Subgroups — C₂×#22612772 #33245180

Order 18688.

5718 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Q₈×Dic₇₃	C₂	1	2	yes	maximal, complemented
2	#22612772	C₂	1	16	yes	maximal, direct product factor
3	C₂×#17988490	C₂	1	6	yes	maximal, complemented
4	C₂×#17988447	C₂	1	3	yes	maximal, complemented
5	C₂×C₄×C₂²⋊Dic₇₃	C₂	1	3	yes	maximal, complemented
6	C₂³×#41×C₇₃	C₂	1	1	yes	maximal, characteristic
7	C₂×Q₈×Dic₇₃	—	2	16	no	—
8	C₂×Q₈×Dic₇₃	C₂²	1	16	yes	complemented
9	C₂²×C₄⋊Dic₇₃	C₂²	1	6	yes	—
10	C₂²×C₂²⋊Dic₇₃	C₂²	1	6	yes	—
11	C₂²×C₂²⋊Dic₇₃	C₂²	1	6	yes	—
12	C₂²×C₄×Dic₇₃	C₂²	1	6	yes	—
13	#17988490	C₂²	1	48	yes	complemented
14	#17988447	C₂²	1	24	yes	complemented
15	C₄×C₂²⋊Dic₇₃	C₂²	1	24	yes	complemented
16	C₂×#14556498	C₂²	1	1	yes	characteristic
17	C₂×#14556498	C₂²	1	3	yes	—
18	C₂²×#41×C₇₃	—	2	16	no	—
19	C₂²×#41×C₇₃	C₄	1	16	yes	complemented
20	C₂²×#41×C₇₃	C₂²	1	8	yes	—
21	C₂³×Q₈×C₇₃	C₂²	1	1	yes	characteristic
22	C₂³×D₄×C₇₃	C₂²	1	3	yes	—
23	C₂⁴×C₄×C₇₃	C₂²	1	3	yes	—
24	C₂³×Dic₇₃	C₂³	1	6	yes	—
25	C₂³×Dic₇₃	C₂³	1	2	yes	—
26	Q₈×Dic₇₃	—	2	64	no	—
27	C₂×C₄⋊Dic₇₃	—	2	24	no	—
28	C₂×C₄⋊Dic₇₃	C₂³	1	24	yes	—
29	C₂×C₂²⋊Dic₇₃	—	2	24	no	—
30	C₂×C₂²⋊Dic₇₃	—	4	12	no	—
31	C₂×C₂²⋊Dic₇₃	C₂³	1	24	yes	—
32	C₂×C₂²⋊Dic₇₃	C₂³	1	24	yes	—
33	C₂×C₄×Dic₇₃	—	2	24	no	—
34	C₂×C₄×Dic₇₃	C₂³	1	24	yes	—
35	#14556498	C₂³	1	4	yes	—
36	#14556498	C₂³	1	12	yes	—
37	C₂⁵×C₇₃	C₂³	1	3	yes	—
38	C₂×#41×C₇₃	—	2	128	no	—
39	C₂×#41×C₇₃	—	2	64	no	—
40	C₂×#41×C₇₃	C₂×C₄	1	64	yes	complemented
41	C₂²×Q₈×C₇₃	C₂×C₄	1	8	yes	—
42	C₂²×Q₈×C₇₃	D₄	1	16	yes	—
43	C₂²×Q₈×C₇₃	C₂³	1	4	yes	—
44	C₂²×D₄×C₇₃	—	2	24	no	—
45	C₂²×D₄×C₇₃	C₂×C₄	1	24	yes	—
46	C₂²×D₄×C₇₃	C₂³	1	12	yes	—
47	C₂³×C₄×C₇₃	—	2	24	no	—
48	C₂³×C₄×C₇₃	C₂×C₄	1	24	yes	—
49	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
50	C₂³×C₄×C₇₃	C₂³	1	12	yes	—
51	C₂³×C₄×C₇₃	C₂³	1	3	yes	—
52	C₂²×Dic₇₃	—	2	4	no	—
53	C₂²×Dic₇₃	—	2	4	no	—
54	C₂²×Dic₇₃	—	2	12	no	—
55	C₂²×Dic₇₃	—	2	12	no	—
56	C₂²×Dic₇₃	—	2	12	no	—
57	C₂²×Dic₇₃	#41	1	8	yes	complemented
58	C₂²×Dic₇₃	C₂⁴	1	12	yes	—
