Subgroups — C₂×#22608006 #32997576

Order 18688.

230 conjugacy classes of subgroups.

The subgroup lattice has 178 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17988408	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#11924003	C₂	1	1	yes	maximal, characteristic
3	#22608006	C₂	1	4	yes	maximal, direct product factor
4	C₂×#364×C₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556531	—	2	1	no	—
6	C₂²×#11924003	C₄	1	1	yes	characteristic
7	C₂²×#11924003	C₄	1	1	yes	characteristic
8	#17988408	—	2	1	no	—
9	#17988408	C₂²	1	2	yes	—
10	C₄×#11924003	C₄	1	2	yes	—
11	C₄×#11924003	C₂²	1	2	yes	—
12	C₂×M₃₂×C₇₃	—	2	1	no	—
13	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
14	#364×C₇₃	—	2	1	no	—
15	#364×C₇₃	C₂²	1	2	yes	—
16	#14556531	—	2	2	no	—
17	#14556531	—	4	1	no	—
18	C₂×#11924003	—	2	1	no	—
19	C₂×#11924003	—	2	1	no	—
20	C₂×#11924003	—	4	1	no	—
21	C₂×#11924003	C₈	1	2	yes	—
22	C₂×#11924003	C₈	1	2	yes	—
23	C₂×#11924003	C₂×C₄	1	2	yes	—
24	C₂×#11924003	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
27	C₂×C₄²×C₇₃	Q₈	1	1	yes	characteristic
28	M₃₂×C₇₃	—	2	2	no	—
29	M₃₂×C₇₃	—	4	1	no	—
30	C₂×C₁₆×C₇₃	—	4	1	no	—
31	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
33	C₄×C₈×C₇₃	Q₈	1	1	yes	characteristic
34	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
35	#11924003	—	2	2	no	—
36	#11924003	—	4	2	no	—
37	#11924003	C₂×C₈	1	4	yes	—
38	C₂²×C₄×C₇₃	—	2	1	no	—
39	C₂²×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
43	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
44	C₂×C₈×C₇₃	M₁₆	1	1	yes	characteristic
45	C₂×C₈×C₇₃	M₁₆	1	1	yes	characteristic
46	C₂×C₈×C₇₃	D₈	1	1	yes	characteristic
47	C₂×C₈×C₇₃	D₈	1	1	yes	characteristic
48	C₂×C₈×C₇₃	Q₁₆	1	1	yes	characteristic
49	C₂×C₈×C₇₃	Q₁₆	1	1	yes	characteristic
50	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
52	C₄²×C₇₃	—	2	1	no	—
53	C₄²×C₇₃	C₄⋊C₄	1	1	yes	characteristic
54	C₄²×C₇₃	C₂×Q₈	1	1	yes	characteristic
55	C₁₆×C₇₃	—	4	2	no	—
56	C₂³×C₇₃	#108	1	1	yes	characteristic
57	C₂×C₄×C₇₃	—	2	4	no	—
58	C₂×C₄×C₇₃	—	2	1	no	—
59	C₂×C₄×C₇₃	—	2	1	no	—
60	C₂×C₄×C₇₃	C₄⋊C₈	1	1	yes	characteristic
61	C₂×C₄×C₇₃	C₄⋊C₈	1	1	yes	characteristic
62	C₂×C₄×C₇₃	#107	1	1	yes	characteristic
63	C₂×C₄×C₇₃	#107	1	1	yes	characteristic
64	C₂×C₄×C₇₃	#108	1	1	yes	characteristic
65	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₈×C₇₃	C₄⋊C₈	1	1	yes	characteristic
67	C₈×C₇₃	C₄⋊C₈	1	1	yes	characteristic
68	C₈×C₇₃	#107	1	1	yes	characteristic
69	C₈×C₇₃	#107	1	1	yes	characteristic
70	C₈×C₇₃	C₂²×C₈	1	1	yes	characteristic
71	C₈×C₇₃	C₂×M₁₆	1	1	yes	characteristic
72	C₈×C₇₃	C₂×D₈	1	1	yes	characteristic
73	C₈×C₇₃	C₂×Q₁₆	1	1	yes	characteristic
74	C₂²×C₇₃	—	2	2	no	—
75	C₂²×C₇₃	#335	1	1	yes	characteristic
76	C₂²×C₇₃	#335	1	1	yes	characteristic
77	C₂²×C₇₃	C₂×#108	1	1	yes	characteristic
78	C₄×C₇₃	—	2	2	no	—
79	C₄×C₇₃	—	2	2	no	—
80	C₄×C₇₃	#335	1	1	yes	characteristic
81	C₄×C₇₃	#335	1	1	yes	characteristic
82	C₄×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
83	C₄×C₇₃	C₂×#107	1	1	yes	characteristic
84	C₂×#1373	—	73	1	no	maximal, Sylow
85	C₂×C₇₃	—	2	1	no	—
86	C₂×C₇₃	—	2	1	no	—
87	C₂×C₇₃	#1373	1	2	yes	complemented
88	C₂×C₇₃	C₂×#335	1	1	yes	characteristic
89	C₂×#364	—	73	1	no	—
90	C₂×#364	D₇₃	1	1	yes	characteristic
91	C₂×C₄×C₁₆	—	73	1	no	—
92	#1373	—	73	4	no	—
93	C₇₃	C₂×#1373	1	1	yes	characteristic, complemented, Sylow
94	C₂×M₃₂	—	2	1	no	—
95	C₂×M₃₂	—	146	1	no	—
96	C₂²×C₁₆	—	73	1	no	—
97	C₂²×C₁₆	—	73	1	no	—
98	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
