Subgroups — C₂²×#17989404 #33999810

Order 18688.

10774 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14556638	C₂	1	2	yes	maximal, complemented
2	C₂×#17989404	C₂	1	48	yes	maximal, direct product factor
3	C₂²×#14556624	C₂	1	6	yes	maximal, complemented
4	C₂²×#14556613	C₂	1	3	yes	maximal, complemented
5	C₂³×Dic₂₉₂	C₂	1	3	yes	maximal, complemented
6	C₂²×#131×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#14556638	—	2	24	no	—
8	C₂×#14556638	C₂²	1	48	yes	complemented
9	C₂²×#11924028	C₂²	1	6	yes	—
10	C₂²×#11924027	—	2	6	no	—
11	C₂³×Dic₁₄₆	C₂²	1	6	yes	—
12	C₂²×Q₈×D₇₃	—	2	2	no	—
13	#17989404	C₂²	1	256	yes	direct product factor
14	C₂×#14556624	C₂²	1	144	yes	complemented
15	C₂×#14556613	C₂²	1	72	yes	complemented
16	C₂²×Dic₂₉₂	C₂²	1	3	yes	—
17	C₂²×Dic₂₉₂	C₂²	1	6	yes	—
18	C₂²×Dic₂₉₂	C₂²	1	72	yes	complemented
19	C₂²×#11924017	C₂²	1	6	yes	—
20	C₂²×D₅₈₄	C₂²	1	1	yes	characteristic
21	C₂²×#41×C₇₃	C₂²	1	1	yes	characteristic
22	C₂×#131×C₇₃	C₂²	1	24	yes	complemented
23	C₂²×M₁₆×C₇₃	C₂²	1	3	yes	—
24	C₂³×C₈×C₇₃	C₂²	1	3	yes	—
25	#14556638	—	2	192	no	—
26	#14556638	C₂³	1	128	yes	complemented
27	C₂×#11924028	—	2	36	no	—
28	C₂×#11924028	C₂³	1	72	yes	—
29	C₂×#11924027	—	2	144	no	—
30	C₂²×Dic₁₄₆	—	2	6	no	—
31	C₂²×Dic₁₄₆	—	2	36	no	—
32	C₂²×Dic₁₄₆	C₂³	1	72	yes	—
33	C₂²×Dic₁₄₆	C₂³	1	6	yes	—
34	C₂×Q₈×D₇₃	—	2	48	no	—
35	C₂²×D₂₉₂	C₂³	1	2	yes	—
36	C₂²×C₂²⋊D₇₃	—	2	6	no	—
37	C₂³×Dic₇₃	—	2	6	no	—
38	C₂²×C₄×D₇₃	—	2	6	no	—
39	#14556624	C₂³	1	384	yes	complemented
40	#14556613	C₂³	1	192	yes	complemented
41	C₂×Dic₂₉₂	C₂³	1	72	yes	—
42	C₂×Dic₂₉₂	C₂³	1	36	yes	—
43	C₂×Dic₂₉₂	C₂³	1	192	yes	complemented
44	C₂×#11924017	C₂³	1	72	yes	—
45	C₂×D₅₈₄	C₂³	1	12	yes	—
46	C₂×#41×C₇₃	D₄	1	12	yes	—
47	C₂×#41×C₇₃	C₂³	1	12	yes	—
48	C₂²×Q₈×C₇₃	D₄	1	1	yes	characteristic
49	C₂²×D₄×C₇₃	D₄	1	3	yes	—
50	C₂³×C₄×C₇₃	C₂³	1	3	yes	—
51	#131×C₇₃	C₂³	1	64	yes	complemented
52	C₂×M₁₆×C₇₃	C₂³	1	36	yes	—
53	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
54	C₂²×C₈×C₇₃	C₂³	1	3	yes	—
55	C₂²×C₈×C₇₃	C₂³	1	36	yes	—
56	C₂³×D₇₃	—	2	2	no	—
57	#11924028	—	2	144	no	—
58	#11924028	C₂⁴	1	96	yes	—
59	#11924027	—	2	384	no	—
60	C₂×Dic₁₄₆	—	2	72	no	—
61	C₂×Dic₁₄₆	—	2	18	no	—
62	C₂×Dic₁₄₆	—	2	144	no	—
63	C₂×Dic₁₄₆	C₂⁴	1	36	yes	—
64	C₂×Dic₁₄₆	C₂⁴	1	96	yes	—
65	Q₈×D₇₃	—	2	128	no	—
66	C₂×D₂₉₂	—	2	6	no	—
67	C₂×D₂₉₂	C₂⁴	1	12	yes	—
68	C₂×C₂²⋊D₇₃	—	2	72	no	—
69	C₂²×Dic₇₃	—	2	6	no	—
70	C₂²×Dic₇₃	—	2	72	no	—
71	C₂×C₄×D₇₃	—	2	72	no	—
72	Dic₂₉₂	C₂⁴	1	48	yes	—
73	Dic₂₉₂	C₂⁴	1	96	yes	—
74	#11924017	C₂⁴	1	96	yes	—
75	D₅₈₄	C₂⁴	1	16	yes	—
76	C₂⁴×C₇₃	C₂×D₄	1	3	yes	—
