Subgroups — C₂²×#17988703 #33950274

Order 18688.

2750 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14556613	C₂	1	1	yes	maximal, characteristic
2	C₂²×#14556599	C₂	1	2	yes	maximal, complemented
3	C₂×#17988703	C₂	1	24	yes	maximal, direct product factor
4	C₂²×#14556517	C₂	1	2	yes	maximal, complemented
5	C₂²×C₈×Dic₇₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#108×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#11924028	C₂²	1	2	yes	—
8	C₂×#14556613	C₄	1	12	yes	complemented
9	C₂×#14556613	C₂²	1	12	yes	—
10	C₂²×Dic₂₉₂	C₄	1	1	yes	characteristic, complemented
11	C₂²×#11924017	C₄	1	2	yes	complemented
12	C₂²×D₅₈₄	C₄	1	1	yes	characteristic, complemented
13	C₂×#14556599	—	2	12	no	—
14	C₂×#14556599	C₂²	1	24	yes	complemented
15	C₂²×#11924011	—	2	2	no	—
16	C₂³×#10596831	C₂²	1	1	yes	characteristic
17	C₂²×C₈×D₇₃	—	2	2	no	—
18	C₂²×C₄×Dic₇₃	C₂²	1	1	yes	characteristic
19	#17988703	C₂²	1	64	yes	direct product factor
20	C₂×#14556517	C₂²	1	24	yes	complemented
21	C₂×C₈×Dic₇₃	C₂²	1	12	yes	complemented
22	C₂²×M₁₆×C₇₃	C₂²	1	2	yes	—
23	C₂³×C₈×C₇₃	C₂²	1	1	yes	characteristic
24	C₂×#108×C₇₃	C₂²	1	12	yes	complemented
25	C₂×#11924028	—	2	6	no	—
26	C₂×#11924028	—	2	6	no	—
27	C₂×#11924028	C₂×C₄	1	12	yes	—
28	C₂×#11924028	C₂³	1	12	yes	—
29	C₂²×Dic₁₄₆	C₂×C₄	1	2	yes	—
30	C₂²×D₂₉₂	C₂×C₄	1	2	yes	—
31	C₂²×C₂²⋊D₇₃	—	2	2	no	—
32	C₂³×Dic₇₃	—	2	1	no	—
33	C₂²×C₄×D₇₃	—	2	2	no	—
34	#14556613	C₂×C₄	1	48	yes	complemented
35	#14556613	C₂³	1	16	yes	—
36	C₂×Dic₂₉₂	C₂×C₄	1	12	yes	complemented
37	C₂×#11924017	C₂×C₄	1	24	yes	complemented
38	C₂×D₅₈₄	C₂×C₄	1	12	yes	complemented
39	#14556599	—	2	48	no	—
40	#14556599	C₂³	1	32	yes	complemented
41	C₂×#11924011	—	2	24	no	—
42	C₂²×#10596831	D₄	1	1	yes	characteristic
43	C₂²×#10596831	D₄	1	1	yes	characteristic
44	C₂²×#10596831	D₄	1	6	yes	—
45	C₂²×#10596831	C₂³	1	6	yes	—
46	C₂×C₈×D₇₃	—	2	24	no	—
47	C₂×C₄×Dic₇₃	—	2	3	no	—
48	C₂×C₄×Dic₇₃	C₂³	1	6	yes	—
49	#14556517	C₂³	1	32	yes	complemented
50	C₈×Dic₇₃	C₂³	1	16	yes	complemented
51	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
52	C₂×M₁₆×C₇₃	—	2	6	no	—
53	C₂×M₁₆×C₇₃	C₂³	1	12	yes	—
54	C₂²×C₈×C₇₃	—	2	2	no	—
55	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
