Subgroups — C₂×#22624960 #34011738

Order 18688.

286 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#11924031	C₂	1	2	yes	maximal
2	#22624960	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₈×#10596831	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11924031	—	2	2	no	—
5	C₄⋊#11924031	—	2	2	no	—
6	C₄⋊#11924031	C₂²	1	4	yes	—
7	C₂×C₄×#10596831	C₄	1	1	yes	characteristic
8	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
9	C₈×#10596831	C₄	1	2	yes	—
10	C₈×#10596831	C₂²	1	2	yes	—
11	C₂×C₄×C₈×C₇₃	C₄	1	1	yes	characteristic
12	C₂×#11924031	—	2	4	no	—
13	C₂×#11924031	—	4	4	no	—
14	C₂²×#10596831	—	2	1	no	—
15	C₂²×#10596831	D₄	1	1	yes	characteristic
16	C₂²×#10596831	Q₈	1	1	yes	characteristic
17	C₄×#10596831	—	2	1	no	—
18	C₄×#10596831	C₂×C₄	1	2	yes	—
19	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
20	C₄×#10596831	D₄	1	1	yes	characteristic
21	C₄×#10596831	Q₈	1	1	yes	characteristic
22	C₄×#10596831	C₂³	1	1	yes	characteristic
23	C₂²×C₈×C₇₃	C₈	1	1	yes	characteristic
24	C₂²×C₈×C₇₃	C₈	1	1	yes	characteristic
25	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₇₃	C₈	1	2	yes	—
27	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
28	#11924031	—	4	4	no	—
29	C₂×#10596831	—	2	4	no	—
30	C₂×#10596831	—	2	2	no	—
31	C₂×#10596831	—	2	2	no	—
32	C₂×#10596831	C₄⋊C₄	1	1	yes	characteristic
33	C₂×#10596831	C₄⋊C₄	1	1	yes	characteristic
34	C₂×#10596831	D₈	1	2	yes	—
35	C₂×#10596831	Q₁₆	1	2	yes	—
36	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
37	C₂×#10596831	C₂×Q₈	1	1	yes	characteristic
38	C₂²×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
39	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
40	C₂²×C₄×C₇₃	M₁₆	1	1	yes	characteristic
41	C₂×C₈×C₇₃	—	2	2	no	—
42	C₂×C₈×C₇₃	C₁₆	1	4	yes	complemented
43	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
44	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
45	C₄²×C₇₃	C₄⋊C₄	1	1	yes	characteristic
46	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
47	C₄²×C₇₃	M₁₆	1	1	yes	characteristic
48	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
49	C₂³×C₇₃	C₄⋊C₈	1	1	yes	characteristic
50	C₂×C₄×C₇₃	—	2	2	no	—
51	C₂×C₄×C₇₃	C₄⋊C₈	1	1	yes	characteristic
52	C₂×C₄×C₇₃	C₄⋊C₈	1	1	yes	characteristic
53	C₂×C₄×C₇₃	C₄⋊C₈	1	1	yes	characteristic
54	C₂×C₄×C₇₃	#107	1	2	yes	—
55	C₂×C₄×C₇₃	#108	1	2	yes	—
56	C₂×C₄×C₇₃	C₂×C₁₆	1	2	yes	—
57	C₂×C₄×C₇₃	M₃₂	1	2	yes	—
58	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
60	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
61	C₈×C₇₃	—	2	2	no	—
62	C₈×C₇₃	C₂×C₁₆	1	4	yes	complemented
63	#10596831	—	2	4	no	—
64	#10596831	—	2	1	no	—
65	#10596831	—	2	1	no	—
66	#10596831	#107	1	2	yes	—
67	#10596831	C₂×D₈	1	1	yes	characteristic
68	#10596831	C₂×Q₁₆	1	1	yes	characteristic
69	C₂²×C₇₃	#335	1	2	yes	—
70	C₂²×C₇₃	C₄⋊C₁₆	1	2	yes	—
71	C₂²×C₇₃	#364	1	2	yes	—
72	C₂²×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
73	C₄×C₇₃	—	2	1	no	—
74	C₄×C₇₃	—	2	1	no	—
75	C₄×C₇₃	#335	1	2	yes	—
76	C₄×C₇₃	C₄⋊C₁₆	1	2	yes	—
77	C₄×C₇₃	C₂×#107	1	1	yes	characteristic
78	C₄×C₇₃	C₂×#108	1	1	yes	characteristic
79	C₄×C₇₃	C₂²×C₁₆	1	1	yes	characteristic
80	C₄×C₇₃	C₂×M₃₂	1	1	yes	characteristic
81	C₂×#1370	—	73	1	no	maximal, Sylow
82	C₂×C₇₃	#1370	1	4	yes	complemented
83	C₂×C₇₃	C₂×#335	1	1	yes	characteristic
84	C₂×C₇₃	C₂×C₄⋊C₁₆	1	1	yes	characteristic
85	C₂×C₇₃	C₂×#364	1	1	yes	characteristic
86	C₂×C₄⋊C₁₆	—	73	2	no	—
87	C₂×C₈²	—	73	1	no	—
88	#1370	—	73	4	no	—