77	#41×C₇₃	C₂×D₄	1	48	yes	—
78	#41×C₇₃	C₂⁴	1	16	yes	—
79	C₂×Q₈×C₇₃	C₂×D₄	1	12	yes	—
80	C₂×D₄×C₇₃	C₂×D₄	1	36	yes	—
81	C₂²×C₄×C₇₃	C₂×D₄	1	3	yes	—
82	C₂²×C₄×C₇₃	C₂×D₄	1	18	yes	—
83	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
84	C₂²×C₄×C₇₃	C₂⁴	1	18	yes	—
85	M₁₆×C₇₃	C₂⁴	1	48	yes	—
86	C₂×C₈×C₇₃	C₂⁴	1	6	yes	—
87	C₂×C₈×C₇₃	C₂⁴	1	18	yes	—
88	C₂×C₈×C₇₃	C₂⁴	1	48	yes	—
89	C₂³×C₇₃	—	2	12	no	—
90	C₂³×C₇₃	C₂²×D₄	1	18	yes	—
91	C₂³×C₇₃	C₂²×D₄	1	1	yes	characteristic
92	C₂²×D₇₃	—	2	12	no	—
93	C₂²×D₇₃	—	4	8	no	—
94	Q₈×C₇₃	C₂²×D₄	1	16	yes	—
95	D₄×C₇₃	C₂²×D₄	1	48	yes	—
96	C₂×C₄×C₇₃	C₂²×D₄	1	3	yes	—
97	C₂×C₄×C₇₃	C₂²×D₄	1	18	yes	—
98	C₂×C₄×C₇₃	C₂²×D₄	1	36	yes	—
99	C₂×C₄×C₇₃	C₂⁵	1	3	yes	—
100	C₂×C₄×C₇₃	C₂⁵	1	12	yes	—
101	C₂²⋊D₇₃	—	2	96	no	—
102	C₂×Dic₇₃	—	2	36	no	—
103	C₂×Dic₇₃	—	2	96	no	—
104	D₂₉₂	—	2	12	no	—
105	D₂₉₂	C₂⁵	1	8	yes	—
106	C₄×D₇₃	—	2	96	no	—
107	Dic₁₄₆	—	2	96	no	—
108	Dic₁₄₆	—	2	36	no	—
109	Dic₁₄₆	C₂⁵	1	24	yes	—
110	C₈×C₇₃	C₂⁵	1	4	yes	—
111	C₈×C₇₃	C₂⁵	1	12	yes	—
112	C₂²×C₇₃	—	2	36	no	—
113	C₂²×C₇₃	#578	1	4	yes	complemented
114	C₂²×C₇₃	C₂³×D₄	1	12	yes	—
115	C₂²×C₇₃	C₂³×D₄	1	3	yes	—
116	C₂×D₇₃	—	2	8	no	—
117	C₂×D₇₃	—	4	24	no	—
118	C₄×C₇₃	C₂³×D₄	1	3	yes	—
119	C₄×C₇₃	C₂³×D₄	1	12	yes	—
120	C₄×C₇₃	C₂⁶	1	1	yes	characteristic
121	Dic₇₃	—	2	24	no	—
122	C₂²×#578	—	73	1	no	maximal, Sylow
123	C₂×C₇₃	—	2	12	no	—
124	C₂×C₇₃	C₂×#578	1	6	yes	complemented
125	C₂×C₇₃	C₂⁴×D₄	1	1	yes	characteristic
126	D₇₃	—	4	8	no	—
127	C₂²×2₋¹⁺⁴	—	73	2	no	—
128	C₂×#578	—	73	48	no	—
129	C₂²×#137	—	73	6	no	—
130	C₂²×#135	—	73	3	no	—
131	C₂³×Q₁₆	—	73	3	no	—
132	C₂²×#131	D₇₃	1	1	yes	characteristic, complemented
133	C₇₃	C₂²×#578	1	1	yes	characteristic, complemented, Sylow
134	C₂×2₋¹⁺⁴	—	73	48	no	—
135	C₂×2₋¹⁺⁴	—	146	24	no	—
136	C₂²×#41	—	73	6	no	—
137	C₂²×#41	—	146	6	no	—
138	C₂²×#41	C₂×D₇₃	1	1	yes	characteristic
139	C₂³×Q₈	—	73	6	no	—
140	C₂³×Q₈	—	146	2	no	—
141	#578	—	73	256	no	—
142	C₂×#137	—	73	144	no	—
143	C₂×#135	—	73	72	no	—
144	C₂²×Q₁₆	—	73	3	no	—
145	C₂²×Q₁₆	—	73	6	no	—
146	C₂²×Q₁₆	—	73	72	no	—
147	C₂²×SD₁₆	—	73	6	no	—
148	C₂²×D₈	—	73	1	no	—
149	C₂×#131	C₂×D₇₃	1	24	yes	complemented
150	C₂²×M₁₆	C₂×D₇₃	1	3	yes	—
151	C₂³×C₈	C₂×D₇₃	1	3	yes	—
152	2₋¹⁺⁴	—	73	128	no	—
153	2₋¹⁺⁴	—	146	192	no	—
154	C₂×#41	—	73	72	no	—
155	C₂×#41	—	146	144	no	—
156	C₂×#41	—	146	36	no	—
157	C₂×#41	D₂₉₂	1	12	yes	—
158	C₂×#41	C₂²×D₇₃	1	12	yes	—
159	C₂²×Q₈	—	73	6	no	—
160	C₂²×Q₈	—	73	72	no	—
161	C₂²×Q₈	—	146	48	no	—
162	C₂²×Q₈	—	146	6	no	—
163	C₂²×Q₈	—	146	36	no	—
164	C₂²×Q₈	D₂₉₂	1	1	yes	characteristic
165	C₂²×D₄	—	73	2	no	—
166	C₂²×D₄	—	146	6	no	—
