Subgroups — C₂²×#17989655 #34032697

Order 18688.

11230 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17989655	C₂	1	48	yes	maximal, direct product factor
2	C₂²×Q₈×#64408	C₂	1	8	yes	maximal, complemented
3	C₂²×#14556670	C₂	1	6	yes	maximal
4	C₂³×Q₈×D₇₃	C₂	1	1	yes	maximal, characteristic
5	#17989655	C₂²	1	256	yes	direct product factor
6	C₂×Q₈×#64408	C₂²	1	192	yes	complemented
7	C₂×#14556670	C₂²	1	144	yes	—
8	C₂²×C₄⋊#64408	C₂²	1	12	yes	—
9	C₂²×C₂²⋊#64408	C₂²	1	4	yes	—
10	C₂²×C₄×#64408	C₂²	1	12	yes	—
11	C₂³×Dic₁₄₆	C₄	1	3	yes	complemented
12	C₂²×Q₈×D₇₃	C₄	1	4	yes	complemented
13	C₂²×Q₈×D₇₃	C₄	1	24	yes	complemented
14	C₂²×Q₈×D₇₃	C₂²	1	4	yes	—
15	C₂²×Q₈×D₇₃	C₂²	1	24	yes	—
16	C₂³×C₄×D₇₃	C₂²	1	3	yes	—
17	C₂³×Q₈×C₇₃	C₄	1	1	yes	characteristic, complemented
18	C₂³×#64408	C₂³	1	8	yes	—
19	Q₈×#64408	C₂³	1	512	yes	complemented
20	#14556670	C₂³	1	384	yes	—
21	C₂×C₄⋊#64408	C₂³	1	144	yes	—
22	C₂×C₂²⋊#64408	C₂³	1	48	yes	—
23	C₂×C₄×#64408	C₂³	1	144	yes	—
24	C₂⁴×D₇₃	C₂³	1	1	yes	characteristic
25	C₂²×Dic₁₄₆	C₂×C₄	1	12	yes	—
26	C₂²×Dic₁₄₆	C₂×C₄	1	72	yes	complemented
27	C₂×Q₈×D₇₃	C₂×C₄	1	48	yes	—
28	C₂×Q₈×D₇₃	C₂×C₄	1	96	yes	complemented
29	C₂×Q₈×D₇₃	C₂×C₄	1	192	yes	complemented
30	C₂×Q₈×D₇₃	C₂³	1	48	yes	—
31	C₂×Q₈×D₇₃	C₂³	1	64	yes	—
32	C₂³×Dic₇₃	C₂×C₄	1	3	yes	—
33	C₂²×C₄×D₇₃	C₂×C₄	1	6	yes	—
34	C₂²×C₄×D₇₃	C₂×C₄	1	36	yes	—
35	C₂²×C₄×D₇₃	C₂³	1	6	yes	—
36	C₂²×C₄×D₇₃	C₂³	1	36	yes	—
37	C₂²×Q₈×C₇₃	C₂×C₄	1	4	yes	—
38	C₂²×Q₈×C₇₃	C₂×C₄	1	24	yes	complemented
39	C₂³×C₄×C₇₃	C₂×C₄	1	3	yes	—
40	C₂²×#64408	—	2	32	no	—
41	C₂²×#64408	C₂⁴	1	48	yes	—
42	C₄⋊#64408	C₂⁴	1	192	yes	—
43	C₂²⋊#64408	C₂⁴	1	64	yes	—
44	C₄×#64408	C₂⁴	1	192	yes	—
45	C₂³×D₇₃	—	2	8	no	—
46	C₂³×D₇₃	C₂²×C₄	1	6	yes	—
47	C₂³×D₇₃	C₂²×C₄	1	1	yes	characteristic
48	C₂³×D₇₃	C₂⁴	1	6	yes	—
49	C₂³×D₇₃	C₂⁴	1	1	yes	characteristic
50	C₂×Dic₁₄₆	C₂²×C₄	1	144	yes	—
51	C₂×Dic₁₄₆	C₂²×C₄	1	192	yes	complemented
52	Q₈×D₇₃	C₂²×C₄	1	192	yes	—
53	Q₈×D₇₃	C₂²×C₄	1	256	yes	complemented
54	Q₈×D₇₃	C₂⁴	1	64	yes	—
