Subgroups — #34034681

Order 18688.

818 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×#17989709	C₂	1	2	yes	maximal
2	#22627093	C₂	1	6	yes	maximal, complemented
3	C₄×#14556688	C₂	1	3	yes	maximal, complemented
4	#22627087	C₂	1	3	yes	maximal, complemented
5	C₄×#14556679	C₂	1	1	yes	maximal, characteristic
6	#17989709	C₄	1	16	yes	complemented
7	C₂×#14556688	C₂²	1	6	yes	—
8	C₂×#14556686	C₂²	1	6	yes	—
9	#17989681	C₂²	1	1	yes	characteristic
10	#17989681	C₂²	1	3	yes	—
11	#17989681	C₂²	1	6	yes	—
12	C₄×#11924040	C₂²	1	6	yes	—
13	C₂×#14556679	C₄	1	1	yes	characteristic
14	C₂×#14556679	C₂²	1	1	yes	characteristic
15	#17989552	C₄	1	3	yes	—
16	#17989552	C₂²	1	3	yes	—
17	C₄×#11924038	C₄	1	3	yes	—
18	C₄×#11924036	C₂²	1	3	yes	—
19	C₄×#11924029	C₄	1	1	yes	characteristic
20	#14556688	C₂×C₄	1	24	yes	complemented
21	#14556686	C₂×C₄	1	24	yes	complemented
22	C₂×#11924040	C₂³	1	6	yes	—
23	C₂×#11924040	C₂³	1	2	yes	—
24	#14556679	C₈	1	8	yes	—
25	#14556679	C₂×C₄	1	8	yes	—
26	C₂×#11924038	C₂×C₄	1	3	yes	—
27	C₂×#11924038	C₂×C₄	1	3	yes	—
28	C₂×#11924036	C₂×C₄	1	3	yes	—
29	C₂×#11924036	C₂³	1	3	yes	—
30	#14556646	C₂×C₄	1	6	yes	—
31	C₄×#10596833	C₂×C₄	1	3	yes	—
32	C₄×#10596833	C₂×C₄	1	3	yes	—
33	C₄×#10596833	C₂³	1	3	yes	—
34	C₄×#10596833	C₂³	1	1	yes	characteristic
35	C₂×#11924029	C₂×C₄	1	1	yes	characteristic
36	Q₈×Dic₇₃	C₈	1	1	yes	characteristic
37	C₄×D₂₉₂	C₈	1	3	yes	—
38	#14556553	C₈	1	3	yes	—
39	#14556549	C₂×C₄	1	3	yes	—
40	C₄²×D₇₃	C₂×C₄	1	3	yes	—
41	C₄×Q₈×C₇₃	C₈	1	1	yes	characteristic
42	#11924040	C₂²×C₄	1	4	yes	—
43	#11924040	C₂²×C₄	1	12	yes	—
44	#11924038	C₄²	1	12	yes	—
45	#11924038	C₂×C₈	1	12	yes	—
46	#11924036	C₂×C₈	1	12	yes	—
47	#11924036	C₂²×C₄	1	12	yes	—
48	C₂×#10596833	C₂²×C₄	1	3	yes	—
49	C₂×#10596833	C₂²×C₄	1	1	yes	characteristic
50	C₂×#10596833	C₂²×C₄	1	3	yes	—
51	C₂×#10596833	C₂⁴	1	1	yes	characteristic
52	#11924029	C₄²	1	4	yes	—
53	#11924029	C₂×C₈	1	4	yes	—
54	C₂×D₂₉₂	C₂×C₈	1	3	yes	—
55	C₂×C₄×D₇₃	C₂×C₈	1	3	yes	—
56	C₂×C₄×D₇₃	C₂×C₈	1	3	yes	—
57	C₂×C₄×D₇₃	C₂²×C₄	1	3	yes	—
58	C₄⋊Dic₇₃	C₂×C₈	1	3	yes	—
59	#11924005	C₂×C₈	1	6	yes	—
60	C₄×Dic₇₃	C₂×C₈	1	3	yes	—