167	C₂²×D₄	D₂₉₂	1	3	yes	—
168	C₂³×C₄	—	146	6	no	—
169	C₂³×C₄	—	146	6	no	—
170	C₂³×C₄	C₂²×D₇₃	1	3	yes	—
171	#137	—	73	384	no	—
172	#135	—	73	192	no	—
173	C₂×Q₁₆	—	73	36	no	—
174	C₂×Q₁₆	—	73	72	no	—
175	C₂×Q₁₆	—	73	192	no	—
176	C₂×SD₁₆	—	73	72	no	—
177	C₂×D₈	—	73	12	no	—
178	#131	C₂²×D₇₃	1	64	yes	complemented
179	C₂×M₁₆	C₂²×D₇₃	1	36	yes	—
180	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
181	C₂²×C₈	C₂²×D₇₃	1	3	yes	—
182	C₂²×C₈	C₂²×D₇₃	1	36	yes	—
183	C₂⁴	—	146	2	no	—
184	C₂⁴	C₂×D₂₉₂	1	3	yes	—
185	#41	—	73	96	no	—
186	#41	—	146	384	no	—
187	#41	—	146	144	no	—
188	#41	C₂×D₂₉₂	1	48	yes	—
189	#41	C₂³×D₇₃	1	16	yes	—
190	C₂×Q₈	—	73	36	no	—
191	C₂×Q₈	—	73	96	no	—
192	C₂×Q₈	—	146	128	no	—
193	C₂×Q₈	—	146	72	no	—
194	C₂×Q₈	—	146	18	no	—
195	C₂×Q₈	—	146	144	no	—
196	C₂×Q₈	C₂×D₂₉₂	1	12	yes	—
197	C₂×D₄	—	73	12	no	—
198	C₂×D₄	—	146	72	no	—
199	C₂×D₄	—	146	6	no	—
200	C₂×D₄	C₂×D₂₉₂	1	36	yes	—
201	C₂²×C₄	—	146	72	no	—
202	C₂²×C₄	—	146	6	no	—
203	C₂²×C₄	—	146	72	no	—
204	C₂²×C₄	C₂×D₂₉₂	1	18	yes	—
205	C₂²×C₄	C₂×D₂₉₂	1	3	yes	—
206	C₂²×C₄	C₂³×D₇₃	1	18	yes	—
207	C₂²×C₄	C₂³×D₇₃	1	1	yes	characteristic
208	Q₁₆	—	73	48	no	—
209	Q₁₆	—	73	96	no	—
210	SD₁₆	—	73	96	no	—
211	D₈	—	73	16	no	—
212	M₁₆	C₂³×D₇₃	1	48	yes	—
213	C₂×C₈	C₂³×D₇₃	1	6	yes	—
214	C₂×C₈	C₂³×D₇₃	1	18	yes	—
215	C₂×C₈	C₂³×D₇₃	1	48	yes	—
216	C₂³	—	2	12	no	—
217	C₂³	—	292	8	no	—
218	C₂³	—	146	12	no	—
219	C₂³	C₂²×D₂₉₂	1	18	yes	—
220	C₂³	C₂²×D₂₉₂	1	1	yes	characteristic
221	Q₈	—	73	24	no	—
222	Q₈	—	146	96	no	—
223	Q₈	—	146	36	no	—
224	Q₈	C₂²×D₂₉₂	1	16	yes	—
225	D₄	—	73	8	no	—
226	D₄	—	146	96	no	—
227	D₄	—	146	12	no	—
228	D₄	C₂²×D₂₉₂	1	48	yes	—
229	C₂×C₄	—	146	36	no	—
230	C₂×C₄	—	146	96	no	—
231	C₂×C₄	—	146	96	no	—
232	C₂×C₄	C₂²×D₂₉₂	1	18	yes	—
233	C₂×C₄	C₂²×D₂₉₂	1	3	yes	—
234	C₂×C₄	C₂²×D₂₉₂	1	36	yes	—
235	C₂×C₄	C₂⁴×D₇₃	1	3	yes	—
236	C₂×C₄	C₂⁴×D₇₃	1	12	yes	—
237	C₈	C₂⁴×D₇₃	1	4	yes	—
238	C₈	C₂⁴×D₇₃	1	12	yes	—
239	C₂²	—	2	36	no	—
240	C₂²	—	292	24	no	—
241	C₂²	—	146	8	no	—
242	C₂²	#17989404	1	4	yes	direct product factor
243	C₂²	C₂³×D₂₉₂	1	12	yes	—
244	C₂²	C₂³×D₂₉₂	1	3	yes	—
245	C₄	—	146	24	no	—
246	C₄	C₂³×D₂₉₂	1	12	yes	—
247	C₄	C₂³×D₂₉₂	1	3	yes	—
248	C₄	C₂⁵×D₇₃	1	1	yes	characteristic
249	C₂	—	2	12	no	—
250	C₂	—	292	8	no	—
251	C₂	C₂×#17989404	1	6	yes	direct product factor
252	C₂	C₂⁴×D₂₉₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×D₂₉₂
commutator, cosocle	C₄×C₇₃	C₂⁶
Frattini	C₄	C₂⁵×D₇₃
Fitting	C₂²×#131×C₇₃	C₂
socle	C₂³×C₇₃	C₂²×D₄
hypercenter	C₂²×#131	D₇₃
hypocenter	C₇₃	C₂²×#578
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂²×#41 ◁ C₂²×#131