99	#364	—	2	1	no	—
100	#364	—	73	2	no	—
101	#364	—	146	1	no	—
102	#364	C₂×D₇₃	1	2	yes	—
103	C₄×C₁₆	—	73	2	no	—
104	C₄×C₁₆	—	73	2	no	—
105	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
106	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
107	C₂×C₄²	Dic₁₄₆	1	1	yes	characteristic
108	M₃₂	—	2	2	no	—
109	M₃₂	—	4	1	no	—
110	M₃₂	—	292	1	no	—
111	M₃₂	—	146	2	no	—
112	C₂×C₁₆	—	4	1	no	—
113	C₂×C₁₆	—	292	1	no	—
114	C₂×C₁₆	—	73	2	no	—
115	C₂×C₁₆	—	73	2	no	—
116	C₂×C₁₆	—	73	2	no	—
117	C₂×C₁₆	—	73	2	no	—
118	C₂×C₁₆	—	146	1	no	—
119	C₂×C₁₆	—	146	1	no	—
120	C₄×C₈	Dic₁₄₆	1	1	yes	characteristic
121	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
122	C₄×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
123	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
124	C₂²×C₄	—	2	1	no	—
125	C₂²×C₄	#11924007	1	1	yes	characteristic
126	C₂×C₈	#11924007	1	1	yes	characteristic
127	C₂×C₈	#11924007	1	1	yes	characteristic
128	C₂×C₈	C₈×D₇₃	1	1	yes	characteristic
129	C₂×C₈	C₈×D₇₃	1	1	yes	characteristic
130	C₂×C₈	#11924011	1	1	yes	characteristic
131	C₂×C₈	#11924011	1	1	yes	characteristic
132	C₂×C₈	#11924013	1	1	yes	characteristic
133	C₂×C₈	#11924013	1	1	yes	characteristic
134	C₂×C₈	#11924018	1	1	yes	characteristic
135	C₂×C₈	#11924018	1	1	yes	characteristic
136	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
137	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
138	C₄²	—	2	1	no	—
139	C₄²	#11924007	1	1	yes	characteristic
140	C₄²	C₂×Dic₁₄₆	1	1	yes	characteristic
141	C₁₆	—	4	2	no	—
142	C₁₆	—	292	2	no	—
143	C₁₆	—	73	4	no	—
144	C₁₆	—	146	2	no	—
145	C₂³	#14556526	1	1	yes	characteristic
146	C₂×C₄	—	2	4	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	1	no	—
149	C₂×C₄	#14556520	1	1	yes	characteristic
150	C₂×C₄	#14556520	1	1	yes	characteristic
151	C₂×C₄	#14556524	1	1	yes	characteristic
152	C₂×C₄	#14556524	1	1	yes	characteristic
153	C₂×C₄	#14556526	1	1	yes	characteristic
154	C₂×C₄	C₂×#11924007	1	1	yes	characteristic
155	C₈	#14556520	1	1	yes	characteristic
156	C₈	#14556520	1	1	yes	characteristic
157	C₈	#14556524	1	1	yes	characteristic
158	C₈	#14556524	1	1	yes	characteristic
159	C₈	C₂×C₈×D₇₃	1	1	yes	characteristic
160	C₈	C₂×#11924011	1	1	yes	characteristic
161	C₈	C₂×#11924013	1	1	yes	characteristic
162	C₈	C₂×#11924018	1	1	yes	characteristic
163	C₂²	—	2	2	no	—
164	C₂²	#17988339	1	1	yes	characteristic
165	C₂²	#17988339	1	1	yes	characteristic
166	C₂²	C₂×#14556526	1	1	yes	characteristic
167	C₄	—	2	2	no	—
168	C₄	—	2	2	no	—
169	C₄	#17988339	1	1	yes	characteristic
170	C₄	#17988339	1	1	yes	characteristic
171	C₄	C₂×#14556520	1	1	yes	characteristic
172	C₄	C₂×#14556524	1	1	yes	characteristic
173	C₂	—	2	1	no	—
174	C₂	—	2	1	no	—
175	C₂	#22608006	1	2	yes	direct product factor
176	C₂	C₂×#17988339	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	#11924013
commutator	C₈×C₇₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₇₃
Fitting	C₂×#364×C₇₃	C₂
socle	C₂²×C₇₃	#335
prime radical	C₂³×C₇₃	#108
cosocle	C₄×C₈×C₇₃	C₂³
hypercenter	C₂×#364	D₇₃
hypocenter	C₇₃	C₂×#1373
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
3. #22608006	C₂	direct (N × H)
87. C₂×C₇₃	#1373	semidirect
93. C₇₃	C₂×#1373	semidirect
175. C₂	#22608006	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×#17988408	2
C₂×C₄×#11924003	2
#22608006	2
C₂×#364×C₇₃	2
C₂×#1373	73

Subgroups — C2×#22608006 #32997576

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22608006 #32997576