Subgroups — C₂²×#17988990 #33955580

Order 18688.

2398 conjugacy classes of subgroups.

The subgroup lattice has 214 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×#17988990	C₂	1	24	yes	maximal, direct product factor
3	C₂²×#14556524	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	#17988990	C₂²	1	64	yes	direct product factor
13	C₂×#14556524	C₂²	1	48	yes	—
14	C₂×C₄×#10596831	C₂²	1	12	yes	—
15	C₂³×Q₈×C₇₃	—	2	1	no	—
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	C₂²	1	2	yes	—
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	Q₈	1	16	yes	complemented
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	#14556524	C₂³	1	64	yes	—
32	C₄×#10596831	C₂³	1	16	yes	—
33	C₂²×Q₈×C₇₃	—	2	12	no	—
34	C₂²×Q₈×C₇₃	—	4	8	no	—
35	C₂³×C₄×C₇₃	—	2	2	no	—
36	C₂³×C₄×C₇₃	D₄	1	1	yes	characteristic
37	C₂³×C₄×C₇₃	D₄	1	1	yes	characteristic
38	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
39	C₄⋊Q₈×C₇₃	—	2	24	no	—
40	C₄⋊Q₈×C₇₃	C₂³	1	16	yes	—
41	C₂×C₄⋊C₄×C₇₃	—	2	12	no	—
42	C₂×C₄⋊C₄×C₇₃	—	2	6	no	—
43	C₂×C₄⋊C₄×C₇₃	C₂³	1	12	yes	—
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	C₂×Q₈	1	48	yes	complemented
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	16	no	—
57	C₂×Q₈×C₇₃	—	4	48	no	—
58	C₂²×C₄×C₇₃	—	2	12	no	—
59	C₂²×C₄×C₇₃	—	4	8	no	—
60	C₂²×C₄×C₇₃	D₈	1	8	yes	—
61	C₂²×C₄×C₇₃	Q₁₆	1	8	yes	—
62	C₂²×C₄×C₇₃	C₂×D₄	1	3	yes	—
63	C₂²×C₄×C₇₃	C₂×D₄	1	6	yes	—
64	C₂²×C₄×C₇₃	C₂×D₄	1	6	yes	—
65	C₂²×C₄×C₇₃	C₂×Q₈	1	8	yes	—
66	C₂²×C₄×C₇₃	C₂⁴	1	3	yes	—
67	C₄⋊C₄×C₇₃	—	2	16	no	—
68	C₄⋊C₄×C₇₃	—	2	12	no	—
69	C₄⋊C₄×C₇₃	C₂⁴	1	8	yes	—
70	C₄²×C₇₃	C₂×D₄	1	12	yes	—
71	C₄²×C₇₃	C₂⁴	1	4	yes	—
72	C₂³×C₇₃	C₄⋊Q₈	1	4	yes	—
73	C₂³×C₇₃	C₂×D₈	1	4	yes	—
74	C₂³×C₇₃	C₂×Q₁₆	1	4	yes	—
75	C₂³×C₇₃	C₂²×D₄	1	3	yes	—
76	Q₈×C₇₃	—	4	32	no	—
77	C₂×C₄×C₇₃	—	2	8	no	—
78	C₂×C₄×C₇₃	—	4	24	no	—
79	C₂×C₄×C₇₃	C₄⋊Q₈	1	12	yes	—
80	C₂×C₄×C₇₃	C₂×D₈	1	24	yes	—
81	C₂×C₄×C₇₃	C₂×Q₁₆	1	24	yes	—
82	C₂×C₄×C₇₃	C₂²×D₄	1	3	yes	—
83	C₂×C₄×C₇₃	C₂²×D₄	1	4	yes	—
84	C₂×C₄×C₇₃	C₂²×D₄	1	4	yes	—
85	C₂×C₄×C₇₃	C₂²×Q₈	1	12	yes	—
86	C₂×C₄×C₇₃	C₂⁵	1	1	yes	characteristic
87	C₂×Dic₇₃	—	2	8	no	—
88	C₂×Dic₇₃	—	4	24	no	—
89	Dic₁₄₆	—	4	32	no	—