59	C₂²×Dic₇₃	C₂⁴	1	4	yes	—
60	C₄⋊Dic₇₃	—	2	48	no	—
61	C₂²⋊Dic₇₃	—	2	48	no	—
62	C₂²⋊Dic₇₃	—	4	24	no	—
63	C₄×Dic₇₃	—	2	48	no	—
64	C₂⁴×C₇₃	—	2	12	no	—
65	C₂⁴×C₇₃	—	2	12	no	—
66	C₂⁴×C₇₃	—	2	6	no	—
67	C₂⁴×C₇₃	—	2	6	no	—
68	C₂⁴×C₇₃	C₂²×C₄	1	12	yes	—
69	C₂⁴×C₇₃	C₂⁴	1	6	yes	—
70	C₂⁴×C₇₃	C₂⁴	1	1	yes	characteristic
71	#41×C₇₃	—	2	256	no	—
72	C₂×Q₈×C₇₃	C₂²⋊C₄	1	64	yes	complemented
73	C₂×Q₈×C₇₃	C₂²×C₄	1	16	yes	—
74	C₂×Q₈×C₇₃	C₂×D₄	1	32	yes	—
75	C₂×D₄×C₇₃	—	2	96	no	—
76	C₂×D₄×C₇₃	—	2	48	no	—
77	C₂×D₄×C₇₃	C₂²×C₄	1	48	yes	—
78	C₂²×C₄×C₇₃	—	2	4	no	—
79	C₂²×C₄×C₇₃	—	2	96	no	—
80	C₂²×C₄×C₇₃	—	2	48	no	—
81	C₂²×C₄×C₇₃	C₂²×C₄	1	4	yes	—
82	C₂²×C₄×C₇₃	C₂²×C₄	1	48	yes	—
83	C₂²×C₄×C₇₃	C₂²×C₄	1	12	yes	—
84	C₂²×C₄×C₇₃	C₂×D₄	1	24	yes	—
85	C₂²×C₄×C₇₃	C₂⁴	1	2	yes	—
86	C₂²×C₄×C₇₃	C₂⁴	1	6	yes	—
87	C₂³×C₇₃	—	2	24	no	—
88	C₂³×C₇₃	—	2	24	no	—
89	C₂³×C₇₃	—	2	24	no	—
90	C₂³×C₇₃	—	2	12	no	—
91	C₂³×C₇₃	—	2	12	no	—
92	C₂³×C₇₃	—	2	6	no	—
93	C₂³×C₇₃	—	2	6	no	—
94	C₂³×C₇₃	—	4	24	no	—
95	C₂³×C₇₃	—	4	24	no	—
96	C₂³×C₇₃	C₂³×C₄	1	12	yes	—
97	C₂³×C₇₃	C₂³×C₄	1	2	yes	—
98	C₂³×C₇₃	C₂²×D₄	1	4	yes	—
99	C₂³×C₇₃	C₂×#41	1	4	yes	—
100	C₂³×C₇₃	H(5,𝔽₂)	1	2	yes	—
101	C₂³×C₇₃	2₋¹⁺⁴	1	2	yes	—
102	C₂³×C₇₃	C₂⁵	1	1	yes	characteristic
103	Q₈×C₇₃	C₂×C₂²⋊C₄	1	64	yes	complemented
104	D₄×C₇₃	—	2	96	no	—
105	C₂×C₄×C₇₃	—	2	8	no	—
106	C₂×C₄×C₇₃	—	2	4	no	—
107	C₂×C₄×C₇₃	—	2	96	no	—
108	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	48	yes	—
109	C₂×C₄×C₇₃	C₂³×C₄	1	4	yes	—
110	C₂×C₄×C₇₃	C₂³×C₄	1	12	yes	—
111	C₂×C₄×C₇₃	C₂²×D₄	1	24	yes	—
112	C₂×Dic₇₃	—	2	16	no	—
113	C₂×Dic₇₃	—	2	4	no	—
114	C₂×Dic₇₃	—	2	4	no	—
115	C₂×Dic₇₃	—	2	12	no	—
116	C₂×Dic₇₃	—	2	12	no	—
117	C₂×Dic₇₃	—	2	12	no	—
118	C₂×Dic₇₃	—	4	24	no	—
119	C₂×Dic₇₃	C₂×#41	1	8	yes	complemented
120	C₂²×C₇₃	—	2	12	no	—
121	C₂²×C₇₃	—	2	12	no	—
122	C₂²×C₇₃	—	2	12	no	—
123	C₂²×C₇₃	—	4	48	no	—
124	C₂²×C₇₃	—	4	12	no	—
125	C₂²×C₇₃	—	4	12	no	—
126	C₂²×C₇₃	C₂²×C₂²⋊C₄	1	4	yes	—
127	C₂²×C₇₃	C₄×#41	1	4	yes	—
128	C₂²×C₇₃	#520	1	4	yes	—
129	C₂²×C₇₃	#548	1	8	yes	—
