Subgroups — C₂×C₄₁×#2074 #23799827

Order 10496.

1062 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#103×C₄₁	C₂	1	2	yes	maximal
2	C₂×C₄₁×#391	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄₁×#382	C₂	1	1	yes	maximal, characteristic
4	C₄₁×#2074	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₄₁×#340	C₂	1	2	yes	maximal
6	C₂×C₄₁×#336	C₂	1	1	yes	maximal, characteristic
7	C₂³×Q₈×C₄₁	C₂²	1	1	yes	characteristic
8	C₂³×C₈×C₄₁	C₂²	1	2	yes	—
9	C₂²×C₄⋊C₄×C₄₁	C₂²	1	2	yes	—
10	C₂²×C₄⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
11	C₂²×C₄²×C₄₁	C₂²	1	1	yes	characteristic
12	C₂×#103×C₄₁	—	2	8	no	—
13	C₂×#103×C₄₁	C₂²	1	8	yes	—
14	C₄₁×#391	—	2	2	no	—
15	C₄₁×#391	C₂²	1	4	yes	—
16	C₄₁×#382	—	2	2	no	—
17	C₄₁×#382	C₂²	1	4	yes	—
18	C₂×#95×C₄₁	—	2	2	no	—
19	C₂×#95×C₄₁	—	2	2	no	—
20	C₄₁×#340	C₂²	1	8	yes	—
21	C₄₁×#336	C₂²	1	4	yes	—
22	C₂²×Q₈×C₄₁	—	2	1	no	—
23	C₂²×Q₈×C₄₁	—	2	2	no	—
24	C₂²×Q₈×C₄₁	—	2	4	no	—
25	C₂²×Q₈×C₄₁	—	2	4	no	—
26	C₂²×Q₈×C₄₁	D₄	1	4	yes	—
27	C₂²×Q₈×C₄₁	C₂³	1	2	yes	—
28	C₂³×C₄×C₄₁	—	2	1	no	—
29	C₂³×C₄×C₄₁	—	2	2	no	—
30	C₂³×C₄×C₄₁	—	2	1	no	—
31	C₂³×C₄×C₄₁	D₄	1	2	yes	—
32	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
33	C₂²×C₈×C₄₁	—	2	4	no	—
34	C₂²×C₈×C₄₁	—	2	4	no	—
35	C₂²×C₈×C₄₁	—	2	4	no	—
36	C₂²×C₈×C₄₁	C₂³	1	4	yes	—
37	C₂×C₄⋊C₄×C₄₁	—	2	4	no	—
38	C₂×C₄⋊C₄×C₄₁	—	2	4	no	—
39	C₂×C₄⋊C₄×C₄₁	—	2	4	no	—
40	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
41	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
42	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
43	C₂×C₄⋊C₄×C₄₁	C₂³	1	4	yes	—
44	C₂×C₄²×C₄₁	—	2	4	no	—
45	C₂×C₄²×C₄₁	D₄	1	2	yes	—
46	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
47	#103×C₄₁	—	2	16	no	—
48	#95×C₄₁	—	2	8	no	—
49	#95×C₄₁	—	2	8	no	—
50	C₂⁴×C₄₁	C₂×D₄	1	1	yes	characteristic
51	C₂×Q₈×C₄₁	—	2	4	no	—
52	C₂×Q₈×C₄₁	—	2	2	no	—
53	C₂×Q₈×C₄₁	—	2	16	no	—
54	C₂×Q₈×C₄₁	—	2	4	no	—
55	C₂×Q₈×C₄₁	—	2	4	no	—
56	C₂×Q₈×C₄₁	—	4	2	no	—
57	C₂×Q₈×C₄₁	—	4	2	no	—
58	C₂×Q₈×C₄₁	—	4	8	no	—
59	C₂×Q₈×C₄₁	C₂×D₄	1	4	yes	—
60	C₂²×C₄×C₄₁	—	2	4	no	—
61	C₂²×C₄×C₄₁	—	2	4	no	—
62	C₂²×C₄×C₄₁	—	2	4	no	—
63	C₂²×C₄×C₄₁	—	2	4	no	—
64	C₂²×C₄×C₄₁	—	2	2	no	—
65	C₂²×C₄×C₄₁	—	2	4	no	—
66	C₂²×C₄×C₄₁	—	2	2	no	—
67	C₂²×C₄×C₄₁	—	4	2	no	—
68	C₂²×C₄×C₄₁	—	4	2	no	—
69	C₂²×C₄×C₄₁	—	4	4	no	—
70	C₂²×C₄×C₄₁	—	4	2	no	—
71	C₂²×C₄×C₄₁	—	4	4	no	—
72	C₂²×C₄×C₄₁	—	4	4	no	—
73	C₂²×C₄×C₄₁	—	4	4	no	—
