Subgroups — C₂×#22626898 #34032683

Order 18688.

2074 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22626898	C₂	1	16	yes	maximal, direct product factor
2	C₂×#17989645	C₂	1	8	yes	maximal, complemented
3	C₂²×#14556665	C₂	1	6	yes	maximal, complemented
4	C₂²×#14556598	C₂	1	1	yes	maximal, characteristic
5	#17989645	C₂²	1	64	yes	complemented
6	C₂×#14556665	C₂²	1	4	yes	—
7	C₂×#14556665	C₂²	1	12	yes	complemented
8	C₂×#14556665	C₂²	1	48	yes	complemented
9	C₂²×#11924031	C₂²	1	12	yes	complemented
10	C₂×#14556598	C₄	1	4	yes	—
11	C₂×#14556598	C₄	1	8	yes	—
12	C₂×#14556598	C₂²	1	4	yes	—
13	C₂×#14556598	C₂²	1	8	yes	—
14	C₂²×#11924012	C₄	1	3	yes	—
15	C₂³×#10596831	C₂²	1	3	yes	—
16	C₂²×#41×C₇₃	C₄	1	1	yes	characteristic
17	#14556665	C₂³	1	16	yes	—
18	#14556665	C₂³	1	48	yes	complemented
19	C₂×#11924031	C₂³	1	8	yes	—
20	C₂×#11924031	C₂³	1	48	yes	complemented
21	#14556598	C₂×C₄	1	16	yes	—
22	#14556598	C₂×C₄	1	32	yes	—
23	#14556598	C₂³	1	16	yes	—
24	C₂×#11924012	C₂×C₄	1	12	yes	—
25	C₂×#11924012	C₂×C₄	1	24	yes	—
26	C₂²×#10596831	C₂×C₄	1	3	yes	—
27	C₂²×#10596831	C₂×C₄	1	12	yes	—
28	C₂²×#10596831	C₂×C₄	1	6	yes	—
29	C₂²×#10596831	C₂³	1	6	yes	—
30	C₂²×#10596831	C₂³	1	1	yes	characteristic
31	C₂²×#10596831	C₂³	1	12	yes	—
32	C₂×#41×C₇₃	C₈	1	4	yes	—
33	C₂×#41×C₇₃	C₈	1	8	yes	—
34	C₂×#41×C₇₃	C₂×C₄	1	4	yes	—
35	C₂×#41×C₇₃	C₂×C₄	1	8	yes	—
36	C₂²×Q₈×C₇₃	C₈	1	1	yes	characteristic
37	C₂²×D₄×C₇₃	C₈	1	3	yes	—
38	C₂³×C₄×C₇₃	C₂×C₄	1	3	yes	—
39	#11924031	C₂⁴	1	16	yes	—
40	#11924012	C₂²×C₄	1	48	yes	—
41	C₂×#10596831	C₂²×C₄	1	2	yes	—
42	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
43	C₂×#10596831	C₂²×C₄	1	12	yes	—
44	C₂×#10596831	C₂²×C₄	1	6	yes	—
45	C₂×#10596831	C₂²×C₄	1	12	yes	—
46	C₂×#10596831	C₂²×C₄	1	24	yes	—
47	C₂×#10596831	C₂⁴	1	1	yes	characteristic
48	C₂×#10596831	C₂⁴	1	2	yes	—
49	C₂×#10596831	C₂⁴	1	12	yes	—
50	C₂⁴×C₇₃	C₂×C₈	1	3	yes	—
51	#41×C₇₃	C₂×C₈	1	16	yes	—
52	#41×C₇₃	C₂×C₈	1	32	yes	—
53	#41×C₇₃	C₂²×C₄	1	16	yes	—
54	C₂×Q₈×C₇₃	C₂×C₈	1	8	yes	—
55	C₂×Q₈×C₇₃	C₂×C₈	1	4	yes	—
56	C₂×D₄×C₇₃	C₂×C₈	1	12	yes	—
57	C₂×D₄×C₇₃	C₂×C₈	1	24	yes	—
58	C₂²×C₄×C₇₃	C₂×C₈	1	3	yes	—
59	C₂²×C₄×C₇₃	C₂×C₈	1	6	yes	—
60	C₂²×C₄×C₇₃	C₂×C₈	1	12	yes	—
61	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₇₃	C₂²×C₄	1	12	yes	—
63	C₂²×C₄×C₇₃	C₂²×C₄	1	6	yes	—
64	C₂³×C₇₃	—	4	6	no	—
65	C₂³×C₇₃	C₂²×C₈	1	1	yes	characteristic
66	C₂³×C₇₃	C₂²×C₈	1	6	yes	—
67	C₂³×C₇₃	C₂²×C₈	1	12	yes	—
68	Q₈×C₇₃	C₂²×C₈	1	16	yes	—