61	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
62	C₄×Dic₇₃	C₂²×C₄	1	3	yes	—
63	C₂×Q₈×C₇₃	C₂×C₈	1	1	yes	characteristic
64	C₄⋊C₄×C₇₃	C₂×C₈	1	3	yes	—
65	C₄²×C₇₃	C₂×C₈	1	3	yes	—
66	C₂²×D₇₃	C₂²×C₈	1	3	yes	—
67	Q₈×C₇₃	C₄×C₈	1	4	yes	—
68	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
69	C₂×C₄×C₇₃	C₂²×C₈	1	3	yes	—
70	C₂×C₄×C₇₃	C₂²×C₈	1	3	yes	—
71	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
72	C₂×Dic₇₃	C₂²×C₈	1	3	yes	—
73	C₂×Dic₇₃	C₂³×C₄	1	1	yes	characteristic
74	D₂₉₂	C₄×C₈	1	12	yes	—
75	C₄×D₇₃	—	4	3	no	—
76	C₄×D₇₃	—	4	3	no	—
77	C₄×D₇₃	C₂×C₄²	1	6	yes	—
78	C₄×D₇₃	C₂²×C₈	1	6	yes	—
79	#10596833	C₂×C₄²	1	6	yes	—
80	#10596833	C₂²×C₈	1	2	yes	—
81	#10596833	C₂²×C₈	1	6	yes	—
82	#10596833	C₂³×C₄	1	2	yes	—
83	C₂²×C₇₃	C₂³×C₈	1	1	yes	characteristic
84	C₂×D₇₃	—	2	3	no	—
85	C₂×D₇₃	—	2	3	no	—
86	C₂×D₇₃	C₂×C₄×C₈	1	6	yes	—
87	C₄×C₇₃	—	2	1	no	—
88	C₄×C₇₃	—	2	3	no	—
89	C₄×C₇₃	C₂×C₄×C₈	1	6	yes	—
90	Dic₇₃	—	2	1	no	—
91	Dic₇₃	—	2	3	no	—
92	Dic₇₃	C₂²×C₄²	1	1	yes	characteristic
93	Dic₇₃	C₂³×C₈	1	1	yes	characteristic
94	#33326	—	73	1	no	maximal, Sylow
95	C₂×C₇₃	C₂²×C₄×C₈	1	1	yes	characteristic
96	C₂×C₇₃	#3407	1	1	yes	characteristic
97	C₂×C₇₃	#3407	1	1	yes	characteristic
98	D₇₃	—	2	6	no	—
99	C₂×#504	—	73	2	no	—
100	C₄×#131	—	73	1	no	—
101	#2108	—	73	6	no	—
102	C₄×M₃₂	—	73	3	no	—
103	C₂×#346	—	73	3	no	—
104	C₇₃	#33326	1	1	yes	characteristic, complemented, Sylow
105	C₂×#131	—	73	1	no	—
106	C₂×#131	—	73	1	no	—
107	C₄×#41	—	73	1	no	—
108	#504	—	73	16	no	—
109	C₂×M₃₂	—	73	6	no	—
110	C₂²×C₁₆	—	73	6	no	—
111	#405	—	73	3	no	—
112	#405	—	73	3	no	—
113	C₄×M₁₆	—	73	3	no	—
114	C₂×C₄×C₈	—	73	3	no	—
115	#346	—	73	1	no	—
116	#346	—	73	3	no	—
117	#346	—	73	6	no	—
118	C₄×C₁₆	—	73	6	no	—
119	C₂×#41	—	73	1	no	—
120	#131	—	73	8	no	—
121	#131	—	73	8	no	—
122	C₂×M₁₆	—	73	3	no	—
123	C₂×M₁₆	—	73	3	no	—
124	C₂²×C₈	—	73	3	no	—
125	C₂²×C₈	—	73	3	no	—
126	C₄×Q₈	—	73	1	no	—
127	C₄×Q₈	#10596842	1	1	yes	characteristic
128	C₄×D₄	—	73	3	no	—
129	C₄×D₄	—	73	3	no	—
130	#117	—	73	3	no	—
131	C₂×C₄²	—	73	3	no	—
132	M₃₂	—	73	24	no	—
133	C₂×C₁₆	—	73	6	no	—
134	C₂×C₁₆	—	73	2	no	—
135	C₂×C₁₆	—	73	24	no	—