74	C₂²×C₄×C₄₁	C₂×D₄	1	4	yes	—
75	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₄₁	#41	1	2	yes	—
78	C₂²×C₄×C₄₁	#41	1	4	yes	—
79	C₂²×C₄×C₄₁	#41	1	4	yes	—
80	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
81	C₂×C₈×C₄₁	—	2	4	no	—
82	C₂×C₈×C₄₁	—	2	4	no	—
83	C₂×C₈×C₄₁	—	2	4	no	—
84	C₂×C₈×C₄₁	—	4	8	no	—
85	C₄⋊C₄×C₄₁	—	2	4	no	—
86	C₄⋊C₄×C₄₁	—	2	4	no	—
87	C₄⋊C₄×C₄₁	—	2	4	no	—
88	C₄⋊C₄×C₄₁	—	4	4	no	—
89	C₄⋊C₄×C₄₁	—	4	2	no	—
90	C₄²×C₄₁	—	2	4	no	—
91	C₄²×C₄₁	—	2	4	no	—
92	C₂³×C₄₁	#121	1	2	yes	—
93	C₂³×C₄₁	#123	1	2	yes	—
94	C₂³×C₄₁	#124	1	2	yes	—
95	C₂³×C₄₁	#135	1	4	yes	—
96	C₂³×C₄₁	#137	1	4	yes	—
97	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
98	Q₈×C₄₁	—	2	8	no	—
99	Q₈×C₄₁	—	4	4	no	—
100	Q₈×C₄₁	—	4	8	no	—
101	C₂×C₄×C₄₁	—	2	4	no	—
102	C₂×C₄×C₄₁	—	2	4	no	—
103	C₂×C₄×C₄₁	—	2	4	no	—
104	C₂×C₄×C₄₁	—	2	4	no	—
105	C₂×C₄×C₄₁	—	4	2	no	—
106	C₂×C₄×C₄₁	—	4	2	no	—
107	C₂×C₄×C₄₁	—	4	8	no	—
108	C₂×C₄×C₄₁	—	4	8	no	—
109	C₂×C₄×C₄₁	—	4	4	no	—
110	C₂×C₄×C₄₁	—	4	2	no	—
111	C₂×C₄×C₄₁	—	4	4	no	—
112	C₂×C₄×C₄₁	—	4	4	no	—
113	C₂×C₄×C₄₁	—	4	4	no	—
114	C₂×C₄×C₄₁	—	8	8	no	—
115	C₂×C₄×C₄₁	—	8	4	no	—
116	C₂×C₄×C₄₁	#121	1	2	yes	—
117	C₂×C₄×C₄₁	#123	1	8	yes	—
118	C₂×C₄×C₄₁	#123	1	2	yes	—
119	C₂×C₄×C₄₁	#124	1	2	yes	—
120	C₂×C₄×C₄₁	#126	1	8	yes	—
121	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
122	C₂×C₄×C₄₁	C₂×#41	1	2	yes	—
123	C₂×C₄×C₄₁	C₂×#41	1	2	yes	—
124	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
125	C₈×C₄₁	—	4	4	no	—
126	C₂×#2074	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
127	C₂²×C₄₁	#397	1	4	yes	—
128	C₂²×C₄₁	#464	1	8	yes	—
129	C₂²×C₄₁	#485	1	8	yes	—
130	C₂²×C₄₁	#488	1	4	yes	—
131	C₂²×C₄₁	#491	1	4	yes	—
132	C₂²×C₄₁	C₂×#121	1	1	yes	characteristic
133	C₂²×C₄₁	C₂×#123	1	1	yes	characteristic
134	C₂²×C₄₁	C₂×#124	1	1	yes	characteristic
135	C₂²×C₄₁	C₂×#135	1	2	yes	—
136	C₂²×C₄₁	C₂×#137	1	2	yes	—
137	C₄×C₄₁	—	4	4	no	—
138	C₄×C₄₁	—	4	4	no	—
139	C₄×C₄₁	—	8	2	no	—
140	C₄×C₄₁	—	8	4	no	—
141	C₄×C₄₁	#397	1	4	yes	—
142	C₄×C₄₁	C₂×#123	1	2	yes	—
143	C₄×C₄₁	C₂×#126	1	2	yes	—
144	C₂²×#103	C₂×C₄₁	1	2	yes	—
145	C₂×#391	C₂×C₄₁	1	1	yes	characteristic
146	C₂×#382	C₂×C₄₁	1	1	yes	characteristic
147	#2074	C₂×C₄₁	1	8	yes	direct product factor
148	C₂×#340	C₂×C₄₁	1	2	yes	—
149	C₂×#336	C₂×C₄₁	1	1	yes	characteristic
150	C₂×C₄₁	#2074	1	8	yes	direct product factor
151	C₂×C₄₁	C₂×#397	1	1	yes	characteristic
152	C₂×C₄₁	C₂×#464	1	2	yes	—
