Subgroups — C₂×C₄×#10621900 #19147773

Order 5504.

1728 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#10621900	C₂	1	1	yes	maximal, characteristic
2	C₄×#10621900	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₄×Dic₈₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#13063290	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₄×Dic₄₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×C₂²⋊D₄₃	C₂	1	2	yes	maximal, complemented
7	C₂×(C₂²⋊D₄₃)⋊C₄	C₂	1	2	yes	maximal, complemented
8	C₂×#13063283	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13063281	C₂	1	2	yes	maximal, complemented
10	C₂²×C₄×Dic₄₃	C₂	1	2	yes	maximal, complemented
11	C₂×C₄²×D₄₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₄×D₄×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×#10621900	C₄	1	16	yes	direct product factor
14	C₂×#10621900	C₂²	1	8	yes	—
15	C₂²×Dic₈₆	C₂²	1	1	yes	characteristic
16	C₂²×C₂²⋊D₄₃	C₂²	1	2	yes	—
17	C₂³×Dic₄₃	C₂²	1	2	yes	—
18	C₂³×Dic₄₃	C₂²	1	2	yes	—
19	C₂²×C₄×D₄₃	C₂²	1	1	yes	characteristic
20	C₂²×C₄×D₄₃	C₂²	1	1	yes	characteristic
21	C₂²×C₄×D₄₃	C₂²	1	1	yes	characteristic
22	C₄×Dic₈₆	C₂²	1	8	yes	complemented
23	#13063290	C₂²	1	8	yes	complemented
24	D₄×Dic₄₃	C₂²	1	8	yes	complemented
25	C₄×C₂²⋊D₄₃	C₂²	1	16	yes	complemented
26	(C₂²⋊D₄₃)⋊C₄	C₂²	1	16	yes	complemented
27	#13063283	C₂²	1	8	yes	complemented
28	#13063281	C₂²	1	16	yes	complemented
29	C₂×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
30	C₂×Dic₄₃⋊C₄	C₂²	1	2	yes	—
31	C₂×Dic₄₃⋊C₄	C₂²	1	2	yes	—
32	C₂×C₂²⋊Dic₄₃	C₂²	1	2	yes	—
33	C₂×(C₂×D₄₃)⋊C₄	C₂²	1	2	yes	—
34	C₂×C₄×Dic₄₃	C₂²	1	1	yes	characteristic
35	C₂×C₄×Dic₄₃	C₂²	1	1	yes	characteristic
36	C₂×C₄×Dic₄₃	C₂²	1	1	yes	characteristic
37	C₂×C₄×Dic₄₃	C₂²	1	2	yes	—
38	C₂×C₄×Dic₄₃	C₂²	1	2	yes	—
39	C₂×C₄×Dic₄₃	C₂²	1	2	yes	—
40	C₂×C₄×Dic₄₃	C₂²	1	16	yes	complemented
41	C₄²×D₄₃	C₂²	1	8	yes	complemented
42	C₂²×D₄×C₄₃	C₂²	1	1	yes	characteristic
43	C₂³×C₄×C₄₃	C₂²	1	2	yes	—
44	C₄×D₄×C₄₃	C₂²	1	8	yes	complemented
45	C₂×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
46	C₂×C₂²⋊C₄×C₄₃	C₂²	1	2	yes	—
47	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
48	C₂⁴×C₄₃	C₂³	1	2	yes	—
49	C₂³×D₄₃	C₂³	1	1	yes	characteristic
50	C₂×D₄×C₄₃	C₂×C₄	1	8	yes	complemented
51	C₂×D₄×C₄₃	C₂³	1	4	yes	—