130	C₂²×C₇₃	#550	1	8	yes	—
131	C₂²×C₇₃	C₂⁴×C₄	1	1	yes	characteristic
132	C₂²×C₇₃	C₂³×D₄	1	2	yes	—
133	C₂²×C₇₃	C₂²×#41	1	2	yes	—
134	C₂²×C₇₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
135	C₂²×C₇₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
136	C₄×C₇₃	—	2	4	no	—
137	C₄×C₇₃	C₂²×C₂²⋊C₄	1	24	yes	—
138	Dic₇₃	—	2	8	no	—
139	Dic₇₃	—	4	12	no	—
140	C₂×#2341	—	73	1	no	maximal, Sylow
141	C₂×C₇₃	—	4	12	no	—
142	C₂×C₇₃	#2341	1	8	yes	complemented
143	C₂×C₇₃	C₂³×C₂²⋊C₄	1	1	yes	characteristic
144	C₂×C₇₃	C₂×C₄×#41	1	1	yes	characteristic
145	C₂×C₇₃	C₂×#520	1	1	yes	characteristic
146	C₂×C₇₃	C₂×#548	1	2	yes	—
147	C₂×C₇₃	C₂×#550	1	2	yes	—
148	C₂³×#41	D₇₃	1	1	yes	characteristic
149	C₂²×C₄×Q₈	—	73	2	no	—
150	#2341	—	73	16	no	—
151	C₂×#390	—	73	6	no	—
152	C₂×#380	—	73	3	no	—
153	C₂×C₄×C₂²⋊C₄	—	73	3	no	—
154	C₇₃	C₂×#2341	1	1	yes	characteristic, complemented, Sylow
155	C₂²×#41	—	2	16	no	—
156	C₂²×#41	Dic₇₃	1	16	yes	complemented
157	C₂²×#41	C₂×D₇₃	1	8	yes	—
158	C₂³×Q₈	C₂×D₇₃	1	1	yes	characteristic
159	C₂³×D₄	C₂×D₇₃	1	3	yes	—
160	C₂⁴×C₄	C₂×D₇₃	1	3	yes	—
161	C₂×C₄×Q₈	—	73	16	no	—
162	C₂×C₄×Q₈	—	146	16	no	—
163	C₂²×C₄⋊C₄	—	73	6	no	—
164	C₂²×C₂²⋊C₄	—	73	6	no	—
165	C₂²×C₂²⋊C₄	—	73	6	no	—
166	C₂²×C₄²	—	73	6	no	—
167	#390	—	73	48	no	—
168	#380	—	73	24	no	—
169	C₄×C₂²⋊C₄	—	73	24	no	—
170	C₂×#95	—	73	1	no	—
171	C₂×#95	—	73	3	no	—
172	C₂⁵	C₂²×D₇₃	1	3	yes	—
173	C₂×#41	—	2	128	no	—
174	C₂×#41	—	2	64	no	—
175	C₂×#41	C₂×Dic₇₃	1	64	yes	complemented
176	C₂²×Q₈	C₂×Dic₇₃	1	8	yes	—
177	C₂²×Q₈	C₂²⋊D₇₃	1	16	yes	—
178	C₂²×Q₈	C₂²×D₇₃	1	4	yes	—
179	C₂²×D₄	—	2	24	no	—
180	C₂²×D₄	C₂×Dic₇₃	1	24	yes	—
181	C₂²×D₄	C₂²×D₇₃	1	12	yes	—
182	C₂³×C₄	—	2	24	no	—
183	C₂³×C₄	—	73	2	no	—
184	C₂³×C₄	—	73	6	no	—
185	C₂³×C₄	C₂×Dic₇₃	1	24	yes	—
186	C₂³×C₄	C₂²×D₇₃	1	12	yes	—
187	C₂³×C₄	C₂²×D₇₃	1	1	yes	characteristic
188	C₂³×C₄	C₂²×D₇₃	1	3	yes	—
189	C₄×Q₈	—	146	64	no	—
190	C₂×C₄⋊C₄	—	73	24	no	—
191	C₂×C₄⋊C₄	—	146	24	no	—
192	C₂×C₂²⋊C₄	—	292	12	no	—
193	C₂×C₂²⋊C₄	—	73	24	no	—
194	C₂×C₂²⋊C₄	—	73	24	no	—