55	C₂²×Dic₇₃	C₂²×C₄	1	6	yes	—
56	C₂²×Dic₇₃	C₂²×C₄	1	36	yes	—
57	C₂×C₄×D₇₃	C₂²×C₄	1	36	yes	—
58	C₂×C₄×D₇₃	C₂²×C₄	1	144	yes	—
59	C₂×C₄×D₇₃	C₂²×C₄	1	72	yes	—
60	C₂×C₄×D₇₃	C₂⁴	1	36	yes	—
61	C₂×C₄×D₇₃	C₂⁴	1	48	yes	—
62	C₂⁴×C₇₃	C₂²×C₄	1	1	yes	characteristic
63	C₂×Q₈×C₇₃	C₂²×C₄	1	48	yes	—
64	C₂×Q₈×C₇₃	C₂²×C₄	1	64	yes	complemented
65	C₂²×C₄×C₇₃	C₂²×C₄	1	6	yes	—
66	C₂²×C₄×C₇₃	C₂²×C₄	1	36	yes	—
67	C₂³×C₇₃	—	2	4	no	—
68	C₂³×C₇₃	C₂³×C₄	1	6	yes	—
69	C₂³×C₇₃	C₂³×C₄	1	1	yes	characteristic
70	C₂²×D₇₃	—	2	4	no	—
71	C₂²×D₇₃	—	2	24	no	—
72	C₂²×D₇₃	—	2	24	no	—
73	C₂²×D₇₃	C₂³×C₄	1	6	yes	—
74	C₂²×D₇₃	C₂³×C₄	1	12	yes	—
75	C₂²×D₇₃	C₂³×C₄	1	3	yes	—
76	C₂²×D₇₃	2₋¹⁺⁴	1	8	yes	—
77	C₂²×D₇₃	C₂⁵	1	4	yes	—
78	C₂²×D₇₃	C₂⁵	1	3	yes	—
79	C₂×#64408	—	2	96	no	—
80	C₂×#64408	C₂⁵	1	32	yes	—
81	Q₈×C₇₃	C₂³×C₄	1	64	yes	—
82	C₂×C₄×C₇₃	C₂³×C₄	1	36	yes	—
83	C₂×C₄×C₇₃	C₂³×C₄	1	48	yes	—
84	C₂×Dic₇₃	C₂³×C₄	1	36	yes	—
85	C₂×Dic₇₃	C₂³×C₄	1	48	yes	—
86	C₄×D₇₃	C₂³×C₄	1	72	yes	—
87	C₄×D₇₃	C₂³×C₄	1	96	yes	—
88	C₄×D₇₃	C₂⁵	1	24	yes	—
89	Dic₁₄₆	C₂³×C₄	1	192	yes	—
90	C₂²×C₇₃	—	2	12	no	—
91	C₂²×C₇₃	#519	1	4	yes	complemented
92	C₂²×C₇₃	C₂⁴×C₄	1	4	yes	—
93	C₂²×C₇₃	C₂⁴×C₄	1	3	yes	—
94	C₂×D₇₃	—	2	12	no	—
95	C₂×D₇₃	—	2	8	no	—
96	C₂×D₇₃	—	2	24	no	—
97	C₂×D₇₃	#519	1	12	yes	complemented
98	C₂×D₇₃	C₂⁴×C₄	1	4	yes	—
99	C₂×D₇₃	C₂⁴×C₄	1	3	yes	—
100	C₂×D₇₃	C₂×2₋¹⁺⁴	1	12	yes	—
101	C₂×D₇₃	C₂⁶	1	1	yes	characteristic
102	#64408	—	2	32	no	—
103	C₄×C₇₃	C₂⁴×C₄	1	24	yes	—
104	Dic₇₃	C₂⁴×C₄	1	24	yes	—
105	C₂²×#519	—	73	1	no	maximal, Sylow
106	C₂×C₇₃	—	2	4	no	—
107	C₂×C₇₃	C₂×#519	1	6	yes	complemented
108	C₂×C₇₃	C₂⁵×C₄	1	1	yes	characteristic
109	D₇₃	—	2	4	no	—
110	D₇₃	C₂×#519	1	6	yes	complemented
111	D₇₃	C₂²×2₋¹⁺⁴	1	2	yes	—
112	C₂⁴×Q₈	—	73	1	no	—
113	C₂×#519	—	73	48	no	—
114	C₂²×C₄×Q₈	—	73	8	no	—
115	C₂²×#117	—	73	6	no	—
116	C₇₃	C₂²×#519	1	1	yes	characteristic, complemented, Sylow
117	C₂³×Q₈	—	73	4	no	—
118	C₂³×Q₈	—	73	4	no	—
119	C₂³×Q₈	—	73	24	no	—
120	C₂³×Q₈	—	73	24	no	—
121	C₂³×Q₈	—	73	3	no	—