52	C₂²×C₄×C₄₃	—	2	8	no	—
53	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
54	C₂²×C₄×C₄₃	C₂³	1	2	yes	—
55	C₂²×C₄×C₄₃	C₂³	1	8	yes	—
56	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
58	C₂×C₂²⋊D₄₃	C₂×C₄	1	16	yes	complemented
59	C₂×C₂²⋊D₄₃	C₂³	1	8	yes	—
60	C₂²×Dic₄₃	—	2	8	no	—
61	C₂²×Dic₄₃	C₂×C₄	1	16	yes	complemented
62	C₂²×Dic₄₃	C₂³	1	2	yes	—
63	C₂²×Dic₄₃	C₂³	1	1	yes	characteristic
64	C₂²×Dic₄₃	C₂³	1	8	yes	—
65	C₂²×Dic₄₃	C₂³	1	1	yes	characteristic
66	C₂²×Dic₄₃	C₂³	1	2	yes	—
67	C₂²×Dic₄₃	C₂³	1	8	yes	—
68	C₂²×Dic₄₃	C₂³	1	1	yes	characteristic
69	#10621900	C₂×C₄	1	64	yes	direct product factor
70	C₂×C₄×D₄₃	—	2	4	no	—
71	C₂×C₄×D₄₃	—	2	4	no	—
72	C₂×C₄×D₄₃	C₂×C₄	1	8	yes	complemented
73	C₂×C₄×D₄₃	C₂³	1	4	yes	—
74	C₂×C₄×D₄₃	C₂³	1	4	yes	—
75	C₂×C₄×D₄₃	C₂³	1	4	yes	—
76	C₂×Dic₈₆	C₂×C₄	1	8	yes	complemented
77	C₂×Dic₈₆	C₂³	1	4	yes	—
78	C₄⋊C₄×C₄₃	C₂³	1	4	yes	—
79	C₂²⋊C₄×C₄₃	C₂³	1	8	yes	—
80	C₄²×C₄₃	C₂³	1	4	yes	—
81	C₂²⋊Dic₄₃	C₂³	1	8	yes	—
82	(C₂×D₄₃)⋊C₄	C₂³	1	8	yes	—
83	C₄⋊Dic₄₃	C₂³	1	4	yes	—
84	Dic₄₃⋊C₄	C₂³	1	8	yes	—
85	Dic₄₃⋊C₄	C₂³	1	8	yes	—
86	C₄×Dic₄₃	C₂³	1	4	yes	—
87	C₄×Dic₄₃	C₂³	1	4	yes	—
88	C₄×Dic₄₃	C₂³	1	4	yes	—
89	C₄×Dic₄₃	C₂³	1	8	yes	—
90	C₄×Dic₄₃	C₂³	1	8	yes	—
91	C₄×Dic₄₃	C₂³	1	8	yes	—
92	C₂³×C₄₃	—	2	4	no	—
93	C₂³×C₄₃	—	2	4	no	—
94	C₂³×C₄₃	C₂²×C₄	1	8	yes	—
95	C₂³×C₄₃	C₂⁴	1	4	yes	—
96	C₂³×C₄₃	C₂⁴	1	1	yes	characteristic
97	C₂²×D₄₃	—	2	2	no	—
98	C₂²×D₄₃	—	2	2	no	—
99	C₂²×D₄₃	C₂²×C₄	1	4	yes	—
100	C₂²×D₄₃	C₂⁴	1	2	yes	—
101	D₄×C₄₃	C₂²×C₄	1	16	yes	complemented
102	C₂×C₄×C₄₃	—	2	2	no	—
103	C₂×C₄×C₄₃	—	2	4	no	—
104	C₂×C₄×C₄₃	—	2	16	no	—
105	C₂×C₄×C₄₃	C₂²×C₄	1	4	yes	—
106	C₂×C₄×C₄₃	#41	1	4	yes	complemented
107	C₂×C₄×C₄₃	C₂⁴	1	4	yes	—
108	C₂×C₄×C₄₃	C₂⁴	1	2	yes	—
109	C₂×C₄×C₄₃	C₂⁴	1	2	yes	—
110	C₂×C₄×C₄₃	C₂⁴	1	2	yes	—
111	C₂²⋊D₄₃	C₂²×C₄	1	32	yes	complemented
112	C₂×Dic₄₃	—	2	2	no	—
113	C₂×Dic₄₃	—	2	4	no	—
114	C₂×Dic₄₃	—	2	16	no	—
115	C₂×Dic₄₃	C₂²×C₄	1	8	yes	—
116	C₂×Dic₄₃	C₂²×C₄	1	4	yes	—
117	C₂×Dic₄₃	C₂²×C₄	1	32	yes	complemented
118	C₂×Dic₄₃	#41	1	4	yes	complemented
119	C₂×Dic₄₃	C₂⁴	1	2	yes	—
120	C₂×Dic₄₃	C₂⁴	1	4	yes	—
121	C₂×Dic₄₃	C₂⁴	1	2	yes	—
122	C₂×Dic₄₃	C₂⁴	1	4	yes	—
123	C₂×Dic₄₃	C₂⁴	1	2	yes	—
124	C₄×D₄₃	—	2	8	no	—
125	C₄×D₄₃	—	2	8	no	—
126	C₄×D₄₃	C₂²×C₄	1	16	yes	complemented