56	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
57	C₂²×C₈×C₇₃	C₂×C₄	1	6	yes	—
58	C₂²×C₈×C₇₃	C₂³	1	6	yes	—
59	#108×C₇₃	C₂³	1	16	yes	complemented
60	C₂³×D₇₃	—	2	2	no	—
61	#11924028	—	2	24	no	—
62	#11924028	—	2	12	no	—
63	#11924028	—	2	12	no	—
64	#11924028	C₂²×C₄	1	24	yes	—
65	#11924028	C₂⁴	1	8	yes	—
66	C₂×Dic₁₄₆	—	2	6	no	—
67	C₂×Dic₁₄₆	C₂²×C₄	1	12	yes	—
68	C₂×D₂₉₂	—	2	6	no	—
69	C₂×D₂₉₂	C₂²×C₄	1	12	yes	—
70	C₂×C₂²⋊D₇₃	—	2	12	no	—
71	C₂×C₂²⋊D₇₃	—	2	12	no	—
72	C₂²×Dic₇₃	—	2	8	no	—
73	C₂²×Dic₇₃	—	2	2	no	—
74	C₂²×Dic₇₃	—	2	6	no	—
75	C₂×C₄×D₇₃	—	2	12	no	—
76	C₂×C₄×D₇₃	—	2	12	no	—
77	Dic₂₉₂	C₂²×C₄	1	16	yes	complemented
78	#11924017	C₂²×C₄	1	32	yes	complemented
79	D₅₈₄	C₂²×C₄	1	16	yes	complemented
80	#11924011	—	2	32	no	—
81	C₂×#10596831	C₂×D₄	1	6	yes	—
82	C₂×#10596831	C₂×D₄	1	6	yes	—
83	C₂×#10596831	C₂×D₄	1	12	yes	—
84	C₂×#10596831	C₂⁴	1	4	yes	—
85	C₈×D₇₃	—	2	32	no	—
86	C₄×Dic₇₃	—	2	6	no	—
87	C₄×Dic₇₃	C₂⁴	1	4	yes	—
88	C₂⁴×C₇₃	#41	1	1	yes	characteristic
89	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
90	C₂²×C₄×C₇₃	C₂²×C₄	1	3	yes	—
91	C₂²×C₄×C₇₃	C₂×D₄	1	3	yes	—
92	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₇₃	#41	1	3	yes	—
94	C₂²×C₄×C₇₃	C₂⁴	1	3	yes	—
95	M₁₆×C₇₃	—	2	12	no	—
96	M₁₆×C₇₃	C₂⁴	1	8	yes	—
97	C₂×C₈×C₇₃	—	2	12	no	—
98	C₂×C₈×C₇₃	C₂²×C₄	1	6	yes	—
99	C₂×C₈×C₇₃	C₂²×C₄	1	6	yes	—
100	C₂×C₈×C₇₃	C₂²×C₄	1	12	yes	—
101	C₂×C₈×C₇₃	C₂⁴	1	4	yes	—
102	C₂³×C₇₃	—	2	4	no	—
103	C₂³×C₇₃	C₄×D₄	1	3	yes	—
104	C₂³×C₇₃	C₄×D₄	1	1	yes	characteristic
105	C₂³×C₇₃	C₂×#41	1	3	yes	—
106	C₂²×D₇₃	—	2	12	no	—
107	C₂²×D₇₃	—	4	8	no	—
108	C₂×C₄×C₇₃	C₄×D₄	1	3	yes	—
109	C₂×C₄×C₇₃	C₄×D₄	1	3	yes	—
110	C₂×C₄×C₇₃	C₄×D₄	1	6	yes	—
111	C₂×C₄×C₇₃	C₂³×C₄	1	3	yes	—
112	C₂×C₄×C₇₃	C₂³×C₄	1	3	yes	—
113	C₂×C₄×C₇₃	C₂²×D₄	1	3	yes	—
114	C₂×C₄×C₇₃	C₂²×D₄	1	3	yes	—
115	C₂×C₄×C₇₃	C₂×#41	1	3	yes	—
116	C₂×C₄×C₇₃	C₂⁵	1	1	yes	characteristic
117	C₂²⋊D₇₃	—	2	24	no	—
118	C₂²⋊D₇₃	—	2	8	no	—
119	C₂×Dic₇₃	—	2	24	no	—
120	C₂×Dic₇₃	—	2	12	no	—
121	C₂×Dic₇₃	—	2	4	no	—
122	D₂₉₂	—	2	12	no	—
123	D₂₉₂	C₂³×C₄	1	8	yes	—
124	C₄×D₇₃	—	2	24	no	—
125	C₄×D₇₃	—	2	8	no	—
126	Dic₁₄₆	—	2	12	no	—
127	Dic₁₄₆	C₂³×C₄	1	8	yes	—
128	C₈×C₇₃	—	2	8	no	—
129	C₈×C₇₃	C₂³×C₄	1	4	yes	—