122	C₂³×Q₈	#64408	1	1	yes	characteristic, complemented
123	C₂⁴×C₄	—	73	3	no	—
124	#519	—	73	256	no	—
125	C₂×C₄×Q₈	—	73	192	no	—
126	C₂×#117	—	73	144	no	—
127	C₂²×C₄⋊C₄	—	73	12	no	—
128	C₂²×C₂²⋊C₄	—	73	4	no	—
129	C₂²×C₄²	—	73	12	no	—
130	C₂⁵	—	73	1	no	—
131	C₂²×Q₈	—	73	48	no	—
132	C₂²×Q₈	—	73	48	no	—
133	C₂²×Q₈	—	73	96	no	—
134	C₂²×Q₈	—	73	64	no	—
135	C₂²×Q₈	—	73	192	no	—
136	C₂²×Q₈	—	73	12	no	—
137	C₂²×Q₈	—	73	72	no	—
138	C₂²×Q₈	C₂×#64408	1	4	yes	—
139	C₂²×Q₈	C₂×#64408	1	24	yes	complemented
140	C₂³×C₄	—	73	3	no	—
141	C₂³×C₄	—	73	6	no	—
142	C₂³×C₄	—	73	36	no	—
143	C₂³×C₄	—	73	36	no	—
144	C₂³×C₄	—	73	6	no	—
145	C₂³×C₄	—	73	8	no	—
146	C₂³×C₄	C₂×#64408	1	3	yes	—
147	C₄×Q₈	—	73	512	no	—
148	#117	—	73	384	no	—
149	C₂×C₄⋊C₄	—	73	144	no	—
150	C₂×C₂²⋊C₄	—	73	48	no	—
151	C₂×C₄²	—	73	144	no	—
152	C₂⁴	—	73	1	no	—
153	C₂⁴	—	73	6	no	—
154	C₂⁴	—	73	6	no	—
155	C₂⁴	—	73	1	no	—
156	C₂⁴	—	146	8	no	—
157	C₂⁴	C₂²×#64408	1	1	yes	characteristic
158	C₂×Q₈	—	73	192	no	—
159	C₂×Q₈	—	73	256	no	—
160	C₂×Q₈	—	73	64	no	—
161	C₂×Q₈	—	73	144	no	—
162	C₂×Q₈	—	73	192	no	—
163	C₂×Q₈	C₂²×#64408	1	48	yes	—
164	C₂×Q₈	C₂²×#64408	1	64	yes	complemented
165	C₂²×C₄	—	73	36	no	—
166	C₂²×C₄	—	73	6	no	—
167	C₂²×C₄	—	73	36	no	—
168	C₂²×C₄	—	73	72	no	—
169	C₂²×C₄	—	73	144	no	—
170	C₂²×C₄	—	73	36	no	—
171	C₂²×C₄	—	73	48	no	—
172	C₂²×C₄	—	73	48	no	—
173	C₂²×C₄	—	146	32	no	—
174	C₂²×C₄	C₂²×#64408	1	36	yes	—
175	C₂²×C₄	C₂²×#64408	1	6	yes	—
176	C₄⋊C₄	—	73	192	no	—
177	C₂²⋊C₄	—	73	64	no	—
178	C₄²	—	73	192	no	—
179	C₂³	—	2	4	no	—
180	C₂³	—	73	8	no	—
181	C₂³	—	73	3	no	—
182	C₂³	—	73	3	no	—
183	C₂³	—	73	4	no	—
184	C₂³	—	73	12	no	—
185	C₂³	—	73	6	no	—
186	C₂³	—	146	24	no	—
187	C₂³	—	146	24	no	—
188	C₂³	—	146	4	no	—
189	C₂³	C₂³×#64408	1	6	yes	—
190	C₂³	C₂³×#64408	1	1	yes	characteristic
191	Q₈	—	73	192	no	—
192	Q₈	C₂³×#64408	1	64	yes	—
193	C₂×C₄	—	73	48	no	—
194	C₂×C₄	—	73	36	no	—
195	C₂×C₄	—	73	24	no	—
196	C₂×C₄	—	73	72	no	—
197	C₂×C₄	—	73	96	no	—
198	C₂×C₄	—	73	32	no	—
199	C₂×C₄	—	146	96	no	—
