Subgroups — C₄×#17988409 #33001369

Order 18688.

330 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17988409	C₂	1	2	yes	maximal
2	C₄×#14556531	C₂	1	1	yes	maximal, characteristic
3	C₄×#14556527	C₂	1	1	yes	maximal, characteristic
4	#22608029	C₂	1	2	yes	maximal
5	C₄×#106×C₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556531	C₂²	1	1	yes	characteristic
7	C₂×#14556531	C₂²	1	1	yes	characteristic
8	C₂×#14556527	C₂²	1	2	yes	—
9	C₄×#11924012	—	2	1	no	—
10	#17988409	C₄	1	8	yes	direct product factor
11	#17988363	C₄	1	2	yes	complemented
12	C₄×#11924003	C₄	1	2	yes	complemented
13	#17988352	—	2	1	no	—
14	C₂×#106×C₇₃	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
16	C₄×C₄⋊C₄×C₇₃	—	2	1	no	—
17	#340×C₇₃	—	2	1	no	—
18	C₂×#11924012	—	2	2	no	—
19	#14556531	—	2	2	no	—
20	#14556531	C₂×C₄	1	4	yes	—
21	C₂×#11924003	C₂×C₄	1	2	yes	—
22	C₂×#11924003	C₂×C₄	1	2	yes	—
23	#14556527	—	2	2	no	—
24	#14556527	C₂×C₄	1	4	yes	—
25	#14556502	—	4	1	no	—
26	C₄×#10596831	—	4	1	no	—
27	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
29	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
30	C₂×C₄²×C₇₃	—	4	1	no	—
31	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
32	#106×C₇₃	—	2	2	no	—
33	#106×C₇₃	C₂×C₄	1	4	yes	—
34	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
35	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
36	C₄×C₈×C₇₃	Q₈	1	1	yes	characteristic
37	C₄×C₈×C₇₃	Q₈	1	1	yes	characteristic
38	#95×C₇₃	—	4	1	no	—
39	#11924012	—	2	4	no	—
40	#11924012	—	4	2	no	—
41	C₂×#10596831	—	4	2	no	—
42	#11924003	—	2	2	no	—
43	#11924003	C₄²	1	4	yes	complemented
44	C₂²×C₄×C₇₃	—	4	2	no	—
45	C₂²×C₄×C₇₃	SD₁₆	1	1	yes	characteristic
46	C₂²×C₄×C₇₃	SD₁₆	1	1	yes	characteristic
47	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
48	C₂×C₈×C₇₃	—	2	2	no	—
49	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
53	C₂×C₈×C₇₃	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₈×C₇₃	#41	1	2	yes	—
55	C₄⋊C₄×C₇₃	—	2	4	no	—
56	C₄⋊C₄×C₇₃	—	4	2	no	—
57	C₄²×C₇₃	—	8	2	no	—
58	C₄²×C₇₃	C₄⋊C₄	1	1	yes	characteristic
59	C₄²×C₇₃	C₄⋊C₄	1	1	yes	characteristic
60	C₄²×C₇₃	SD₁₆	1	1	yes	characteristic
61	C₄²×C₇₃	SD₁₆	1	1	yes	characteristic
62	C₂³×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
63	C₂×C₄×C₇₃	—	2	2	no	—
64	C₂×C₄×C₇₃	—	4	1	no	—
65	C₂×C₄×C₇₃	—	4	4	no	—
66	C₂×C₄×C₇₃	—	8	2	no	—
67	C₂×C₄×C₇₃	—	8	2	no	—
68	C₂×C₄×C₇₃	#106	1	1	yes	characteristic
69	C₂×C₄×C₇₃	#106	1	1	yes	characteristic
70	C₂×C₄×C₇₃	#106	1	1	yes	characteristic
71	C₂×C₄×C₇₃	#106	1	1	yes	characteristic
72	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
74	C₂×C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
75	C₂×C₄×C₇₃	#135	1	2	yes	—
76	C₈×C₇₃	—	2	1	no	—
77	C₈×C₇₃	—	2	1	no	—
78	C₈×C₇₃	C₂×C₄²	1	1	yes	characteristic
79	C₈×C₇₃	#117	1	1	yes	characteristic
80	C₈×C₇₃	C₄×Q₈	1	2	yes	—
81	#10596831	—	4	4	no	—
82	C₂²×C₇₃	—	2	1	no	—
83	C₂²×C₇₃	—	2	1	no	—
84	C₂²×C₇₃	C₂×#106	1	1	yes	characteristic
85	C₂²×C₇₃	C₄×SD₁₆	1	2	yes	—
86	C₄×C₇₃	—	2	1	no	—
87	C₄×C₇₃	—	2	1	no	—
88	C₄×C₇₃	—	2	1	no	—
89	C₄×C₇₃	—	8	4	no	—
90	C₄×C₇₃	#365	1	2	yes	complemented
91	C₄×C₇₃	C₄×C₄⋊C₄	1	1	yes	characteristic
92	C₄×C₇₃	#427	1	1	yes	characteristic
93	C₄×C₇₃	C₄×SD₁₆	1	2	yes	—
94	C₄×#365	—	73	1	no	maximal, Sylow
95	C₂×C₇₃	—	2	2	no	—
96	C₂×C₇₃	C₄×#106	1	1	yes	characteristic
97	C₂×C₇₃	C₂×#365	1	1	yes	characteristic
98	C₂×C₇₃	#2155	1	1	yes	characteristic
99	C₂×#365	—	73	2	no	—
100	C₄×M₃₂	—	73	1	no	—
101	C₄×#108	—	73	1	no	—
102	C₄×#106	D₇₃	1	1	yes	characteristic
103	#1383	—	73	2	no	—
104	C₇₃	C₄×#365	1	1	yes	characteristic, complemented, Sylow
105	C₂×M₃₂	—	73	1	no	—
106	C₂×M₃₂	—	73	1	no	—
107	C₂×#108	—	73	2	no	—
108	C₂×#106	C₂×D₇₃	1	2	yes	—
109	C₄×M₁₆	—	146	1	no	—