127	Dic₈₆	C₂²×C₄	1	16	yes	complemented
128	C₂²×C₄₃	—	2	16	no	—
129	C₂²×C₄₃	—	2	4	no	—
130	C₂²×C₄₃	—	2	4	no	—
131	C₂²×C₄₃	C₂³×C₄	1	8	yes	—
132	C₂²×C₄₃	C₂³×C₄	1	2	yes	—
133	C₂²×C₄₃	C₂×#41	1	2	yes	—
134	C₂²×C₄₃	C₂×#41	1	2	yes	—
135	C₂²×C₄₃	C₂⁵	1	1	yes	characteristic
136	C₂×D₄₃	—	2	8	no	—
137	C₂×D₄₃	—	2	2	no	—
138	C₂×D₄₃	—	2	2	no	—
139	C₂×D₄₃	C₂³×C₄	1	4	yes	—
140	C₄×C₄₃	—	2	2	no	—
141	C₄×C₄₃	—	2	4	no	—
142	C₄×C₄₃	C₂³×C₄	1	4	yes	—
143	C₄×C₄₃	C₂×#41	1	4	yes	complemented
144	Dic₄₃	—	2	2	no	—
145	Dic₄₃	—	2	4	no	—
146	Dic₄₃	C₂³×C₄	1	8	yes	—
147	Dic₄₃	C₂³×C₄	1	4	yes	—
148	Dic₄₃	C₂×#41	1	4	yes	complemented
149	C₂×C₄×#41	—	43	1	no	maximal, Sylow
150	C₂×C₄₃	—	2	8	no	—
151	C₂×C₄₃	C₄×#41	1	4	yes	complemented
152	C₂×C₄₃	C₂⁴×C₄	1	1	yes	characteristic
153	C₂×C₄₃	C₂²×#41	1	1	yes	characteristic
154	C₂×C₄₃	C₂²×#41	1	1	yes	characteristic
155	D₄₃	—	2	4	no	—
156	C₂²×#41	—	43	1	no	—
157	C₄×#41	—	43	16	no	—
158	C₂×C₄×Q₈	—	43	1	no	—
159	C₂×C₄×Q₈	—	43	1	no	—
160	C₂×C₄×D₄	—	43	1	no	—
161	C₂×C₄×D₄	—	43	2	no	—
162	C₂×C₄×D₄	—	43	2	no	—
163	C₂×C₄×D₄	D₄₃	1	1	yes	characteristic, complemented
164	C₂×#117	—	43	1	no	—
165	C₂×#117	—	43	2	no	—
166	C₂²×C₄²	—	43	2	no	—
167	C₂²×C₄²	—	43	1	no	—
168	C₄₃	C₂×C₄×#41	1	1	yes	characteristic, complemented, Sylow
169	C₂×#41	—	43	8	no	—
170	C₂×#41	—	43	16	no	—
171	C₂²×Q₈	—	43	1	no	—
172	C₂²×D₄	—	43	2	no	—
173	C₂²×D₄	C₂×D₄₃	1	1	yes	characteristic
174	C₂³×C₄	—	43	2	no	—
175	C₂³×C₄	—	43	1	no	—
176	C₂³×C₄	—	43	2	no	—
177	C₂³×C₄	—	43	1	no	—
178	C₂³×C₄	—	43	1	no	—
179	C₂³×C₄	C₂×D₄₃	1	2	yes	—
180	C₄×Q₈	—	43	8	no	—
181	C₄×Q₈	—	43	8	no	—
182	C₄×D₄	—	43	8	no	—
183	C₄×D₄	—	43	16	no	—
184	C₄×D₄	—	43	16	no	—
185	C₄×D₄	C₂×D₄₃	1	8	yes	complemented
186	#117	—	43	8	no	—
187	#117	—	43	16	no	—
188	C₂×C₄⋊C₄	—	43	2	no	—
189	C₂×C₄⋊C₄	—	43	2	no	—
190	C₂×C₄⋊C₄	—	43	1	no	—
191	C₂×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
192	C₂×C₂²⋊C₄	—	43	2	no	—
193	C₂×C₂²⋊C₄	—	43	2	no	—
194	C₂×C₂²⋊C₄	C₂×D₄₃	1	2	yes	—
195	C₂×C₄²	—	43	2	no	—
196	C₂×C₄²	—	43	1	no	—
197	C₂×C₄²	—	43	2	no	—
198	C₂×C₄²	—	43	1	no	—
199	C₂×C₄²	—	43	16	no	—
200	C₂×C₄²	—	43	2	no	—
201	C₂×C₄²	—	43	1	no	—
202	C₂×C₄²	—	43	8	no	—
203	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
204	C₂⁴	—	43	1	no	—
205	C₂⁴	C₂²×D₄₃	1	2	yes	—
