344	D₈	—	2	16	no	—
345	D₈	C₂×D₄×C₄₁	1	8	yes	complemented
346	M₁₆	—	2	4	no	—
347	M₁₆	—	2	16	no	—
348	M₁₆	C₂⁴×C₄₁	1	8	yes	—
349	C₂×C₈	—	2	4	no	—
350	C₂×C₈	—	2	16	no	—
351	C₂×C₈	C₂×D₄×C₄₁	1	2	yes	—
352	C₂×C₈	C₂×D₄×C₄₁	1	2	yes	—
353	C₂×C₈	C₂×D₄×C₄₁	1	4	yes	—
354	C₂×C₈	C₂⁴×C₄₁	1	4	yes	—
355	C₄⋊C₄	—	4	4	no	—
356	C₄²	—	2	1	no	—
357	C₄²	C₂⁴×C₄₁	1	2	yes	—
358	C₂³	—	2	4	no	—
359	C₂³	—	2	4	no	—
360	C₂³	—	4	8	no	—
361	C₂³	—	4	2	no	—
362	C₂³	—	4	2	no	—
363	C₂³	—	8	4	no	—
364	C₂³	—	16	4	no	—
365	C₂³	H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
366	Q₈	—	2	4	no	—
367	Q₈	—	2	8	no	—
368	Q₈	—	4	8	no	—
369	Q₈	—	4	4	no	—
370	Q₈	C₂²×D₄×C₄₁	1	8	yes	—
371	D₄	—	2	8	no	—
372	D₄	—	2	8	no	—
373	D₄	—	2	4	no	—
374	D₄	—	2	8	no	—
375	D₄	—	4	16	no	—
376	D₄	—	4	8	no	—
377	D₄	—	4	4	no	—
378	D₄	—	8	8	no	—
379	D₄	C₂²×D₄×C₄₁	1	8	yes	—
380	C₂×C₄	—	2	4	no	—
381	C₂×C₄	—	2	4	no	—
382	C₂×C₄	—	2	2	no	—
383	C₂×C₄	—	2	8	no	—
384	C₂×C₄	—	2	8	no	—
385	C₂×C₄	—	2	4	no	—
386	C₂×C₄	—	4	8	no	—
387	C₂×C₄	—	4	16	no	—
388	C₂×C₄	—	4	2	no	—
389	C₂×C₄	—	4	2	no	—
390	C₂×C₄	C₂²×D₄×C₄₁	1	2	yes	—
391	C₂×C₄	C₂²×D₄×C₄₁	1	2	yes	—
392	C₂×C₄	H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
393	C₂×C₄	C₂⁵×C₄₁	1	1	yes	characteristic
394	C₈	—	2	8	no	—
395	C₈	C₂²×D₄×C₄₁	1	4	yes	—
396	C₈	C₂²×D₄×C₄₁	1	4	yes	—
397	C₂²	—	2	2	no	—
398	C₂²	—	2	8	no	—
399	C₂²	—	4	8	no	—
400	C₂²	—	4	4	no	—
401	C₂²	—	8	4	no	—
402	C₂²	—	8	4	no	—
403	C₂²	—	16	2	no	—
404	C₂²	—	16	8	no	—
405	C₂²	D₄²×C₄₁	1	1	yes	characteristic
406	C₂²	D₄²×C₄₁	1	1	yes	characteristic
407	C₂²	C₂×H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
408	C₄	—	2	4	no	—
409	C₄	—	2	8	no	—
410	C₄	—	4	4	no	—
411	C₄	D₄²×C₄₁	1	2	yes	—
412	C₄	C₂³×D₄×C₄₁	1	2	yes	—
413	C₂	—	2	1	no	—
414	C₂	—	2	1	no	—
415	C₂	—	4	8	no	—
416	C₂	—	8	4	no	—
417	C₂	C₄₁×#3291	1	2	yes	direct product factor
418	C₂	C₂×D₄²×C₄₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₄₁	D₄²
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₄₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄₁ ◁ C₂²×C₄×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₄₁×#577	C₂	semidirect
2. C₄₁×#3291	C₂	direct (N × H)
3. C₂×C₄₁×#492	C₂	semidirect
4. C₂×C₄₁×#471	C₂	semidirect
5. C₂×C₄₁×#456	C₂	semidirect
6. C₂×C₄₁×#453	C₂	semidirect
7. C₂×C₄₁×#443	C₂	semidirect
9. C₂×H(5,𝔽₂)×C₄₁	C₂²	semidirect
11. C₄₁×#577	C₂²	semidirect
13. C₂×#137×C₄₁	C₂²	semidirect
19. C₂²×SD₁₆×C₄₁	C₂²	semidirect
20. C₂×#131×C₄₁	C₂²	semidirect
