Subgroups — #34012111

Order 18688.

678 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14556665	C₂	1	2	yes	maximal
2	#22624970	C₂	1	4	yes	maximal, complemented
3	C₂²×C₈×Dic₇₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#14556665	—	2	4	no	—
5	C₂×#14556665	—	2	4	no	—
6	C₂×#14556665	C₄	1	8	yes	complemented
7	C₂²×#11924031	C₂²	1	4	yes	—
8	C₂³×#10596831	C₂²	1	1	yes	characteristic
9	C₂²×C₄×Dic₇₃	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₇₃	—	2	4	no	—
11	C₂×C₈×Dic₇₃	C₄	1	2	yes	—
12	C₂×C₈×Dic₇₃	C₂²	1	2	yes	—
13	C₂³×C₈×C₇₃	C₄	1	1	yes	characteristic
14	#14556665	—	2	16	no	—
15	C₂×#11924031	—	2	4	no	—
16	C₂×#11924031	—	2	4	no	—
17	C₂×#11924031	C₂×C₄	1	8	yes	complemented
18	C₂³×Dic₇₃	C₈	1	2	yes	—
19	C₂²×#10596831	C₂×C₄	1	4	yes	—
20	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
21	C₂²×#10596831	D₄	1	4	yes	—
22	C₂²×#10596831	D₄	1	2	yes	—
23	C₂²×#10596831	Q₈	1	2	yes	—
24	C₂²×#10596831	C₂³	1	1	yes	characteristic
25	C₂×C₄×Dic₇₃	—	2	2	no	—
26	C₂×C₄×Dic₇₃	—	2	2	no	—
27	C₂×C₄×Dic₇₃	C₈	1	2	yes	—
28	C₂×C₄×Dic₇₃	C₂×C₄	1	2	yes	—
29	C₈×Dic₇₃	—	2	4	no	—
30	C₈×Dic₇₃	—	2	4	no	—
31	C₂³×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₇₃	—	2	2	no	—
33	C₂²×C₈×C₇₃	—	2	2	no	—
34	C₂²×C₈×C₇₃	C₈	1	4	yes	—
35	C₂²×C₈×C₇₃	C₂×C₄	1	2	yes	—
36	#11924031	—	2	8	no	—
37	C₂²×Dic₇₃	—	2	4	no	—
38	C₂²×Dic₇₃	—	2	8	no	—
39	C₂²×Dic₇₃	C₂×C₈	1	4	yes	—
40	C₂×#10596831	C₄²	1	2	yes	—
41	C₂×#10596831	C₂²⋊C₄	1	2	yes	—
42	C₂×#10596831	C₂²⋊C₄	1	8	yes	—
43	C₂×#10596831	C₄⋊C₄	1	2	yes	—
44	C₂×#10596831	C₄⋊C₄	1	8	yes	—
45	C₂×#10596831	C₂²×C₄	1	2	yes	—
46	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
47	C₂×#10596831	C₂×D₄	1	2	yes	—
48	C₂×#10596831	C₂×Q₈	1	1	yes	characteristic
49	C₄×Dic₇₃	—	2	4	no	—
50	C₄×Dic₇₃	—	2	2	no	—
51	C₄×Dic₇₃	—	2	2	no	—
52	C₂⁴×C₇₃	C₂×C₈	1	1	yes	characteristic
53	C₂²×C₄×C₇₃	C₄²	1	2	yes	—
54	C₂²×C₄×C₇₃	C₂²⋊C₄	1	2	yes	—
55	C₂²×C₄×C₇₃	C₄⋊C₄	1	2	yes	—
56	C₂²×C₄×C₇₃	C₂×C₈	1	2	yes	—
57	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
58	C₂²×C₄×C₇₃	M₁₆	1	2	yes	—
59	C₂²×C₄×C₇₃	M₁₆	1	2	yes	—
60	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₈×C₇₃	—	2	8	no	—
62	C₂×C₈×C₇₃	—	2	2	no	—
63	C₂×C₈×C₇₃	—	2	2	no	—
64	C₂×C₈×C₇₃	C₂×C₈	1	4	yes	—
65	C₂³×C₇₃	—	2	4	no	—
66	C₂³×C₇₃	C₄×C₈	1	2	yes	—
67	C₂³×C₇₃	C₂²⋊C₈	1	2	yes	—
68	C₂³×C₇₃	C₄⋊C₈	1	2	yes	—
69	C₂³×C₇₃	C₂²×C₈	1	1	yes	characteristic
70	C₂×C₄×C₇₃	#95	1	4	yes	—
71	C₂×C₄×C₇₃	C₄×C₈	1	2	yes	—
72	C₂×C₄×C₇₃	#97	1	4	yes	—
73	C₂×C₄×C₇₃	C₂²⋊C₈	1	4	yes	—
74	C₂×C₄×C₇₃	C₂²⋊C₈	1	2	yes	—
75	C₂×C₄×C₇₃	C₄⋊C₈	1	4	yes	—