Complements

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

N	H ≅ G/N	type
1. C₂²×#14556638	C₂	semidirect
2. C₂×#17989404	C₂	direct (N × H)
3. C₂²×#14556624	C₂	semidirect
4. C₂²×#14556613	C₂	semidirect
5. C₂³×Dic₂₉₂	C₂	semidirect
6. C₂²×#131×C₇₃	C₂	semidirect
8. C₂×#14556638	C₂²	semidirect
13. #17989404	C₂²	direct (N × H)
14. C₂×#14556624	C₂²	semidirect
15. C₂×#14556613	C₂²	semidirect
18. C₂²×Dic₂₉₂	C₂²	semidirect
22. C₂×#131×C₇₃	C₂²	semidirect
26. #14556638	C₂³	semidirect
39. #14556624	C₂³	semidirect
40. #14556613	C₂³	semidirect
43. C₂×Dic₂₉₂	C₂³	semidirect
51. #131×C₇₃	C₂³	semidirect
113. C₂²×C₇₃	#578	semidirect
124. C₂×C₇₃	C₂×#578	semidirect
132. C₂²×#131	D₇₃	semidirect
133. C₇₃	C₂²×#578	semidirect
149. C₂×#131	C₂×D₇₃	semidirect
178. #131	C₂²×D₇₃	semidirect
242. C₂²	#17989404	direct (N × H)
251. C₂	C₂×#17989404	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×#14556638	2
C₂×#17989404	2
C₂²×#14556624	2
C₂²×#14556613	2
C₂³×Dic₂₉₂	2
C₂²×#131×C₇₃	2
C₂²×#578	73

Subgroups — C22×#17989404 #33999810

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#17989404 #33999810