69	D₄×C₇₃	C₂²×C₈	1	48	yes	—
70	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
71	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
72	C₂×C₄×C₇₃	C₂²×C₈	1	12	yes	—
73	C₂×C₄×C₇₃	C₂²×C₈	1	6	yes	—
74	C₂×C₄×C₇₃	C₂²×C₈	1	24	yes	—
75	C₂×C₄×C₇₃	C₂²×C₈	1	12	yes	—
76	C₂×C₄×C₇₃	C₂³×C₄	1	2	yes	—
77	C₂×C₄×C₇₃	C₂³×C₄	1	12	yes	—
78	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
79	#10596831	C₂³×C₄	1	1	yes	characteristic
80	#10596831	C₂³×C₄	1	2	yes	—
81	#10596831	C₂³×C₄	1	12	yes	—
82	#10596831	C₂⁵	1	1	yes	characteristic
83	C₂²×C₇₃	—	2	2	no	—
84	C₂²×C₇₃	—	2	12	no	—
85	C₂²×C₇₃	—	4	12	no	—
86	C₂²×C₇₃	C₂³×C₈	1	2	yes	—
87	C₂²×C₇₃	C₂³×C₈	1	12	yes	—
88	C₂²×C₇₃	C₂³×C₈	1	1	yes	characteristic
89	C₄×C₇₃	C₂³×C₈	1	2	yes	—
90	C₄×C₇₃	C₂³×C₈	1	1	yes	characteristic
91	C₄×C₇₃	C₂³×C₈	1	12	yes	—
92	C₄×C₇₃	C₂⁴×C₄	1	1	yes	characteristic
93	C₂×#3407	—	73	1	no	maximal, Sylow
94	C₂×C₇₃	—	2	2	no	—
95	C₂×C₇₃	—	2	12	no	—
96	C₂×C₇₃	#3407	1	2	yes	complemented
97	C₂×C₇₃	C₂⁴×C₈	1	1	yes	characteristic
98	C₂²×#131	—	73	1	no	—
99	#3407	—	73	16	no	—
100	C₂×#504	—	73	8	no	—
101	C₂²×M₃₂	—	73	6	no	—
102	C₇₃	C₂×#3407	1	1	yes	characteristic, complemented, Sylow
103	C₂²×#41	#64408	1	1	yes	characteristic
104	C₂×#131	—	73	4	no	—
105	C₂×#131	—	73	8	no	—
106	C₂×#131	—	73	8	no	—
107	C₂×#131	—	73	4	no	—
108	C₂²×M₁₆	—	73	3	no	—
109	C₂³×C₈	—	73	3	no	—
110	#504	—	73	64	no	—
111	C₂×M₃₂	—	73	4	no	—
112	C₂×M₃₂	—	73	12	no	—
113	C₂×M₃₂	—	73	48	no	—
114	C₂²×C₁₆	—	73	12	no	—
115	C₂×#41	#10596833	1	4	yes	—
116	C₂×#41	#10596833	1	8	yes	—
117	C₂×#41	C₂×#64408	1	8	yes	—
118	C₂×#41	C₂×#64408	1	4	yes	—
119	C₂²×Q₈	#10596833	1	1	yes	characteristic
120	C₂²×D₄	#10596833	1	3	yes	—
121	C₂³×C₄	C₂×#64408	1	3	yes	—
122	#131	—	73	16	no	—
123	#131	—	73	32	no	—
124	#131	—	73	16	no	—
125	C₂×M₁₆	—	73	24	no	—
126	C₂×M₁₆	—	73	12	no	—
127	C₂²×C₈	—	73	1	no	—
128	C₂²×C₈	—	73	3	no	—
129	C₂²×C₈	—	73	12	no	—
130	C₂²×C₈	—	73	6	no	—
131	C₂²×C₈	—	73	6	no	—
132	C₂²×C₈	—	73	12	no	—
133	M₃₂	—	73	16	no	—
134	M₃₂	—	73	48	no	—
135	C₂×C₁₆	—	73	8	no	—
136	C₂×C₁₆	—	73	48	no	—
137	C₂⁴	C₂×#10596833	1	3	yes	—
138	#41	C₂×#10596833	1	32	yes	—
139	#41	C₂×#10596833	1	16	yes	—
140	#41	C₂²×#64408	1	16	yes	—
141	C₂×Q₈	C₂×#10596833	1	8	yes	—
142	C₂×Q₈	C₂×#10596833	1	4	yes	—
143	C₂×D₄	C₂×#10596833	1	12	yes	—
144	C₂×D₄	C₂×#10596833	1	24	yes	—
145	C₂²×C₄	C₂×#10596833	1	3	yes	—
146	C₂²×C₄	C₂×#10596833	1	6	yes	—
147	C₂²×C₄	C₂×#10596833	1	12	yes	—