130	C₈×C₇₃	C₂³×C₄	1	4	yes	—
131	#10596831	C₂²×D₄	1	4	yes	—
132	#10596831	C₂²×D₄	1	4	yes	—
133	C₂²×C₇₃	—	2	6	no	—
134	C₂²×C₇₃	—	2	6	no	—
135	C₂²×C₇₃	#443	1	4	yes	complemented
136	C₂²×C₇₃	C₂×C₄×D₄	1	3	yes	—
137	C₂²×C₇₃	C₂×C₄×D₄	1	3	yes	—
138	C₂²×C₇₃	C₂²×#41	1	1	yes	characteristic
139	C₂×D₇₃	—	2	8	no	—
140	C₂×D₇₃	—	4	24	no	—
141	C₄×C₇₃	C₂×C₄×D₄	1	3	yes	—
142	C₄×C₇₃	C₂×C₄×D₄	1	3	yes	—
143	C₄×C₇₃	C₂⁴×C₄	1	1	yes	characteristic
144	C₄×C₇₃	C₂³×D₄	1	1	yes	characteristic
145	Dic₇₃	—	2	8	no	—
146	Dic₇₃	—	2	8	no	—
147	C₂²×#443	—	73	1	no	maximal, Sylow
148	C₂×C₇₃	—	2	1	no	—
149	C₂×C₇₃	—	2	3	no	—
150	C₂×C₇₃	C₂×#443	1	6	yes	complemented
151	C₂×C₇₃	C₂²×C₄×D₄	1	1	yes	characteristic
152	D₇₃	—	4	8	no	—
153	C₂²×#135	—	73	1	no	—
154	C₂²×#131	—	73	2	no	—
155	C₂×#443	—	73	24	no	—
156	C₂²×#108	D₇₃	1	1	yes	characteristic, complemented
157	C₂²×#104	—	73	2	no	—
158	C₂²×C₄×C₈	—	73	1	no	—
159	C₇₃	C₂²×#443	1	1	yes	characteristic, complemented, Sylow
160	C₂²×#41	—	73	2	no	—
161	C₂×#135	—	73	12	no	—
162	C₂×#135	—	73	12	no	—
163	C₂²×Q₁₆	—	73	1	no	—
164	C₂²×SD₁₆	—	73	2	no	—
165	C₂²×D₈	—	73	1	no	—
166	C₂×#131	—	73	24	no	—
167	C₂×#131	—	146	12	no	—
168	C₂²×M₁₆	—	146	2	no	—
169	C₂²×M₁₆	C₂×D₇₃	1	2	yes	—
170	C₂³×C₈	—	73	1	no	—
171	C₂³×C₈	—	146	2	no	—
172	C₂³×C₈	C₂×D₇₃	1	1	yes	characteristic
173	C₂²×C₄²	—	73	1	no	—
174	#443	—	73	64	no	—
175	C₂×#108	C₂×D₇₃	1	12	yes	complemented
176	C₂×#104	—	73	24	no	—
177	C₂×C₄×C₈	—	73	12	no	—
178	C₂×#41	—	73	12	no	—
179	C₂×#41	—	73	12	no	—
180	C₂×#41	—	146	6	no	—
181	C₂×#41	—	146	6	no	—
182	C₂²×Q₈	—	73	2	no	—
183	C₂²×D₄	—	73	2	no	—
184	C₂²×D₄	—	146	2	no	—
185	C₂³×C₄	—	146	1	no	—
186	C₂³×C₄	—	146	2	no	—
187	C₂³×C₄	C₂²×D₇₃	1	1	yes	characteristic
188	#135	—	73	48	no	—
189	#135	—	73	16	no	—
190	C₂×Q₁₆	—	73	12	no	—
191	C₂×SD₁₆	—	73	24	no	—
192	C₂×D₈	—	73	12	no	—
193	#131	—	73	32	no	—
194	#131	—	146	48	no	—
195	C₂×M₁₆	—	2	6	no	—
196	C₂×M₁₆	—	146	24	no	—
197	C₂×M₁₆	C₂²×D₇₃	1	12	yes	—
198	C₂²×C₈	—	2	2	no	—
199	C₂²×C₈	—	73	6	no	—
200	C₂²×C₈	—	73	1	no	—
201	C₂²×C₈	—	73	1	no	—
202	C₂²×C₈	—	73	6	no	—
203	C₂²×C₈	—	146	24	no	—
204	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
205	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