153	C₂×C₄₁	C₂×#485	1	2	yes	—
154	C₂×C₄₁	C₂×#488	1	1	yes	characteristic
155	C₂×C₄₁	C₂×#491	1	1	yes	characteristic
156	C₂³×Q₈	C₂²×C₄₁	1	1	yes	characteristic
157	C₂³×C₈	C₂²×C₄₁	1	2	yes	—
158	C₂²×C₄⋊C₄	C₂²×C₄₁	1	2	yes	—
159	C₂²×C₄⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
160	C₂²×C₄²	C₂²×C₄₁	1	1	yes	characteristic
161	C₂×#103	—	2	8	no	—
162	C₂×#103	C₂²×C₄₁	1	8	yes	—
163	#391	—	2	2	no	—
164	#391	C₂²×C₄₁	1	4	yes	—
165	#382	—	2	2	no	—
166	#382	C₂²×C₄₁	1	4	yes	—
167	C₂×#95	—	2	2	no	—
168	C₂×#95	—	2	2	no	—
169	#340	C₂²×C₄₁	1	8	yes	—
170	#336	C₂²×C₄₁	1	4	yes	—
171	C₄₁	C₂×#2074	1	1	yes	characteristic, direct product factor, Sylow
172	C₂²×Q₈	—	2	1	no	—
173	C₂²×Q₈	—	2	2	no	—
174	C₂²×Q₈	—	2	4	no	—
175	C₂²×Q₈	—	2	4	no	—
176	C₂²×Q₈	D₄×C₄₁	1	4	yes	—
177	C₂²×Q₈	C₂³×C₄₁	1	2	yes	—
178	C₂³×C₄	—	2	1	no	—
179	C₂³×C₄	—	2	2	no	—
180	C₂³×C₄	—	2	1	no	—
181	C₂³×C₄	D₄×C₄₁	1	2	yes	—
182	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
183	C₂²×C₈	—	2	4	no	—
184	C₂²×C₈	—	2	4	no	—
185	C₂²×C₈	—	2	4	no	—
186	C₂²×C₈	C₂³×C₄₁	1	4	yes	—
187	C₂×C₄⋊C₄	—	2	4	no	—
188	C₂×C₄⋊C₄	—	2	4	no	—
189	C₂×C₄⋊C₄	—	2	4	no	—
190	C₂×C₄⋊C₄	—	2	2	no	—
191	C₂×C₄⋊C₄	—	2	1	no	—
192	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
193	C₂×C₄⋊C₄	C₂³×C₄₁	1	4	yes	—
194	C₂×C₄²	—	2	4	no	—
195	C₂×C₄²	D₄×C₄₁	1	2	yes	—
196	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
197	#103	—	2	16	no	—
198	#95	—	2	8	no	—
199	#95	—	2	8	no	—
200	C₂⁴	C₂×D₄×C₄₁	1	1	yes	characteristic
201	C₂×Q₈	—	2	4	no	—
202	C₂×Q₈	—	2	2	no	—
203	C₂×Q₈	—	2	16	no	—
204	C₂×Q₈	—	2	4	no	—
205	C₂×Q₈	—	2	4	no	—
206	C₂×Q₈	—	4	2	no	—
207	C₂×Q₈	—	4	2	no	—
208	C₂×Q₈	—	4	8	no	—
209	C₂×Q₈	C₂×D₄×C₄₁	1	4	yes	—
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	—	2	4	no	—
212	C₂²×C₄	—	2	4	no	—
213	C₂²×C₄	—	2	4	no	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	—	2	4	no	—
216	C₂²×C₄	—	2	2	no	—
217	C₂²×C₄	—	4	2	no	—
218	C₂²×C₄	—	4	2	no	—
219	C₂²×C₄	—	4	4	no	—
220	C₂²×C₄	—	4	2	no	—
221	C₂²×C₄	—	4	4	no	—
222	C₂²×C₄	—	4	4	no	—
223	C₂²×C₄	—	4	4	no	—
224	C₂²×C₄	C₂×D₄×C₄₁	1	4	yes	—
225	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
226	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
227	C₂²×C₄	#41×C₄₁	1	2	yes	—
228	C₂²×C₄	#41×C₄₁	1	4	yes	—
229	C₂²×C₄	#41×C₄₁	1	4	yes	—
230	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
231	C₂×C₈	—	2	4	no	—
232	C₂×C₈	—	2	4	no	—
233	C₂×C₈	—	2	4	no	—
