Subgroups — C₂²×#17988937 #33955173

Order 18688.

2398 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14556591	C₂	1	1	yes	maximal, characteristic
2	C₂×#17988937	C₂	1	24	yes	maximal, direct product factor
3	C₂²×#14556515	C₂	1	4	yes	maximal
4	C₂²×C₄×#10596831	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄⋊Q₈×C₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂³×Dic₁₄₆	—	2	1	no	—
7	C₂³×#10596831	C₂²	1	2	yes	—
8	C₂×#14556591	—	2	6	no	—
9	C₂×#14556591	C₂²	1	12	yes	—
10	C₂²×C₄⋊Dic₇₃	—	2	1	no	—
11	C₂²×C₄⋊Dic₇₃	C₂²	1	2	yes	—
12	#17988937	C₂²	1	64	yes	direct product factor
13	C₂×#14556515	C₂²	1	48	yes	—
14	C₂×C₄×#10596831	C₂²	1	12	yes	—
15	C₂³×Q₈×C₇₃	C₂²	1	2	yes	—
16	C₂×C₄⋊Q₈×C₇₃	—	2	6	no	—
17	C₂×C₄⋊Q₈×C₇₃	C₂²	1	12	yes	—
18	C₂²×C₄⋊C₄×C₇₃	—	2	1	no	—
19	C₂²×C₄⋊C₄×C₇₃	—	2	1	no	—
20	C₂²×C₄²×C₇₃	C₂²	1	1	yes	characteristic
21	C₂²×Dic₁₄₆	—	2	12	no	—
22	C₂²×Dic₁₄₆	—	4	8	no	—
23	C₂³×Dic₇₃	—	2	2	no	—
24	C₂²×#10596831	—	2	8	no	—
25	C₂²×#10596831	C₂³	1	12	yes	—
26	#14556591	—	2	24	no	—
27	#14556591	C₂³	1	16	yes	—
28	C₂×C₄⋊Dic₇₃	—	2	6	no	—
29	C₂×C₄⋊Dic₇₃	—	2	12	no	—
30	C₂×C₄⋊Dic₇₃	C₂³	1	12	yes	—
31	#14556515	C₂³	1	64	yes	—
32	C₄×#10596831	C₂³	1	16	yes	—
33	C₂²×Q₈×C₇₃	—	2	6	no	—
34	C₂²×Q₈×C₇₃	—	2	16	no	—
35	C₂²×Q₈×C₇₃	C₂³	1	12	yes	—
36	C₂³×C₄×C₇₃	—	2	2	no	—
37	C₂³×C₄×C₇₃	D₄	1	1	yes	characteristic
38	C₂³×C₄×C₇₃	D₄	1	1	yes	characteristic
39	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
40	C₄⋊Q₈×C₇₃	—	2	24	no	—
41	C₄⋊Q₈×C₇₃	C₂³	1	16	yes	—
42	C₂×C₄⋊C₄×C₇₃	—	2	12	no	—
43	C₂×C₄⋊C₄×C₇₃	—	2	12	no	—
44	C₂×C₄²×C₇₃	D₄	1	6	yes	—
45	C₂×C₄²×C₇₃	C₂³	1	6	yes	—
46	C₂×Dic₁₄₆	—	2	16	no	—
47	C₂×Dic₁₄₆	—	4	48	no	—
48	C₂²×Dic₇₃	—	2	12	no	—
49	C₂²×Dic₇₃	—	4	8	no	—
50	C₂×#10596831	—	2	24	no	—
51	C₂×#10596831	C₂⁴	1	8	yes	—
52	C₄⋊Dic₇₃	—	2	16	no	—
53	C₄⋊Dic₇₃	—	2	12	no	—
54	C₄⋊Dic₇₃	C₂⁴	1	8	yes	—