206	C₂²×C₈	C₄×D₇₃	1	6	yes	—
207	C₂²×C₈	C₂²×D₇₃	1	6	yes	—
208	C₂×C₄²	—	73	6	no	—
209	C₂×C₄²	—	146	3	no	—
210	#108	C₂²×D₇₃	1	16	yes	complemented
211	#104	—	73	32	no	—
212	C₄×C₈	—	73	16	no	—
213	C₂⁴	—	146	2	no	—
214	C₂⁴	#11924029	1	1	yes	characteristic
215	#41	—	73	24	no	—
216	#41	—	73	8	no	—
217	#41	—	146	24	no	—
218	#41	—	146	12	no	—
219	#41	—	146	12	no	—
220	C₂×Q₈	—	73	12	no	—
221	C₂×Q₈	—	146	6	no	—
222	C₂×D₄	—	73	12	no	—
223	C₂×D₄	—	146	12	no	—
224	C₂×D₄	—	146	12	no	—
225	C₂×D₄	—	146	6	no	—
226	C₂²×C₄	—	146	8	no	—
227	C₂²×C₄	—	146	2	no	—
228	C₂²×C₄	—	146	6	no	—
229	C₂²×C₄	—	146	12	no	—
230	C₂²×C₄	—	146	12	no	—
231	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
232	C₂²×C₄	C₂×C₄×D₇₃	1	3	yes	—
233	C₂²×C₄	D₄×D₇₃	1	3	yes	—
234	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
235	C₂²×C₄	#11924029	1	3	yes	—
236	C₂²×C₄	C₂³×D₇₃	1	3	yes	—
237	Q₁₆	—	73	16	no	—
238	SD₁₆	—	73	32	no	—
239	D₈	—	73	16	no	—
240	M₁₆	—	2	12	no	—
241	M₁₆	—	146	32	no	—
242	M₁₆	C₂³×D₇₃	1	8	yes	—
243	C₂×C₈	—	2	12	no	—
244	C₂×C₈	—	73	6	no	—
245	C₂×C₈	—	73	6	no	—
246	C₂×C₈	—	73	12	no	—
247	C₂×C₈	—	73	4	no	—
248	C₂×C₈	—	146	32	no	—
249	C₂×C₈	C₂×C₄×D₇₃	1	6	yes	—
250	C₂×C₈	C₂×C₄×D₇₃	1	6	yes	—
251	C₂×C₈	C₂×C₄×D₇₃	1	12	yes	—
252	C₂×C₈	C₂³×D₇₃	1	4	yes	—
253	C₄²	—	73	4	no	—
254	C₄²	—	146	6	no	—
255	C₂³	—	2	4	no	—
256	C₂³	—	292	8	no	—
257	C₂³	—	146	12	no	—
258	C₂³	#14556553	1	3	yes	—
259	C₂³	#14556553	1	1	yes	characteristic
260	C₂³	C₂×#11924029	1	3	yes	—
261	Q₈	—	73	8	no	—
262	Q₈	—	146	12	no	—
263	D₄	—	73	8	no	—
264	D₄	—	146	8	no	—
265	D₄	—	146	24	no	—
266	D₄	—	146	12	no	—
267	C₂×C₄	—	146	24	no	—
268	C₂×C₄	—	146	4	no	—
269	C₂×C₄	—	146	12	no	—
270	C₂×C₄	—	146	8	no	—
271	C₂×C₄	—	146	24	no	—
272	C₂×C₄	#14556553	1	3	yes	—
273	C₂×C₄	#14556553	1	3	yes	—
274	C₂×C₄	#14556553	1	6	yes	—
275	C₂×C₄	C₂²×C₄×D₇₃	1	3	yes	—
276	C₂×C₄	C₂²×C₄×D₇₃	1	3	yes	—
277	C₂×C₄	C₂×D₄×D₇₃	1	3	yes	—
278	C₂×C₄	C₂×D₄×D₇₃	1	3	yes	—
279	C₂×C₄	C₂×#11924029	1	3	yes	—
280	C₂×C₄	C₂⁴×D₇₃	1	1	yes	characteristic
281	C₈	—	2	8	no	—
282	C₈	—	73	4	no	—
283	C₈	—	73	4	no	—
284	C₈	C₂²×C₄×D₇₃	1	4	yes	—
285	C₈	C₂²×C₄×D₇₃	1	4	yes	—
286	C₂²	—	2	6	no	—
287	C₂²	—	2	6	no	—
288	C₂²	—	292	24	no	—