234	C₂×C₈	—	4	8	no	—
235	C₄⋊C₄	—	2	4	no	—
236	C₄⋊C₄	—	2	4	no	—
237	C₄⋊C₄	—	2	4	no	—
238	C₄⋊C₄	—	4	4	no	—
239	C₄⋊C₄	—	4	2	no	—
240	C₄²	—	2	4	no	—
241	C₄²	—	2	4	no	—
242	C₂³	#121×C₄₁	1	2	yes	—
243	C₂³	#123×C₄₁	1	2	yes	—
244	C₂³	#124×C₄₁	1	2	yes	—
245	C₂³	#135×C₄₁	1	4	yes	—
246	C₂³	#137×C₄₁	1	4	yes	—
247	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
248	Q₈	—	2	8	no	—
249	Q₈	—	4	4	no	—
250	Q₈	—	4	8	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	4	8	no	—
258	C₂×C₄	—	4	8	no	—
259	C₂×C₄	—	4	4	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	4	no	—
262	C₂×C₄	—	4	4	no	—
263	C₂×C₄	—	4	4	no	—
264	C₂×C₄	—	8	8	no	—
265	C₂×C₄	—	8	4	no	—
266	C₂×C₄	#121×C₄₁	1	2	yes	—
267	C₂×C₄	#123×C₄₁	1	8	yes	—
268	C₂×C₄	#123×C₄₁	1	2	yes	—
269	C₂×C₄	#124×C₄₁	1	2	yes	—
270	C₂×C₄	#126×C₄₁	1	8	yes	—
271	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
272	C₂×C₄	C₂×#41×C₄₁	1	2	yes	—
273	C₂×C₄	C₂×#41×C₄₁	1	2	yes	—
274	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
275	C₈	—	4	4	no	—
276	C₂²	C₄₁×#397	1	4	yes	—
277	C₂²	C₄₁×#464	1	8	yes	—
278	C₂²	C₄₁×#485	1	8	yes	—
279	C₂²	C₄₁×#488	1	4	yes	—
280	C₂²	C₄₁×#491	1	4	yes	—
281	C₂²	C₂×#121×C₄₁	1	1	yes	characteristic
282	C₂²	C₂×#123×C₄₁	1	1	yes	characteristic
283	C₂²	C₂×#124×C₄₁	1	1	yes	characteristic
284	C₂²	C₂×#135×C₄₁	1	2	yes	—
285	C₂²	C₂×#137×C₄₁	1	2	yes	—
286	C₄	—	4	4	no	—
287	C₄	—	4	4	no	—
288	C₄	—	8	2	no	—
289	C₄	—	8	4	no	—
290	C₄	C₄₁×#397	1	4	yes	—
291	C₄	C₂×#123×C₄₁	1	2	yes	—
292	C₄	C₂×#126×C₄₁	1	2	yes	—
293	C₂	C₄₁×#2074	1	8	yes	direct product factor
294	C₂	C₂×C₄₁×#397	1	1	yes	characteristic
295	C₂	C₂×C₄₁×#464	1	2	yes	—
296	C₂	C₂×C₄₁×#485	1	2	yes	—
297	C₂	C₂×C₄₁×#488	1	1	yes	characteristic
298	C₂	C₂×C₄₁×#491	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₄₁	C₂×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₁
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₄₁ ◁ C₂²×C₄²×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₄₁×#2074	C₂	direct (N × H)
126. C₂×#2074	C₄₁	direct (N × H)
147. #2074	C₂×C₄₁	direct (N × H)
150. C₂×C₄₁	#2074	direct (N × H)
171. C₄₁	C₂×#2074	direct (N × H)
293. C₂	C₄₁×#2074	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×#103×C₄₁	2
C₂×C₄₁×#391	2
C₂×C₄₁×#382	2
C₄₁×#2074	2
C₂×C₄₁×#340	2
C₂×C₄₁×#336	2
C₂×#2074	41

Subgroups — C2×C41×#2074 #23799827

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄₁×#2074 #23799827