Subgroups — C₂×#22624954 #34011714

Order 18688.

286 conjugacy classes of subgroups.

The subgroup lattice has 168 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	#22624954	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	C₈	1	1	yes	characteristic
15	C₂²×#10596831	C₈	1	1	yes	characteristic
16	C₂²×#10596831	D₄	1	1	yes	characteristic
17	C₂²×#10596831	Q₈	1	1	yes	characteristic
18	C₄×#10596831	C₈	1	2	yes	—
19	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
20	C₄×#10596831	C₂×C₄	1	2	yes	—
21	C₄×#10596831	D₄	1	1	yes	characteristic
22	C₄×#10596831	Q₈	1	1	yes	characteristic
23	C₄×#10596831	C₂³	1	1	yes	characteristic
24	C₂²×C₈×C₇₃	—	2	1	no	—
25	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₇₃	—	2	1	no	—
27	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
28	#11924031	—	4	4	no	—
29	C₂×#10596831	—	2	2	no	—
30	C₂×#10596831	—	2	2	no	—
31	C₂×#10596831	C₁₆	1	4	yes	complemented
32	C₂×#10596831	C₄⋊C₄	1	1	yes	characteristic
33	C₂×#10596831	C₄⋊C₄	1	1	yes	characteristic
34	C₂×#10596831	C₂×C₈	1	2	yes	—
35	C₂×#10596831	C₂×C₈	1	2	yes	—
36	C₂×#10596831	SD₁₆	1	4	yes	—
37	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
38	C₂×#10596831	C₂×Q₈	1	1	yes	characteristic
39	C₂²×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
40	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
41	C₂²×C₄×C₇₃	M₁₆	1	1	yes	characteristic
42	C₂×C₈×C₇₃	—	2	4	no	—
43	C₂×C₈×C₇₃	—	2	2	no	—
44	C₄²×C₇₃	C₄⋊C₄	1	1	yes	characteristic
45	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
46	C₄²×C₇₃	M₁₆	1	1	yes	characteristic
47	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
48	C₂³×C₇₃	C₄⋊C₈	1	1	yes	characteristic
49	C₂×C₄×C₇₃	—	2	2	no	—
50	C₂×C₄×C₇₃	C₄⋊C₈	1	1	yes	characteristic
51	C₂×C₄×C₇₃	C₄⋊C₈	1	1	yes	characteristic
52	C₂×C₄×C₇₃	C₄⋊C₈	1	1	yes	characteristic
53	C₂×C₄×C₇₃	#106	1	2	yes	—
54	C₂×C₄×C₇₃	#108	1	2	yes	—
55	C₂×C₄×C₇₃	C₂×C₁₆	1	2	yes	—
56	C₂×C₄×C₇₃	M₃₂	1	2	yes	—
57	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
59	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
60	C₈×C₇₃	—	2	4	no	—
61	#10596831	—	2	1	no	—
62	#10596831	—	2	1	no	—
63	#10596831	—	2	2	no	—
64	#10596831	#106	1	2	yes	—
65	#10596831	C₂×C₁₆	1	4	yes	complemented
66	#10596831	C₂×SD₁₆	1	2	yes	—
67	C₂²×C₇₃	#334	1	2	yes	—
68	C₂²×C₇₃	C₄⋊C₁₆	1	2	yes	—
69	C₂²×C₇₃	#364	1	2	yes	—
70	C₂²×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
71	C₄×C₇₃	—	2	1	no	—
72	C₄×C₇₃	—	2	1	no	—
73	C₄×C₇₃	#334	1	2	yes	—
74	C₄×C₇₃	C₄⋊C₁₆	1	2	yes	—
75	C₄×C₇₃	C₂×#106	1	1	yes	characteristic
76	C₄×C₇₃	C₂×#108	1	1	yes	characteristic
77	C₄×C₇₃	C₂²×C₁₆	1	1	yes	characteristic
78	C₄×C₇₃	C₂×M₃₂	1	1	yes	characteristic
79	C₂×#1367	—	73	1	no	maximal, Sylow
80	C₂×C₇₃	#1367	1	4	yes	complemented
81	C₂×C₇₃	C₂×#334	1	1	yes	characteristic
82	C₂×C₇₃	C₂×C₄⋊C₁₆	1	1	yes	characteristic
83	C₂×C₇₃	C₂×#364	1	1	yes	characteristic
84	C₂×C₄⋊C₁₆	—	73	2	no	—
85	C₂×C₈²	—	73	1	no	—
86	#1367	—	73	4	no	—