289	C₂²	—	146	8	no	—
290	C₂²	#17988703	1	4	yes	direct product factor
291	C₂²	C₂×#14556553	1	3	yes	—
292	C₂²	C₂×#14556553	1	3	yes	—
293	C₂²	C₂²×#11924029	1	1	yes	characteristic
294	C₄	—	146	8	no	—
295	C₄	—	146	8	no	—
296	C₄	C₂×#14556553	1	3	yes	—
297	C₄	C₂×#14556553	1	3	yes	—
298	C₄	C₂³×C₄×D₇₃	1	1	yes	characteristic
299	C₄	C₂²×D₄×D₇₃	1	1	yes	characteristic
300	C₂	—	2	1	no	—
301	C₂	—	2	3	no	—
302	C₂	—	292	8	no	—
303	C₂	C₂×#17988703	1	6	yes	direct product factor
304	C₂	C₂²×#14556553	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₇₃
commutator	C₄×C₇₃	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₇₃
Fitting	C₂²×#108×C₇₃	C₂
socle	C₂³×C₇₃	C₄×D₄
prime radical	C₂²×#14556613	C₂
cosocle	C₂×C₄×C₇₃	C₂⁵
hypercenter	C₂²×#108	D₇₃
hypocenter	C₇₃	C₂²×#443
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂²×#108

Complements

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

N	H ≅ G/N	type
2. C₂²×#14556599	C₂	semidirect
3. C₂×#17988703	C₂	direct (N × H)
4. C₂²×#14556517	C₂	semidirect
5. C₂²×C₈×Dic₇₃	C₂	semidirect
6. C₂²×#108×C₇₃	C₂	semidirect
8. C₂×#14556613	C₄	semidirect
10. C₂²×Dic₂₉₂	C₄	semidirect
11. C₂²×#11924017	C₄	semidirect
12. C₂²×D₅₈₄	C₄	semidirect
14. C₂×#14556599	C₂²	semidirect
19. #17988703	C₂²	direct (N × H)
20. C₂×#14556517	C₂²	semidirect
21. C₂×C₈×Dic₇₃	C₂²	semidirect
24. C₂×#108×C₇₃	C₂²	semidirect
34. #14556613	C₂×C₄	semidirect
36. C₂×Dic₂₉₂	C₂×C₄	semidirect
37. C₂×#11924017	C₂×C₄	semidirect
38. C₂×D₅₈₄	C₂×C₄	semidirect
40. #14556599	C₂³	semidirect
49. #14556517	C₂³	semidirect
50. C₈×Dic₇₃	C₂³	semidirect
59. #108×C₇₃	C₂³	semidirect
77. Dic₂₉₂	C₂²×C₄	semidirect
78. #11924017	C₂²×C₄	semidirect
79. D₅₈₄	C₂²×C₄	semidirect
135. C₂²×C₇₃	#443	semidirect
150. C₂×C₇₃	C₂×#443	semidirect
156. C₂²×#108	D₇₃	semidirect
159. C₇₃	C₂²×#443	semidirect
175. C₂×#108	C₂×D₇₃	semidirect
210. #108	C₂²×D₇₃	semidirect
290. C₂²	#17988703	direct (N × H)
303. C₂	C₂×#17988703	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×#14556613	2
C₂²×#14556599	2
C₂×#17988703	2
C₂²×#14556517	2
C₂²×C₈×Dic₇₃	2
C₂²×#108×C₇₃	2
C₂²×#443	73

Subgroups — C22×#17988703 #33950274

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#17988703 #33950274