90	#10596831	C₂²×Q₈	1	16	yes	complemented
91	C₂²×C₇₃	#500	1	16	yes	complemented
92	C₂²×C₇₃	C₂×C₄⋊Q₈	1	6	yes	—
93	C₂²×C₇₃	C₂²×D₈	1	6	yes	—
94	C₂²×C₇₃	C₂²×Q₁₆	1	6	yes	—
95	C₂²×C₇₃	C₂³×D₄	1	1	yes	characteristic
96	C₄×C₇₃	—	4	8	no	—
97	C₄×C₇₃	C₂×C₄⋊Q₈	1	6	yes	—
98	C₄×C₇₃	C₂²×D₈	1	8	yes	—
99	C₄×C₇₃	C₂²×Q₁₆	1	8	yes	—
100	C₄×C₇₃	C₂³×Q₈	1	2	yes	—
101	Dic₇₃	—	4	8	no	—
102	C₂²×#500	—	73	1	no	maximal, Sylow
103	C₂×C₇₃	C₂×#500	1	12	yes	complemented
104	C₂×C₇₃	C₂²×C₄⋊Q₈	1	1	yes	characteristic
105	C₂×C₇₃	C₂³×D₈	1	1	yes	characteristic
106	C₂×C₇₃	C₂³×Q₁₆	1	1	yes	characteristic
107	C₂²×C₄⋊Q₈	—	73	1	no	—
108	C₂²×C₄⋊Q₈	D₇₃	1	1	yes	characteristic
109	C₂×#500	—	73	24	no	—
110	C₂²×#107	—	73	4	no	—
111	C₂²×C₄×C₈	—	73	1	no	—
112	C₇₃	C₂²×#500	1	1	yes	characteristic, complemented, Sylow
113	C₂³×Q₈	—	2	1	no	—
114	C₂³×Q₈	—	146	1	no	—
115	C₂³×C₈	—	73	2	no	—
116	C₂×C₄⋊Q₈	—	2	6	no	—
117	C₂×C₄⋊Q₈	—	73	12	no	—
118	C₂×C₄⋊Q₈	—	146	6	no	—
119	C₂×C₄⋊Q₈	C₂×D₇₃	1	12	yes	—
120	C₂²×C₄⋊C₄	—	2	1	no	—
121	C₂²×C₄⋊C₄	—	73	2	no	—
122	C₂²×C₄⋊C₄	—	146	1	no	—
123	C₂²×C₄⋊C₄	C₂×D₇₃	1	2	yes	—
124	C₂²×C₄²	C₂×D₇₃	1	1	yes	characteristic
125	#500	—	73	64	no	—
126	C₂×#107	—	73	48	no	—
127	C₂×C₄×C₈	—	73	12	no	—
128	C₂²×Q₈	—	2	12	no	—
129	C₂²×Q₈	—	4	8	no	—
130	C₂²×Q₈	—	292	8	no	—
131	C₂²×Q₈	—	146	12	no	—
132	C₂³×C₄	—	2	2	no	—
133	C₂³×C₄	—	146	2	no	—
134	C₂³×C₄	C₂²⋊D₇₃	1	1	yes	characteristic
135	C₂³×C₄	C₂²⋊D₇₃	1	1	yes	characteristic
136	C₂³×C₄	C₂²×D₇₃	1	1	yes	characteristic
137	C₂²×C₈	—	73	16	no	—
138	C₂²×C₈	—	73	12	no	—
139	C₄⋊Q₈	—	2	24	no	—
140	C₄⋊Q₈	—	73	16	no	—
141	C₄⋊Q₈	—	146	24	no	—
142	C₄⋊Q₈	C₂²×D₇₃	1	16	yes	—
143	C₂×C₄⋊C₄	—	2	12	no	—
144	C₂×C₄⋊C₄	—	2	6	no	—
145	C₂×C₄⋊C₄	—	73	12	no	—
146	C₂×C₄⋊C₄	—	146	6	no	—
147	C₂×C₄⋊C₄	—	146	12	no	—
148	C₂×C₄⋊C₄	C₂²×D₇₃	1	12	yes	—
149	C₂×C₄²	C₂²⋊D₇₃	1	6	yes	—
150	C₂×C₄²	C₂²×D₇₃	1	6	yes	—
151	#107	—	73	64	no	—
152	C₄×C₈	—	73	16	no	—
153	C₂⁴	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
154	C₂×Q₈	—	2	16	no	—
155	C₂×Q₈	—	4	48	no	—
156	C₂×Q₈	—	292	48	no	—
157	C₂×Q₈	—	146	16	no	—
158	C₂²×C₄	—	2	12	no	—
159	C₂²×C₄	—	4	8	no	—
160	C₂²×C₄	—	292	8	no	—
161	C₂²×C₄	—	146	12	no	—
162	C₂²×C₄	#11924013	1	8	yes	—