195	C₂×C₂²⋊C₄	—	146	24	no	—
196	C₂×C₄²	—	73	24	no	—
197	C₂×C₄²	—	146	24	no	—
198	#95	—	73	4	no	—
199	#95	—	73	12	no	—
200	C₂⁴	—	2	12	no	—
201	C₂⁴	—	2	12	no	—
202	C₂⁴	—	2	6	no	—
203	C₂⁴	—	2	6	no	—
204	C₂⁴	C₂²×Dic₇₃	1	12	yes	—
205	C₂⁴	C₂³×D₇₃	1	6	yes	—
206	C₂⁴	C₂³×D₇₃	1	1	yes	characteristic
207	#41	—	2	256	no	—
208	C₂×Q₈	C₂²⋊Dic₇₃	1	64	yes	complemented
209	C₂×Q₈	C₂²×Dic₇₃	1	16	yes	—
210	C₂×Q₈	C₂×C₂²⋊D₇₃	1	32	yes	—
211	C₂×D₄	—	2	96	no	—
212	C₂×D₄	—	2	48	no	—
213	C₂×D₄	C₂²×Dic₇₃	1	48	yes	—
214	C₂²×C₄	—	2	4	no	—
215	C₂²×C₄	—	2	96	no	—
216	C₂²×C₄	—	2	48	no	—
217	C₂²×C₄	—	73	8	no	—
218	C₂²×C₄	—	73	4	no	—
219	C₂²×C₄	—	73	12	no	—
220	C₂²×C₄	—	146	12	no	—
221	C₂²×C₄	—	146	12	no	—
222	C₂²×C₄	—	146	12	no	—
223	C₂²×C₄	—	146	4	no	—
224	C₂²×C₄	—	146	4	no	—
225	C₂²×C₄	C₂²×Dic₇₃	1	4	yes	—
226	C₂²×C₄	C₂²×Dic₇₃	1	48	yes	—
227	C₂²×C₄	C₂²×Dic₇₃	1	12	yes	—
228	C₂²×C₄	C₂×C₂²⋊D₇₃	1	24	yes	—
229	C₂²×C₄	C₂³×D₇₃	1	2	yes	—
230	C₂²×C₄	C₂³×D₇₃	1	6	yes	—
231	C₄⋊C₄	—	146	48	no	—
232	C₂²⋊C₄	—	292	24	no	—
233	C₂²⋊C₄	—	146	48	no	—
234	C₄²	—	146	48	no	—
235	C₂³	—	2	12	no	—
236	C₂³	—	2	12	no	—
237	C₂³	—	2	6	no	—
238	C₂³	—	2	6	no	—
239	C₂³	—	2	24	no	—
240	C₂³	—	2	24	no	—
241	C₂³	—	2	24	no	—
242	C₂³	—	4	24	no	—
243	C₂³	—	4	24	no	—
244	C₂³	C₂³×Dic₇₃	1	12	yes	—
245	C₂³	C₂³×Dic₇₃	1	2	yes	—
246	C₂³	C₂²×C₂²⋊D₇₃	1	4	yes	—
247	C₂³	#41×D₇₃	1	4	yes	—
248	C₂³	#14556637	1	2	yes	—
249	C₂³	#14556638	1	2	yes	—
250	C₂³	C₂⁴×D₇₃	1	1	yes	characteristic
251	Q₈	C₂×C₂²⋊Dic₇₃	1	64	yes	complemented
252	D₄	—	2	96	no	—
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	2	8	no	—
255	C₂×C₄	—	2	96	no	—
256	C₂×C₄	—	292	24	no	—
257	C₂×C₄	—	73	8	no	—
258	C₂×C₄	—	146	12	no	—
259	C₂×C₄	—	146	12	no	—
260	C₂×C₄	—	146	12	no	—
261	C₂×C₄	—	146	4	no	—
262	C₂×C₄	—	146	4	no	—
263	C₂×C₄	—	146	16	no	—
264	C₂×C₄	C₂×C₂²⋊Dic₇₃	1	48	yes	—
265	C₂×C₄	C₂³×Dic₇₃	1	4	yes	—
266	C₂×C₄	C₂³×Dic₇₃	1	12	yes	—
267	C₂×C₄	C₂²×C₂²⋊D₇₃	1	24	yes	—
268	C₂²	—	2	12	no	—
269	C₂²	—	2	12	no	—
270	C₂²	—	2	12	no	—
271	C₂²	—	4	48	no	—
272	C₂²	—	4	12	no	—