55	C₂⁴×C₇₃	C₂×D₄	1	1	yes	characteristic
56	C₂×Q₈×C₇₃	—	2	12	no	—
57	C₂×Q₈×C₇₃	—	2	48	no	—
58	C₂×Q₈×C₇₃	—	4	24	no	—
59	C₂×Q₈×C₇₃	C₂⁴	1	8	yes	—
60	C₂²×C₄×C₇₃	—	2	12	no	—
61	C₂²×C₄×C₇₃	—	4	8	no	—
62	C₂²×C₄×C₇₃	SD₁₆	1	8	yes	—
63	C₂²×C₄×C₇₃	Q₁₆	1	8	yes	—
64	C₂²×C₄×C₇₃	C₂×D₄	1	3	yes	—
65	C₂²×C₄×C₇₃	C₂×D₄	1	6	yes	—
66	C₂²×C₄×C₇₃	C₂×D₄	1	6	yes	—
67	C₂²×C₄×C₇₃	#41	1	8	yes	—
68	C₂²×C₄×C₇₃	C₂⁴	1	3	yes	—
69	C₄⋊C₄×C₇₃	—	2	16	no	—
70	C₄⋊C₄×C₇₃	—	2	16	no	—
71	C₄²×C₇₃	C₂×D₄	1	12	yes	—
72	C₄²×C₇₃	C₂⁴	1	4	yes	—
73	C₂³×C₇₃	#124	1	4	yes	—
74	C₂³×C₇₃	C₂×SD₁₆	1	4	yes	—
75	C₂³×C₇₃	C₂×Q₁₆	1	4	yes	—
76	C₂³×C₇₃	C₂²×D₄	1	3	yes	—
77	Q₈×C₇₃	—	2	16	no	—
78	Q₈×C₇₃	—	4	24	no	—
79	C₂×C₄×C₇₃	—	2	8	no	—
80	C₂×C₄×C₇₃	—	4	24	no	—
81	C₂×C₄×C₇₃	#124	1	12	yes	—
82	C₂×C₄×C₇₃	C₂×SD₁₆	1	24	yes	—
83	C₂×C₄×C₇₃	C₂×Q₁₆	1	24	yes	—
84	C₂×C₄×C₇₃	C₂²×D₄	1	3	yes	—
85	C₂×C₄×C₇₃	C₂²×D₄	1	4	yes	—
86	C₂×C₄×C₇₃	C₂²×D₄	1	4	yes	—
87	C₂×C₄×C₇₃	C₂×#41	1	12	yes	—
88	C₂×C₄×C₇₃	C₂⁵	1	1	yes	characteristic
89	C₂×Dic₇₃	—	2	8	no	—
90	C₂×Dic₇₃	—	4	24	no	—
91	Dic₁₄₆	—	4	32	no	—
92	#10596831	—	2	8	no	—
93	C₂²×C₇₃	#487	1	16	yes	complemented
94	C₂²×C₇₃	C₂×#124	1	6	yes	—
95	C₂²×C₇₃	C₂²×SD₁₆	1	6	yes	—
96	C₂²×C₇₃	C₂²×Q₁₆	1	6	yes	—
97	C₂²×C₇₃	C₂³×D₄	1	1	yes	characteristic
98	C₄×C₇₃	—	4	8	no	—
99	C₄×C₇₃	C₂×#124	1	6	yes	—
100	C₄×C₇₃	C₂²×SD₁₆	1	8	yes	—
101	C₄×C₇₃	C₂²×Q₁₆	1	8	yes	—
102	C₄×C₇₃	C₂²×#41	1	2	yes	—
103	Dic₇₃	—	4	8	no	—
104	C₂²×#487	—	73	1	no	maximal, Sylow
105	C₂×C₇₃	C₂×#487	1	12	yes	complemented
106	C₂×C₇₃	C₂²×#124	1	1	yes	characteristic
107	C₂×C₇₃	C₂³×SD₁₆	1	1	yes	characteristic
108	C₂×C₇₃	C₂³×Q₁₆	1	1	yes	characteristic
109	C₂²×C₄⋊Q₈	—	73	1	no	—
110	C₂²×C₄⋊Q₈	D₇₃	1	1	yes	characteristic
111	C₂×#487	—	73	24	no	—
112	C₂²×#103	—	73	4	no	—
113	C₂²×C₄×C₈	—	73	1	no	—
114	C₇₃	C₂²×#487	1	1	yes	characteristic, complemented, Sylow
115	C₂³×Q₈	—	146	1	no	—