21. C₂×C₄⋊Q₈×C₄₁	C₂²	semidirect
23. C₄₁×#492	C₂²	semidirect
24. C₄₁×#471	C₂²	semidirect
25. C₄₁×#456	C₂²	semidirect
26. C₄₁×#453	C₂²	semidirect
27. C₄₁×#443	C₂²	semidirect
28. C₂×#108×C₄₁	C₂²	semidirect
29. C₂×#104×C₄₁	C₂²	semidirect
30. C₂×#101×C₄₁	C₂²	semidirect
31. C₂×#100×C₄₁	C₂²	semidirect
32. C₂×C₄×C₈×C₄₁	C₂²	semidirect
38. H(5,𝔽₂)×C₄₁	C₂³	semidirect
51. #137×C₄₁	C₂³	semidirect
56. #135×C₄₁	D₄	semidirect
59. C₂×Q₁₆×C₄₁	D₄	semidirect
66. C₂×SD₁₆×C₄₁	D₄	semidirect
67. C₂×SD₁₆×C₄₁	C₂³	semidirect
69. C₂×D₈×C₄₁	D₄	semidirect
71. #131×C₄₁	C₂³	semidirect
77. C₄⋊Q₈×C₄₁	C₂³	semidirect
82. #108×C₄₁	C₂³	semidirect
83. #104×C₄₁	C₂³	semidirect
84. #101×C₄₁	C₂³	semidirect
85. #100×C₄₁	C₂³	semidirect
86. C₄×C₈×C₄₁	C₂³	semidirect
128. Q₁₆×C₄₁	C₂×D₄	semidirect
133. SD₁₆×C₄₁	C₂×D₄	semidirect
135. D₈×C₄₁	C₂×D₄	semidirect
187. C₂×#3291	C₄₁	direct (N × H)
204. C₂×#577	C₂×C₄₁	semidirect
205. #3291	C₂×C₄₁	direct (N × H)
206. C₂×#492	C₂×C₄₁	semidirect
207. C₂×#471	C₂×C₄₁	semidirect
208. C₂×#456	C₂×C₄₁	semidirect
209. C₂×#453	C₂×C₄₁	semidirect
210. C₂×#443	C₂×C₄₁	semidirect
215. C₂×C₄₁	#3291	direct (N × H)
218. C₂×H(5,𝔽₂)	C₂²×C₄₁	semidirect
220. #577	C₂²×C₄₁	semidirect
222. C₂×#137	C₂²×C₄₁	semidirect
228. C₂²×SD₁₆	C₂²×C₄₁	semidirect
229. C₂×#131	C₂²×C₄₁	semidirect
230. C₂×C₄⋊Q₈	C₂²×C₄₁	semidirect
232. #492	C₂²×C₄₁	semidirect
233. #471	C₂²×C₄₁	semidirect
234. #456	C₂²×C₄₁	semidirect
235. #453	C₂²×C₄₁	semidirect
236. #443	C₂²×C₄₁	semidirect
237. C₂×#108	C₂²×C₄₁	semidirect
238. C₂×#104	C₂²×C₄₁	semidirect
239. C₂×#101	C₂²×C₄₁	semidirect
240. C₂×#100	C₂²×C₄₁	semidirect
241. C₂×C₄×C₈	C₂²×C₄₁	semidirect
242. C₄₁	C₂×#3291	direct (N × H)
248. H(5,𝔽₂)	C₂³×C₄₁	semidirect
261. #137	C₂³×C₄₁	semidirect
266. #135	D₄×C₄₁	semidirect
269. C₂×Q₁₆	D₄×C₄₁	semidirect
276. C₂×SD₁₆	D₄×C₄₁	semidirect
277. C₂×SD₁₆	C₂³×C₄₁	semidirect
279. C₂×D₈	D₄×C₄₁	semidirect
281. #131	C₂³×C₄₁	semidirect
287. C₄⋊Q₈	C₂³×C₄₁	semidirect
292. #108	C₂³×C₄₁	semidirect
293. #104	C₂³×C₄₁	semidirect
294. #101	C₂³×C₄₁	semidirect
295. #100	C₂³×C₄₁	semidirect
296. C₄×C₈	C₂³×C₄₁	semidirect
338. Q₁₆	C₂×D₄×C₄₁	semidirect
343. SD₁₆	C₂×D₄×C₄₁	semidirect
345. D₈	C₂×D₄×C₄₁	semidirect
417. C₂	C₄₁×#3291	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×C₄₁×#577	2
C₄₁×#3291	2
C₂×C₄₁×#492	2
C₂×C₄₁×#471	2
C₂×C₄₁×#456	2
C₂×C₄₁×#453	2
C₂×C₄₁×#443	2
C₂×#3291	41

Subgroups — C2×C41×#3291 #23837303

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄₁×#3291 #23837303