76	C₂×C₄×C₇₃	C₄⋊C₈	1	2	yes	—
77	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
78	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
81	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
82	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
83	C₂×Dic₇₃	—	2	16	no	—
84	C₂×Dic₇₃	—	2	4	no	—
85	C₂×Dic₇₃	—	2	4	no	—
86	C₂×Dic₇₃	M₃₂	1	8	yes	complemented
87	C₈×C₇₃	—	2	4	no	—
88	#10596831	#95	1	4	yes	—
89	#10596831	C₂×C₂²⋊C₄	1	2	yes	—
90	#10596831	C₂×C₄⋊C₄	1	2	yes	—
91	C₂²×C₇₃	—	2	4	no	—
92	C₂²×C₇₃	—	2	4	no	—
93	C₂²×C₇₃	—	2	4	no	—
94	C₂²×C₇₃	#336	1	4	yes	—
95	C₂²×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
96	C₂²×C₇₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
97	C₂²×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
98	C₂²×C₇₃	C₂×M₃₂	1	4	yes	—
99	C₄×C₇₃	#336	1	4	yes	—
100	C₄×C₇₃	C₂×#95	1	1	yes	characteristic
101	C₄×C₇₃	C₂×#97	1	1	yes	characteristic
102	C₄×C₇₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
103	C₄×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
104	Dic₇₃	—	2	8	no	—
105	#12544	—	73	1	no	maximal, Sylow
106	C₂×C₇₃	—	2	4	no	—
107	C₂×C₇₃	C₂×#336	1	1	yes	characteristic
108	C₂×C₇₃	C₄×M₃₂	1	2	yes	—
109	C₂×C₇₃	#2112	1	2	yes	—
110	C₂×C₇₃	#2150	1	2	yes	—
111	C₂²×M₃₂	—	73	2	no	—
112	C₂²×C₄×C₈	—	73	1	no	—
113	#1374	—	73	4	no	—
114	C₇₃	#12544	1	1	yes	characteristic, complemented, Sylow
115	C₂³×C₈	—	73	1	no	—
116	C₂³×C₈	#64408	1	1	yes	characteristic
117	C₂²×C₄²	—	73	1	no	—
118	C₂×M₃₂	—	73	8	no	—
119	C₂×M₃₂	—	146	4	no	—
120	C₂×M₃₂	—	146	4	no	—
121	C₂²×C₁₆	—	73	4	no	—
122	C₂×C₄×C₈	—	73	2	no	—
123	C₂×C₄×C₈	—	73	2	no	—
124	C₂×C₄×C₈	—	146	4	no	—
125	C₂³×C₄	—	73	2	no	—
126	C₂³×C₄	C₂×#64408	1	1	yes	characteristic
127	C₂²×C₈	—	2	2	no	—
128	C₂²×C₈	—	2	2	no	—
129	C₂²×C₈	—	73	4	no	—
130	C₂²×C₈	—	73	4	no	—
131	C₂²×C₈	—	73	2	no	—
132	C₂²×C₈	—	73	2	no	—
133	C₂²×C₈	—	73	1	no	—
134	C₂²×C₈	—	73	1	no	—
135	C₂²×C₈	#10596833	1	4	yes	—
136	C₂²×C₈	C₂×#64408	1	2	yes	—
137	C₂×C₄²	—	73	2	no	—
138	C₂×C₄²	—	73	2	no	—
139	C₂×C₄²	—	146	2	no	—
140	C₂×C₄²	—	146	2	no	—
141	M₃₂	—	146	16	no	—
142	C₂×C₁₆	—	73	8	no	—
143	C₂×C₁₆	—	146	4	no	—
144	C₂×C₁₆	—	146	4	no	—
145	C₄×C₈	—	146	4	no	—
146	C₄×C₈	—	146	4	no	—
147	C₂⁴	#11924036	1	1	yes	characteristic
148	C₂²×C₄	—	73	4	no	—
149	C₂²×C₄	—	146	8	no	—
150	C₂²×C₄	—	146	4	no	—
151	C₂²×C₄	C₄×#64408	1	2	yes	—
152	C₂²×C₄	C₂²⋊#64408	1	2	yes	—
153	C₂²×C₄	C₄⋊#64408	1	2	yes	—
154	C₂²×C₄	C₂×#10596833	1	2	yes	—
155	C₂²×C₄	#11924036	1	1	yes	characteristic
156	C₂²×C₄	#11924037	1	2	yes	—
157	C₂²×C₄	#11924038	1	2	yes	—