136	#97	—	73	6	no	—
137	C₄×C₈	—	73	3	no	—
138	C₄×C₈	—	73	1	no	—
139	C₄×C₈	—	73	3	no	—
140	C₄×C₈	—	73	3	no	—
141	#41	—	73	4	no	—
142	#41	—	73	4	no	—
143	C₂×Q₈	C₂×#10596842	1	1	yes	characteristic
144	C₂×D₄	—	73	3	no	—
145	C₂²×C₄	—	73	3	no	—
146	C₂²×C₄	—	73	3	no	—
147	C₂²×C₄	—	73	3	no	—
148	M₁₆	—	73	12	no	—
149	M₁₆	—	73	12	no	—
150	C₂×C₈	—	73	1	no	—
151	C₂×C₈	—	73	3	no	—
152	C₂×C₈	—	73	3	no	—
153	C₂×C₈	—	73	1	no	—
154	C₂×C₈	—	73	12	no	—
155	C₂×C₈	—	73	12	no	—
156	C₄⋊C₄	—	73	3	no	—
157	C₄⋊C₄	C₂×#10596842	1	3	yes	—
158	C₂²⋊C₄	—	73	6	no	—
159	C₄²	—	73	3	no	—
160	C₄²	—	73	1	no	—
161	C₄²	—	73	3	no	—
162	C₄²	C₂×#10596842	1	3	yes	—
163	C₁₆	—	73	4	no	—
164	C₁₆	—	73	12	no	—
165	C₂³	—	73	3	no	—
166	Q₈	C₄×#10596842	1	4	yes	—
167	D₄	—	73	12	no	—
168	C₂×C₄	—	292	3	no	—
169	C₂×C₄	—	292	3	no	—
170	C₂×C₄	—	73	1	no	—
171	C₂×C₄	—	73	6	no	—
172	C₂×C₄	—	73	6	no	—
173	C₂×C₄	—	73	1	no	—
174	C₂×C₄	—	73	3	no	—
175	C₂×C₄	C₂²×#10596842	1	1	yes	characteristic
176	C₂×C₄	C₂²×#10596842	1	3	yes	—
177	C₂×C₄	C₂²×#10596842	1	3	yes	—
178	C₈	—	73	2	no	—
179	C₈	—	73	2	no	—
180	C₈	—	73	6	no	—
181	C₈	—	73	6	no	—
182	C₂²	—	73	6	no	—
183	C₂²	—	146	3	no	—
184	C₂²	—	146	3	no	—
185	C₂²	C₂³×#10596842	1	1	yes	characteristic
186	C₄	—	2	3	no	—
187	C₄	—	2	1	no	—
188	C₄	—	73	1	no	—
189	C₄	—	73	1	no	—
190	C₄	—	146	1	no	—
191	C₄	—	146	3	no	—
192	C₄	C₂×C₄×#10596842	1	6	yes	—
193	C₂	—	146	6	no	—
194	C₂	C₂²×C₄×#10596842	1	1	yes	characteristic
195	C₂	#22627185	1	1	yes	characteristic
196	C₂	#22627186	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#10596842
commutator	C₂×C₇₃	C₂²×C₄×C₈
Fitting	C₄×Q₈×C₇₃	C₈
socle	C₂²×C₇₃	C₂³×C₈
prime radical	C₂×#11924029	C₂×C₄
cosocle	C₂×#10596833	C₂⁴
hypercenter	C₄×Q₈	#10596842
hypocenter	C₇₃	#33326
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
2. #22627093	C₂	semidirect
3. C₄×#14556688	C₂	semidirect
4. #22627087	C₂	semidirect
6. #17989709	C₄	semidirect
20. #14556688	C₂×C₄	semidirect
21. #14556686	C₂×C₄	semidirect
104. C₇₃	#33326	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17989709	2
#22627093	2
C₄×#14556688	2
#22627087	2
C₄×#14556679	2
#33326	73