206	#41	—	43	64	no	—
207	C₂×Q₈	—	43	8	no	—
208	C₂×Q₈	—	43	4	no	—
209	C₂×D₄	—	43	16	no	—
210	C₂×D₄	—	43	8	no	—
211	C₂×D₄	C₄×D₄₃	1	8	yes	complemented
212	C₂×D₄	C₂²×D₄₃	1	4	yes	—
213	C₂²×C₄	—	2	8	no	—
214	C₂²×C₄	—	43	16	no	—
215	C₂²×C₄	—	43	1	no	—
216	C₂²×C₄	—	43	1	no	—
217	C₂²×C₄	—	43	2	no	—
218	C₂²×C₄	—	43	8	no	—
219	C₂²×C₄	—	43	8	no	—
220	C₂²×C₄	—	43	4	no	—
221	C₂²×C₄	—	43	2	no	—
222	C₂²×C₄	—	43	1	no	—
223	C₂²×C₄	—	43	8	no	—
224	C₂²×C₄	—	43	4	no	—
225	C₂²×C₄	—	43	4	no	—
226	C₂²×C₄	—	86	4	no	—
227	C₂²×C₄	—	86	8	no	—
228	C₂²×C₄	—	86	4	no	—
229	C₂²×C₄	C₂²×D₄₃	1	2	yes	—
230	C₂²×C₄	C₂²×D₄₃	1	8	yes	—
231	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
232	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
233	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
234	C₄⋊C₄	—	43	8	no	—
235	C₄⋊C₄	—	43	8	no	—
236	C₄⋊C₄	—	43	4	no	—
237	C₄⋊C₄	C₂²×D₄₃	1	4	yes	—
238	C₂²⋊C₄	—	43	8	no	—
239	C₂²⋊C₄	—	43	8	no	—
240	C₂²⋊C₄	C₂²×D₄₃	1	8	yes	—
241	C₄²	—	43	4	no	—
242	C₄²	—	43	8	no	—
243	C₄²	—	43	4	no	—
244	C₄²	—	43	8	no	—
245	C₄²	—	43	4	no	—
246	C₄²	—	43	8	no	—
247	C₄²	C₂²×D₄₃	1	4	yes	—
248	C₂³	—	2	4	no	—
249	C₂³	—	2	4	no	—
250	C₂³	—	43	4	no	—
251	C₂³	—	43	2	no	—
252	C₂³	—	86	2	no	—
253	C₂³	—	86	2	no	—
254	C₂³	C₂×C₄×D₄₃	1	8	yes	—
255	C₂³	C₂³×D₄₃	1	4	yes	—
256	C₂³	C₂³×D₄₃	1	1	yes	characteristic
257	Q₈	—	43	16	no	—
258	D₄	—	43	32	no	—
259	D₄	C₂×C₄×D₄₃	1	16	yes	complemented
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	2	16	no	—
263	C₂×C₄	—	43	4	no	—
264	C₂×C₄	—	43	8	no	—
265	C₂×C₄	—	43	32	no	—
266	C₂×C₄	—	43	4	no	—
267	C₂×C₄	—	43	2	no	—
268	C₂×C₄	—	43	2	no	—
269	C₂×C₄	—	43	4	no	—
270	C₂×C₄	—	43	16	no	—
271	C₂×C₄	—	43	4	no	—
272	C₂×C₄	—	43	2	no	—
273	C₂×C₄	—	86	8	no	—
274	C₂×C₄	—	86	16	no	—
275	C₂×C₄	—	86	4	no	—
276	C₂×C₄	—	86	2	no	—
277	C₂×C₄	—	86	8	no	—
278	C₂×C₄	C₂×C₄×D₄₃	1	4	yes	—
279	C₂×C₄	#10621900	1	4	yes	direct product factor
280	C₂×C₄	C₂³×D₄₃	1	4	yes	—
281	C₂×C₄	C₂³×D₄₃	1	2	yes	—
282	C₂×C₄	C₂³×D₄₃	1	2	yes	—
283	C₂×C₄	C₂³×D₄₃	1	2	yes	—
284	C₂²	—	2	16	no	—
285	C₂²	—	2	4	no	—
286	C₂²	—	2	4	no	—
287	C₂²	—	43	4	no	—
288	C₂²	—	86	8	no	—
289	C₂²	—	86	2	no	—
290	C₂²	—	86	2	no	—
291	C₂²	C₂²×C₄×D₄₃	1	8	yes	—
292	C₂²	C₂²×C₄×D₄₃	1	2	yes	—
293	C₂²	#41×D₄₃	1	2	yes	—
294	C₂²	C₂×#10621900	1	2	yes	—