89	C₇₃	C₂×#1370	1	1	yes	characteristic, complemented, Sylow
90	C₂²×C₁₆	—	146	2	no	—
91	C₂×C₄×C₈	—	73	1	no	—
92	C₂×C₄×C₈	—	73	1	no	—
93	C₂×C₄×C₈	#64408	1	1	yes	characteristic
94	C₄⋊C₁₆	—	73	4	no	—
95	C₄⋊C₁₆	—	146	2	no	—
96	C₈²	—	73	2	no	—
97	C₈²	—	73	2	no	—
98	C₂²×C₈	—	73	1	no	—
99	C₂²×C₈	—	73	1	no	—
100	C₂²×C₈	—	146	1	no	—
101	C₂²×C₈	#10596833	1	1	yes	characteristic
102	C₂²×C₈	#10596833	1	1	yes	characteristic
103	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
104	C₂×C₁₆	—	292	4	no	—
105	C₂×C₁₆	—	146	4	no	—
106	C₄×C₈	—	73	1	no	—
107	C₄×C₈	—	73	1	no	—
108	C₄×C₈	—	73	1	no	—
109	C₄×C₈	—	73	1	no	—
110	C₄×C₈	—	73	2	no	—
111	C₄×C₈	—	146	1	no	—
112	C₄×C₈	#10596833	1	2	yes	—
113	C₄×C₈	C₂×#64408	1	2	yes	—
114	C₂²×C₄	C₄⋊#64408	1	1	yes	characteristic
115	C₂²×C₄	C₂×#10596833	1	1	yes	characteristic
116	C₂²×C₄	#11924038	1	1	yes	characteristic
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	73	2	no	—
119	C₂×C₈	—	73	2	no	—
120	C₂×C₈	—	73	1	no	—
121	C₂×C₈	—	73	1	no	—
122	C₂×C₈	—	73	1	no	—
123	C₂×C₈	—	73	1	no	—
124	C₂×C₈	—	146	4	no	—
125	C₂×C₈	—	146	2	no	—
126	C₂×C₈	—	146	2	no	—
127	C₂×C₈	#11924031	1	4	yes	complemented
128	C₂×C₈	C₂×#10596833	1	2	yes	—
129	C₂×C₈	C₂×#10596833	1	2	yes	—
130	C₄²	C₄⋊#64408	1	1	yes	characteristic
131	C₄²	C₂×#10596833	1	1	yes	characteristic
132	C₄²	#11924038	1	1	yes	characteristic
133	C₄²	C₂²×#64408	1	1	yes	characteristic
134	C₁₆	—	292	4	no	—
135	C₂³	C₄⋊#10596833	1	1	yes	characteristic
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	C₄⋊#10596833	1	1	yes	characteristic
138	C₂×C₄	C₄⋊#10596833	1	1	yes	characteristic
139	C₂×C₄	C₄⋊#10596833	1	1	yes	characteristic
140	C₂×C₄	#14556660	1	2	yes	—
141	C₂×C₄	#14556662	1	2	yes	—
142	C₂×C₄	C₂×#11924031	1	2	yes	—
143	C₂×C₄	#14556665	1	2	yes	—
144	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
145	C₂×C₄	C₂²×#10596833	1	1	yes	characteristic
146	C₂×C₄	C₂×#11924038	1	1	yes	characteristic
147	C₈	—	2	2	no	—
148	C₈	—	73	1	no	—
149	C₈	—	73	2	no	—
150	C₈	—	73	1	no	—
151	C₈	—	146	4	no	—
152	C₈	—	146	1	no	—
153	C₈	—	146	1	no	—
154	C₈	C₂×#11924031	1	4	yes	complemented
155	C₂²	#17989450	1	2	yes	—
156	C₂²	C₄⋊#11924031	1	2	yes	—
157	C₂²	#17989509	1	2	yes	—
158	C₂²	C₂×C₄⋊#10596833	1	1	yes	characteristic
159	C₄	—	2	1	no	—
160	C₄	—	2	1	no	—
161	C₄	#17989450	1	2	yes	—
162	C₄	C₄⋊#11924031	1	2	yes	—
163	C₄	C₂×#14556660	1	1	yes	characteristic
164	C₄	C₂×#14556662	1	1	yes	characteristic
165	C₄	C₂²×#11924031	1	1	yes	characteristic
166	C₄	C₂×#14556665	1	1	yes	characteristic
167	C₂	#22624960	1	4	yes	direct product factor
168	C₂	C₂×#17989450	1	1	yes	characteristic
169	C₂	C₂×C₄⋊#11924031	1	1	yes	characteristic
170	C₂	C₂×#17989509	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₄⋊#64408
commutator	C₄×C₇₃	C₂²×C₁₆
Frattini	C₄²	C₂²×#64408
Fitting	C₂×C₄×C₈×C₇₃	C₄
socle, prime radical	C₂³×C₇₃	C₄⋊C₈
cosocle	C₄×#10596831	C₂³
hypercenter	C₂×C₄×C₈	#64408
hypocenter	C₇₃	C₂×#1370
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #22624960	C₂	direct (N × H)
42. C₂×C₈×C₇₃	C₁₆	semidirect
62. C₈×C₇₃	C₂×C₁₆	semidirect
82. C₂×C₇₃	#1370	semidirect
89. C₇₃	C₂×#1370	semidirect
127. C₂×C₈	#11924031	semidirect
154. C₈	C₂×#11924031	semidirect
167. C₂	#22624960	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×C₄⋊#11924031	2
#22624960	2
C₂×C₈×#10596831	2
C₂×#1370	73

Subgroups — C2×#22624960 #34011738

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22624960 #34011738