163	C₂²×C₄	#11924018	1	8	yes	—
164	C₂²×C₄	C₂×C₂²⋊D₇₃	1	3	yes	—
165	C₂²×C₄	C₂×C₂²⋊D₇₃	1	6	yes	—
166	C₂²×C₄	C₂×C₂²⋊D₇₃	1	6	yes	—
167	C₂²×C₄	Q₈×D₇₃	1	8	yes	—
168	C₂²×C₄	C₂³×D₇₃	1	3	yes	—
169	C₂×C₈	—	73	48	no	—
170	C₂×C₈	—	73	8	no	—
171	C₄⋊C₄	—	2	16	no	—
172	C₄⋊C₄	—	2	12	no	—
173	C₄⋊C₄	—	73	8	no	—
174	C₄⋊C₄	—	146	12	no	—
175	C₄⋊C₄	—	146	16	no	—
176	C₄⋊C₄	C₂³×D₇₃	1	8	yes	—
177	C₄²	C₂×C₂²⋊D₇₃	1	12	yes	—
178	C₄²	C₂³×D₇₃	1	4	yes	—
179	C₂³	#14556589	1	4	yes	—
180	C₂³	C₂×#11924013	1	4	yes	—
181	C₂³	C₂×#11924018	1	4	yes	—
182	C₂³	C₂²×C₂²⋊D₇₃	1	3	yes	—
183	Q₈	—	4	32	no	—
184	Q₈	—	292	32	no	—
185	C₂×C₄	—	2	8	no	—
186	C₂×C₄	—	4	24	no	—
187	C₂×C₄	—	292	24	no	—
188	C₂×C₄	—	146	8	no	—
189	C₂×C₄	#14556589	1	12	yes	—
190	C₂×C₄	C₂×#11924013	1	24	yes	—
191	C₂×C₄	C₂×#11924018	1	24	yes	—
192	C₂×C₄	C₂²×C₂²⋊D₇₃	1	3	yes	—
193	C₂×C₄	C₂²×C₂²⋊D₇₃	1	4	yes	—
194	C₂×C₄	C₂²×C₂²⋊D₇₃	1	4	yes	—
195	C₂×C₄	C₂×Q₈×D₇₃	1	12	yes	—
196	C₂×C₄	C₂⁴×D₇₃	1	1	yes	characteristic
197	C₈	—	73	16	no	—
198	C₂²	#17988990	1	16	yes	direct product factor
199	C₂²	C₂×#14556589	1	6	yes	—
200	C₂²	C₂²×#11924013	1	6	yes	—
201	C₂²	C₂²×#11924018	1	6	yes	—
202	C₂²	C₂³×C₂²⋊D₇₃	1	1	yes	characteristic
203	C₄	—	4	8	no	—
204	C₄	—	292	8	no	—
205	C₄	C₂×#14556589	1	6	yes	—
206	C₄	C₂²×#11924013	1	8	yes	—
207	C₄	C₂²×#11924018	1	8	yes	—
208	C₄	C₂²×Q₈×D₇₃	1	2	yes	—
209	C₂	C₂×#17988990	1	12	yes	direct product factor
210	C₂	C₂²×#14556589	1	1	yes	characteristic
211	C₂	C₂³×#11924013	1	1	yes	characteristic
212	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×#500
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×#17988990	C₂	direct (N × H)
12. #17988990	C₂²	direct (N × H)
24. C₂²×#10596831	Q₈	semidirect
50. C₂×#10596831	C₂×Q₈	semidirect
90. #10596831	C₂²×Q₈	semidirect
91. C₂²×C₇₃	#500	semidirect
103. C₂×C₇₃	C₂×#500	semidirect
112. C₇₃	C₂²×#500	semidirect
198. C₂²	#17988990	direct (N × H)
209. C₂	C₂×#17988990	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×#17988990	2
C₂²×#14556524	2
C₂²×C₄×#10596831	2
C₂²×C₄⋊Q₈×C₇₃	2
C₂²×#500	73

Subgroups — C22×#17988990 #33955580

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#17988990 #33955580