110	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
111	C₄×C₄⋊C₄	—	2	1	no	—
112	#365	—	73	8	no	—
113	#346	—	73	2	no	—
114	C₄×C₁₆	—	73	2	no	—
115	#341	—	146	1	no	—
116	#340	—	2	1	no	—
117	C₂×M₁₆	—	146	2	no	—
118	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
119	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
120	C₂×C₄⋊C₄	—	2	2	no	—
121	C₂×C₄²	—	4	1	no	—
122	C₂×C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
123	M₃₂	—	73	4	no	—
124	M₃₂	—	146	2	no	—
125	C₂×C₁₆	—	73	2	no	—
126	C₂×C₁₆	—	73	2	no	—
127	#108	—	73	4	no	—
128	#108	—	146	2	no	—
129	#106	—	2	2	no	—
130	#106	C₄×D₇₃	1	4	yes	—
131	#97	—	292	1	no	—
132	C₄×C₈	—	292	1	no	—
133	C₄×C₈	Dic₁₄₆	1	1	yes	characteristic
134	C₄×C₈	Dic₁₄₆	1	1	yes	characteristic
135	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
136	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
137	#95	—	4	1	no	—
138	C₂²×C₄	—	4	2	no	—
139	C₂²×C₄	#11924016	1	1	yes	characteristic
140	C₂²×C₄	#11924016	1	1	yes	characteristic
141	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
142	M₁₆	—	292	2	no	—
143	M₁₆	—	146	4	no	—
144	C₂×C₈	—	2	2	no	—
145	C₂×C₈	—	292	2	no	—
146	C₂×C₈	#11924007	1	1	yes	characteristic
147	C₂×C₈	#11924007	1	1	yes	characteristic
148	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
149	C₂×C₈	C₂×C₄×D₇₃	1	2	yes	—
150	C₂×C₈	C₂×Dic₁₄₆	1	1	yes	characteristic
151	C₂×C₈	#11924028	1	2	yes	—
152	C₄⋊C₄	—	2	4	no	—
153	C₄⋊C₄	—	4	2	no	—
154	C₄²	—	8	2	no	—
155	C₄²	#11924007	1	1	yes	characteristic
156	C₄²	#11924007	1	1	yes	characteristic
157	C₄²	#11924015	1	1	yes	characteristic
158	C₄²	#11924015	1	1	yes	characteristic
159	C₁₆	—	73	4	no	—
160	C₁₆	—	146	2	no	—
161	C₂³	C₂×#11924016	1	1	yes	characteristic
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	4	1	no	—
164	C₂×C₄	—	4	4	no	—
165	C₂×C₄	—	8	2	no	—
166	C₂×C₄	—	8	2	no	—
167	C₂×C₄	#14556522	1	1	yes	characteristic
168	C₂×C₄	#14556522	1	1	yes	characteristic
169	C₂×C₄	#14556522	1	1	yes	characteristic
170	C₂×C₄	#14556522	1	1	yes	characteristic
171	C₂×C₄	C₂×#11924007	1	1	yes	characteristic
172	C₂×C₄	C₄×C₂²⋊D₇₃	1	2	yes	—
173	C₂×C₄	C₂×#11924015	1	1	yes	characteristic
174	C₂×C₄	#14556611	1	2	yes	—
175	C₈	—	2	1	no	—
176	C₈	—	2	1	no	—
177	C₈	—	292	4	no	—
178	C₈	C₄²×D₇₃	1	1	yes	characteristic
179	C₈	#14556548	1	1	yes	characteristic
180	C₈	C₄×Dic₁₄₆	1	2	yes	—
181	C₂²	—	2	1	no	—
182	C₂²	—	2	1	no	—
183	C₂²	C₂×#14556522	1	1	yes	characteristic
184	C₂²	C₄×#11924016	1	2	yes	—
185	C₄	—	2	1	no	—
186	C₄	—	2	1	no	—
187	C₄	—	2	1	no	—
188	C₄	—	8	4	no	—
189	C₄	#17988409	1	2	yes	direct product factor
190	C₄	C₄×#11924007	1	1	yes	characteristic
191	C₄	#17988623	1	1	yes	characteristic
192	C₄	C₄×#11924015	1	2	yes	—
193	C₂	—	2	2	no	—
194	C₂	C₄×#14556522	1	1	yes	characteristic
195	C₂	C₂×#17988409	1	1	yes	characteristic
196	C₂	#22611703	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14556522
commutator	C₈×C₇₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₇₃
Fitting	C₄×#106×C₇₃	C₂
socle	C₂²×C₇₃	C₂×#106
prime radical	C₂³×C₇₃	C₂×SD₁₆
cosocle	C₂²×C₈×C₇₃	C₂³
hypercenter	C₄×#106	D₇₃
hypocenter	C₇₃	C₄×#365
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#106

Complements

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

N	H ≅ G/N	type
10. #17988409	C₄	direct (N × H)
11. #17988363	C₄	semidirect
12. C₄×#11924003	C₄	semidirect
43. #11924003	C₄²	semidirect
90. C₄×C₇₃	#365	semidirect
104. C₇₃	C₄×#365	semidirect
189. C₄	#17988409	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×#17988409	2
C₄×#14556531	2
C₄×#14556527	2
#22608029	2
C₄×#106×C₇₃	2
C₄×#365	73

Subgroups — C4×#17988409 #33001369

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#17988409 #33001369