148	C₂²×C₄	C₂²×#64408	1	12	yes	—
149	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
150	C₂²×C₄	C₂²×#64408	1	6	yes	—
151	M₁₆	—	73	48	no	—
152	C₂×C₈	—	73	2	no	—
153	C₂×C₈	—	73	12	no	—
154	C₂×C₈	—	73	12	no	—
155	C₂×C₈	—	73	24	no	—
156	C₂×C₈	—	73	12	no	—
157	C₂×C₈	—	73	2	no	—
158	C₂×C₈	—	73	1	no	—
159	C₂×C₈	—	73	6	no	—
160	C₂×C₈	—	73	1	no	—
161	C₁₆	—	73	16	no	—
162	C₂³	—	4	6	no	—
163	C₂³	C₂²×#10596833	1	1	yes	characteristic
164	C₂³	C₂²×#10596833	1	6	yes	—
165	C₂³	C₂²×#10596833	1	12	yes	—
166	Q₈	C₂²×#10596833	1	16	yes	—
167	D₄	C₂²×#10596833	1	48	yes	—
168	C₂×C₄	C₂²×#10596833	1	1	yes	characteristic
169	C₂×C₄	C₂²×#10596833	1	2	yes	—
170	C₂×C₄	C₂²×#10596833	1	12	yes	—
171	C₂×C₄	C₂²×#10596833	1	6	yes	—
172	C₂×C₄	C₂²×#10596833	1	24	yes	—
173	C₂×C₄	C₂²×#10596833	1	12	yes	—
174	C₂×C₄	C₂³×#64408	1	2	yes	—
175	C₂×C₄	C₂³×#64408	1	12	yes	—
176	C₂×C₄	C₂³×#64408	1	1	yes	characteristic
177	C₈	—	73	1	no	—
178	C₈	—	73	1	no	—
179	C₈	—	73	2	no	—
180	C₈	—	73	12	no	—
181	C₂²	—	2	2	no	—
182	C₂²	—	2	12	no	—
183	C₂²	—	4	12	no	—
184	C₂²	C₂³×#10596833	1	2	yes	—
185	C₂²	C₂³×#10596833	1	12	yes	—
186	C₂²	C₂³×#10596833	1	1	yes	characteristic
187	C₄	C₂³×#10596833	1	2	yes	—
188	C₄	C₂³×#10596833	1	1	yes	characteristic
189	C₄	C₂³×#10596833	1	12	yes	—
190	C₄	C₂⁴×#64408	1	1	yes	characteristic
191	C₂	—	2	2	no	—
192	C₂	—	2	12	no	—
193	C₂	#22626898	1	2	yes	direct product factor
194	C₂	C₂⁴×#10596833	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂³×#64408
commutator	C₂×C₇₃	C₂⁴×C₈
Frattini	C₄	C₂⁴×#64408
Fitting, prime radical	C₂²×#41×C₇₃	C₄
socle	C₂²×C₇₃	C₂³×C₈
cosocle	#10596831	C₂⁵
hypercenter	C₂²×#41	#64408
hypocenter	C₇₃	C₂×#3407
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×#41

Complements

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

N	H ≅ G/N	type
1. #22626898	C₂	direct (N × H)
2. C₂×#17989645	C₂	semidirect
3. C₂²×#14556665	C₂	semidirect
5. #17989645	C₂²	semidirect
7. C₂×#14556665	C₂²	semidirect
8. C₂×#14556665	C₂²	semidirect
9. C₂²×#11924031	C₂²	semidirect
18. #14556665	C₂³	semidirect
20. C₂×#11924031	C₂³	semidirect
96. C₂×C₇₃	#3407	semidirect
102. C₇₃	C₂×#3407	semidirect
193. C₂	#22626898	direct (N × H)

Maximal subgroups

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

subgroup	index
#22626898	2
C₂×#17989645	2
C₂²×#14556665	2
C₂²×#14556598	2
C₂×#3407	73

Subgroups — C2×#22626898 #34032683

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22626898 #34032683