273	C₂²	—	4	12	no	—
274	C₂²	C₂²×C₂²⋊Dic₇₃	1	4	yes	—
275	C₂²	#41×Dic₇₃	1	4	yes	—
276	C₂²	#17989070	1	4	yes	—
277	C₂²	#17989244	1	8	yes	—
278	C₂²	#17989256	1	8	yes	—
279	C₂²	C₂⁴×Dic₇₃	1	1	yes	characteristic
280	C₂²	C₂³×C₂²⋊D₇₃	1	2	yes	—
281	C₂²	C₂×#41×D₇₃	1	2	yes	—
282	C₂²	C₂×#14556637	1	1	yes	characteristic
283	C₂²	C₂×#14556638	1	1	yes	characteristic
284	C₄	—	2	4	no	—
285	C₄	—	292	12	no	—
286	C₄	—	146	8	no	—
287	C₄	C₂²×C₂²⋊Dic₇₃	1	24	yes	—
288	C₂	—	4	12	no	—
289	C₂	#22612772	1	8	yes	direct product factor
290	C₂	C₂³×C₂²⋊Dic₇₃	1	1	yes	characteristic
291	C₂	C₂×#41×Dic₇₃	1	1	yes	characteristic
292	C₂	C₂×#17989070	1	1	yes	characteristic
293	C₂	C₂×#17989244	1	2	yes	—
294	C₂	C₂×#17989256	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂³×D₇₃
commutator	C₂²×C₇₃	C₂⁴×C₄
Frattini	C₂³	C₂⁴×D₇₃
Fitting, prime radical	C₂³×#41×C₇₃	C₂
socle	C₂⁴×C₇₃	C₂⁴
cosocle	C₂³×C₇₃	C₂⁵
hypercenter	C₂³×#41	D₇₃
hypocenter	C₇₃	C₂×#2341
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂⁴ ◁ C₂³×#41

Complements

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

N	H ≅ G/N	type
1. C₂²×Q₈×Dic₇₃	C₂	semidirect
2. #22612772	C₂	direct (N × H)
3. C₂×#17988490	C₂	semidirect
4. C₂×#17988447	C₂	semidirect
5. C₂×C₄×C₂²⋊Dic₇₃	C₂	semidirect
8. C₂×Q₈×Dic₇₃	C₂²	semidirect
13. #17988490	C₂²	semidirect
14. #17988447	C₂²	semidirect
15. C₄×C₂²⋊Dic₇₃	C₂²	semidirect
19. C₂²×#41×C₇₃	C₄	semidirect
40. C₂×#41×C₇₃	C₂×C₄	semidirect
57. C₂²×Dic₇₃	#41	semidirect
72. C₂×Q₈×C₇₃	C₂²⋊C₄	semidirect
103. Q₈×C₇₃	C₂×C₂²⋊C₄	semidirect
119. C₂×Dic₇₃	C₂×#41	semidirect
142. C₂×C₇₃	#2341	semidirect
154. C₇₃	C₂×#2341	semidirect
156. C₂²×#41	Dic₇₃	semidirect
175. C₂×#41	C₂×Dic₇₃	semidirect
208. C₂×Q₈	C₂²⋊Dic₇₃	semidirect
251. Q₈	C₂×C₂²⋊Dic₇₃	semidirect
289. C₂	#22612772	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×Q₈×Dic₇₃	2
#22612772	2
C₂×#17988490	2
C₂×#17988447	2
C₂×C₄×C₂²⋊Dic₇₃	2
C₂³×#41×C₇₃	2
C₂×#2341	73

Subgroups — C2×#22612772 #33245180

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22612772 #33245180