295	C₂²	C₂⁴×D₄₃	1	1	yes	characteristic
296	C₄	—	2	2	no	—
297	C₄	—	2	4	no	—
298	C₄	—	43	4	no	—
299	C₄	—	43	8	no	—
300	C₄	—	43	4	no	—
301	C₄	—	86	4	no	—
302	C₄	—	86	2	no	—
303	C₄	C₂²×C₄×D₄₃	1	4	yes	—
304	C₄	C₂×#10621900	1	4	yes	direct product factor
305	C₂	—	2	8	no	—
306	C₂	—	86	4	no	—
307	C₂	C₄×#10621900	1	4	yes	direct product factor
308	C₂	C₂³×C₄×D₄₃	1	1	yes	characteristic
309	C₂	C₂×#41×D₄₃	1	1	yes	characteristic
310	C₂	C₂²×#10621900	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄₃
commutator	C₂×C₄₃	C₂⁴×C₄
Frattini	C₂²	C₂⁴×D₄₃
Fitting	C₂×C₄×D₄×C₄₃	C₂
socle	C₂³×C₄₃	C₂⁴
hypercenter	C₂×C₄×D₄	D₄₃
hypocenter	C₄₃	C₂×C₄×#41
LCS	G ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
2. C₄×#10621900	C₂	direct (N × H)
3. C₂×C₄×Dic₈₆	C₂	semidirect
4. C₂×#13063290	C₂	semidirect
5. C₂×D₄×Dic₄₃	C₂	semidirect
6. C₂×C₄×C₂²⋊D₄₃	C₂	semidirect
7. C₂×(C₂²⋊D₄₃)⋊C₄	C₂	semidirect
8. C₂×#13063283	C₂	semidirect
9. C₂×#13063281	C₂	semidirect
10. C₂²×C₄×Dic₄₃	C₂	semidirect
11. C₂×C₄²×D₄₃	C₂	semidirect
12. C₂×C₄×D₄×C₄₃	C₂	semidirect
13. C₂×#10621900	C₄	direct (N × H)
22. C₄×Dic₈₆	C₂²	semidirect
23. #13063290	C₂²	semidirect
24. D₄×Dic₄₃	C₂²	semidirect
25. C₄×C₂²⋊D₄₃	C₂²	semidirect
26. (C₂²⋊D₄₃)⋊C₄	C₂²	semidirect
27. #13063283	C₂²	semidirect
28. #13063281	C₂²	semidirect
40. C₂×C₄×Dic₄₃	C₂²	semidirect
41. C₄²×D₄₃	C₂²	semidirect
44. C₄×D₄×C₄₃	C₂²	semidirect
50. C₂×D₄×C₄₃	C₂×C₄	semidirect
58. C₂×C₂²⋊D₄₃	C₂×C₄	semidirect
61. C₂²×Dic₄₃	C₂×C₄	semidirect
69. #10621900	C₂×C₄	direct (N × H)
72. C₂×C₄×D₄₃	C₂×C₄	semidirect
76. C₂×Dic₈₆	C₂×C₄	semidirect
101. D₄×C₄₃	C₂²×C₄	semidirect
106. C₂×C₄×C₄₃	#41	semidirect
111. C₂²⋊D₄₃	C₂²×C₄	semidirect
117. C₂×Dic₄₃	C₂²×C₄	semidirect
118. C₂×Dic₄₃	#41	semidirect
126. C₄×D₄₃	C₂²×C₄	semidirect
127. Dic₈₆	C₂²×C₄	semidirect
143. C₄×C₄₃	C₂×#41	semidirect
148. Dic₄₃	C₂×#41	semidirect
151. C₂×C₄₃	C₄×#41	semidirect
163. C₂×C₄×D₄	D₄₃	semidirect
168. C₄₃	C₂×C₄×#41	semidirect
185. C₄×D₄	C₂×D₄₃	semidirect
211. C₂×D₄	C₄×D₄₃	semidirect
259. D₄	C₂×C₄×D₄₃	semidirect
279. C₂×C₄	#10621900	direct (N × H)
304. C₄	C₂×#10621900	direct (N × H)
307. C₂	C₄×#10621900	direct (N × H)

Subgroups — C2×C4×#10621900 #19147773

Conjugacy classes of subgroups

Complements

Subgroups — C₂×C₄×#10621900 #19147773