200	C₂×C₄	C₂³×#64408	1	36	yes	—
201	C₂×C₄	C₂³×#64408	1	48	yes	—
202	C₂²	—	2	12	no	—
203	C₂²	—	73	12	no	—
204	C₂²	—	73	12	no	—
205	C₂²	—	73	1	no	—
206	C₂²	—	73	3	no	—
207	C₂²	—	73	4	no	—
208	C₂²	—	146	8	no	—
209	C₂²	—	146	24	no	—
210	C₂²	—	146	12	no	—
211	C₂²	#17989655	1	4	yes	direct product factor
212	C₂²	C₂⁴×#64408	1	4	yes	—
213	C₂²	C₂⁴×#64408	1	3	yes	—
214	C₄	—	73	24	no	—
215	C₄	—	146	32	no	—
216	C₄	C₂⁴×#64408	1	24	yes	—
217	C₂	—	2	4	no	—
218	C₂	—	73	2	no	—
219	C₂	—	73	6	no	—
220	C₂	—	146	4	no	—
221	C₂	C₂×#17989655	1	6	yes	direct product factor
222	C₂	C₂⁵×#64408	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×#64408
commutator	C₂×C₇₃	C₂⁵×C₄
Frattini	C₂	C₂⁵×#64408
Fitting	C₂³×Q₈×C₇₃	C₄
socle	C₂³×C₇₃	C₂³×C₄
prime radical	C₂⁴×D₇₃	C₂³
cosocle	C₂×D₇₃	C₂⁶
hypercenter	C₂³×Q₈	#64408
hypocenter	C₇₃	C₂²×#519
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂³×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×#17989655	C₂	direct (N × H)
2. C₂²×Q₈×#64408	C₂	semidirect
5. #17989655	C₂²	direct (N × H)
6. C₂×Q₈×#64408	C₂²	semidirect
11. C₂³×Dic₁₄₆	C₄	semidirect
12. C₂²×Q₈×D₇₃	C₄	semidirect
13. C₂²×Q₈×D₇₃	C₄	semidirect
17. C₂³×Q₈×C₇₃	C₄	semidirect
19. Q₈×#64408	C₂³	semidirect
26. C₂²×Dic₁₄₆	C₂×C₄	semidirect
28. C₂×Q₈×D₇₃	C₂×C₄	semidirect
29. C₂×Q₈×D₇₃	C₂×C₄	semidirect
38. C₂²×Q₈×C₇₃	C₂×C₄	semidirect
51. C₂×Dic₁₄₆	C₂²×C₄	semidirect
53. Q₈×D₇₃	C₂²×C₄	semidirect
64. C₂×Q₈×C₇₃	C₂²×C₄	semidirect
91. C₂²×C₇₃	#519	semidirect
97. C₂×D₇₃	#519	semidirect
107. C₂×C₇₃	C₂×#519	semidirect
110. D₇₃	C₂×#519	semidirect
116. C₇₃	C₂²×#519	semidirect
122. C₂³×Q₈	#64408	semidirect
139. C₂²×Q₈	C₂×#64408	semidirect
164. C₂×Q₈	C₂²×#64408	semidirect
211. C₂²	#17989655	direct (N × H)
221. C₂	C₂×#17989655	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#17989655	2
C₂²×Q₈×#64408	2
C₂²×#14556670	2
C₂³×Q₈×D₇₃	2
C₂²×#519	73

Subgroups — C22×#17989655 #34032697

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#17989655 #34032697