158	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
159	C₂×C₈	—	2	2	no	—
160	C₂×C₈	—	2	8	no	—
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	—	73	8	no	—
163	C₂×C₈	—	73	8	no	—
164	C₂×C₈	—	73	2	no	—
165	C₂×C₈	—	73	2	no	—
166	C₂×C₈	—	73	2	no	—
167	C₂×C₈	—	73	2	no	—
168	C₂×C₈	—	73	2	no	—
169	C₂×C₈	—	73	1	no	—
170	C₂×C₈	—	73	1	no	—
171	C₂×C₈	C₂×#10596833	1	4	yes	—
172	C₄²	—	146	2	no	—
173	C₄²	—	146	4	no	—
174	C₄²	—	146	2	no	—
175	C₁₆	—	146	8	no	—
176	C₂³	—	2	4	no	—
177	C₂³	C₄×#10596833	1	2	yes	—
178	C₂³	#14556648	1	2	yes	—
179	C₂³	#14556658	1	2	yes	—
180	C₂³	C₂×#11924036	1	1	yes	characteristic
181	C₂×C₄	—	73	8	no	—
182	C₂×C₄	—	146	16	no	—
183	C₂×C₄	—	146	4	no	—
184	C₂×C₄	—	146	4	no	—
185	C₂×C₄	(C₂×#64408)⋊C₄	1	4	yes	—
186	C₂×C₄	C₄×#10596833	1	2	yes	—
187	C₂×C₄	#14556646	1	4	yes	—
188	C₂×C₄	C₂²⋊#10596833	1	4	yes	—
189	C₂×C₄	#14556648	1	2	yes	—
190	C₂×C₄	C₄⋊#10596833	1	4	yes	—
191	C₂×C₄	#14556658	1	2	yes	—
192	C₂×C₄	C₂×C₄×#64408	1	1	yes	characteristic
193	C₂×C₄	C₂×C₂²⋊#64408	1	1	yes	characteristic
194	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
195	C₂×C₄	C₂²×#10596833	1	1	yes	characteristic
196	C₂×C₄	C₂×#11924037	1	1	yes	characteristic
197	C₂×C₄	C₂×#11924038	1	1	yes	characteristic
198	C₈	—	2	4	no	—
199	C₈	—	73	2	no	—
200	C₈	—	73	4	no	—
201	C₈	—	73	2	no	—
202	C₂²	—	2	4	no	—
203	C₂²	—	2	4	no	—
204	C₂²	—	2	4	no	—
205	C₂²	#17989453	1	4	yes	—
206	C₂²	C₂×C₄×#10596833	1	1	yes	characteristic
207	C₂²	C₂×#14556648	1	1	yes	characteristic
208	C₂²	C₂×#14556658	1	1	yes	characteristic
209	C₂²	#17989643	1	4	yes	—
210	C₄	—	146	8	no	—
211	C₄	#17989453	1	4	yes	—
212	C₄	C₂×(C₂×#64408)⋊C₄	1	1	yes	characteristic
213	C₄	C₂×#14556646	1	1	yes	characteristic
214	C₄	C₂×C₂²⋊#10596833	1	1	yes	characteristic
215	C₄	C₂×C₄⋊#10596833	1	1	yes	characteristic
216	C₂	—	2	4	no	—
217	C₂	C₂×#17989453	1	1	yes	characteristic
218	C₂	#22626137	1	2	yes	—
219	C₂	#22626160	1	2	yes	—
220	C₂	#22626265	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#64408
commutator	C₂²×C₇₃	C₂×C₄×C₈
Fitting	C₂³×C₈×C₇₃	C₄
socle	C₂³×C₇₃	C₂²×C₈
prime radical	C₂⁴×C₇₃	C₂×C₈
cosocle	C₂²×#10596831	C₂³
hypercenter	C₂³×C₈	#64408
hypocenter	C₇₃	#12544
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₈

Complements

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

N	H ≅ G/N	type
2. #22624970	C₂	semidirect
6. C₂×#14556665	C₄	semidirect
17. C₂×#11924031	C₂×C₄	semidirect
86. C₂×Dic₇₃	M₃₂	semidirect
114. C₇₃	#12544	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#14556665	2
#22624970	2
C₂²×C₈×Dic₇₃	2
#12544	73