116	C₂³×Q₈	C₂×D₇₃	1	2	yes	—
117	C₂³×C₈	—	73	2	no	—
118	C₂×C₄⋊Q₈	—	2	6	no	—
119	C₂×C₄⋊Q₈	—	73	12	no	—
120	C₂×C₄⋊Q₈	—	146	6	no	—
121	C₂×C₄⋊Q₈	C₂×D₇₃	1	12	yes	—
122	C₂²×C₄⋊C₄	—	2	1	no	—
123	C₂²×C₄⋊C₄	—	2	1	no	—
124	C₂²×C₄⋊C₄	—	73	2	no	—
125	C₂²×C₄⋊C₄	—	146	1	no	—
126	C₂²×C₄²	C₂×D₇₃	1	1	yes	characteristic
127	#487	—	73	64	no	—
128	C₂×#103	—	73	48	no	—
129	C₂×C₄×C₈	—	73	12	no	—
130	C₂²×Q₈	—	2	6	no	—
131	C₂²×Q₈	—	2	16	no	—
132	C₂²×Q₈	—	292	8	no	—
133	C₂²×Q₈	—	146	12	no	—
134	C₂²×Q₈	C₂²×D₇₃	1	12	yes	—
135	C₂³×C₄	—	2	2	no	—
136	C₂³×C₄	—	146	2	no	—
137	C₂³×C₄	C₂²⋊D₇₃	1	1	yes	characteristic
138	C₂³×C₄	C₂²⋊D₇₃	1	1	yes	characteristic
139	C₂³×C₄	C₂²×D₇₃	1	1	yes	characteristic
140	C₂²×C₈	—	73	12	no	—
141	C₂²×C₈	—	146	8	no	—
142	C₄⋊Q₈	—	2	24	no	—
143	C₄⋊Q₈	—	73	16	no	—
144	C₄⋊Q₈	—	146	24	no	—
145	C₄⋊Q₈	C₂²×D₇₃	1	16	yes	—
146	C₂×C₄⋊C₄	—	2	12	no	—
147	C₂×C₄⋊C₄	—	2	12	no	—
148	C₂×C₄⋊C₄	—	73	12	no	—
149	C₂×C₄⋊C₄	—	146	6	no	—
150	C₂×C₄⋊C₄	—	146	12	no	—
151	C₂×C₄²	C₂²⋊D₇₃	1	6	yes	—
152	C₂×C₄²	C₂²×D₇₃	1	6	yes	—
153	#103	—	73	64	no	—
154	C₄×C₈	—	73	16	no	—
155	C₂⁴	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
156	C₂×Q₈	—	2	12	no	—
157	C₂×Q₈	—	2	48	no	—
158	C₂×Q₈	—	4	24	no	—
159	C₂×Q₈	—	292	48	no	—
160	C₂×Q₈	—	146	16	no	—
161	C₂×Q₈	C₂³×D₇₃	1	8	yes	—
162	C₂²×C₄	—	2	12	no	—
163	C₂²×C₄	—	4	8	no	—
164	C₂²×C₄	—	292	8	no	—
165	C₂²×C₄	—	146	12	no	—
166	C₂²×C₄	#11924015	1	8	yes	—
167	C₂²×C₄	#11924018	1	8	yes	—
168	C₂²×C₄	C₂×C₂²⋊D₇₃	1	3	yes	—
169	C₂²×C₄	C₂×C₂²⋊D₇₃	1	6	yes	—
170	C₂²×C₄	C₂×C₂²⋊D₇₃	1	6	yes	—
171	C₂²×C₄	#11924027	1	8	yes	—
172	C₂²×C₄	C₂³×D₇₃	1	3	yes	—
173	C₂×C₈	—	73	8	no	—
174	C₂×C₈	—	146	24	no	—
175	C₄⋊C₄	—	2	16	no	—
176	C₄⋊C₄	—	2	16	no	—
177	C₄⋊C₄	—	73	8	no	—
178	C₄⋊C₄	—	146	12	no	—
179	C₄⋊C₄	—	146	16	no	—
180	C₄²	C₂×C₂²⋊D₇₃	1	12	yes	—
181	C₄²	C₂³×D₇₃	1	4	yes	—
182	C₂³	#14556578	1	4	yes	—