87	C₇₃	C₂×#1367	1	1	yes	characteristic, complemented, Sylow
88	C₂²×C₁₆	—	146	2	no	—
89	C₂×C₄×C₈	—	73	1	no	—
90	C₂×C₄×C₈	—	73	1	no	—
91	C₂×C₄×C₈	#64408	1	1	yes	characteristic
92	C₄⋊C₁₆	—	73	4	no	—
93	C₄⋊C₁₆	—	146	2	no	—
94	C₈²	—	73	2	no	—
95	C₈²	—	73	2	no	—
96	C₂²×C₈	—	2	1	no	—
97	C₂²×C₈	—	73	1	no	—
98	C₂²×C₈	—	73	1	no	—
99	C₂²×C₈	—	73	1	no	—
100	C₂²×C₈	—	73	1	no	—
101	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
102	C₂×C₁₆	—	292	4	no	—
103	C₂×C₁₆	—	146	4	no	—
104	C₄×C₈	—	2	1	no	—
105	C₄×C₈	—	73	1	no	—
106	C₄×C₈	—	73	1	no	—
107	C₄×C₈	—	73	2	no	—
108	C₄×C₈	—	73	2	no	—
109	C₄×C₈	—	73	1	no	—
110	C₄×C₈	—	73	1	no	—
111	C₄×C₈	C₂×#64408	1	2	yes	—
112	C₂²×C₄	C₄⋊#64408	1	1	yes	characteristic
113	C₂²×C₄	#11924036	1	1	yes	characteristic
114	C₂²×C₄	#11924037	1	1	yes	characteristic
115	C₂×C₈	—	2	4	no	—
116	C₂×C₈	—	2	2	no	—
117	C₂×C₈	—	73	2	no	—
118	C₂×C₈	—	73	4	no	—
119	C₂×C₈	—	73	4	no	—
120	C₂×C₈	—	73	1	no	—
121	C₂×C₈	—	73	1	no	—
122	C₂×C₈	—	73	2	no	—
123	C₂×C₈	—	73	1	no	—
124	C₂×C₈	—	73	1	no	—
125	C₂×C₈	—	146	2	no	—
126	C₂×C₈	—	146	2	no	—
127	C₄²	C₄⋊#64408	1	1	yes	characteristic
128	C₄²	#11924036	1	1	yes	characteristic
129	C₄²	#11924037	1	1	yes	characteristic
130	C₄²	C₂²×#64408	1	1	yes	characteristic
131	C₁₆	—	292	4	no	—
132	C₂³	#14556658	1	1	yes	characteristic
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	#14556658	1	1	yes	characteristic
135	C₂×C₄	#14556658	1	1	yes	characteristic
136	C₂×C₄	#14556658	1	1	yes	characteristic
137	C₂×C₄	#14556659	1	2	yes	—
138	C₂×C₄	#14556662	1	2	yes	—
139	C₂×C₄	#14556664	1	2	yes	—
140	C₂×C₄	#14556666	1	2	yes	—
141	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
142	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
143	C₂×C₄	C₂×#11924037	1	1	yes	characteristic
144	C₈	—	2	4	no	—
145	C₈	—	73	4	no	—
146	C₈	—	73	2	no	—
147	C₈	—	73	2	no	—
148	C₈	—	146	2	no	—
149	C₈	—	146	1	no	—
150	C₈	—	146	1	no	—
151	C₂²	#17989449	1	2	yes	—
152	C₂²	#17989508	1	2	yes	—
153	C₂²	#17989510	1	2	yes	—
154	C₂²	C₂×#14556658	1	1	yes	characteristic
155	C₄	—	2	1	no	—
156	C₄	—	2	1	no	—
157	C₄	#17989449	1	2	yes	—
158	C₄	#17989508	1	2	yes	—
159	C₄	C₂×#14556659	1	1	yes	characteristic
160	C₄	C₂×#14556662	1	1	yes	characteristic
161	C₄	C₂×#14556664	1	1	yes	characteristic
162	C₄	C₂×#14556666	1	1	yes	characteristic
163	C₂	#22624954	1	4	yes	direct product factor
164	C₂	C₂×#17989449	1	1	yes	characteristic
165	C₂	C₂×#17989508	1	1	yes	characteristic
166	C₂	C₂×#17989510	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×#1367
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. #22624954	C₂	direct (N × H)
31. C₂×#10596831	C₁₆	semidirect
65. #10596831	C₂×C₁₆	semidirect
80. C₂×C₇₃	#1367	semidirect
87. C₇₃	C₂×#1367	semidirect
163. C₂	#22624954	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
#22624954	2
C₂×C₈×#10596831	2
C₂×#1367	73

Subgroups — C2×#22624954 #34011714

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22624954 #34011714