183	C₂³	C₂×#11924015	1	4	yes	—
184	C₂³	C₂×#11924018	1	4	yes	—
185	C₂³	C₂²×C₂²⋊D₇₃	1	3	yes	—
186	Q₈	—	2	16	no	—
187	Q₈	—	4	24	no	—
188	Q₈	—	292	32	no	—
189	C₂×C₄	—	2	8	no	—
190	C₂×C₄	—	4	24	no	—
191	C₂×C₄	—	292	24	no	—
192	C₂×C₄	—	146	8	no	—
193	C₂×C₄	#14556578	1	12	yes	—
194	C₂×C₄	C₂×#11924015	1	24	yes	—
195	C₂×C₄	C₂×#11924018	1	24	yes	—
196	C₂×C₄	C₂²×C₂²⋊D₇₃	1	3	yes	—
197	C₂×C₄	C₂²×C₂²⋊D₇₃	1	4	yes	—
198	C₂×C₄	C₂²×C₂²⋊D₇₃	1	4	yes	—
199	C₂×C₄	C₂×#11924027	1	12	yes	—
200	C₂×C₄	C₂⁴×D₇₃	1	1	yes	characteristic
201	C₈	—	146	8	no	—
202	C₂²	#17988937	1	16	yes	direct product factor
203	C₂²	C₂×#14556578	1	6	yes	—
204	C₂²	C₂²×#11924015	1	6	yes	—
205	C₂²	C₂²×#11924018	1	6	yes	—
206	C₂²	C₂³×C₂²⋊D₇₃	1	1	yes	characteristic
207	C₄	—	4	8	no	—
208	C₄	—	292	8	no	—
209	C₄	C₂×#14556578	1	6	yes	—
210	C₄	C₂²×#11924015	1	8	yes	—
211	C₄	C₂²×#11924018	1	8	yes	—
212	C₄	C₂²×#11924027	1	2	yes	—
213	C₂	C₂×#17988937	1	12	yes	direct product factor
214	C₂	C₂²×#14556578	1	1	yes	characteristic
215	C₂	C₂³×#11924015	1	1	yes	characteristic
216	C₂	C₂³×#11924018	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂×C₂²⋊D₇₃
commutator, cosocle	C₂×C₄×C₇₃	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₇₃
Fitting	C₂²×C₄⋊Q₈×C₇₃	C₂
socle, prime radical	C₂⁴×C₇₃	C₂×D₄
hypercenter	C₂²×C₄⋊Q₈	D₇₃
hypocenter	C₇₃	C₂²×#487
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
2. C₂×#17988937	C₂	direct (N × H)
12. #17988937	C₂²	direct (N × H)
93. C₂²×C₇₃	#487	semidirect
105. C₂×C₇₃	C₂×#487	semidirect
114. C₇₃	C₂²×#487	semidirect
202. C₂²	#17988937	direct (N × H)
213. C₂	C₂×#17988937	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×#14556591	2
C₂×#17988937	2
C₂²×#14556515	2
C₂²×C₄×#10596831	2
C₂²×C₄⋊Q₈×C₇₃	2
C₂²×#487	73

Subgroups — C22×#17988937 #33955173

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#17988937 #33955173